Time reversal symmetry breaking superconductors

The paper reviews recent work on time reversal symmetry (TRS) breaking superconductors. The family of TRS breaking superconductors is growing relatively fast, with many of its newly discovered members being non-centrosymmetric. However, many of the superconductors which possess center of inversion also break TRS. The TRS is often identified by means of the muon spin relaxation effect ($\mu$SR) or/and the Kerr effect. Both methods effectively measure the spontaneous appearance of the bulk magnetic field below superconducting transition temperature. One of the systems most carefully studied so far is Sr$_2$RuO$_4$ believed to be spin triplet chiral p-wave superconductor. This compound provides an example of the material whose many band, multi-condensate modelling has enjoyed a number of successes. We discuss in some details the properties of the material. Among them is the polar Kerr effect, which understanding has resulted in the discovery of the novel mechanism of the phenomenon. The mechanism is universal and thus applicable to all systems with multiorbital character of states at the Fermi energy.


I. INTRODUCTION
Discovery of the high temperature superconductors 1 a few decades ago started vivid and still ongoing experimental and theoretical races to uncover their secrets.This breakthrough in the research of superconductivity was preceded by the discovery of CeCu 2 Si 2 , the first heavy fermion superconductor 2 and other members of this family UBe 13 3 and UPt 3 4 .
The concomitant development of new technologies of material synthesis, and curiosity these findings have evoked, resulted in the numerous discoveries of simple, like MgB 2 5 , and more complex families of superconducting materials including Sr 2 RuO 4 6 the perovskite superconductor without copper.A lot of excitement have evoked dicoveries of systems superconducting at high pressures.These findings culminated in recent measurements of superconducting materials with transition temperature T c approaching not so cold winter day temperature 7,8 .
Novel superconductors are often denoted as exotic 9,10 or (in more modern language) as unconventional.This last word sometimes is used in relation to superconductors in which other than electron -phonon pairing mechanism is operative 11 .Slightly more formally, unconventional chaeacter of superconductivity means that not only the gauge U(1) symmetry is spontaneously broken below the transition temperature as in all superconductors but also other symmetries are broken.Among them the time reversal symmetry (TRS) plays a special role.
Recently, the increased interest is observed in the detection and understanding the TRS breaking and its relation to other symmetries.In the last few years a number of time reversal superconductors with chiral 12 and other order parameters breaking the time reversal symmetry have been identified.The importance of chiral superconductors is in part related to the fact that they have been predicted to host Majorana particles and other quantum states of interest 13 .As an example of such system we consider Sr 2 RuO 4 and present some of its properties and our approach to model the material.We concentrate on the theory of Kerr effect in this material paying special attention to its multi-orbital character.This is because the recent discovery of the novel 14,15 intrinsic mechanism of the Kerr effect operating in strontium ruthenate and possibly other systems.
The paper is organised as follows.In the next section (II) we review the earlier and recent experimental discoveries of superconductors which break TRS.Our studies of various aspects of superconductivity in Sr 2 RuO 4 are reviewed in Section (III).The multiorbital/multiband mechanism of the Kerr effect is introduced in Section (IV) with application to strontium ruthenate.We end up with few general observations and the summary of the paper in Section (V).

II. TIME REVERSAL SYMMETRY BREAKING IN SUPERCONDUCTORS
The detection of the time reversal symmetry (TRS) breaking necessarily means the unconventional nature of superconductivity.This is so, becouse in many superconductors the U(1) gauge symmetry spontaneous breaking is often accompanied by additional symmetries: time reversal or spatial.For example, the d-wave character of the order parameter symmetry in high temperature superconductors (HTS), as confirmed again recently with a new dielectric resonator method 16 , provides a case of spatial symmetry breaking.The crystals of HTS have a four-fold symmetry axis perpendicular to their ab plane but the d-wave order parameter ∆(k) = ∆ 0 (T )[cos(k x a) − cos(k y a)] features only the two-fold one.

