On the Remarkable Superconductivity of FeSe and its Close Cousins

Emergent electronic phenomena in iron-based superconductors have been at the forefront of condensed matter physics for more than a decade. Much has been learned about the origin and intertwined roles of ordered phases, including nematicity, magnetism, and superconductivity, in this fascinating class of materials. In recent years focus has been centered on the peculiar and highly unusual properties of FeSe and its close cousins. This family of materials has attracted considerable attention due to the discovery of unexpected superconducting gap structures, a wide range of superconducting critical temperatures, and evidence for nontrivial band topology, including associated spin-helical surface states and vortex-induced Majorana bound states. Here, we review superconductivity in iron chalcogenide superconductors, including bulk FeSe, doped bulk FeSe, FeTe$_{1-x}$Se$_x$, intercalated FeSe materials, and monolayer FeSe and FeTe$_{1-x}$Se$_x$ on SrTiO$_3$. We focus on the superconducting properties, including a survey of the relevant experimental studies, and a discussion of the different proposed theoretical pairing scenarios. In the last part of the paper, we review the growing recent evidence for nontrivial topological effects in FeSe-related materials, focusing again on interesting implications for superconductivity.

After a dozen years of research into iron-based superconductivity (FeSC), a good deal has been learned about the phenomenology and microscopic origins of this fascinating phenomenon that can be generally agreed upon. Since the bulk electron-phonon interaction is rather weak, the mechanism for pairing is almost certainly electronic, and related to the intermediate-to-strong local Coulomb interactions in these materials 1-4 . Because the Fermi surface takes the form of small electron-and holelike pockets centered at high symmetry points, and the bare interactions are repulsive, the most likely superconducting states change sign between pockets (Fig. 3). Because the nesting between electron and hole pockets is often particularly strong, most systems are believed to be of so-called s ± character, where the sign change occurs between the gap amplitudes on the Γ-centered hole pockets and the M -centered electron pockets 5,6 . Significant gap anisotropy also exists due to the multiorbital character of Fermi surface sheets, and to the necessity of minimizing the Coulomb interaction 7,8 .
Strong pair scattering between the electron pockets exists as well, implying that the d-wave attraction is also strong, and competes with the s-wave 9 . Transitions between competing superconducting states, as well as timereversal symmetry breaking admixtures of the two 10 , or of purely s-wave amplitudes with different phases on different pockets 11,12 are therefore possible. The existence of sign-changing superconducting order has now been relatively well established in some systems via observations of the neutron resonance 13 and disorder properties [14][15][16][17][18][19][20][21] . While the exact gap structures, as well as observations of time-reversal symmetry breaking, remain controversial, the general principles outlined above are generally accepted.
Most of the consensus described above was developed in the context of the Fe-pnictide superconductors, particularly the 122 systems, but the Fe-chalcogenide materials present a completely new set of questions, and even pose challenges to the central paradigm of superconductivity established for the pnictides. The strength of electronic correlations and spin-orbit coupling is expected to be higher in the chalcogenides, and may be responsible for the remarkable behavior of bulk FeSe, where superconductivity condenses out of a strongly nematic state with no magnetic long-range order. Modifying the 8K superconductor FeSe in almost any way, including pressure, intercalation, or deposition of a monolayer film on a substrate, produces a high-T c superconductor. The new states engendered by these modifications are thought to be related to one another, and indeed some have remarkably similar Fermi surface structures, notably lacking hole bands at the Fermi level. Why such systems, in violation of the central paradigm established apparently quite generally for Fe-pnictides, should have the highest critical temperatures of the FeSC family, is the central current question of iron-based superconductivity research.
Finally, a wave of recent measurements and theories have offered considerable evidence for topological superconductivity in the FeTe 1−x Se x system, holding out the prospect of creating and manipulating Majorana bound states for quantum computation in these materials. Together with the rough consensus on many aspects of Fe-pnictide superconductivity, the challenges posed by these and other discoveries in the Fe-chalcogenide family suggest that the time is ripe to review developments in this field. Following reviews of mostly experimental results 22,23 , a review focussed on bulk FeSe 24 , and recent specialized reviews of topological aspects 25,26 , we attempt here to synthesize what has been learned about the Fe-chalcogenide superconductors, with an emphasis on the superconducting state. Our goal is to elucidate which new theoretical ideas have been stimulated by experimental discoveries, and highlight remaining open questions in the field.
The paper is structured as follows: In Sec. II, we first give an overview of FeSe itself, with an emphasis on its unusual band structure created by strong correlations and the implications of the tiny, highly nematic pockets at the Fermi surface. In Sec. III, we discuss what is known about bulk FeSe's magnetic properties, a knowledge of which is essential to understand the spin fluctuation pairing interaction, and discuss measurements in the superconducting state that provide information on the highly anisotropic gap. The spin fluctuation theory of pairing is introduced, and modifications required to explain the "orbitally selective" pairing reported in this system are explained. Next, in Sec. IV, we discuss the remarkable effects of pressure and chemical pressure (via S doping on the Se site) on the FeSe phase diagram. We then consider in Sec. V the FeSe monolayer system on SrTiO 3 (STO) substrate, with the highest T c in the FeSC family. We discuss various ideas that have been put forward to understand the mechanism of electron doping by the substrate, and its effect on superconductivity. In Sec. VI, we consider FeSe intercalated with alkali atoms, organic molecules, and LiOH, all of which are high-T c materials, and at least some of which share similar electronic properties to the monolayer on STO. Finally in the last Section VII, we review the recent theoretical proposals and experimental evidence for non-trivial band topology in some FeSCs. We also discuss reports of topological superconductivity in these materials.

A. Iron pnictides
We begin by briefly reviewing the essential ingredients in the recipe for an iron-based superconductor 1-3,28-32 . The original Fe-based superconductors, LaFePO and Fdoped LaFeAsO, were discovered by H. Hosono 33,34 , with structures containing square lattices of Fe atoms with pnictogen As placed in out-of-plane positions above and below the Fe plane, such that there are two inequivalent As per unit cell. They were quickly noted to resemble other materials classes of unconventional superconductors such as cuprates and heavy fermion systems in exhibiting electronic correlations which play a significant role in emergent ordered phases such as magnetism, nematicity and superconductivity 35,36 . Several other materials with similar iron planes were discovered in short order, including Ba(Fe 1−x Co x ) 2 As 2 , Ba 1−x K x Fe 2 As 2 , BaFe 2 (As 1−x P x ) 2 , and LiFeAs. Like cuprates, their band structures are quite twodimensional, but the parent compounds of the Fe-based superconductors are metallic rather than insulating. Instead of the large Fermi surfaces seen in cuprates at optimal doping, Fe-based systems display small Fermi surface pockets centered at high symmetry points. These pockets have almost pure Fe-d-character (pnictide and chalcogenide p-states are typically several eV from the Fermi level), but the d-orbital content winds around each pocket, as depicted in Fig. 1 (a,b). Note that the two inequivalent As atoms implies that the correct 2-Fe Brillouin zone is one-half the size of the reference 1-Fe zone.
The d-spins on the Fe sites are not strongly localized in character, but to understand the low-lying magnetic states in these systems it is frequently convenient to examine the effective exchanges J ij between spins on sites i, j. Calculations and experiments 37 both suggest that the nearest neighbor Fe exchange J 1 is of the same order of magnitude as the next nearest neighbor exchange J 2 , due to the strong overlap of the pnictide p orbtials with the next nearest neighbor Fe. This unusual situation is responsible for magnetic ordering in a stripelike pattern with wave vector (π, 0) in the one-Fe zone. In most Fepnictides, stripe magnetic order is dominant in the doping range near 6 electrons per Fe, with other magnetic orders, e.g. Néel order, double stripe order, and various C 4 symmetric phases often close by in energy [38][39][40][41][42][43][44][45] . In special situations, these states are observed condense in small parts of the phase diagram, but the stripe order is generally dominant. As the ordered magnetic state is weakened by doping, a competing superconducting dome emerges at lower temperatures ( Fig. 1 (c)).
At higher temperatures near the edge of the magnetic phase boundary, an electronic nematic phase forms where the crystal structure is very slightly orthorhombic, but the electronic responses are found to be highly anisotropic. Many authors have identified the nematic phase as the natural consequence of the competing spin fluctuations in a J 1 − J 2 spin-modelà la Chandra-Coleman and Larkin 37,46 , and essentially the same Ising nematic state is obtained in an itinerant picture 47,48 . On the other hand, one must be careful, because such a transition can in principle be driven also by orbital or lattice degrees of freedom, since the same symmetry is broken by all effects 48 . For example, some authors have used the lack of long-range magnetic order to argue for a nematic transition driven by orbital fluctuations rather than spin 49 . We do not review these arguments here, but refer the interested reader to the literature.

B. How FeSe is different from pnictides
In the last few years, the field of Fe-based superconductivity has been driven largely by studies of bulk FeSe and its close cousins, including FeTe 1−x Se x , doped or "dosed" FeSe, thin layers of FeSe or FeTe 1−x Se x on SrTiO 3 (STO) substrates, and a number of intercalated FeSe compounds. Reasons for the focused attention on this class of systems include improved sample control 55 and a series of surprising discoveries that remain topics of considerable current controversy; 1) peculiar nematic effects including highly anisotropic electronic properties, in the absence of long-range magnetic order; 2) unusual lowenergy electronic structure compared to other FeSCs, 3) tunable superconducting critical transition temperatures T c , and 4) evidence for topologically non-trivial bands and associated topological superconductivity.
In this work we concentrate on the Fe chalcogenides, with the focus on the FeSe system and its cousin materials created by replacing Se by sulfur or tellurium, electron-doping by intercalation, and preparation of thin films and application of pressure. FeSe with excess Fe was discovered to be an 8K superconductor in 2008 56 , but a cold vapor deposition technique was required to reliably make high-quality, stoichiometric, crystals 55 . A summary of interesting properties of this compound has been provided in earlier reviews 24,57,58 , reviews on monolayer FeSe can be found in Refs. 59-62. As for FeTe, it turns out to be the most stable compound of the 11 chalcogenides in its pristine form 22 , which may, together with the presence of interstitial Fe, be responsible for the difficulty of making homogeneous samples doped with Se away from the FeTe point. FeTe exhibits a double stripe magnetic structure, appears to be more strongly correlated than the other FeSC (see Fig. 2 (d)), and is expected to have larger spin-orbit coupling (SOC).
As opposed to most other FeSCs, FeSe develops no static magnetic order at ambient pressure. At high temperature, the crystal is tetragonal, but makes a transition to an orthorhombic structure below 90K, and undergoes no further ordering until the superconducting transition at 8-9K. The entire phase immediately below the structural transition is referred to as the nematic phase, displaying very strongly anisotropic responses to external fields although the change in the lattice constant at the transition is only about 0.1%. The reasons for the absence of static magnetism and the microscopic origin and nature of the dominant nematic order are the subject of considerable debate, which we do not attempt to discuss or resolve here.
Angle-resolved photoemission spectroscopy (ARPES), scanning tunneling microscopy (STM) and quantum oscillations (QO) show that FeSe bulk crystals exhibit tiny hole and electron pockets at the Fermi surface quite different from other FeSCs and very different from the results of standard first-principles calculations. We discuss the spectroscopic data in Section III A below. The exact description of the low-energy electronic structure and Fermi surface is under intense debate at the time of writing. However, a few qualitative aspects are clear. First of all, the correlation effects appear to be quite strong relative to the pnictides, a conclusion reflected in a number of observables. The effective masses in the different orbital channels, extracted by comparing a variety of observables 53,54 , are found to be substantially larger for Fe-chalcogenides than for Fe-pnictides, see Fig.  2, a trend captured quite well by dynamical mean field theory (DMFT) in general and studied for FeSe, e.g. in Refs. 50-52, 63, and 64. Secondly, the mass renormalizations appear to be more strongly orbitally differentiated in these materials, with the largest renormalizations occurring in the d xy case 65 . Finally, the size and shape of the pockets is strongly renormalized, in a manner not captured by DMFT. The pocket shrinkage relative to DFT is observed in virtually all Fe-based superconductors, but is particularly severe in FeSe. Several authors have recently pointed out that renormalizations of this type can be obtained only with a nonlocal treatment of the self-energy [66][67][68] .
One aspect of the band structure of bulk FeSe which is particularly remarkable is the fact that the Fermi energies of hole and electron pockets (band extrema) in the low temperature phase near T c are quite small, of order 5-10meV (see Sec. III A below). By contrast, typical Fermi energies in pnictides are ∼ O (50 meV) This implies that the O(2 meV) superconducting gaps observed spectroscopically are nearly as large as the Fermi energies, an observation that has led to the search for effects characteristic of the BCS-BEC crossover regime. While the most straightforward consequences of BEC behavior, a characteristically broadened specific heat transition and a pseudogap, are not observed, there are significant anomalies in transport and NMR that have lent credence to the suggestion 69 . It is also true that the properties of a multiband system in the BCS-BEC crossover regime are not well studied; while there are many predictions for single band systems, and a number of two-band calculations, few appear to be appropriate for FeSC with both hole and electron bands simultaneously (see, however, Ref. 70). This is of course a crucial distinction, since the chemical potential will be pinned or nearly so in a compensated system, suppressing canonical BEC crossover effects. The discussions surrounding this fascinating possibility were reviewed recently in Ref. 23.
In Sec. III C, we review spectroscopic data from measurements capable of determining the superconducting gap structure. Since superconductivity condenses out of a C 2 symmetric nematic normal state, it is not surprising that the gap function determined in experiment reflects this symmetry breaking. The degree of anisotropy, however, is very surprising; the momentum structure of the superconducting gap is extremely distorted relative to C 4 -symmetry despite the tiny orthorhombicity of the underlying crystal structure. This interesting property of the superconducting gap has given rise to a variety of different theoretical suggestions for the origin of the gap structure. We regard it as a healthy development, largely driven by FeSCs, that theoretical models are competing to best describe such measured gap "details", as opposed merely to overall symmetry properties. We now sketch some of these theoretical approaches.

C. Theoretical approaches to pairing
A model of the electronic structure for the FeSCs often employed for theoretical calculations is a multiband tight binding model with the kinetic energy term [71][72][73] where c † i σ creates an electron in Wannier orbital with spin σ. Note that is an orbital index with ∈ (1, . . . , 5) corresponding to the states which have dominating character of the five Fe 3d orbitals (d xy , d x 2 −y 2 , d xz , d yz , d 3z 2 −r 2 ). Extensions of such models to include the p orbitals of the pnictogen or chalcogen atoms are sometimes used, although usually not needed to describe the low energy properties. To the kinetic energy is added a Hubbard-Kanamori (general on-site) interaction, where U is the usual Hubbard interaction between opposite spins, J is the Hund's rule exchange, U is the bare interorbital interaction, J is a pair hopping term and n i σ = c † i σ c i σ (n i = n i ↑ + n i ↓ ) denotes the (total) density operator. The parameters U , U , J, J are related in the case of spin rotational invariance by U = U − 2J, and J = J , i.e. the two quantities U and J/U fix the interactions 6,74 .
For a given choice of the parameters, one must search for a superconducting instability. Since U, U , J, J are all repulsive, mean field theory does not initially appear to be a useful approach, since bare interactions are repulsive. We discuss below in Sec. V D 2 some interesting recent results which show that this is not necessarily true in the multiorbital pairing case, particularly if spin-orbit coupling is strong. Nevertheless, the most reliable way to find an attractive pair channel is to study the effective interaction vertex in the Cooper channel generated by the exchange of particle-hole pairs.
One popular approximation to this effective vertex goes under the name of random phase approximation (RPA), and dates back to the ideas of Schrieffer 75 . The pairing vertex is proportional to the generalized particlehole susceptibility in the paramagnetic state 71 where we have adopted the shorthand notation k ≡ (k, ω n ) for the momentum and frequency. The weight factors M are given by where the a ν (k) are the matrix elements of the unitary transformation that diagonalizes the kinetic energy. The Green's function describing band µ is given by Calculating the interacting susceptibility within RPA, where bubble diagrams are included, one gets and FIG. 3. Schematic pictures of candidate superconducting order parameters depicted in 2-Fe Brillouin zone for a tetragonal system. Gaps depicted by the thickness of the green (∆ > 0) and orange (∆ < 0) lines on simple Fermi surface pockets. Conventional s± state is driven by strong pair scattering between inner hole pocket(s) and electron pockets. Subsequent states depicted do not have hole pockets at Fermi level. d-wave is driven by scattering between electron pockets, "incipient s± is driven by resonantly enhanced scattering processes connecting incipient hole band states with the electron states at the Fermi level (Sec. V D 3), and "bonding-antibonding" s± is driven by scattering between electron pockets supplemented by strong hybridization.
the pairing vertices in the orbital and band basis, respectively. HereŪ c is the analog of the interaction in the charge channel and χ RPA 0 is the corresponding charge susceptibility in RPA approximation 71 . The susceptibility is then approximated by the static susceptibility, i.e. at zero frequency, and the appearance of a superconducting instability can be sought by solving the linearized gap equation for the eigenvalues λ i and the eigenvectors g i (k). The mathematics of the pairing interaction in the multiorbital system are straightforward but not completely transparent. Nevertheless, it is relatively easy to anticipate what kinds of gap structures may be favored in a given situation by examining the structure of the generalized susceptibility. Scattering processes , k → , k are favored if they nest an electron with a hole pocket or, more generally, connect Fermi surface segments with opposite Fermi velocity. Interorbital scattering processes are suppressed relative to intraorbital ones 8,74 . For a Fermi surface like that of Fig. 1 (a), which shows a standard pnictide like Fermi surface at k z = 0, obvious scattering processes include that at (π, 0) from the d yz section of the inner Γ-centered hole pocket to the d yz section of the electron pocket at M X , which drives the s ± interaction leading to the first state shown in Fig. 3 (a). This is roughly speaking the situation in BaFe 2 As 2 , and explains the fact that the inner hole pocket and electron pocket gaps are the largest.
In FeSe, however, stronger interactions renormalize the inner hole band downward, leaving only a very small vestige of the outer hole pocket, and shrink the electron pockets correspondingly, so the s ± process is less favored. When electron doping effects are included, the hole pockets can disappear completely from the Fermi level, but leave a residual interaction with the incipient outer hole band (see Sec. V D 3; state also depicted in Fig. 3 (c)). Competing with the e-h process is the d xy → d xy process between the electron pockets, a scattering vector parallel to (π, π) but smaller in magnitude, which drives both the d-wave and, including hybridization between the two electron pockets, the bonding-antibonding s ± with gap sign change between the electron pockets also shown in Fig. 3 (d).

III. BULK FESE
A. Electronic structure of FeSe From a general perspective, the electronic structure of FeSe is very similar to other FeSCs in the sense that the states at the Fermi level are mostly of Fe-d character, where states of d xy and d xz/yz symmetry dominate at the Fermi level. This picture was established initially theoretically within DFT, and also been verified experimentally. However, the electronic structure of FeSe is more complex than anticipated, and because band energy scales are very small, it evolves with temperature even more than typical FeSCs 76,77 . One important issue that has received too little attention relates to the d xy band that sometimes results in a Γ-centered hole pocket in FeSC, e.g. in LiFeAs. According to ARPES, it appears that a band of d xy character does not cross the Fermi level while ab-initio calculations predict the existence of such a Fermi surface sheet 72 , i.e. realistic models for the electronic structure cannot be derived from those ab-initio calculations.
Indirect measurements of the electronic structure as a function of temperature are magnetotransport investigations where a sharp increase of the resistance below T s 78,79 was found. This was subsequently interpreted as changes in the carrier density and the mobility in the orthorhombic phase 78,[80][81][82][83][84] which in this view point towards drastic changes of the low energy properties and the Fermi surface.
To account for such changes in the electronic structure upon entering the orthorhombic phase, it has been proposed that an orbital order term with a sign change of the orbital splittings from (0, 0) to (π, 0), should be present in the kinetic energy. Here ∆ b and ∆ s are the values of a bond order and site order term 20,85,86 .
In addition, an orbital ordering term in the d xy orbital is allowed by symmetry and has been included in some more recent works 87,88 . From a LDA+U perspective an off-diagonal orbital order term that lowers the overall symmetry has been found to be the ground state, a result that would allow a band hybridization such that the Y pocket is lifted 89 . The electronic structure has been measured by a number of ARPES investigations which have been reviewed briefly above 58,90,91 . From a theoretical perspective, the spectral function is measured in these experiments to a good approximation. The spectral function is turn related to the retarded Green's function where E k is bare kinetic energy as for example described by the eigenvalues of Eq. (1) and electronic correlations (or other scattering processes) are parametrized by the self energy Σ(k, iω n ). We give here a summary of the findings, which are based on the identification of the peak of the spectral function, together with a polarization analysis. At high temperatures in the tetragonal phase, ARPES measurements find one holelike cylinder at the Γ point of d xz /d yz character (see Fig. 4 (a)). Additionally, there is one holelike band with the same orbital character which does not cross the Fermi level. This band is split from the other holelike band due to a spin-orbit coupling of a couple of meV and therefore pushed below the Fermi surface 92,93 At the X and Y points in the 1-Fe Brillouin zone, two electronlike cylinders of d yz /d xy , respectively d xz /d xy , character are present, as expected from DFT. Even in the tetragonal phase, the sizes of the Fermi surface pockets are significantly smaller than predicted from DFT 77,94 , and additionally, the predicted holelike band of d xy character exhibits a significant band shift downward in energy, such that it does not cross the Fermi level at all. The need for models of the electronic structure consistent with ARPES data has lead to the proposal of "engineered" models for the electronic structure where the hoppings in Eq. (1) have been adjusted to match experimental findings of the spectral positions of the observed bands 20,95,96 .
Upon entering the nematic phase, the Γ pocket is elongated and a modification of relative weights of d xz /d yz character occurs. The splitting to the other holelike band has been estimated as ∼ 10meV 97 (15 meV in Refs. 98 and 99), which must be interpreted as the combined effect of SOC and a splitting due to orbital ordering. The latter effect can be modelled by suitable choice of the orbital order terms in Eq. (9). One electronlike sheet at the X-point becomes peanut-shape like, as seen by experiment, while the Y -pocket remains quite elongated along y. Experimentally, the shape, orbital character, and even the existence of the Y -Fermi pockets is controversial.
Early ARPES experiments on FeSe were done on twinned samples, making it difficult to separate nematicity from the effect of averaging over twin domains. Subsequent measurements were performed on detwinned crystals and were able to resolve the electronic structure on one orthorhombic domain, see Fig. 4 (c,d). Those measurements could only observe one of the the two crossed "peanut-shaped" electron pockets at the X point 97,100-102 ; similar conclusions were reported recently using nano ARPES 103 within individual nematic domains. Various explanations for this dramatic consequence of nematicity were offered, including selection rules specific to ARPES 101 , strong correlation effects rendering some of the electronic states less coherent 20 , unusual band shifts and hybridization of bands 104,105 ; some data of these experiments are shown in Fig. 4 (e-g). Finally, evidences for an additional band splitting was reported, which could be due to magnetism or SOC on the surface 106 . These issues may seem rather arcane to the newcomer to the field, but they are rather essential to the goal of understanding how correlations exactly effect the electronic structure that is essential for deducing the pairing interaction.
Recently, evidence for the second pocket at the Y point have been reported in the literature as well 108,109 suggest- ing that a detailed understanding of this issue might also be related to shifts of the d xz and d yz orbital states in the orthorhombic state. Depending on the assignments of peaks in the measured spectral function to bands at the X-point, a large splitting of 50 meV 100,104,107,110 or much smaller splitting of 10 meV 98,101 has been deduced while in both scenarios the sign of the splitting is reversed between the Γ point and the X point 97 , similar to findings in Ba122 111 . The four branches of oscillation frequencies observed in quantum oscillations of the resistivity 83,112,113 correspond to the extremely small areas of the Fermi surface sheets covering only few percent of the Brillouin zone. Estimates for the Sommerfeld coefficient using the areas and effective masses from these investigations are in agreement with specific heat data 112,114,115 , presented in Fig. 9 (g). By assigning certain oscillation frequencies to two cylinders (from hybridized electronlike bands in the 2 Fe zone) this is consistent with the presence of the Y pocket, while an assignment of the frequencies to maximal and minimal areas of one corrugated cylinder would agree with the absence of the Y pocket. STM measurements 20,108 (see Fig. 5), which are able to measure within a single domain, agree in the size and shape of the Γ centered Fermi pocket and the X pocket with the deductions of the ARPES measurements. As expected from the layered structure of FeSe, only a weak k z dispersion is found, rendering the pockets as weakly corrugated cylinders 107,116 .
Since the FeSe system exhibits a large temperature range where the nematic state is stable, it can be used to test a number of theoretical scenarios describing nematicity in FeSC. The nematic state in FeSe exhibits the lower (orthorhombic) symmetry via an Ising nematic type order parameter 48 . Since conventional softening of the lattice orthorhombicity via a static linear coupling was excluded as the sole driving force for the structural transition early on 118 , spin, orbital/electronic charge instabilities of the electronic structure remain as candidates for driving the transition. These have been studied extensively by Raman scattering measurements [118][119][120][121][122] and time resolved ARPES 123,124 . A frequently encountered argument in favor of orbital/charge fluctuations begins by noting that while in FeSe lattice distortion, elastic softening and elasto-resistivity measurements associated with the structural transition at T s , are comparable to other FeSC 73 , nuclear magnetic resonance (NMR) and inelastic neutron scattering measurements do not detect sizable low energy spin fluctuations above T s (as, e.g. in Ba122) 125,126 . However, we argue below in Sec. III B that there is a simple explanation for the apparent "lack" of high temperature fluctuations, so the spin nematic explanation cannot be ruled out on this basis. -data -sim. The interacting electron gas with multiple orbital degrees of freedom can be unstable against unequal occupation of the d xz and the d yz orbital. Taking into account nearest neighbor Coulomb interactions of strength V , it turns out that the general changes in low temperature electronic structure can be well described by a mean field approach already 85,86 , giving rise to the orbital order terms described in Eq. (9). An alternative explanation for differing signs of orbital order on Γ and X, Y has been revealed by a renormalization group analysis, where a solution of this type was shown to be driven by the d-wave orbital channel 127 . Another theoretical approach 27 involves starting from anisotropic spin fluctuations in the nematic phase, parametrizing them by a bosonic spin-fluctuation propagator of the form Here ω 0 is a constant while ω X/Y sf (T ) = ω 0 (1 + T /T θ ) is the characteristic energy scale of spin modes at the X and Y point of the Brillouin zone. Such an approach, using Eq. (13) to calculate the interband self-energyà la Ortenzi et al. 66 , is also capable of qualitatively capturing the evolution of the electronic structure upon entering the nematic state, while at the same time capture the basic properties of the Fermi surface shrinking despite neglect of the full momentum dependence of the spin-fluctuation propagator 27 . In the same framework, the shifts of bands relative to DFT results can be explained and strongly anisotropic spin fluctuations might be crucial to understand the superconducting order parameter 96,128 .
In addition to band shifts and orbital order, another important effect of the one-particle self-energy is to create decoherence, i.e. a reduction in quasiparticle spectral weight. The effective mass m * is strongly renormalized, and the quasiparticle weight on the Fermi surface (FS) deviates from unity, although the latter is considerably more difficult to measure. It is well known that these effects depend significantly on orbital channel, and that d xy orbital states are generally the most strongly correlated 50,53,54,65,129 , see Fig. 2 (d). In the nematic phase, the renormalizations of the d xz and d yz orbitals will generally be different. Whether the ARPES data imply a strongly differentiated coherence between the d xz and the d yz orbital 96 , e.g. to understand the absence of the Y pocket, is currently controversial. Experimental investigations have proposed as an alternative explanation strong shifts of the bands close to the Y point together with an orbital hybridization such that the Y pocket is diminished or a gap opens 104,105 . Within a tight-binding approach, it is however difficult to construct a hybridization term that will lift the Y pocket entirely away from the Fermi surface, while preserving all symmetries of the crystal.
A combination of decoherence of d xy states (without significant d xz,yz decoherence), nematic order, spin-orbit coupling and/or surface hybridization has been proposed to account for the observed configuration of the bands at the X point 88 . The second pocket (in the 1 Fe zone appearing at the Y point) is in this scenario of dominant d xy character everywhere in the nematic state (instead of d xz ) and difficult to observe spectroscopically due to its decoherence.
Finally, recent experimental measurements using neutron scattering and X ray diffraction and an analysis of the pair distribution function found that short range orthorhombic distortions even at high temperatures can account for the experimental findings 130,131 . Possible theoretical scenarios for this observation might be locally induced nematicity by impurities 132 or local displacements of the atomic positions from its ideal tetragonal symmetry positions which are found to be stabilized within a DFT calculation using large elementary cells of overall tetragonal symmetry. The latter scenario was also found account for unusual band shifts at the Γ point 133 , in particular the suppression of the xy band. However, in this formulation it is not a priori clear what makes the electronic structure of FeSe special.

Long range order
Unlike most Fe-pnictide parent compounds, bulk FeSe does not order magnetically at ambient pressure, but magnetic order occurs with the application of relatively small pressures (Sec. IV), suggesting that ordered magnetism is "nearby", and that strong magnetic correlations play a significant role. The statement that no long-range order exists is based primarily on neutron diffraction experiments 136,137 , which observe no magnetic Bragg peaks. However, more exotic magnetic states with quadrupolar magnetic order, resulting from competition of various dipolar states, have been proposed 134,138 . Such order would only be visible indirectly in a typical diffraction experiment.
To explain the background of such proposals, we note that magnetism in FeSCs has frequently been discussed in terms of a Heisenberg model with localized spins, which indeed can describe the spin-wave modes in the observed ordered phases. To account for the lack of dipolar order and aspects of the low-energy spin modes in FeSe, the Hamiltonian must contain bilinear and biquadratic couplings of spin operators S i , Here j = i+δ n , and δ n connects site i and its n-th nearest neighbor sites. Frustration among competing magnetic states was proposed to explain the absence of magnetic order by Glasbrenner et al. 139 , who compared the energies of various stripe and Néel states within DFT, and showed that they were within a few meV of each other for FeSe, whereas in Ba122 and other pnictides, the simple (π, 0) stripe state was lower in energy than competing states by a large margin. They then discussed the competition among these states within a localized spin model with biquadratic exchange, and showed that estimates from ab-initio approaches of the coefficients J, K in Eq. (16) put FeSe near a multicritical point in the magnetic phase diagram where several stripelike states were nearly degenerate. It was argued that under these circumstances, quantum fluctuations would prevent ordering. Intriguingly, Glasbrenner et al. also noted that all such states were consistent with the observed nematic order, suggesting that the robust nematic order observed in FeSe was also a consequence of these magnetic fluctuations. Other groups have sought explanations in starting from the same spin model, but argued that further frustration in the biquadratic couplings including K n up to n = 3 can explain the absence of dipolar magnetism in FeSe 134 and stabilize quadrupolar order, see phase diagram in Fig. 6 (a). To our knowledge, there is no definitive evidence for such order in experiment. The suppression of the biquadratic couplings upon application of pressure on FeSe should make the magnetic order reap- pear as proposed recently 138 . Similar ideas were put forward independently in a description of FeSe as a paradigmatic quantum paramagnet 140 . Finally, this type of argument was also advanced in the context of the monolayer FeSe when calculating Boltzmann weighted spectra for different spiral magnetic configurations of similar energy 141 .