A. Detection methods of TRS breaking
There are a few etsblished methods to detect existence of TRS breaking (TRSB) in superconductors.These are muon spin rotation and relaxation (µSR), polarised neutron scattering, detection of circular currents and polar Kerr effect 17 .Recently, there appeared different additional proposals to detect the TRSB state experimentally.On of them is by laser pulse probe 18 , when one expects changes of the condensate properties around the laser heated spot.The modifications depend on the symmetry of the order parameter.Depending if it is of s-wave or p-wave one expects different patterns of the magnetic field, which can be measured.Still another proposal relies on detection of the diamagnetic response of multi-band superconductors 19 .According to the authors the current induced magnetic flux response could in principle be used to to detect breaking of TRS in the ground state.
The method most often applied to detect breaking of TRS at the onset to the superconducting state is muon spin rotation and relaxation (µSR).were found not only to show a superconductivity, but also the one with TRS breaking superconducting order parameter.Neither the mechanism nor the superconducting state in the system is uniquely identified.We mention, without deeper discussion, the observations indicating TRS breaking near the pseudogap phase of HTS 22 , which might be caused by the recently discovered 23 slowly fluctuating magnetic fields of intra-unit-cell origin in the same region of the phase diagram.
Superconductors which break time reversal symmetry may also have distinctive response to other disturbances like local temperature bias ∆T (r), etc.It has been found 24 that the thermoelectric response of TRS breaking superconductors to ∆T (r) depends on the thermally induced magnetic field with a profile that is sensitive to the presence of domain walls and anisotropy of superconducting states.If the heating process is non-stationary the time dependent B field produces an electric field.As a result there appears a charge imbalance in different bands.In view of the puzzles related to understanding thermopower in superconductors [25][26][27][28] and hybrid structures 29,30 this is an interesting result worth of deeper analysis.
The µSR technique is a local, very sensitive probe 31,32 detecting tiny magnetic fields in the bulk of superconductors.It appears below the superconducting transition temperature.
The beam of spin-polarised muons is directed towards the sample.The individual, typically positive, muons injected into the sample stop at the random point due to the loss of energy by electrostatic interactions.These interactions preserve the direction of spin polarisation of muons.After the time comparable to their lifetime (≈ 2.2µs) the muons in the sample decay via week interaction process into positron and two neutrinos.The positron is preferably emitted in the direction parallel towards the muon spin.Thus the angular distribution of the positrons reflects the angular distribution of the muons.Existence of, even tiny, magnetic fields (of order of 0.1 G) at the stopping site of the muon induces the Larmor precession of its spin, what changes the measured distribution of positrons.The application of this technique to study unconventional superconductors have been recently reviewed 33 .
To see how the Kerr effect is related to the frequency dependent Hall transport coefficient where n is the complex refraction coefficient.On the other hand the polar Kerr angle (2) can be shown 35 to read in the high frequency regime (ω > ω ab ) and for light frequencies smaller than the in-plane plasma frequency ω ab .The experimental details related to the measurements of the polar Kerr effect in TRS breaking superconductors, together with the relevant examples of the temperature dependence of θ K , have been presented by Kapitulnik and co-workers 17,36 .The polar Kerr effect measures the rotation of the linearly polarised light reflected from the surface at normal incidence.The proper analysis of the Kerr effect requires careful consideration of the symmetries.Due to the delicate relation between reciprocity and time reversal symmetries a number of proposals turned out to be not valid as discussed recently 37 .For example, the gyrotropy, being a result of natural optical activity of the materials can not lead to the non-zero Kerr response.