Spin fluctuations in normal state
Hints to the microscopic origins of magnetic correlations in FeSe can be found in the complex temperature and momentum dependence of magnetic fluctuations, and its imprints on the nematic state and superconductivity, see Sec. III C 5 below. Experimentally, these correlations have been studied using NMR and inelastic neutron scattering experiments 137,142,[144][145][146][147][148] , with the latter summarized in Fig. 7. In the spin nematic scenario, these fluctuations are argued to drive the nematic order, eventually leading to a divergence of the nematic susceptibility. Alternatively, nematicity is proposed to arise through orbital or charge fluctuations 118,147 . Recall the NMR data on FeSe 125,147 are rather different from the Fe-pnictides, where a strong upturn in the spin-lattice relaxation time 1/(T 1 T ) beginning well above T s is taken to signal the onset of strong spin fluctuations at high temperature. In FeSe, this upturn is visible only below T s and just above T c , suggesting rather weak spin fluctuations in the vicinity of the nematic transition, and leading to suggestions of the primacy of orbital fluctuations.
On the other hand, the spin fluctuations of bulk FeSe have been measured in detail using inelastic neutron scattering on powder samples 149 and (twinned) crystals 137,142,150 revealing the complex dependence on temperature and momentum transfer. Magnetic fluctuations of stripe-type and Néel-type were detected 142 , revealing a transfer of spectral weight at energies 60 meV away from Néel-type fluctuations as temperature decreases and the system enters the nematic phase, see Fig. 7 (c). A large fluctuating moment of ∼ 5.1 µ 2 B /Fe, corresponding to an effective spin of S ∼ 0.74 was estimated 142 which is almost unchanged from high tem-peratures T > T s to very low temperatures as evidenced by the local susceptibility presented in Fig. 7 (e). The overall bandwidth is found to be smaller than in 122type FeSC systems, and a sizeable low energy spectral weight grows below T s 137 which agrees with findings from NMR 125,146,147 and is in line with the proposal of competition between stripe-type and Néel-type magnetic ordering vectors as suggested also by Raman spectroscopy 151 . The presence of spin fluctuations at low energies indicates the proximity of the system to a magnetically ordered state which can be realized by tuning the system with pressure.
In the context of the inelastic neutron data, the early NMR results on FeSe 125,147 that suggested weak spin fluctuations (Sec. II) present above T s and seemed to point to an orbital fluctuation-driven nematic transition should be re-examined. It is important to remember that the spin-lattice relaxation is local, i.e. 1/(T 1 T ) ∝ Im q χ (q, ω)/ω, i.e. spin fluctuations at all wavelengths contribute. Inelastic neutron scattering measurements show quite different temperature dependence for Néel (π, π) and stripe (π, 0) spin fluctuations, such that at low energies ω, Néel fluctuations dominate at high T , whereas stripe fluctuations dominate at low T as shown in Fig. 7 (c). Both are summed in 1/(T 1 T ) and since one is increasing and one decreasing with T can give a relatively flat total T dependence 95 as observed in experiment. The conclusion is that the stripelike fluctuations are indeed present, and are additionally strongly enhanced above T s , just as in, e.g. Ba122; the difference with the Fe-pnictides is the existence of the strong fluctuations at other wavevectors at higher temperatures near the nematic transition.
Since most inelastic neutron experiments have been performed on twinned crystals, it has been difficult to test whether or not the nematic state creates a significant anisotropy of spin fluctuations. In order to discriminate in a neutron scattering experiment between fluctuations with momentum transfer (π, 0) and (0, π), one needs to (a) place a reasonably large amount of the sample material into the beam and (b) simultaneously detwin the crystals. This has been achieved by gluing many (small) crystals of FeSe on large crystals of Ba-122 and mechan- ically detwin the Ba-122 (see Fig. 7 (f)) such that also the detwinning ratio of the FeSe crystals could be observed using elastic neutron scattering. Then, by measuring the nominal signal from momentum transfer of (π, 0) and (0, π), one can correct the data and extract the spin fluctuations with momentum transfer of (0, π), finding no measurable signal at low energies 143 ( Fig. 7 (g,h)). This approach is restricted to the energy window where the Ba-122 has a gapped spin fluctuation spectrum.
Thus any theory of FeSe must account for the extremely anisotropic spin fluctuation in the nematic normal state of FeSe. Any calculation of the spin-fluctuation spectrum from standard first principles methods then has the drawback that the low temperature nematic phase is not correctly captured and any non-magnetic tetragonal calculation yields a C 4 symmetric spin fluctuation spectrum as shown in Fig. 8 (a,b) where the corresponding dynamic structure factor shows low energy weight at (π, 0), but also at (0, π). More phenomenological approaches indeed yield anisotropic spin fluctuations for example by construction of bosonic propagators of anisotropic spin modes 27 which can be justified from a derivation of an effective action including quartic terms in the nematic order 154 , or by assuming strongly orbitally selective quasiparticles with reduced coherence 153 ( Fig.   8 (c), resembling the measured spectrum on a twinned crystal as shown in Fig. 7 (b)); reduced coherence in the d xy orbital has also been found to be necessary to explain nematic fluctuations in Fe 1+y Te 1−x Se x 155 . In Ref. 154, it was also found that a Fermi surface with nesting between states of different orbital character (as realized in FeSe) favors nematic order, while magnetism is usually favored by nesting between states of the same orbital character, suggested as a factor in the absence of magnetic order in FeSe. It is unclear at present whether the proposal of changes in the orbital content 156 of the Fermi surface in the nematic state, together with strong d xy decoherence, can account for the extreme spin fluctuation anisotropy unless tuned very close to the magnetic instability 157 . Models of localized spins are capable to describe the magnetic phase diagram of FeSe and predict C 4 symmetric spin fluctuations 134 ( Fig. 6(b)) or a strongly anisotropic fluctuation spectrum 135 , see Fig. 6 (c-e), resembling the INS data on twinned crystals presented in Fig. 7  (c) Structure factor from a tight binding model using reduced coherence of some orbital channels plotted along a high symmetry directions. For the twinned result, an average over two orthorhombic domains was performed. 153 .

C. Superconducting gap
The superconducting transition temperature of bulk FeSe is about 8 − 9 K. The symmetry and structure of the superconducting order parameter, in particular the existence of minima or nodes in the gap, determine the density of low energy quasiparticle excitations, and thereby the form of low-temperature power laws in thermodynamic and transport properties. The gap structure also indirectly reflects the form of the pairing interaction.
Here we review various measurements that provide information on gap structure, theories of T c and pairing, and what conclusions may be drawn.

Thermodynamic probe of quasiparticle excitations
Thermodynamic measurements such as specific heat and magnetic penetration depth probe low energy excitations, and became more reliable once high quality crystals of FeSe with large RRR (residual resistance ratio) were available 55,162,163 . However, the existence of quasiparticles at arbitrarily small energies, i.e. whether or not bulk FeSe has true gap nodes, as opposed to deep gap minima, has been answered differently by a number of studies. Initial measurements of the London penetration depth λ L by Kasahara et al. 117 reported a quasilinear temperature dependence for T T c , consistent with gap nodes, and µSR results were also claimed to be consistent with a nodal superconductor 160 ( Fig. 9 (c)). However other measurements observed a small spectral gap 158,159 ( Fig. 9 (a,b)).
The jump of the specific heat at the transition temperature T c of ∆C/γ n T c = 1.65 ( Fig. 9 (f)) seems to indicate a moderate to strong coupling superconductor 114 because of the deviation from the expected magnitude of a BCS superconductor, particularly since the large gap anisotropy tends to reduce rather than enhance this ratio. Some more recent investigations on the specific heat tried to extract the order parameter on the different bands 115,161,164-166 by fitting procedures. Measurements of field-angle dependent specific heat 167 found evidence for three distinct superconducting gaps, where the two smallest appear to be anisotropic and the smallest possibly nodal. The specific heat studies of Refs. 115, 168-170 tend to assign nodal superconductivity to FeSe ( Fig. 9 (g)), while Ref. 171 comes to the conclusion that the system is fully gapped. Reports of thermal conductivity are similarly split on the issue of a true gap: some observe fully gapped 161,172 and other claim nodal 117 behavior.
Several authors have attempted to grapple with these apparent conflicts 170,173 , by pointing out differences in low-temperature behavior according to small variations in growth techniques. In Ref. 174, the authors performed an STM study close to and far away from twin boundaries, pointing out that a full gap existed over rather large distance scales near the boundary, while the bulk was nodal. The authors attributed the full gap to the onset of a time-reversal symmetry breaking mixture of two irreducible representations in the pairing near twin boundaries. While this behavior is not reproduced in all STM studies 170 , it suggests that the density of twins, which in turn depends on sample preparation, could control thermodynamic properties at very low temperatures.
In any case, the theoretical implication is fairly clear: the observed strong sensitivity of the low-energy gap to disorder and twin structure almost certainly reflects an order parameter with accidental nodes or near nodes, i.e. not enforced by symmetry. The nematic phase of FeSe exhibits an orthorhombic crystal symmetry; thus the superconducting order parameter is necessarily a mixture of the corresponding tetragonal Brillouin zone harmonics, e.g. s and d. Thus, no symmetry protected nodal positions are expected, and small variations of the gap structure are possible due to differences in the sample preparation such as the local Fe:Se ratio, twin density, or internal stress.
Shallow nodes or near-nodes are then consistent, crudely speaking, with a near-degeneracy of s-and dwave pairing in the reference tetragonal system. It is important to note that the existence or nonexistence of a true spectral gap in the system is perhaps not the most important issue. On the other hand, if the gap is indeed formed due to the growth of a second irreducible repre- sentation near defects, this could be an important hint to the structure of the intrinsic pairing interaction.
One way to decide this issue is to probe the superconducting state with controlled disorder. A rapid suppression of T c upon introduction of (nonmagnetic) impurities or detecting a bound state in STS close to such an impurity is usually taken as evidence for a sign change of the order parameter. At present it is not clear if nonmagnetic impurities in FeSe are pairbreaking or not. In Fig. 9(g), specific heat data on four samples from the Karlsruhe group are shown. If one interprets the lower T c sample as the most disordered, as would be usual in an unconventional superconductor, the anticorrelation of T c with the residual Sommerfeld coefficient at T → 0 could be interpreted as a node-lifting phenomenon, where the spectral gap opens as disorder averages the order parameter; this effect has been established in Fe-pnictides as characteristic of accidental nodes 175 . A proton irradiation study also claimed to observe node-lifting induced by disorder 176 . Investigations on the field-dependence of the thermal conductivity 117,161 and specific heat 115,169 came to similar conclusions; for details see Ref. 23. On the other hand, penetration depth measurements by Teknowijoyo et al. 159 with controlled low-energy electron irradiation that creates Frenkel pairs of defects provided evidence that T c increases with increasing disorder (See Fig.  9(b). These authors considered various explanations for this remarkable result, including local enhancement of spin fluctuation pairing by impurities 177 and competition of superconductivity and nematic order 178 , but this question remains open.
In summary, the numerous studies agree on the point that FeSe exhibits a strongly anisotropic superconducting order parameter. Nodes, if these are detected, might be lifted easily by external manipulations or due to disorder 179 although the critical temperature does not seem to be very sensitive to such effects 170 . In addition, the appearance of a feature in the specific heat at very low temperatures 170 seems to be present in some samples only.

STM/ARPES measurements of gap structure
Measurement of the superconducting gap in momentum space is possible using ARPES, where the pullback of the spectral function in the superconducting state is used to obtain maps of the gap function ∆ k on the Fermi surface. STM measurements and the subsequent analysis of the quasiparticle interference makes use of the large partial density of states at saddle points of the Bogoliubov dispersion, and allows one to trace back the Fermi surface and measure the spectroscopic gap in the superconducting state. The latter experimental technique relies on the interference of quasiparticles scattered by disorder, and is additionally capable of detecting the phase of the superconducting order parameter 20 . From a theoretical perspective, spectroscopy near impurities can reveal the properties of the superconducting order parameter. A non-magnetic scatterer in a superconductor with no sign change of the order parameter cannot create impurity resonances within the superconducting gap 14 . In a system with a sign-changing gap, the spectral position of the bound state depends on the specific value of the impurity potential of the scatterer and might not be easy to detect. More recently a method that does not rely on bound states has been proposed 19,183 . It relies on the analysis of the antisymmetrized and (partially) integrated QPI signal, i.e. the Fourier transformed conductance maps as measured in STM. Other approaches to detect the sign change of the order parameter are based on similar mathematical properties of the tunneling conductance and interference effects [184][185][186][187] .
The tunneling spectra as measured on pristine surfaces of thin films 188,189 and crystals 117 exhibit a V shaped structure revealing a strongly anisotropic superconducting order parameter, see Fig. 11(a,f). A full, but small gap has recently been detected with high resolution measurements 20,171 . Bulk FeSe is orthorhombic, and the concomitant anisotropy of electronic structure and superconducting order parameter has been revealed by the observation of elongated vortices in thin films 190 and bulk single crystals 174 of FeSe. A detailed mapping of the order parameter on the Γ centered and X-centered Fermi surface was performed with high-resolution Bogoliubov quasiparticle interference 20 ; these authors find a highly anisotropic gap which has deep gap minima along the k x axis for the X pocket and the k y axis for the Γ pocket. This result is consistent with an ARPES measurement reporting a significant 2-fold anisotropy on the Γ pocket in lightly sulfur doped FeSe 180 , expected to have very similar properties as pristine FeSe since it is still deep in the nematic phase. The same findings for the Γ pocket 102,109,181 and the X pocket 109 were also reported subsequently by ARPES measurements on bulk FeSe itself. The data are summarized in Fig. 10, which shows that the ARPES measurements consistently find strongly anisotropic gaps with maxima on the flat sides of the elliptic Γ pocket. The order parameters between the holelike and electronlike Fermi surface sheets are also of opposite sign, as evidenced by the antisymmetrized tunneling conductance 20 , see Fig. 11 (e). This is consistent with the observation of nonmagnetic impurity bound states in this system 191 .

Orbital selective pairing
Theoretical investigations into the superconducting paring interaction and ground state order parameter for bulk FeSe suffer from several problems: As outlined in the previous section, FeSe seems to be more correlated (c) Topograph centered on a typical impurity with overlay of local structure (red x: Se atoms, yellow + Fe sites) 20 (d) Typical ρ− QPI map with integration area (black circle) corresponding to interpocket scattering that has been used to obtain the momentum integrated ρ− in (e) showing clean signature of a sign changing order parameter and no agreement to simulation data for non sign-changing order parameter (red curve) 20 . (f) High resolution differential conductance spectrum dI/dV (E) exhibiting two energy scales of the maximum energy gap two bands ∆max,α and ∆max, 20 . (g) Summary of the measured k-space structure of the energy gaps of FeSe for the two pockets ∆α and ∆ which exhibit a strong anisotropy and a sign change (red/blue color) 20 .
than other FeSC (such as the Fe pnictides, Fig. 2); thus reliable methods or phenomenological models describing the strongly correlated electronic structure are needed. Moreover, FeSe is highly nematic which also strongly modifies the superconducting state, i.e. a theoretical calculation needs to also include effects of the nematic order parameter. However, models for the electronic structure based on ab-initio methods are either derived from the tetragonal state, or need to begin with a stripelike magnetic ground state, which differs from the true low energy state of FeSe at ambient pressure. A way out it to use a model-based approach which starts from the electronic structure that agrees with the experimentally observed one, i.e. a tight binding parametrization with as few hopping elements 72 as possible, and fit these to agree with the positions and orbital content of the electronic structure as observed by ARPES and STM. Such an approach has been used already in the context of the cuprates 192 , for the Fe-pnictides [193][194][195] and was also adopted for the case of FeSe 20,96 . Starting from an electronic structure including a nematic distortion in the form of an orbital ordering term, see Eq. (9), and examining superconducting instabilities within the spin-fluctuation pairing approach (see Sec. II C) yields a strong mixture of harmonics of s-wave and d-wave character 95 .
Examining the effects of electronic correlation in more detail, it has been established that the FeSC (and therefore also FeSe) can be understood as a "Hund's metal", as first proposed in Ref. 53. The presence of the Hund's coupling in the interaction Hamiltonian, Eq. (2), leads to enhanced correlations and effective masses, tendencies to electronic configurations with high local spin and (most importantly for what follows here) to a differentiation, or "selectivity" of electronic correlation strength depending on the orbital character 54,64,196 . This renormalization is gradually enhanced as the electronic filling (in the corresponding orbital channel) approaches half filling (5 electrons/iron) where in a one band system the Mott transition would occur, but strongly modifies properties of the metallic state even for most of the Fe-based systems discussed here, which are quite far from this doping. This physics can be understood theoretically with the slave-spin mean field approach 196 or DMFT 50,51,53,63 . Orbital selectivity is clearly manifest in the FeSC 54,197 , including the Fe-chalcogenides, and has a clear connection to the nematicity in FeSe 108,198 . Most of the theoretical approaches to pairing in FeSe which will be reviewed in the following incorporate the basic fingerprints of the Hund's metal state and are therefore connected to the normal state electronic properties in this system as well.
In the QPI and subsequently ARPES analysis of the gap structure of FeSe, a strongly anisotropic order parameter was observed; it was pointed out by Sprau et al. 20 that its magnitude as function of Fermi surface angle follows the orbital content of the d yz orbital, suggesting the conclusion that the superconducting pairing is dominated by electrons in this orbital. Given the small nematic splitting of the electronic structure, this effect cannot a priori be explained by a pure spin-fluctuation scenario, i.e. such a calculation would give small anisotropy and/or small magnitude of the gap 96,128 , unless one tunes extremely close to the magnetic instability in an RPA approach where the spin fluctuations acquire a very nonlinear dependence on the interaction parameters 157 . Elec- Driven by orbital selective spin fluctuations (red and green arrows), the electronic structure exhibits shifts; pairing mainly mediated by dyz spin fluctuations yields a gap structure comparable to experiment, as shown in (b), while orbital splitting in the electronic structure only will lead to much smaller anisotropy 128 . Note that the theory of Ref. 156 is based on a Fermi surface at kz = π only, allowing for a much larger variation of the orbital content, such that a strongly anisotropic gap is easier to achieve. (c) Calculating the spin susceptibility and the charge susceptibility in a framework for higher order many-body effects yields a strong orbital dependence and favors an anisotropic s± state for FeSe at ambient pressure 49 . (d) Modifications of the orbital content on the holelike pocket due to nematic order Φ h , (e) same on the electronlike pocket at the X point, leading to a strongly anisotropic gap structure when pairing in the dxy channel is suppressed 156 . tronic correlations parameterized by a self-energy in Eq. (11) lead to band renormalizations and broadening of the spectral function, but also to reduced coherence of the electronic states. Usually, the self-energy is expanded in powers of frequency near the Fermi level by introducing of the quasiparticle weight as given in Eq. (15) such that the interacting Green's function on the real axis can be parameterized at small frequencies as with the quasiparticle weight Z k (Eq. (15)) and a broadening Γ k which is given by the imaginary part of the self-energy. In a multiorbital, multiband system, there are effects which cannot be described with the parameterization in Eq. (17). First, the intrinsic momentum dependence of the self-energy can induce non-local effects such as relative band shifts which turn out to be important for the electronic structure of FeSC 66-68 , and second, in general Σ(k, ω) is a matrix in orbital space which can induce different quasiparticle weights for different states at the Fermi level. For such an orbitally selective electron gas 53,54,129 , where the quasiparticle weights of the different orbitals are not identical, one expects that the quasiparticle weight at the Fermi surface of band µ acquires a "trivial" momentum dependence due to the matrix elements a µ (k F ) connecting orbital and band space, as well as one arising through correlations reflected in the orbital quasiparticle weight Z , such that on the Fermi The shifts of the eigenenergies can be captured in a phenomenological model that matches the band energies of the real material (as found experimentally), but the orbitally selective reduction of quasiparticle coherence 53,54,129 also needs to be incorporated. In the nematic state, this can also lead to a distinction of the d yz orbital and the d xz orbital correlations. To achieve the strongly anisotropic order parameter in FeSe from a theoretical calculation, one needs (1) strongly reduced coherence of the d xy orbital, as expected from many theoretical investigations within dynamical mean field theory (DMFT) or slave spin calculations for FeSC in general and FeSe in particular 199 . In addition, (2) a strong splitting of the pairing interaction in d yz and d xz channel is required, because both orbital components are detected at the Fermi level (for the holelike pocket 102,200 ). The second effect can be achieved by making the assumption that the d xz states are much less coherent than the d yz states 20,96 and doing a modified spin-fluctuation pairing calculation. One should in principle calculate quasiparticle weights in each orbital channel, requiring an approximation to the full self-energy, or self-consistency within renormalized mean field theories like DMFT or slave-spin theory 199,201 , but in practice the Z have been mostly free (fit) parameters so far (see Ref. 202 for an exception). However, in the spirit of Fermi liquid theory, one might employ an approach where one considers one experiment to fix the phenomenological parameters and then uses the same model to predict other observables such as penetration depth 160,203 and specific heat 203 . It turns out that the low-T , low-energy susceptibility as calculated from such a correlated model, where the Z are chosen to fit the QPI-derived gap structure in Ref. 20 yields a strong (π, 0) contribution, but essentially no (0, π) contribution at low energies 153 (Fig. 8 (c)). The low energy, low-T magnetic excitations have recently been measured in detwinned FeSe via neutron scattering, finding no (0, π) over a low energy range 143 , see Fig. 7 (f-k), in agreement with the prediction. As the temperature is raised, nematic order vanishes at T s , such that Z xz = Z yz . Such a theory then naturally explains the observed transfer of spectral weight 142,153 from (π, 0) to both (0, π) and (π, π), see Fig. 7 (c).
At the same time the decoherence, as parametrized by orbitally distinct quasiparticle weights Z xy < Z xz < Z yz can account for (a) the strongly anisotropic scattering properties on impurities in the nematic state 108 (Fig. 5) and (b) the difficulties to detect the Y pocket in spectroscopic probes 101,108 because the corresponding d xy , d xz states would be incoherent and thus the spectral weight, as calculated from Eq. (17), would be small as well. Similar arguments should hold for the QPI data, since the scattering amplitude from an impurity may be expressed via the Fourier transform of the modulations of the density of states, calculated using the T-matrix formalism by 14,108 Again, the quasiparticle weight enters quadratically through the Green's functionsĜ R (k, ω) characterizing the homogeneous system, which in the multiband case are matrices, as is the impurityT -matrix. The original choice of Z used to fit the gap structure worked well to explain the evolution of the normal state QPI at temperatures just above T c 108 , see Fig. 5 (a). Recently, alternative explanations of the latter two experimental results were brought forward in terms of effects of the three dimensional electronic structure giving rise to some averaging in the sum over k on the r.h.s of Eq. (18) 204 or possible lifting of the Y pocket due to band hybridizations 104,105 . However, the latter explanation has been questioned by a theoretical modelling of the expected spectral functions when accounting for an electronic structure that has a nematic order parameter in the d xz /d yz channel and the d xy channel and additionally hybridizations due to spin-orbit coupling are taken into account 88 . In the quest to find the microscopic origin of the strongly anisotropic order parameter, there have also been other theoretical proposals: Kang et al. examined the effects of a modified orbital content on the Fermi surface which, together with strongly correlated states in the d xy orbital channel and an induced anisotropy in the pairing interaction from small nematic splitting of the electronic structure, can also account qualitatively for the observed structure of the order parameter 156 , see Fig. 12 (d-e). As mentioned in the previous section, a phenomenological model of strongly anisotropic spin fluctuations, i.e. orbitally selective spin fluctuations can explain the modifications of the electronic structure in FeSe at low temperatures, thus giving rise to the Fermi surface shrinkage and nematic distortion 27,154 , see Fig.  12 (a). Subsequently, it was shown that the same spin fluctuations can also lead to a strongly anisotropic superconducting order parameter since these provide the strongly anisotropic pairing interaction 128 , see Fig. 12 (b). It is our belief that ultimately these two alternative approaches 96,128 are quite similar in spirit to the orbitally selective Z-factor approach described above; the equations solved are ultimately the same, but the required anisotropic pairing interaction incorporated in somewhat different ways. These approaches differ from that of Ref. 156, where anisotropy in the spin fluctuation spectrum must arise entirely from d xz /d yz orbital content anisotropy in the nematic state. This effect seems unlikely to explain the dramatic measured anisotropy in the spin fluctuation spectrum reported in Ref. 143.
Including vertex corrections in a calculation of the superconducting instabilities, a strongly anisotropic order parameter on the electron and hole pockets was obtained 49 , while the pairing interactions become anisotropic due to the creation of the orbital order from the intraorbital vertex corrections, see Fig. 12 (c). Another proposal assumes the existence of a nematic quantum spin liquid in FeSe that exhibits strongly anisotropic spin fluctuations. Taking additionally into account the spectral imbalance between d xy orbitals and d xz /d yz via orbital dependent Kondo-like couplings, one indeed finds a strongly anisotropic superconducting order parameter comparable to experiment 135 . detected 206 . Therefore, the experimental results have been interpreted in terms of the system's proximity to the BCS-BEC crossover regime where the Cooper pairs form already at T pair > T c and then eventually condense into a superfluid state at T c . In this limit, the size of the Cooper pairs is much smaller than the average distance between electrons, i.e. the product of the Fermi wavevector and the coherence length is small k F ξ 1. Consequences of proximity to the crossover regime in a one-band model are well known: that the chemical potential µ becomes negative and larger in magnitude than the energy gap, such that the band shows a back-bending instead of an opening of a gap when entering the superconducting state. As mentioned in the Introduction, however, the generalization to electron-hole multiband systems is not obvious because the chemical potential may be pinned with temperature, or nearly so, due to the presence of a band with opposite curvature. Thus, the pseudogap due to preformed pairs above T c that is characteristic of the one-band crossover regime, as well as the characteristic broadening of thermodynamic transitions, may not occur 207 . This indeed seems to be the case for FeSe and Fe(Se,S) 206 . However, the occurrence of a large diamagnetic response in weak magnetic fields above T c beyond the expected signature of Gaussian superconducting fluctuations yielding the Aslamasov-Larkin susceptibility might be a signature of preformed Cooper pairs 69 . For a more detailed discussion of these intriguing observations and the behavior of superconducting FeSe in high magnetic field, the reader is referred to a recent review 23 .
In fact, the prospect of observing phenomena characteristic of the BCS/BEC crossover was first raised in the Fe 1+y Te 1−x Se x system in measurements of the electronic structure using ARPES that found a very shallow holelike band that just crosses the Fermi level 208,209 and evidences for gap opening at higher temperatures 209 . Note that the superconducting gap in this system is larger than in FeSe and additionally, the chemical potential can be tuned by chemical doping using excess Fe 208 . Interestingly, also an electronlike band above the Fermi level at the Γ point has been detected which participates in pairing and shows effects of a pseudogap at temperatures of few Kelvin above T c 210 . The consequences of small Fermi energies in this system have been less comprehensively explored than in FeSe and FeSe,S due to the materials difficulties, and also because of the overwhelming interest in the topological properties of Fe 1+y Te 1−x Se x , see Sec. VII.

Spin fluctuations in superconducting state
In the superconducting state, a clear spin-resonance is observed around the stripe-type wave vectors 137 (see Fig.  7 (a,d), later confirmed in an experiment on detwinned FeSe; Fig. 7 (g)), consistent with a spin-fluctuationmediated superconducting pairing mechanism. Polar-ized neutron measurements found that low-energy magnetic fluctuations, including the superconducting resonance, are mainly along the c-axis 150 . The spin resonance in the superconducting state just at the (π, 0) momentum transfer vector was detected 143,153 (Fig. 7 (g)) and its dependence on field analysed 211 , pointing towards a sign-changing order parameter. From the perspective of the theoretical conclusions of this experimental finding, there are a number of proposals that actually predict or assume strongly anisotropic spin fluctuations to mediate superconducting pairing: the proposal of an orbitally selective spin-fluctuation mechanism 128 ; the nematic quantum spin liquid 135 ; and more exotic proposals on quadrupolar magnetic order which exhibits magnetic fluctuations in the dipole channel 134,212 (Fig. 6 (b)); and the picture of itinerant electrons exhibiting orbitally selective decoherence 96 , see Fig. 8.
Many of these theoretical proposals are effective lowenergy theories; it is important to have a phenomenology that can demonstrate a good agreement with inelastic neutron data, including the temperature -and energydependendent transfers of spectral weights and evolution of the strong spin fluctuation anisotropy up to the nematic temperature T s and at least over energies scales of ∼ 50meV or so in order to have some predictive power for superconductivity. To our knowledge only the last of the examples given 96,153 has been compared sufficiently closely with experiment to make this claim.

IV. EFFECTS OF PHYSICAL AND CHEMICAL PRESSURE
A. FeSe under pressure As discussed above, interest in FeSe among all Febased superconductors was initially muted because of the difficulty of preparing stoichiometric crystals, as well as the relatively low 8K T c . However it was soon realized that large changes in T c can be obtained by applying pressure 217-219 , up to a maximum of nearly 40 K at p of order 10 GPa. This remarkable enhancement was not initially associated with any change in magnetic order, but there were early hints from NMR that pressure strongly enhanced spin fluctuations 125 , consistent with the enhanced T c . Subsequently, Bendele et al. 220 used AC magnetization and muon spin rotation measurements to argue that the kink in T c vs. p that had been observed earlier at ∼ 1 GPa was in fact due to the onset of some kind of magnetic order. Conceptually the discovery of magnetism "nearby" the ambient pressure point was important, because it rendered less likely claims that FeSe was dramatically different from other Fe-based superconductors due both to the absence of magnetic order and to the lack of significant rise of spin fluctuation intensity in (T 1 T ) −1 at the nematic transition (as, e.g. in BaFe 2 As 2 ). With this and subsequent measurements of magnetism in the pressure phase diagram 213,216,[221][222][223] , it became clear that special aspects of the FeSe electronic structure were frustrating or suppressing long-range magnetic order of the usual type 139,140 in the parent compound. With a small amount of pressure, this special frustrating condition is relieved, allowing spin fluctuations and T c to grow, suppressing nematic order and leading eventually to a magnetically ordered state. The situation is summarized in the phase diagram of Fig. 13 (a).

B. FeSe under chemical pressure: S substituion
The similarity of phase evolution under chemical pressure to physical hydrostatic or near-hydrostatic pressure was noted already in the BaFe 2 As 2 system using P substitution for As. Similarly, in FeSe, we may expect that substituting the isovalent but smaller S atom on the Se site will act as chemical pressure. Indeed, as seen in Fig.  13 (b), the nematic order is supressed with S doping more or less as with pressure, with an apparent vanishing of nematic order at around S concentration of x = 0.17 accompanied with unusual properties in the resistivity 224,225 close to the quantum critical point (QCP). On the other hand, substitution of S does not appear to stabilize any long-range magnetic order. Perhaps equally interestingly, the superconducting critical temperature is not enhanced at the critical point, suggesting that nematic fluctuations themselves are irrelevant for superconductivity in this particular system, a conclusion that has also been drawn recently from an ab-initio study of FeSe 226 . Paul and Garst have pointed out that lattice effects should cut off the divergence of the nematic susceptibility, so this absence of a peak in T c is, in that light, not surprising 227 .
In general, the interplay of nematic and superconducting effects are subtle in Fe-based systems and may be dependent on details. In FeSe,S, the Karlsruhe group reported a surprising lack of coupling between the orthorhombic a, b axis lattice constant splitting (∝ nematic order) and superconductivity in FeSe, in stark contrast to Ba-122, where the splitting was suppressed below T c , indicating competition of the two orders. In FeSe there was no effect at all on a − b 55 at T c . When FeSe was doped with S by the same group in Ref. 168, a − b was found to increase as T was lowered below T c , indicating a cooperative effect of superconductivity and nematicity in these samples, see Fig. 14 (a,b). The reason for this difference between the two canonical Fe-based families is not clear at this writing 228 ; from a theoretical point of view, details of the relative orientation of Fermi surface distortion and gap function, along with orbital degrees of freedom, may govern this behavior 229 .
To compare how chemical pressure and external pressure affect the FeSe system, several groups have subjected FeSe,S samples at different dopings to external pressure 216,233 . For example, the Tokyo-Kyoto group has measured the comprehensive phase diagram shown in Fig. 13 (c). The apparently inimical effect of the S-substitution on magnetism already mentioned above was shown explicitly, with the shrinking of the pressureinduced SDW phase. The similar effects of pressure and S-substitution on nematic order are also confirmed: the weaker the nematic transition temperature T s in x, the smaller its extent in pressure.

C. Diminishing correlations
Given the large discrepancies discussed above between DFT and DMFT electronic structure calculations in the parent compound FeSe, it is interesting to investigate the evolution of the correlations assumed responsible for these dramatic renormalizations with physical and chemical pressure. In both cases, one naively expects that the compression of the lattice should result in a decrease of the effective degree of correlation, simply because the kinetic energy (hoppings) in such a situation should be enhanced due to the decreased distance between the ions, whereas the effects on local interaction parameters like U and J should be smaller.
The first systematic study of the electronic structure changes with S doping was performed with ARPES by Watson et al. 230 , who reported results roughly in line with expectations. Increasing S substitution was found to enlarge the Fermi surface pockets, and increase the Fermi velocities, albeit by a relatively small amount, see Fig. 14(c). While the pocket size thus changes to become closer to (but still smaller than) DFT predictions, the d xy band predicted in DFT studies at the Fermi level remained 50 meV below in the ARPES study, and thus never plays a role in the chemical pressure phase diagram. At the same time, these authors pointed out a systematic decrease in the d xz /d yz orbital ordering as the structural transition temperature fell (Fig. 13). Finally, the apparent weakening of correlations also led to a Lifshitz transition around x = 0.12 as the inner d xz /d yz hole pocket, pushed below the Fermi level in the parent compound, reappeared. While this evolution was largely confirmed in a subsequent Shubnikov-de Haas study by Coldea et al. 231 , these authors reported the disappearance at x = 0.19 of the small oscillation frequency associated with the outer d xz /d yz hole pocket (see Fig. 14 (c)), and interpreted it as a second Lifshitz transition where the outer cylinder pinched off at k z = 0 to form Z-centered 3D pockets.

D. Abrupt change in gap symmetry in tetragonal phase
The evolution of the superconductivity with S substitution is clearly of great importance, since it provides a clue to how the changing electronic structure, including the disappearance of the nematic order, affects the pairing. While T c itself is relatively insensitive to these changes, Sato et al. 214 reported evidence that the superconducting gap undergoes a dramatic change at a concentration around x = 0.17, very close to the disappearance of nematic order.
We begin by summarizing what is known about the gap structure in the nematic phase of FeS 1−x S x . At x = 0, as discussed in Sec. III, the gap structure measured on the hole and electron pocket detectable by spectroscopy at the Fermi surface is highly anisotropic and nematic. As shown in Fig. 10(a), 3% sulfur does not change the gap structure significantly, at least on the α pocket where it was measured, so it seems likely that the gap structure for small nematicity is quite similar to FeSe. There are no other direct measurements of the gap in FeSe,S of which we are aware at higher doping, so information about gap structure has been deduced mostly from thermodynamics.
As shown in Fig. 14 (e), the superconducting state specific heat is consistent with a zero residual value in the nematic phase, while jumping to a rather large residual value in the tetragonal phase. Since the gap in FeSe is known to be highly anisotropic, with nodes or nearnodes, it is tempting to attribute any such residual term to disorder. There are several reasons to reject this explanation, however. First, STM topographs on these samples suggest that they are very clean, inconsistent with residual Sommerfeld coefficients γ s /γ n of O(1) 232 . Second, there is no reason to expect a discontinuous change in the disorder itself at the nematic transition, so one would have to postulate that the superconducting state undergoes a transition making it fundamentally more sensitive to disorder. Since the system evolves already out of a state with nodes or near-nodes, and an established gap sign change 20 , it is far from compelling that a transition, e.g. from s ± to d-wave would cause such an abrupt enhancement of the residual density of states, which is clearly present, as shown also directly by STM 232 (Fig. 14 (g)). Finally, it is expected that the S dopants, away from the Fe plane, act as relatively weak scatterers, and could in any case not give rise to a γ s /γ n of O(1).
Traditionally, the existence of a finite residual κ/T as T → 0 is taken as an indication that the unconventional superconductor in question has line nodes. However, this is normally a signature of quasi-universal transport 234 , as in the canonical d-wave case, where κ/T in the limit of weak disorder is a constant independent of the quasiparticle relaxation time (the limiting low-T value is not quite universal in an s ± state, but the nonzero κ/T does remain even in the limit of vanishing disorder 235 ). This helps explain why for low x, the FeSe,S material displays a small thermal conductivity (Fig. 14 (f)) that stays roughly constant over several low dopings. On the other hand, it is also seen that the x=0.20 doping residual κ/T jumps significantly to a large fraction of the normal state value, inconsistent with the usual "universality" arising from superconducting gaps with line nodes.
In the STM data reproduced in Fig. 14 (g), it is clear that changes in the superconducting state are not confined to the finite value of the residual DOS that occurs near x = 0.17. In addition, it is seen from the drop in coherence peak energies that the gap is becoming abruptly smaller in magnitude. This, together with the fully developed Volovik effect observed in Ref. 214 in the tetragonal phase, led the authors of Refs. 214 and 232 to conclude that as the system becomes tetragonal, the gap structure beyond the nematic critical point was becoming even more anisotropic than it was known to be for FeSe.