B. Superconductors with TRS broken state
Table (I) gives a summary of TRS breaking materials together with some of their properties.In the table we indicate the method(s) used to detect TRS, the superconducting transition temperature and an important crystallographic aspect, namely the existence or not of the centre of inversion.The systems with the center of inversion are known as centrosymmetric (C) and those without it as non-centrosymmetric (NC).The informations on the superconducting order parameter are mostly scarce.They are subject to ongoing intensive research.Some of the compounds mentioned in the Table have been known for many years, while others have been discovered only recently.Here we shall add a few remarks concerning the properties of some materials and/or their families.For more comprehensive discussion we direct the reader to the original cited literature.
The superconductivity in URu 2 Si 2 has been discovered 38 in 1986, but most of the work has been devoted to the 17.5K anomaly observed in this material and termed "hidden" order.The identification of it is the subject of the on-going vigorous work and debate 39 .
One of the first examples where the broken time reversal symmetry has been detected [40][41][42] is UPt 3 .This superconductor has a reach phase diagram on the temperature -magnetic field plane with three different phases (named A, B and C) characterised by the unique nodal structures of the superconducting order parameter 43 .It is well established that the TRS is broken in one the phases.
The effect of disorder on the TRS broken state is not known in details yet.Typically the disorder seem to decrease the internal fields.In two alloy series Pr  der parameter in elemental Re superconductor and its superconducting compounds are still unknown.However, the lack of TRS in Re compounds seem to be crucially related to the presence of this element.To further explore the role of Re in the ReT M compounds the binary Re 1−x Mo x alloy series have been prepared and shown 46 to be superconducting for all concentrations x of Mo with the highest T c = 12.4K for x ≈ 0.6.Depending on x different crystallographic structures have been found.Among them the non-centrosymmetric α-Mn has been obtained.Superconductivity in non -centrosymmetric materials has been reviewed recently 70 .

III. SR 2 RUO 4 : PUZZLES, SOLUTIONS AND STILL OPEN ISSUES
The superconductivity in strontium ruthenate has been discovered 25 years ago and its understanding still presents a challenge.The title of the recent review "Even odder after twenty-three years: the superconducting order parameter puzzle of Sr 2 RuO 4 " very well summarizes 71 the state of the art in our understanding of its superconducting properties.
The normal state seem to be of the Fermi liquid variety and well behaved, although the issues related to the strength of electron correlations and spin-orbit coupling are not quite clear [72][73][74][75] .Beside the above mentioned, many excellent reviews on all aspects related to strontium ruthenate exist 12,[76][77][78][79] .Thus we shall concentrate on a few less common issues, namely observation and understanding of the Kerr effect in the material, recent experimental signatures of the multi-band and multi-condensate features.
After the discovery of the superconductivity in Sr 2 RuO 4 , the material being a crystallographically identical to high temperature cuprate superconductors but without copper, the hope was that its understanding will shed light on the latter systems.This was supported by the observation that at low temperature strontium ruthenate is a metal and behaves as an anisotropic and possibly correlated but otherwise well defined Fermi liquid 80 .Later on it turned out that Sr 2 RuO 4 is much more complicated with three bands being in play hosting probably spin triplet p-wave superconductivity.The first hint on the unconventional character of superconductivity has been provided by the extreme sensitivity of T c on non-magnetic disorder 81 .
The Fermi surface of the materials consists of three sheets known as α, β and γ.The first two are of one dimensional origin and result from hybridised d xz and d yz Ru orbitals, while the last one from d xy orbitals.This variety of orbitals has resulted in a discussion about the active bands and the role various orbitals play in the superconducting state [82][83][84] .
To understand experimental results showing the existence of line nodes, or at least deep minima in the quasiparticle density of states it has been proposed 83 that a full gap exists in the active band, which is that derived from d xy orbitals and the line nodes develop in passive bands by the interband proximity effect.The multiband aspects of superconductivity in this system, however, are of different variety from that considered in other materials [85][86][87] , albeit some similarities can be found.The discussion about the role of active vs. passive bands continues [88][89][90] .It is not unrelated to the studies of spin fluctuations in the material and the relative role of Coulomb interactions in different orbitals, mentioned earlier.
Kallin and Berlinsky finishing their review on chiral superconductors 12 write one would also like to have detailed information of where the low-lying excitations in Sr 2 RuO 4 are in momentum space [...].Probably the recent experiments 91 provide, at least partial, answer to that issue.Thus, even though the origin of spin triplet pairing in strontium ruthenate is still under debate, the discovery of the low energy modes 91 seem to give novel argument in favour of the early proposals 92,93 pointing out at the ferromagnetic spin fluctuations operating in the material and being responsible for its superconducting instability.Spin fluctuations have been studied in numerous papers [94][95][96][97] without clear cut conclusions.The relative importance of the ferro-vs.antiferromagnetic fluctuation has also been probed 98 .The conclusion of the recent paper 99 studying this issue by means of polarized inelastic neutron scattering is that spin fluctuations alone are not enough to generate a triplet state.This is in opposition to the results presented by Akebi et al. 91 and calls for further detailed analysis.