E. Bogoliubov Fermi surface scenario
One intriguing solution to this puzzle was put forward in Ref. 236, where it was suggested that the system might naturally make a transition into a topological state that manifested a so-called Bogoliubov Fermi surface, a locus of points in k-space that supported zero energy excitations at low temperature in the superconducting state. Note this manifold has the same dimension as that of the underlying normal state Fermi surface, i.e. is a 2D patch in a system of three spatial dimensions, etc., as distinct from an unconventional superconductor with line or point nodes. This state was a generalization to spin-1/2 multiband systems of an idea of Agterberg and collaborators 237,238 for a system of paired j = 3/2 fermions. The conditions for the existence of this topological transition are that the pairing be in the even parity channel, with dominant intraband spin singlet gaps (e.g. ∆ 1 and ∆ 2 for a 2-band systems) together with SOC-induced triplet interband component δ. The latter amplitude is assumed to spontaneously break time reversal symmetry in spin space, analogous to the A1 phase of the 3 He superfluid 239 . The authors show that the Pfaffian of the system is proportional to |∆ 1 (k)||∆ 2 (k)| − δ 2 , such that the change from trivial to topologically nontrivial, accompanied by the creation of the Bogoliubov Fermi surface, occurs when the Pfaffian changes sign from positive to negative. Although δ is expected to be small, the topological transition can be achieved due to the nodal (or near-nodal) structure of the interband gaps.
The existence of a Bogoliubov Fermi surface in this system would naturally explain why a relatively clean superconductor can support a finite density of quasiparticles, as reflected in the residual Sommerfeld coefficient and the differential conductance seen in STM. Clearly, independent verification of the assumed time reversal symmetry breaking is needed before such an explanation can be accepted above more conventional ones, but the idea is intriguing. Very recently, a signature of TRSB in tetragonal FeSe,S from µSR was reported 23  Despite the unusual features of the alkali-intercalates, interest in the FeSe system was relatively muted until the discovery in 2012 of high temperature superconductivity in monolayer FeSe films epitaxially grown on SrTiO 3 by the group of Qi-Kun Xue 243 . Within a relatively short period, it was established a) that the superconducting gap magnitude was much larger than bulk FeSe or FeSe films grown on other substrates 244 , such as graphite; b) that 2 monolayers were either not or rather weakly superconducting 243,245 ; c) that the gap closing temperature of the best films (according to ARPES) was in the neighborhood of 65K 246 , the highest T c measured in the Fe-based systems to that date; and d) that the electronic structure was more similar to the alkali-doped intercalates (and LiOH intercalates, discovered later, see Fig.  15), in the sense that the Γ-centered hole band had a maximum ∼ 80meV below the Fermi level, such that the Fermi surface consisted of electron pockets only. Fig. 16 (b) shows the epitaxial structure of the film used originally to obtain a transport T c four times higher than bulk FeSe (8K); and (c) the subsequent ARPES gap closing temperature of 65K measured on similar samples 246 . The T c enhancement in monolayer FeS films on STO is not observed 247 and the same was found when FeSe is deposited on graphene 244 or Bi 2 Se 3 248 . Taken together, these observations point to a unique high-T c superconducting system based on FeSe where the substrate SrTiO 3 plays an essential role. Exactly what that role is, is still being debated. Here we sketch and extend the discussion of this question in the excellent review of Huang and Hoffman 61 .

Electronic structure and electron doping
ARPES measurements have elucidated in great detail how the high-T c superconducting state evolves out of the as-grown sample, and how the requisite electronic structure evolves with it 246 . When one starts the annealing process, the Γ-centered hole band is at the Fermi level, as in a typical Fe-pnictide, but by the final stage it has been pushed 80 meV below, and the electron pockets have correspondingly enlarged. The Fermi surface therefore consists only of electron pockets at the M points (Fig. 15). Note that the Fermi surface obtained by standard DFT calculations is significantly different from ARPES, even if the system is electron doped "by hand" using rigid band shift or virtual cluster methods, and even accounting for the strained lattice constant imposed by the STO 61,243 . However, strain-modified hopping parameters were proposed to qualitatively account for the band structure in the monolayer FeSe 249 .
Initially it was believed that Se vacancies created in the film itself during the annealing process might electrondope the film, but these appear to induce a hole doping effect instead 250 . More recently, attention has focused on the doping of the STO layer by O vacancies in various configurations. In DFT studies of the FeSe-STO interface, it has been speculated that the O vacancies give rise to a T c enhancement due to a surface reconstruction 251 or suppression of an incipient monolayer spin density wave 252 . A problem with such calculations is that unphysically large O vacancy concentrations appear to be required to suppress the position of the hole band sufficiently, suggesting that electron correlation effects play an important role, consistent with conclusions for the bulk FeSe material. Indirect evidence against O vacancy doping scenarios comes also from measurements of FeSe on anatase TiO 2 (001) surfaces, where high-T c superconductivity was deduced by large STM gaps similar to FeSe/STO (001). In this system, direct imaging of O vacancies was shown to give a concentration much too small to account for the doping level observed 253 , and variation of O vacancy content did not affect the gap.
Other theoretical approaches to the charge transfer problem focus on the novel properties of the STO itself, in particular due to large work-function mismatch. In this picture, the strong coupling to long-wavelength polar phonons generated in the depletion region by the nearly ferroelectric character of STO can enhance superconductivity 254 . Charge transfers of the required magnitude can be obtained by this mechanism, but the details of the renormalized band structure of the interface was not addressed in this work.

Structure of the interface
The calculations and analyses above assumed a single layer of FeSe deposited on the TiO 2 terminated layer of SrTiO 3 . However, as pointed out by Huang and Hoffman 61 , the fabrication process does not necessarily result in such a simple structure. Several groups provided evidence for a reconstructed interface that creates a TiO x double layer at the interface 255,256 . This may certainly aid in the charge transfer process to the FeSe, but suffers from the same requirement as discussed above that the absolute number of O vacancies required, of O(50%), appears to be too large. Zhao et al. 257 performed scanning electron transmission (STEM) imaging together with electron energy loss spectroscopy (EELS) and concluded that, in addition to the double TiO x layer, a Se layer in proximity to the FeSe was necessary to ex- plain observations. This is difficult to understand because the annealing process is intended specifically to remove excess Se. A further proposal came from Sims et al. 258 , who suggested on the basis of STM, STEM and DFT calculations that an interlayer close to Ti 1.5 O 2 (excess Ti) "floats" between the FeSe and the STO, weakly van der Waals coupled to both, and provides part of the requisite doping to shift down the central hole band.

Transition temperature
While monolayer FeSe on STO has frequently been cited as exhibiting the highest critical temperature in the FeSCs, it is important to examine this claim critically. Normal published accounts of superconductivity require proof of a) zero resistance and b) Meissner effect, as well as, ideally, c) other measures of an energy gap closing. Most of the measurements of T c in these systems have been of type c), rather than a),b), due to the difficulties inherent in the low dimensionality, as well as air sensitivity of the samples. ARPES, which measures the onset of an energy gap in the one-particle spectrum rather than a phase coherent state, has generally reported the highest T c 's 243,246,[259][260][261] , between 60-70K, see Fig. 16 (c). Most attempts to cap the samples to avoid the air sensitivity problem have apparently led to sample degradation, such that ex situ transport measurements have generally yielded considerably lower zero resistance T c 's in the 20-30K range, with 40-50K onset values. Very recently, a zero resistance T c of 46 K was reported for monolayer films on LaAlO 3 substrates 262 . The single in situ transport result, reporting a T c of 109K with a 4-probe "fork" measurement 263 , has not been reproduced.
Magnetization measurements 264-266 generally report high onset temperatures consistent with ARPES, but have very broad transitions, and significant suppression of the magnetization does not occur until lower temperatures near the zero resistance T c 's of the ex situ transport zero resistance measurements. Still, it is possible that these lower T c 's are due to extrinsic experimental difficulties. An outstanding problem is therefore to prove that long-range or quasi-long range superconducting phase coherence really does set in at the higher (60-70K) temperatures with probes of true superconducting order rather than gap closing.

B. Dosing of FeSe surface
The question of why T c , interpreted optimistically, is so much higher for the FeSe monolayers on STO than either FeSe itself, or, for that matter, all the other Fe-pnictides and chalcogenides, led to speculation that the high levels of electron-doping, perhaps related to the special Fermi surface structures shown in Fig. 15 might be responsible. This led to new attempts to enhance T c via doping by novel means. Utilizing a technique pioneered by Damascelli for the cuprates 267 , potassium atoms were deposited on the surface of films or crystals of FeSe 268-270 , sufficient to increase the electron doping of the surface layers. Critical temperatures of the surface layer as high as 40 K were reported, roughly the same as bulk FeSe maximum T c under pressure, and similar to FeSe intercalates (see below). Similar results were reported utlilizing ionic liquid gating 271 . Alkali dosing experiments were performed also on FeSe/STO monolayers, with about 10% additional electron dosing leading to a jump of about 10K to about 70K 272 . The authors of Ref. 272 associated the initial rise of T c in crystals upon electron doping to a Lifshitz transition when the central hole pocket disappears, and the rise in the more highly doped FeSe/STO monolayers as due to a second Lifshitz transition when a central electron pocket appeared, as shown in Fig. 16(d).

C. Replica bands and phonons
A potentially important clue to the physics of these systems, and the influence of the substrate, was found in ARPES measurements 260 , which identified in second derivative spectra clear "replica bands", "shadow" copies of bands both at Γ and M shifted rigidly downward in energy by ∼ 100 meV. These experiments were interpreted by the authors of Ref. 260 as implying the presence of a strong interaction of FeSe electrons with phonons, probably originating from the substrate. It was furthermore argued that the electron-phonon interaction must be rather strongly peaked in the forward direction q = 0 to explain this observation. The connection with the unusually high T c was more or less circumstantial, but shortly thereafter it was proposed that coupling primarily to such a forward scattering phonon could provide a natural explanation for the high T c , since T c was found to vary linearly rather than exponentially with the electron-phonon coupling in this extreme case 273 .
It is important to note that such forward scattering phonons are potentially interesting for two reasons. First, electron-phonon superconductivity and unconventional superconductivity based, e.g. on spin fluctuations generally compete, because the former usually rely on strong repulsive interband interactions. Conventional electronphonon coupling vary slowly with momentum transfer, and therefore suppress the interband interaction leading to pairing. Forward scattering phonons, on the other hand, provide an attractive intraband pairing and therefore should add to the total attraction leading to pairing. Secondly, the electron-phonon interaction by itself may lead to an unusually large T c due to the linear dependence on the coupling 273 ; however this requires that the repulsion seemingly present in all other FeSC would be negligible in this system, which seems unlikely. In other words, forward scattering allows phonons to assist spin fluctuations, but may not necessarily amplify the usual electron-phonon mechanism in any qualitative way.
STM and ARPES measurements have both reported a full superconducting gap in the monolayer system, of order 20 meV, drastically different both in magnitude and in isotropy relative to bulk FeSe. There are two coherence-like peaks in the STM spectrum 243 (see Fig.  17 (a)), not unlike other Fe-based systems; however in this case, the hole band is presumed not to participate in superconductivity. Furthermore, the two electron pockets at M in ARPES do not appear to hybridize 260 , so that the double peak is unlikely to be explained by two isotropic gaps on these bands. The most likely scenario is that the two energy scales indicate two independent maxima on the electron pockets (minima do not lead to peaks in the STM spectrum, at least within BCS theory), as indeed measured by ARPES 275 .
Despite the apparent effect of phonons on the ARPES measurements, electron-phonon interactions in the FeSe are likely too weak to alone explain a T c of 70K or above [276][277][278] (a calculation that finds a much higher T ph c than others under some rather generous assumptions is given in Ref. 279). Thus a "plain" s-wave from attractive interactions alone seems improbable, even if soft STO phonons play a role 260,280 . The forward scattering scenario for electron-phonon processes 260,273 then implies that phonons cannot contribute significantly to the interband interaction.

D. Pairing state in monolayers
1. e-pocket only pairing: d-and bonding-antibonding s-wave By itself, the interband spin fluctuation interaction due to pair scattering between electron pockets, considered in the 1-Fe zone, should lead to nodeless d-wave (since χ(q, ω) will be roughly peaked at the momentum connecting the electron pockets) 281,282 . The double maximum in the gap function found by ARPES 275 does not arise from conventional spin fluctuation pairing theory for this system 96 , however, so forward scattering phonons could potentially not only boost this mechanism, but also contribute to the observed anisotropy of the d-wave gap on the electron pockets.
An alternative explanation entirely within the spinfluctuation approach, but incorporating the orbitalselective renormalizations of the dynamical susceptibility as described in Sec. III C 3 was given in Ref. 96. Simply suppressing the d xy orbital weight Z xy , with nearly negligible renormalizations of the d xz/yz weights, consistent with the nearly absent evidence for nematicity in this system, was sufficient to produce the double maximum of the d-wave gap on the electron pockets at the correct energies ( Fig. 17 (c)). Recently, spin-fluctuation mediated pairing was examined using a full-bandwidth Eliashberg approach, finding a d-wave instability as well, but claiming that the high critical temperature cannot be obtained 283 . The existence of such a d-wave gap was also deduced phenomenologically from STM measurements on thin films of FeSe on SrTiO 3 with a step edge 284 .
In the 2-Fe zone, the two electron ellipses overlap and may hybridize due to orbital mixing or SOC. The former is forbidden by symmetry in the monolayer, but SOC may play a role. In the case of large hybridization from either source, the bonding-antibonding s-wave state between two electron pockets 1 is expected to be stabilized by repulsive interactions 285 , but ARPES has not observed any hybridization of the two electron pockets, suggesting that these effects are small 260 . This point is also relevant for the discussion of nodes on a d-wave gap. A d x 2 −y 2 state defined in the 1-Fe zone has nodes along the (0,0)-(π, π) direction, which does not intersect the electron pockets. However, to the extent the two elliptic pockets hybridize and split, nodes will be forced on the hybridized inner and outer electron sheets. Note that these nodes on the Fermi surface are not required by symmetry as in the 122 crystals 286 , and have a narrow angular range proportional to the magnitude of the hybridization, and hence are sometimes referred to as "quasinodes" 287 . In the monolayer system, ARPES and STM report a full gap, as discussed above; so if such quasinodes exist, they must contribute a negligible amount of phase space for low-energy excitations.

SOC driven pair states
The spin-orbit coupling is not particularly strong in the 3d Fe-based superconductors, but it is sufficient to create band splittings of order tens of meV in the lowenergy band structure, which can lead to some important effects. Furthermore, it has been found to be particularly strong in the 11 materials 92,288 . Direct evidence for the significance of SOC for superconductivity comes from the discovery of magnetic susceptibility anisotropy (χ xx = χ yy = χ zz ) in the neutron spinresonance 150,289,290 , which is generally understood to exist due to a coherence factor depending on the sign change of the superconducting gap below T c . This type of spin response anisotropy in the superconducting state can in principle be captured qualitatively merely by incorporating SOC in the electronic structure but ignoring it in the superconducting pairing itself 291 . Similar approaches to incorporating SOC in superconductivity were adopted in treatments of spin fluctuation pairing in 122 and 111 systems, to the extent that the SOC-induced hybridization of bands at high-symmetry points on the electron pockets was included 193,287 . This effect was in fact found to be rather small. However, it has been argued that in the strongly electron-doped systems, the hybridization of the electron pockets is more significant due to stronger SOC 11,287,292,293 .
It is clear in general that the pairing problem itself is also influenced by SOC. Since the L · S interaction preserves time reversal symmetry, one should pair states in the pseudospin basisà la Ng and Sigrist 294 . Including these effects in realistic multiband models with repulsive interactions can be quite cumbersome, but was implemented by Scherer and Andersen, who also found that within the traditional RPA approach, the effects of SOC on the gap structure were actually negligible 295 . Recall, however, that this method considers pairing only between time-reversed states at the Fermi level belonging to the same band.
In an alternate approach, SOC is included in the onebody Hamiltonian treated in the k · p approximation, which preserves crystal symmetry near the M points, and the Hubbard-Kanamori interaction is projected onto these low-energy states and decomposed in mean field theory. In one-band interacting Hamiltonians with repulsive interactions, such a procedure cannot lead to a stable pair state, and one is forced to compute the effective interaction that leads to unconventional pairing in the usual way 4 . In multiband models, although all bare interactions U, J, U , J are repulsive, attractive interactions are found in certain channels for some choices of the interaction parameters. For example, in the Vafek-Chubukov model 296 for two orbitals and two hole pockets, the "A 2g " spin-triplet state corresponds to interaction constant (U − J)/2, which may under physically reasonable circumstances become negative. Here ψ r,σ = (d † yz,σ (r), −d † xz,σ (r)), τ i , s i are Pauli matrices in orbital and spin space, respectively. It is easy to check that there is no Cooper logarithm driving a superconducting instability in such cases, so a finite threshold value is required for the coupling. However, the effect of even infinitesimal SOC is found to induce a Cooper log 296 , leading to the suggestion that such exotic pair states occur "naturally" in Fe-based superconductors despite interorbital pairing, which would normally be suppressed because it generically requires significant pairing of electrons of states away from the Fermi energy. In the case of electron pocket only systems, the even parity states stabilized in mean field 293,297 are essentially those proposed by Khodas and Chubukov 285 , but of course contain admixtures of spin-triplet components due to SOC.
Agterberg and co-workers have followed a similar approach, considering a phenomenological pairing interaction purporting to describe spin-fluctuations of Néel type, which in the 2-Fe zone correspond to small-q scattering processes 292,298 . These then scatter pairs between folded electron ellipses. Spin-orbit coupling hybridizes these ellipses as expected, but for values consistent with upper bounds set by ARPES 299 may produce "naturally" a true nodeless d-wave state, consistent with experiment. A disadvantage is that the theory, which studies only two pockets, does not apparently distinguish the d-wave from the bonding-antibonding s-state that should occur at sufficiently large SOC.
A final proposal for e-pocket only systems studies the exchange of spin and orbital fluctuations, going beyond the usual RPA to include Aslamosov-Larkin type vertex corrections, claiming a conventional s-wave ground state 300 .