A. Modelling of strontium ruthenate
In view of the controversies about the microscopic mechanism of superconductivity in strontium ruthenae we have modelled the system by using precise knowledge related to its spectrum and assumed the phenomenological interaction parameters leading to the p-wave superconducting state in all three bands.The details have been described in the number of papers [100][101][102][103][104][105] and I shall not repeat the details here.The Hamiltonian of the system written in the orbital basis reads where m and m ′ refer to the three Ru One of the puzzling behaviours of strontium ruthenate has been found in measurements of the electronic spin susceptibility in external magnetic field parallel and perpendicular to the ab plane of the material.The spin susceptibility of spin-triplet superconductors is known to have matrix (3×3) structure with entries depending on the direction of the d -vector.
For standard spin-triplet odd parity chiral state the d-vector is of the form 108,109 and is directed along crystallographic c-axis, while for the helical states of the variety both observed a constant susceptibility below T c consistent with chiral triplet paring state with d along c-axis.On the other hand the measurements by Murakawa et al. 112 in magnetic field parallel to the c-axis also observed spin susceptibility below T c of the same value as in the normal state.This immediately indicates that either the superconducting state is not chiral, or B field induces the phase transition from the chiral to helical state.Using the three band model with relatively small but realistic spin-orbit coupling we have argued 113,114 that the d-vector rotates and the phase transition is expected.The calculated entropy jump at low temperature has been found to be very small, so the transition could be undetected in specific heat experiments.

B. Horizontal or vertical line nodes?
Small angle neutron scattering studies 115 provide support to the anisotropic multiband superconducting state with gap nodes or at least deep minima.The authors inferred the multiband behaviour from the superconducting anisotropy in Sr 2 RuO 4 which is hardly temperature dependent but increases for higher fields (≥ 1 T).This paper found evidence from vortex lattice distortion that the intrinsic superconducting anisotropy between the c axis and the Ru-O basal plane being of the order 60 is in agreement with that measured 116 by the ratio of coherence length ξ ab /ξ c ≈ 60, but exceeds the magnetic field anisotropy H ab c2 /H c c2 ≈ 20.In line with the above findings is the work of Kallin and co-workers who have shown 117 that if there are no horizontal nodes in the superconducting order parameter of Sr 2 RuO 4 so it is particularly difficult to reconcile chiral-p-wave order with residual thermal conductivity data.The model (5) leads to the horizontal nodes in the superconducting order parameter.
The order parameter has a number of intra-and interorbital components.They are written as for c(= d xy ) orbitals and, showing the existence of vertical line nodes in Sr 2 RuO 4 provide a good reason for further analysis of the gap nodes as it is not clear if the vertical anisotropy of the gaps as given by ( 9), together with the warping of the essentially cylindrical Fermi surface is enough to understand the measurements.For recent group -theoretical analysis of the gap nodes in tetragonal crystal see 119 .