incipient band s± pairing
Although the Γ-centered hole band observed by ARPES lies ∼ 80meV below the Fermi level and is usually neglected in pairing studies, a few authors have discussed the possibility of pairing in the "incipient s ± " state, driven by the conventional spin fluctuation interaction between the hole band and the electron band centered at M . Naively, such a state is disfavored by energetic arguments 1 . On the other hand, Bang 301 and Chen et al. 302 revisited these arguments and found that in the presence of robust Fermi surface-based superconductivity (e.g. a phonon attraction in the electron pockets of the monolayer) this state could be strongly favored, consistent with the findings of Lee and co-workers 260,280,303 .
This scenario does not address the central question of why superconductivity appears to be stronger in situations with electron-like Fermi surface pockets only. However, Linscheid et al. 304 pointed out that if one includes the dynamics of the spin fluctuation interaction, high-T c pairing in a traditional s ± state with incipient hole band could be understood. Consider a situation with a constant interband pairing interaction between an electron band crossing the Fermi surface and an incipient hole band; without an intraband attraction, no robust superconductivity can be produced 70,302 .
On the other hand, if the interaction is calculated selfconsistently, moving the hole band below the Fermi level can enhance the interband pairing because the paramagnon spectrum is peaked at a finite energy ω sf ∼ 50-100meV. Thus within the incipient band s ± picture, a "sweet spot" in the pairing interaction can be obtained where the hole band extremum is a comparable distance below the Fermi level 304 . This scenario has not yet been confirmed within a realistic multiorbital framework; non-selfconsistent Eliashberg calculations using multiple bands find only a weak enhancement of T c due to incipient pairing 283 .

E. Impurity experiments
The previous section mostly reviews proposals for pairing states in the FeSe monolayer that involve sign changes of the order parameter over the Fermi surface, ultimately due to the repulsive electronic interactions. On the other hand, there is some evidence that the system does not have a sign-changing gap. In STM measurements by Fan et al. 305 T c and the gap were reported to be suppressed only by magnetic impurities, as one might indeed expect from a "plain" s-wave SC. These arguments, if correct, would also rule out states of the "bonding-antibonding s-wave" type 1 . However, the impurities in these studies were adatoms on the surface of the monolayer rather than atoms substituting in the layer itself, and it is possible that the potentials produced in the Fe plane were simply too weak to produce bound states. In a subsequent study where various impurities were incorporated into the monolayer, bound states were observed for certain nominally nonmagnetic atoms 182 , suggesting a sign change in the superconducting order parameter.

A. Alkali-intercalated FeSe
The apparent paradox of high-temperature spin fluctuation driven pairing in electron-pocket only systems was raised first in the context of the alkali-intercalated FeSe materials, discovered in 2010 306 , which nominally correspond to the chemical formula AFe 2 Se 2 , with A=K,Rb,Cs. To this date, the superconducting samples of these materials are available only in mixed-phase form and have relatively low superconducting volume fractions. There is considerable evidence from STM and xrays that the superconductivity exists only in 3D filamentary form 307 . These systems nevertheless excited considerable interest both because of their proximity to unusual high-moment block antiferromagnetic phases, and because ARPES 308 measurements on KFe 2 Se 2 reported that there were no Γ-centered hole pockets at the Fermi level (although a small electron pocket pocket is found near the Z point near the top of the Brillouin zone). At Γ, the hole band maximum is ∼ 50 meV below the Fermi level. An example of one of the ARPES-determined Fermi surfaces of these materials is shown in Fig. 15  (c). While there appears to be some spectral weight at the center of the zone, the centroid of the band is found to be well beneath the Fermi level.
Several workers recognized early on that despite the missing hole pockets, repulsive interactions at the Fermi level existed between electron Fermi surface pockets, which could lead to d-wave pairing with critical temperatures 281,282 of roughly the same order as in the usual s ± hole-electron pocket scattering scenario. As discussed above, the expected hybridization of the two electron bands in the proper 122 body centered tetragonal crystal symmetry leads to two roughly concentric electron Fermi surface sheets at the M point in the 2-Fe Brillouin zone, leading also to the possibility of the bonding-antibonding s wave state, as in the monolayer. In the 122 structure, however, such a state was found to be subdominant to d-wave pairing due to the relatively weak hybridization found in first principles calculations for KFe 2 Se 2 287 . However, the bonding-antibonding swave remains an interesting candidate in part because these systems are apparently intrinsically multiphase; therefore one may question the conventional electronic structure derived from ARPES results, which rests on the picture of a metallic, filamentary phase embedded in an insulating background, and the assumption that averaging over a micron size domain provides a reliable description of the intrinsic properties of this phase.
Inelastic neutron scattering measurements 309 agree rather well with the wave vector ∼ (π, π/2) for the neutron resonance found in calculations of Ref. 282, corresponding to scattering between the sides of the electron pockets centered at (π, 0) and (0, π) in the 1-Fe zone. Such an interorbital scattering process was found to lead to a d-wave ground state, which appears to disagree with the absence of nodes on the small Z-centered pocket observed by ARPES 310 . It may be difficult given current momentum and energy resolution to reliably measure the spectral function pullback below T c on an extremely small Fermi pocket, however. In addition, Pandey et al. 311 argued that the bonding-antibonding s wave state would also support a resonance at roughly the same wave vector observed in experiment. This explanation is natural in 2D, since the bonding-antibonding state represents a folded d-wave state in the 2-Fe zone stabilized by hybridization, it is less obvious in 3D for the bct crystal structure of these systems. Furthermore, such a scenario requires significant hybridization 285 that is not present, at least in DFT 287 .
The uncertainties associated with this system, particularly the materials issues, have left the question of pairing unresolved. Taking the nodeless Z-pocket as a given, another solution was proposed by Nica et al. 312 as a way of understanding both neutron and ARPES experiments, by constructing a pair function that builds both s and dwave symmetry into different orbital channels, such that different symmetry channels effectively dominate different Fermi surface sheets. The resulting orbitally mixed state was shown to be the leading candidate within a t − J 1 − J 2 mean field calculation over some range of parameters.

B. Organic intercalates
The origin of the higher T c in the alkali-intercalated FeSe remains unclear, but after their discovery one obvious explanation was simply the enhanced FeSe layer spacing, possibly by enhancing two-dimensionality and therefore Fermi surface nesting. Intercalation of larger spacer molecules between the layers, initially organic molecular complexes, was achieved shortly thereafter [313][314][315][316][317] . These materials indeed had higher T c 's, up to 46K, but were extremely air-sensitive and only powders were available, so there is no ARPES data on either Li 0.56 (NH 2 ) 0.53 (NH 3 ) 1. 19 Fe 2 Se 2 with T c = 39 K 315 or Li 0.6 (NH 2 ) 0.2 (NH 3 ) 0.8 Fe 2 Se 2 with T c = 44 K 313 . Noji et al. 316 reported a wide variety of FeSe intercalates, along with a strong correlation of T c with inter-FeSe layer spacing, with a quasilinear increase between 5 to 9Å, after which T c saturated between 9 to 12Å. This tendency was plausibly explained by Guterding et al. 318 within spin-fluctuation pairing theory as due to a combination of doping and changes in nesting with increasing twodimensionality. Recently, Shimizu et al. 319 extended this work with a detailed discussion of the doping dependence of T c in the organic intercalate Li x (C 3 N 2 H 10 ) 0.37 FeSe. While the ammoniated FeSe intercalates are fascinating, their air sensitivity prevented many important experimental probes and limited their utility.

C. LiOH intercalates
As mentioned above, there is some similarity between the low-energy band structures of several electron-doped FeSe materials a Fermi surface without Γ-centered hole pockets, including the monolayers on STO and the alkalidoped intercalates, already discussed above. There is a third class of air-stable FeSe intercalates that fits into this category 242 , the lithium iron selenide hydroxides, reported in Refs. 321 and 322 ( Fig. 15 (a)).
While the surface of the alkali-intercalated FeSe does not cleave easily, and aside from a full gap it is difficult to discern distinct features 323 , the STM spectra of FeSe monolayers and LiOH-intercalated FeSe show a striking similarity, with both exhibiting double coherence peaks with roughly the same large gap/small gap ratio 320 , both with extremely large inferred gap-T c ratios of order 8 (see Fig. 17). Du et al. 320 attributed the two peaks to gaps on two hybridized electron pockets, in contrast to the interpretation of Ref. 275, which proposed two separate maxima on each unhybridized electron pocket in the case of FeSe monolayer (see Fig. 17 (c)). Note that within BCS theory, only gap maxima, not minima, produce peak structures in the density of states.
The further observation of an in-gap impurity resonance at a native (Fe-centered) defect site 320 suggests  96 ; the two maxima in ∆ give rise to two saddle points in the quasiparticle dispersion, consistent with two coherence peaks in the spectra.
that the gap is sign changing. This conclusion was bolstered by a subsequent study of phase-sensitive quasiparticle interference (QPI) by the same group 21 , who found a strong single-sign antisymmetrized conductance between the two gap energies, a signature of sign-changing gap structure 19 . They concluded that the LiOH intercalate system has a sign-changing gap, but could not distinguish reliably between a binding-nonbinding s ± state and a nodeless d-wave state.
As discussed in Sec. V D 2, treatments of pairing in the orbital basis including SOC have come to the conclusion that additional superconducting states beyond the d-wave and bonding-antibonding s-wave may play a role in the electron-pocket only systems if SOC is important 293,297 . These exotic pair states are either not present in the conventional Fermi surface based approaches, or correspond to strongly subdominant pairing channels. Specifically, in the context of LiOHintercalated FeSe, Eugenio and Vafek 293 proposed that the ground state of this system could be an interorbital spin triplet state, citing as evidence the two-peak structure seen in STM at positive bias (see Fig. 17; this structure has alternative explanations, as discussed in Sec. V). Gaps away from the Fermi surface are characteristic of interband pairing amplitudes induced by the interorbital interaction 293 .
The full mean-field phase diagram in the presence of SOC was worked out by Böker et al. 297 , and is partially displayed in Fig. 18(c) together with the order parameters at selected values of the SOC (d). For nonzero SOC, the superconducting order parameter is a definite parity combination of spin singlet and spin triplet states. In the weak SOC limit, a dominant spin singlet and small spin triplet gap yielding a state essentially equivalent to the bonding-antibonding s and quasinodeless d identified in the usual spin fluctuation approach. For stronger SOC, however, the superconducting order parameter evolves into a combination of spin singlet and dominant spin triplet gaps in each state. In the (Li 1x Fe x )OHFeSe system, the even parity A 1g and B 2g pairing states with dominant spin triplet component appear to be consistent with available experiments indicating a full gap, including current quasiparticle interference data 297 . The A 1g state shown in Fig. 3 (c) is slightly favored. The spin-singlet dominated A 1g and B 2g -states in this scenario without strong spin fluctuations (mean field approximation) are not consistent with at least one of the existing experiments. The states with dominant spin triplet pairing may be, and Böker et al. proposed ways to detect them with spin polarized quantum interference measurements 297 .
The end result of this analysis is not clear: on the one hand, it is striking that these exotic states requiring interorbital pairing are obtained in a "natural" way from mean field theory. On the other hand, on general grounds the conventional effective spin fluctuation interaction should be significantly stronger and drive pairing in the nodeless B 2g , incipient A 1g , or (singlet) bondingantibonding A 1g channels ( Fig.3 (b-d)). Experiments probing the presence of a spin triplet component would therefore be of the greatest interest. At the same time, it should be possible to make progress theoretically to treat the "exotic" states on the same footing with the "conventional" ones within a generalized spin fluctuation pairing theory that incorporates the SOC into the pairing interaction, and allows for pairing away from the Fermi surface. Only such an approach will be able to ultimately decide whether the exotic pair states are energetically favorable. worth briefly reviewing the basic properties of this material, since they are not particularly "basic", and hence important to keep in mind. Fe 1+y Te 1−x Se x has played a prominent role throughout the "iron age", despite the need to invest considerable effort to control and optimize sample quality, as well as to decipher the roles of the structural and magnetic order, possible phase separation, and excess Fe ions. These issues, and many others, have been reviewed in several review papers dedicated specifically to this material. [324][325][326] The excess Fe ions, indicated by y in Fe 1+y Te 1−x Se x , lead to a partial occupation of the second Fe site in the crystal, and complicate the characterization of the material since these sites disorder, dope, and locally magnetize the system. For a recent review of the role of Fe non-stoichiometries and annealing effects in Fe 1+y Te 1−x Se x , we refer to Ref. 325. In Fig. 19(a), we display the phase diagram of Fe 1+y Te 1−x Se x as mapped out by bulk magnetization measurements in conjunction with (elastic/inelatic) neutron scattering techniques 327 . The compound stays metallic for all x. At x = 0, (non-superconducting) Fe 1+y Te supports a bicollinear antiferromagnetic phase for y 0.12, where magnetic helical order exists for y beyond 12%. For a detailed recent discussion of the magnetic properties and the associated spin excitations of FeTe we refer to Ref. 326. Here we focus on the superconducting properties, where as seen from Fig. 19(a), with enough Se substitution for Te, superconductivity emerges with maximum T c ∼ 14.5K at optimal doping near the 50/50 composition, FeTe 0.5 Se 0.5 . We stress that even at this composition level, excess Fe ions can play an important role in suppressing T c and inducing local magnetism. 325,326,[328][329][330] Unless explicitly addressed, the discussion below relates to nominally excess-Fe-free (annealed) samples. Even for such samples, however, it is well-known e.g. from STM studies that significant electronic and superconducting spatial inhomogeneity remains since Se and Te sit at random lattice positions, as seen from the STM topograph in Fig. 19(b) [331][332][333][334][335] .
From a theoretical perspective, FeTe 1−x Se x is a challenging material to address. Overall there is substantial evidence that FeTe is among the strongest correlated materials of the FeSCs. DFT+DMFT calculations comparing local moments and mass renormalization across the "iron family" locates FeTe as the most strongly correlated compound, see Fig. 2 (d). 53 The mass renormalization is orbital selective due to the Hund's coupling, featuring the strongest correlations in the d xy -dominated bands. Such results seem consistent with a non-nesting-driven magnetic ordering, bad metal behavior, local moment behavior, and the detection of orbital selectivity 65,336 ; more details are discussed in Sec. II B. As discussed above, remnants of this orbital selectivity seem to survive all the way to FeSe, and it appears therefore likely that similar correlated physics is present throughout the phase diagram, Fig. 19(a), of this fascinating material.
The electronic structure of Fe 1+y Te 1−x Se x has been thoroughly investigated, for example, by various spectroscopic probes. 208,209,[337][338][339][340][341] Focusing on the composition close to x ∼ 0.5, and minimal amount of excess Fe ions, the Fermi surface consists of two small hole pockets around Γ and two (also small) electron pockets around the M -point of the BZ. ARPES studies indicate that whether the inner (smallest) hole pocket crosses the Fermi level or not, depends sensitively on the exact values of x and y. 208,210,341 At low temperatures, the superconducting density of states spectrum in optimal T c -samples features a fully gapped state with prominent coherence peaks located close to ±2meV, as seen from Fig. 19(c) 215,342 . From STM measurements, additional shoulders can be identified in the tunneling conductance at lower energies between 1 and 2 meV, presumably related to the detailed gap structure around the largest Fermi sheets 343 . However, a consensus regarding the detailed momentum structure of the superconducting gaps around both hole and electron pockets has not yet been achieved. An early ARPES report claimed an isotropic 4 meV gap on the electron pockets 344 , whereas most other spectroscopic probes point to a maximum gap of around 2 meV. The hole pocket at Γ is known to host a 2 meV gap [208][209][210]344 . Early STM studies using magnetic field dependence of the QPI concluded that FeTe 0.55 Se 0.45 displays sign changes in the superconducting gap between electron and hole pockets 215 , which was recently confirmed by a phase sensitive measurement 187 . Finally, we note that the small Fermi energy E F of order a few meV, and the correspondingly large ratio for ∆/E F ∼ 0.1 − 0.5, has given rise to several studies of potential BCS-BEC crossover physics in Fe 1+y Te 1−x Se x 208-210 , see Sec. III C 4.
Below, we focus on recent nontrivial topological aspects relevant for FeTe 0.55 Se 0.45 and related compounds. Growing evidence points to these materials belonging to a rare class of intrinsic topological superconductors, and in fact may constitute the first known high-temperature topological superconductors. The fact that electron correlations are also substantial makes the matter even more intriguing. We stress that in the current context topological superconductivity refers mainly to superconducting surface states able to host Majorana zero modes (MZM). The surface states consist of a single Dirac cone generated by the nontrivial bulk band structure, and superconductivity is generated on the surface by proximity to the bulk superconducting electrons. The resulting surface state is topologically nontrivial in the sense, that, it should host a single zero energy Majorana fermion in the core of every vortex present. These MZMs are char-acterized by being their own antiparticles, and in addition obey non-Abelian quantum statistics [345][346][347] ; they could in principle be useful for fault-tolerant topological quantum computing. MZMs have been previously realized in, for example, spin-orbit coupled semiconductor nanowires 348,349 , topological insulators 350 , and ferromagnetic adatom chains 351 , all, however, proximitized to conventional s-wave superconductors. These platforms are required to operate at very low temperatures because the superconducting gap is small, and the relevant topological gap which protects the MZMs from random external perturbations is tiny. Therefore FeSC, with their relatively large T c and superconducting gap, could provide a superior platform. On the other hand, for quantum computation applications, one needs to manipulate the MZMs, especially exchange the position of two of them (braiding) to make use of the non-Abelian statistics. For the low dimensional systems with MZMs at the endpoints of chains, this might be done, for example, through a socalled T -junction 352 if the non-topological and topological phase can be tuned, while for MZMs in vortex cores of unconventional superconductors one is faced with the considerable challenge of moving sizable vortex objects to achieve braiding 353 .

B. Theoretical proposals for topological bands
From DFT calculations it was discovered that several bulk FeSC materials could exhibit topologically nontrivial band structures due to band inversion along the Γ − Z direction of the Brillouin zone 354 Fig. 21(b). The generated bulk bandinversion supports topological spin-helical Dirac surface states protected by time-reversal symmetry, positioned inside the spin-orbit-induced gap and centered at the Γ point of the BZ for (001) surfaces, as seen from Fig. 21(g). This process of generating nontrivial topological surface states is similar to the generation of topological surface states in 3D strong topological insulators, but for metallic FeTe 0.5 Se 0.5 the surface states necessarily overlap with bulk states. DFT calculations point to similar nontrivial topological electron states being present also in LiFeAs, as seen from Fig. 21(c). 354,355 Nontrivial topological band structures have also been proposed for monolayers of FeSe on STO 249 and thin films of FeTe 1−x Se x 357 . Hao and Hu performed a theoretical study of the band structure of single-layer FeSe, including the effect of lattice distortion from substrate strain 249 . It was found that in principle a parity-breaking substrate can both suppress the holelike band at Γ and induces a gap at the M point. Provided that the SOC is large enough compared to the tensile strain-induced gap at M , a topological nontrivial Z 2 phase can be stabilized from a band-inversion at M , with associated helical edge states 249 . The band structure of FeSe/STO was also theoretically studied under the additional assumption of checkerboard antiferromagnetic Fe moment ordering 358,359 . In this case, SOC induces a topological gap centered M slightly below the Fermi level, which supports quantum spin-Hall edge states protected by the combined symmetry of time-reversal and a discrete (primitive) lattice translation 358 .
In addition to the above topological "M -point"scenarios, Wu et al. 357 proposed a "Γ-point"-scenario for the generation of nontrivial topological bands in monolayer FeTe 1−x Se x . In this case, by adjusting lattice constants, particularly the anion height (with respect to the Fe plane), it was shown theoretically how a nontrivial Z 2 topological phase can arise by band inversion at the Γ point for FeTe 1−x Se x monolayer films, x < 0.7. 357 In this mechanism, it is a smaller hybridization between Fe d xy orbitals and Se/Te p z orbitals caused by an enhanced anion height, that leads to a band-inversion at Γ by pushing the p z /d xy -dominated electron band far enough down in energy to mix with the hole bands. In the resulting band structure, the inverted parity-exchanged hole (electron) bands acquire p z /d xy (d xz /d yz ) orbital weight. For further details about the generations of topological bands in FeSCs monolayers, we refer to Refs. 249 and 357 and recent reviews in Refs. 25 and 62. While the scenarios for band-inversion discussed above are intriguing and important, the predictions from DFT studies deserve further scrutiny when applied to ironchalcogenides.
This is due to the significant electron interactions and their associated band renormalizations. Standard methods assuming momentum independent self-energies describe the orbital-dependent band squeezing, but in reality nonlocal self-energy effects will further distort the DFT band structure and the final dispersion [66][67][68] . Certainly for FeSe, as discussed at length in Sec. III A, the link between the experimentally extracted low-energy band structure and the DFT-derived bands, remain unclear at present. In this light, for the proposal of nontrivial band topology e.g. in bulk systems, it seems particularly crucial to determine whether the relevant p z -dominated band indeed disperses enough between Γ−Z to instigate a band-inversion, and whether the induced surface states can be relevant near the Fermi level. Thus, it is of crucial importance at this point to turn to experiments, and check for experimental evidence for band inversion and topologically nontrivial surface states.

C. Experimental evidence for topological bands
In this section we turn to the experimental evidence for nontrivial topological bands in FeSCs. As discussed above, several ARPES studies 210,272,354 have addressed the band structure and the superconducting gaps in FeTe 0.55 Se 0.45 , but the acceleration of research in topological aspects of this material was kick-started by the work of Zhang et al. 356 published in 2018. By use of high-resolution ARPES and spin-resolved ARPES, Zhang et al. succeeded in detecting Dirac cone dispersive states near the Fermi level, with associated momentumdependent spin polarization. This result is shown in Fig.21 (d-f). This discovery is consistent with the theoretical prediction of spin-helical (001) surface states near the Γ point of the surface BZ, as shown in the surface projected band structure of Fig. 21(g) 180,354,357 . The Fermi level is not guaranteed to fall inside the band gap near Γ, and additional bulk states reside at the Fermi level in other regions of the BZ, complicating the ex-perimental and theoretical analysis. Zhang et al. additionally focused on the EDC spectra near Γ at T < T c , and found an isotropic 1.8 meV superconducting gap on this band. This was interpreted as superconductivity induced on the helical surface states by the bulk trivial bands, providing an example of self-proximitized topological superconductivity 356 .
The discovery of spin-polarized topological surface states in FeTe 0.55 Se 0.45 , has been followed up by proposals of additional nontrivial topological states in related systems. For example, LiFeAs was shown theoretically to possess similar topological insulator-like surfaces states as discussed above, and ARPES studies of Co-doped LiFeAs have found evidence thereof. 355 Ref. 355 pointed out the additional existence of topological Dirac semimetal states from DFT studies. The bulk 3D Dirac cone associated with the semimetal states remain ungapped due to crystal symmetry, and produce surface states detectable e.g. on (001) surfaces. Laser ARPES experiments on LiFeAs with varying degrees of Co doping to tune the Fermi level, found evidence for both topological insulator-like surface states and Dirac semimetal-like surface states, both with some degree of spin-polarization as determined from spinresolved ARPES, and in agreement with their proposed nature of topological surface states. 355 Topolog-ical Dirac semimetal states are also proposed to exist in FeTe 0.55 Se 0.45 , and could lead to novel types of bulk topological superconductivity. 355,360,361 As stated above, the identification of low energy states observed in ARPES as protected topological surface states is complicated by the fact that they invariably overlap with bulk states, and often exist in bands with tiny Fermi surfaces. Additionally, one has to rely on comparison to DFT studies, which are challenged in obtaining quantitatively correct band structures, particularly for the iron-chalcoginides. In this respect, a recent ARPES study on FeTe 0.55 Se 0.45 investigated the dependence of the photocurrent on incident photon energy 362 . In contrast to DFT results, this photoemission study did not detect any highly dispersive band along the Γ-Z direction, but nevertheless did find evidence for bandinversion along this direction. Specifically, the normalized ARPES intensity as a function of k z revealed pronounced oscillations with maxima (minima) at Γ (Z). From an analysis of the relevant dipole selection rules, this behavior is consistent with a change of the parity eigenvalue (i.e. a change from d-like to p-like orbital character) along the Γ-Z path. 362 Thus, the k z -dependence of the ARPES intensity points to bulk band inversion, and an overall picture consistent with a topological nature of the low-energy surface states in FeTe 0.55 Se 0. 45 362 .
Regarding the proposal of nontrivial band topology in FeSe/STO monolayers only a few experimental studies have hinted at the possible existence of the required edge states. For example, Wang et al. 358 using both STS and ARPES measurements studied the spectral weight inside the SOC gap for both [100] and [110] edges in the FeSe/STO monolayer, and reported evidence for dispersing modes near both edges, consistent with the existence of topological edge states. These, however, were relatively far below the Fermi level. A more recent study focused on domain walls between different nematic domains in multi-layer FeSe/STO. 363 Two experimental studies have addressed the electronic spectroscopic properties of FeTe 1−x Se x monolayers grown on STO, and interpreted their measurements in terms of possible topological bands. 272,364 Shi et al. 272 performed an ARPES study of monolayers of FeTe 1−x Se x /STO for varying x, and found a systematic decrease of the band gap at Γ, i.e. holelike (electronlike) pockets that move up (down) with increasing Te content x. As usual, ARPES can only detect the unoccupied bands through possible thermally excited electrons, visualized via division of the measured data by the Fermi-Dirac distribution function. Through this procedure, Shi et al. 272 found evidence for gap-closing around x ∼ 33%, a prerequisite for possible nontrivial band inversion, but no direct experimental evidence for topological electronic states. More recently, Peng et al. 364 continued this line of research by a comprehensive energy-and polarization dependent ARPES study of FeTe 1−x Se x /STO monolayers for samples in a wide range of x. Overall, for samples with x 0.21 the location and polarization dependence of the bands are consistent with the "Γ-point" topological band-inversion scenario discussed above, with a band gap approximately 20 meV below the Fermi level. 357,364 Convincing evidence for nontrivial topological states, however, requires also e.g. detection of symmetry-protected edge states. Using STS measurements near both [100] and [110] edges, some LDOS enhancement could be identified around -50 meV near the edges. 364 This result is consistent with associated DFT calculations including topological edge states, but still inconclusive in terms of the topological nature of the edge states.

D. Topological superconductivity
The possibility of topological nontrivial band inversion in bulk FeSCs along the Γ − Z direction close to the Fermi level discussed above, raises the possibility of surface-induced topological superconductivity arising from the proximity effect between bulk superconductivity and spin-helical topological surface states. This idea originates from Fu and Kane, who proposed that topological superconductivity can be realized on the surface of a topological insulator in proximity to a conventional s-wave superconductor 365 . In this setup, superconductivity induced in the Dirac cone spin-helical surface states may resemble a two-dimensional p x + ip y -like pairing state, which preserves time-reversal symmetry and exhibits topological characteristics, including MZM bound states in the center of vortex cores 366 . A bulk 2D p x +ip y superconductor is well known to support a Chern number which, when nonzero, ensures the presence of chiral edge modes and the possibility of trapping a single MZM per superconducting vortex. Notably, the MZMs in vortices follow a 1 + 1 = 0 rule, since they are protected by a Z 2 invariant given by the product of the Chern number and the value of the vorticity modulo 2 367 .
For the current discussion, mainly focussed on FeTe 0.55 Se 0.45 and e.g. Co-doped LiFeAs, where any topological surface states necessarily overlap with bulk metallic bands, important questions arise concerning the detailed self-proximity mechanism and stability of the possible resulting topological surface superconductivity. Experimentally, superconductivity in the surface states seems confirmed in the sense that a full momentumindependent gap was detected on the surface band of FeTe 0.55 Se 0.45 . 356 Theoretically, the stability of topological surface superconductivity was investigated by Xu et al. 368 by studying the nature of superconductivity on (001) surfaces within an effective low-energy eight-band model relevant to the Γ and Z points, with input parameters based on a fit to DFT calculations. From this model, the phase diagram shown in Fig. 22 (a) of the stability of topological surface superconductivity could be established as a function of chemical potential and the amplitude of the bulk superconducting gap. As expected, the topological superconductivity is stable in a finite range of chemical potential, but Ref. 368 also pointed out two additional properties both evident from Fig. 22 (a): 1) surface topological superconductivity is suppressed if the bulk superconductivity becomes too strong due to pairing of surface states with the bulk states, and 2) for a regime of parameters, a topological surface phase transition can take place as a function of temperature with topological superconductivity only at the lowest T and normal (nontrivial) superconductivity at higher T .
The bulk Dirac point pointed out by Zhang et al. 355 could also have interesting consequences for superconductivity, if properly tuned near the Fermi level, for example doping of LiFeAs with cobalt. By analogy to theoretical studies of possible superconducting pairing states in other (doped) Dirac semimetals 361 it was suggested that semimetal Dirac cones could give rise to topological superconductivity. Both bulk and surface topological superconductivity has been considered, depending on the nature of the pairing on the bulk Dirac semimetal Fermi surfaces 355 . Furthermore, the presence of the bulk 3D Dirac cones led to a recent theoretical proposal of dispersive 1D helical MZMs inside the vortex cores of (singlet, s-wave) superconducting systems doped close to such Dirac semimetal points, protected by C 4 crystalline symmetry 369 . This scenario is qualitatively distinct from the 0D MZMs localized at the ends of c-axis aligned flux lines discussed further below. Two recent theoretical works are also relevant for this discussion, by providing a topological classification of vortex Majorana modes in doped (s-wave) superconducting 3D Dirac semimetals 370,371 . A recent STM experiment on Codoped LiFeAs did not observe pronounced zero-energy bound state in the vortex cores, which puts constraints on the existence of MZMs in this material, but does not completely eliminate the possibility of the existence of vortex MZMs (perhaps extended along the flux line) as there is still a finite density of states at zero energy 372 .
Finally, a theoretical study has investigated the possibility of intrinsic topological superconductivity in FeSe/STO monolayers 373 . Specifically, Hao and Shen performed a classification of the allowed pairing symmetries relevant to this system, and computed the superconducting phase diagram based on a phenomenological attractive pairing model 373 . In short, this allowed to theoretically identify a leading odd-parity topological s-wave pairing state with spin-triplet, orbital-singlet structure.

E. Experimental evidence for Majorana zero modes: defect states
The initial experimental observation of robust zeroenergy states in FeSCs came from STM measurements near interstitial iron impurities on the surface of nearoptimally doped Fe 1+y Te 1−x Se x . Within samples containing 0.5% (T c = 12K) and 0.1% (T c = 14K) excess Fe ions, Yin et al., 342 located individual interstitial Fe ions on the surface and reported a zero-bias-centered conductance peak at these impurity sites, see Fig. 22 (b,c). Within experimental resolution, the peak was found to remain centered at zero bias as a function of both STM tip position and application of c-axis applied magnetic fields up to 8T. Furthermore, it was measured to extend uniformly in space with a length scale of order 3-4Å, and decrease in amplitude (but not split) when proximate to other impurity bound states 342 . These peculiar properties are not characteristics of standard in-gap bound states of FeSCs arising from magnetic or nonmagnetic impurities [374][375][376][377][378] . Intriguingly, similar robust zeroenergy conductance peaks have been recently detected near Fe adatoms, deposited on top of the stoichiometric materials LiFeAs and PbTaSe 2 379 , and on monolayers of FeSe/STO and FeTe 0.5 Se 0.5 /STO 380 .
What is the origin of these seemingly robust non-split zero-bias conductance peaks discussed above? A recent theoretical work suggested that interstitial iron ions may induce so-called quantum anomalous vortices, which in conjunction with effective p-wave pairing on the surface, host MZMs in their center 381 . By including both impurity-induced SOC, and exchange coupling with the magnetic impurity ion, it was shown theoretically that vortices can be nucleated by the iron moment 381 . Notably, the vortices are stabilized even in the absence of external magnetic fields, hence the name "quantum anomalous". In the presence of topological surface superconductivity, the quantum anomalous vortices support MZMs at their center, thus providing a possible explanation of the STM results reported in Refs. 342, 379, and 380.
Recently, Fan et al. 382 performed a comprehensive STM study of the variability of the conductance near different Fe adatoms deposited on the surface of FeTe 0.55 Se 0. 45 . In agreement with the finding by Yin et al. 342 robust zero-bias conductance peaks exists near some of the Fe adatoms, a finding interpreted in favor of the quantum anomalous vortex scenario 381 . In addition, however, a fraction of the Fe adatoms was shown to feature more standard bound states similar to Yu-Shiba-Rusinov (YSR) impurity states with finite energy bound state energies. Interestingly, some of these YSR bound states can be reversibly manipulated (and irreversibly manipulated by moving the adatom position) into zeroenergy peaks (ZEPs) by changing the tip-sample distance. It is known from other STM studies of YSR states, that the tip can exert a force on adatoms, and thereby alter the exchange coupling between the impurity moment and the conduction electrons, resulting in a tip-induced shift of the YSR bound state energies. 383 The data from Fan et al. 382 reveals the existence of a critical coupling necessary for generating the ZEPs, a result again discussed in Ref. 382 in terms of impurity-induced vortices and MZMs. Lastly, we mention a recent STM study of sub-surface impurity states in FeTe 0.55 Se 0.45 , reporting on another kind of tip-tunable in-gap states 384 . As shown in Ref. 384, some bound states accidentally appear to be located at zero-energy, but "disperse" with the tip position, a property shown to be consistent with a local tip-induced gating of the impurity levels in low-density systems. Zero-energy localized states have also been recently detected by STM at the ends of 1D atomic line defects in 2D single unit-cell thick FeTe 0.5 Se 0.5 films grown on STO(001) substrates 211 . This system exhibits superconductivity below 65 K, and a fully gapped spectrum with two large identifiable gaps of 10.5 meV and 18 meV. The line defects consist of unidirectional lines of missing Te and/or Se atoms at the top layer, as determined by topographic images 211 . Chen et al. 211 studied line defects of 15 and 8 missing Te/Se atoms, and inferred from their spectroscopic characteristics that the most likely explanation for the emergence of zero-energy end states is topological MZMs. However, since unit-cell thick FeTe 0.5 Se 0.5 /STO is likely topologically trivial, it was suggested that the missing atoms themselves induce the necessary ingredients for local nontrivial topological states 211 . Microscopically, this could include locally enhanced Rashba SOC, or induced chain magnetism. Two recent theoretical works explore both possibilities, i.e. local topological "Rashba-chains" and associated local chain-induced odd-parity spin-triplet pairing 385 , and topological antiferromagnetic line defects 386 .  [387][388][389][390][391][392][393][394] . The topic is complicated e.g. due to the sensitiv-ity to instrumental resolution, unknown effects of impurity pinning sites 333 , contributions from bulk states, quasi-particle poisoning, sample inhomogeneity, disordered vortex lattices, and the presence of other topologically trivial but contaminating low-energy vortex cores states. The latter are the well-known Caroli-de Gennes Matricon (CdGM) 395 states which exist quite generally in the cores of superconductors, and tend to produce a broad peak centered at zero energy simply because the energy separation between CdGM states, ∆ 2 /E F , may be significantly smaller than the instrumental resolution 396 . Only in the so-called quantum limit where ∆/E F becomes large enough, can one expect to see discrete wellseparated CdGM states. An advantage of searching for Majorana modes in FeSCs is that indeed for these materials ∆/E F can be rather large, and therefore finite-energy CdGM states should be distinguishable from a potential zero-energy MZMs 397, 398 .
In FeTe 0.55 Se 0.45 , STM reports have identified a range of different CdGM states depending on which specific vortex was probed 388,393,399 . Interestingly, however, some vortex cores feature a conductance peak centered exactly at zero bias as shown in Fig. 23 (b,c), and is pinned to zero energy over a spatial range away from the vortex core center 387,389,392,393 . This zero-energy peak constitutes an important fingerprint of MZMs associated with topological superconductivity on the surface of FeTe 0.55 Se 0. 45 387 . Importantly, it was found from highresolution STM measurements by Machida et al. 389 that not all vortex cores host zero modes, and the fraction of those that do is inversely proportional to the applied magnetic field strength, with approximately 80% (10%) probability of detecting zero energy states at B = 1T (B = 6T), as shown in Fig. 24. The variability of the low-energy states and the presence of possible MZMs ap- pear to be unrelated to disorder sites or local Te/Se concentration variations 389,392 . The STM study presented in Ref. 393 utilized the spatial variability of the vortex electronic structure to identify two classes of vortices distinguished by a half-integer level shift between the in-gap vortex states. In agreement with model calculations, a level sequence of 0, 1, 2, 3, . . . versus 1 2 , 3 2 , 5 2 , . . . in terms of ∆ 2 /E F is indeed expected for topologically nontrivial and trivial bands, respectively 393 .
Recently, based on theoretical simulations of an effective low-energy Hamiltonian it was suggested that Majorana hybridization in conjunction with a realistic distribution of disordered vortex sites, offers an explanation of the decreasing number of ZEPs with increasing magnetic field 389,400 . Earlier theoretical studies investigated the role of random Se/Te substitution on the vortex bound states, finding insignificant effects from this kind of isovalent disorder 401 . It was shown how disorder in vortex locations is important for smearing out oscillations in the field dependence of the density of zero-energy peaks. Ref. 400 also compared the statistics of the lowest energy peaks in vortices without ZEP between experiment and simulations, providing evidence for the scenario of random Majorana hybridizations causing the decrease of ZEP. More recently, an alternate explanation was proposed, namely that the surface hosts two distinct phases competing for Dirac surface states in FeTe 0.55 Se 0. 45 . 402 In this picture, remnant magnetic interstitial moments aligned by an external magnetic field may stabilize regions of half quantum anomalous Hall phases, supporting standard vortices without MZMs, and other regions of topological superconductivity hosting MZMs in their vortices. 402 This scenario offers a prediction of chiral Majorana modes located at the domain wall between these two spatially distinct phases.
In (Li 0.84 Fe 0.16 )OHFeSe, STM studies have also detected zero-bias centered conductance peaks inside the vortex cores 391,403 . In this case, however, zero-energy modes were found only at free vortices, i.e. not pinned by dimer-like impurity sites, on FeSe surfaces. Similar to the discussion above, these zero-energy conductance peaks were interpreted as evidence for MZMs, a conclusion backed up by ARPES measurements and DFT calculations. The photoemission measurements found some spectral weight near the BZ center, which was interpreted as surface Dirac cone states, but no evidence of superconductivity could be detected on these surface bands. However, as further discussed in Ref. 403, (Li 0.84 Fe 0.16 )OHFeSe has the favorable property that the FeSe layers are stoichiometric and T c (ξ) is higher (shorter) than those in FeTe 0.55 Se 0.45 by roughly a factor of four, making the ZEPs 1) correlated to free vortices, and 2) less sensitive to high magnetic fields.
An important characteristic of MZMs is the socalled quantized Majorana conductance of 2e 2 /h, 404,405 a property recently verified in topological semiconductor nanowires 406 . The quantized property is a direct consequence of the particle-antiparticle equivalence of MZMs. If Majorana-induced resonant Andreev reflection takes place between the Majorana bound state in the vortex cores and the normal STM tip, the conductance should reach 2e 2 /h in the ideal T = 0 case, independent of the tunnel coupling. This conclusion holds for the case of a single conducting contributing channel. Two recent lowtemperature experimental STM studies on FeTe 0.55 Se 0.45 and (Li 0.84 Fe 0.16 )OHFeSe both reported on experimental evidence for such quantized 2e 2 /h conductance. 391,394 More specifically, as seen from Fig. 23 (d-f) it was observed that upon decreasing the tip-sample distance, and thereby increasing the tunnel conductance, the zero-bias peak appears to reach a saturation plateau close to 60% or 90% of 2e 2 /h for FeTe 0.55 Se 0.45 and (Li 0.84 Fe 0.16 )OHFeSe, respectively 391,394 . The sub-gap conductance, however, exhibits a significant amount of background weight, and the potential influence of other channels in contributing to the final conductance appears unclear at present.
In addition to the interesting developments summarized above for chalcogenide systems, recent experimental evidence was reported for MZMs in the vortices of iron-pnictide superconductors. Specifically, Liu et al. 407 used both ARPES and STM to perform a spectroscopic study of CaKFe 4 As 4 , and found evidence for superconducting Dirac surface states and vortex core MZMs. In this material, the origin of nontrivial topology is suggested from DFT+DMFT calculations to arise from band inversion along Γ − Z, catalyzed from an additional folding of the BZ along z due to glide-mirror symmetry breaking along the c-axis. 407 There has also been a recent theoretical proposal that the iron-pnictide 112-material Ca 1−x La x FeAs 2 may be a topological superconductor. 408 Thus, while more experimental studies of topological aspects of CaKFe 4 As 4 , Ca 1−x La x FeAs 2 , and LiFeAs are clearly desirable, at present it seems possible that the realm of topological high-T c superconductivity may extend into the iron-pnictides as well.

G. One-dimensional dispersive Majorana modes
The above discussion of zero-dimensional MZMs hosted by induced topological superconductivity on surfaces can be traced back to the original theoretical proposal by Fu and Kane, also discussed briefly above 365 . That work, however, suggested another realization of Majorana fermions by use of a π-junction between two ordinary superconductors deposited on a topological insulator, generating a one-dimensional wire for helical Majorana fermions 365 . As a consequence, if π phase shift domain walls could be generated on the surface of topologi-cal FeSCs, 1D Majorana modes could exist along the domain walls. Above, we briefly mentioned an STM study of nematic domain walls in 20 unit-cell thick FeSe on STO, interpreted in the light of 1D dispersing topological edge states 363 . Unlike monolayers, the multilayer films are known to feature strong electronic nematicity. Specifically, Yuan et al. 363 grew 20 unit cell thick FeSe films on top of STO substrates, and studied the electronic states near domain walls between two distinct nematic regions. The resulting STS data found evidence for edge-induced zero-energy states localized to the domain walls, and interpreted them in terms of topological edge modes 363 .
Another recent STM study managed to identify a certain type of crystalline domain walls associated to halfunit cell shifts of the Se atoms on the surface of bulk FeTe 0.55 Se 0.45 , and measured almost flat dI/dV conductance spectra at low bias (inside the superconducting gap) at the domain wall, as opposed to fully gapped conductance spectra away from the domain wall 409 . This peculiar conductance behavior was not observed near e.g. step edges in FeTe 0.55 Se 0.45 , or at twin-domain walls of topologically trivial FeSe. Therefore, it was proposed that the flat dI/dV curves constitute spectroscopic evidence for linearly dispersing helical Majorana modes generated by the half-unit-cell-shifted domain walls 409 .

H. Higher-order topological states
Another possibility for generating 1D helical Majorana modes in FeTe 0.55 Se 0.45 was first discussed theoretically by Zhang et al. 410 , who predicted the emergence of socalled higher-order superconducting topology with associated 1D localized helical Majorana hinge states between (001) and (100) or (010) surfaces. An n'th order topological phase hosted in a D-dimensional system features (D − n)-dimensional topological edge states. [411][412][413][414] For example, a 2nd order topological superconductor hosts Majorana corner and hinge modes as opposed to standard edge or surface modes in 2D and 3D, respectively. Such topological MZMs are only detectable by probes able to selectively pick out sample corners or hinges. The theoretical analysis by Zhang et al. 410 "hinges" on the band inversion along Γ − Z, and standard s ± superconductivity in the bulk with opposite sign of the pairing gap between the Γ and M points, ∆(k) = ∆ 0 + ∆ 1 (cos(k x ) + cos(k y )) 410 . The latter property is necessary for generating opposite signs of the order parameters on (001) and (100) surfaces, producing an effective π-shifted domain wall at the hinge of the two surfaces. Notably, no external magnetic field is required for the generation of higherorder MZMs. A recent experiment probing the edges of exfoliated flakes of FeTe 0.55 Se 0.45 samples found evidence for such zero energy hinge states 415 . More specifically, by draping suitable contacts over the sides of the sample, normal metal/superconductor junctions were created on the hinges between (001) and (100) surfaces, and a pronounced zero-bias conductance peak could be measured only for junctions in direct contact with the putative hinge modes. 415 Precautions were made to separate contact effects from intrinsic topological modes existing in FeTe 0.55 Se 0.45 samples, leading to the conclusion that the zero-bias conductance peak was direct evidence for topological Majorana hinge modes, and thereby higherorder topological superconductivity in FeTe 0.55 Se 0. 45 415 .
More recently, higher-order topological phases have also been discussed theoretically in the context of Majorana corner modes in 2D superconductors coexisting with suitable magnetic structures 416 . For example, a proposed theoretical setup consists of a monolayer of FeTe 0.55 Se 0.45 , experiencing the magnetic exchange field in proximity to a layer of FeTe exhibiting bicollinear antiferromagnetism 416 . This magnetic structure allows for corners of ferromagnetic and antiferromagnetic edges. From this property, in addition to standard s-wave superconductivity and within an effective band model that includes only part of the band structure (near Γ), it was found that indeed the magnetic order instigates topological edges in the form of an effective nanowire Hamiltonian, known to support Majorana end states, resulting in the present case in Majorana corner modes. 416 A related theoretical study, also explored conditions for stabilizing Majorana corner modes in FeTe 1−x Se x monolayers, but with time-reversal symmetry breaking from an external in-plane magnetic field, creating distinct edges being either parallel or perpendicular to the in-plane field. 417 Future experimental studies will hopefully pursue these interesting proposals for higher-order MZMs in FeSC systems.

VIII. CONCLUSIONS
We have tried here to review the background necessary for the reader to understand the debate over superconductivity in the Fe-chalcogenide materials, what is known about the superconducting state, as well as some theoretical ideas that have been put forward in this context. It is clear that relatively strong electron correlation (quite possibly involving nonlocal effects that have not yet been treated systematically) as well as spin-orbit coupling play important roles and distinguish these materials, at least in degree, from their pnictide counterparts. We have abandoned the attempt to cover many important and fascinating aspects of the normal state of these materials in favor of doing a reasonably thorough job on the superconducting state. Even with our more limited goals, we have necessarily been forced to leave out many important contributions, an omission for which we apologize to the authors so neglected.
Here we attempt to summarize our personal view of the important open questions in this field. First, let us assume that the standard paradigm is correct, that spin fluctuations due to repulsive Coulomb interactions are primarily responsible for pairing in both Fe-pnictides and Fe-chalcogenides, and that the differences arise primarily because of the heightened degree of correlation and perhaps strength of spin-orbit coupling in the latter. If this is in fact true, how does one explain the fact that higher T c 's seem to exist in systems without hole pockets? Note that we refer here not only to the monolayer system, where it has been plausibly argued that substrateinduced phonons can bootstrap the spin fluctuation interaction, but also to the e-doped FeSe intercalates with T c 's above 40K, where no such effect is obviously present. No convincing explanation involving realistic materialsspecific parameters for this phenomenon has yet been put forward.
The proposals for phonon-assisted T c 's in the monolayer systems have stimulated a renewed interest in the role of the lattice in Fe-based superconductors generally. While early estimates of T c 's to be expected from electron-phonon coupling suggested that this physics could be neglected, these questions need to be revisited. The obvious question is whether, and under what circumstances, phonons can assist spin fluctuations to enhance T c . Naively, this can happen only if they are of forward scattering character, since otherwise they drive s ++ pairing that competes with s ± and d-wave pair channels. Nevertheless, it will be interesting to perform materialsspecific calculations including both phonons and spin fluctuations on the same footing.
The role of the lattice is also interesting with regard to normal state physics. For example, the role of nematic fluctuations near the nematic critical point in promoting superconductivity has been questioned, with the suggestion that these fluctuations do not diverge due to a lattice cutoff. At present, we have no material-specific theory that can explain even the balance of cooperation vs. competition between nematic order and superconductivity, which appears to be distinctly different in the Fe-chalcogenides compared to the Fe-pnictides. Going beyond phenomenological theories of pairing due to nematic fluctuations is a current challenge.
µSR experiments have reported signals of time reversal symmetry breaking in both Fe-pnictides and chalcogenides. More detailed studies are needed to distinguish between TRSB states with macroscopic spontaneous currents, and nonchiral complex admixtures that create only local, impurity induced current. Theoretical studies need to make clear predictions for the signals expected for various states from µSR and Kerr measurements, including the disorder dependence, and additionally find ways to distinguish between type-I and type II TSRB order discussed above.
The recent theoretical and experimental studies of topologically nontrivial effects in FeSCs highlight a new exciting direction within this area of research. It is remarkable that band-inversion far off the Fermi-level from DFT predictions, fortuitously gets shifted down to energy scales relevant for superconductivity. While the "smoking gun" proof of topological effects might still be argued to be missing, there certainly exists mounting evidence in its favor at present. In particular the detection of MZMs inside vortices by several different STM groups points to nontrivial band topology and associated self-proximitized topological surface superconductivity. This is remarkable since intrinsic topological superconductors are considered rare, and the fact that they may inherently exist within a family of correlated materials exhibiting unconventional bulk superconductivity, makes the development all the more noteworthy. Of course there are many unsolved questions and we still lack quantitative analysis of most experimental results in terms of realistic material-specific models.
Some current open topics for FeSCs and nontrivial topological superconductivity refer e.g. to the questions of MZM variability in the vortex cores. Why do only some vortices host a MZM on the surface of FeTe 1−x Se x , and why do apparently no vortices host MZM in Codoped LiFeAs. In fact, the latter compound seems particularly elusive regarding its potential topological properties. At this point is seems unclear exactly how bulk and surface states intertwine in the final superconducting condensate. Another open question refers to the robustness of the spin-helical surface or edge states in these systems. Are they topologically protected from basic deterioration? In this regard future experiments able to test, for example, for backscattering blockade would be highly desirable. Furthermore, the generation of defect centers seemingly favorable for MZMs is unresolved; why do point-like Fe ions apparently support MZMs, why do some domain walls stabilize π-shifted regions, and how does strong correlations and local induced magnetism enter the game? While useful theories exist for several of these open points, it is nevertheless also clear that at present we lack quantitative models. These and many more exciting questions may hopefully constitute some of the many research directions pursued in the near future. Thus, even though iron-based superconductors have kept the community busy for more than a decade at present, we have not yet understood all their fascinating electronic properties, and most likely we have not yet unlocked all their secrets.