C. Surface magnetic fields
As already indicated, the µSR and the polar Kerr effect experiments point towards appearance of the spontaneous magnetic fields inside the chiral strontium ruthenate superconductor.According to the expectations for such superconductors at their surfaces, at domain walls and near the impurities 120 one should observe small magnetic field.Despite many experimental efforts none of the measurements has found such surface magnetic fields [121][122][123] .For example in the paper 121 the authors used sensitive scanning Hall bar and superconducting quantum interference device microscopies and did not detect expected supercurrents.Negative results have been obtained in similar measurements for Sr 2 RuO 4 4 and PrOs 4 Sb 12 68,122 and suggested that the size of the chiral domains might be much smaller than expected.The paper 123 imposes an upper limit of ±2.5 mG on the magnitude of spontaneous magnetic fields at the well-defined edges of a mesoscopic disk.It is important to note that despite lack of surface fields the superconductivity-related time-reversal symmetry-breaking fields in the bulk have been observed by muon spin rotation and Kerr effect (see Table ( Judging from the value of the Kerr angle the fields seem to be really small.
On the theory side there appeared a number of papers trying to find reasons of such behaviour [124][125][126] .Recent studies 127 have shown that the surface flux pattern in chiral superconductors is not a universal feature, but instead it depends on many parameters describing the system.As a consequence the magnitude of the expected magnetic fields may differ from case to case and be smaller than expected earlier 128 .

IV. UNDERSTANDING THE KERR EFFECT IN SR
Shortly after the measurements of the Kerr signal in Sr 2 RuO 4 a number of theoretical papers appeared trying to understand its origin, magnitude and temperature dependence.
For references and a critical discussion see 35,129 .As noted earlier the existence of the Kerr signal is intimately related to the anomalous frequency dependent Hall effect σ H (ω). Due to the time reversal symmetry breaking state in superconductors one expects appearance of spontaneous magnetic fields and thus the Hall effect.The latter transport coefficient, however, requires that the charge in an electric field directed along, say x direction, to move also in y direction as in standard Hall effect in a B field.In the presence of magnetic field the perpendicular (i.e. in the y diection) motion is related to the Lorentz force acting on However, in the present case the chiral state responsible for time reversal breaking state is a result of internal interactions between the electrons.Thus the required force has to result from effective internal interactions inducing the chiral k x ± ik y state.This, however is impossible in the Galilean invariant system.This argument about the absence of such skew scattering in one -band Galilean invariant system has been put forward by Read and Green 130 .Good discussion of different constraints related to the observation of Kerr effect can be found in 37 .Later on on we shall present more formal argument showing vanishing of the Hall conductivity in one band chiral superconductor.
It is important to remind that the breaking of time reversal symmetry is a necessary but not sufficient condition for the observation of the Kerr effect.This is also true for magnetic systems, where spin-orbit interaction is a required additional ingredient 131 for the Hall effect to exist.It turns out that similar situation is observed in the time reversal symmetry breaking superconductors.To observe the Kerr effect additional factors have to contribute.In the literature there were various mechanism discussed.These included particle -hole asymmetry 132 , the order parameter collective mode response 133 , the impurity scattering leading to non-trivial vertex corrections and multiorbtal effects and the final size of the laser spot.
The ac Hall effect in chiral superconductors require breaking of the translational invariance and this is achieved by impurities as proposed originally by Goryo 134 and later elaborated by Lutchyn and coworkers 35,129 .Other possibility is provided by the many orbital or many band models.Such novel, earlier not considered mechanism of the Kerr effect has been independently proposed by two groups 14,15 .
The simple, formal argument on the absence of ac Hall conductivity σ H (ω) has been given by Taylor and Kallin 14 .These authors have noted that σ H (ω) is given by the antisymmetric part of the current -current correlation function π xy (q, ω) The correlator itself can be written in terms of the velocity matrices vx and vx and matrices of the Green functions Ĝ of the superconducting system as It follows that σ H vanishes for diagonal velocity matrices vx and vy as they commute with the Green function matrices.The critical overview of various attempts to calculate Kerr effect can be found in 135 .
Two different approaches have been used and two different models sharing, however, the multiband/multiorbital character have been studied by the two groups 14,15 proposing the novel mechanism of the Kerr effect.The results [136][137][138] provide a novel description of the anomalous ac Hall conductivity and are valid for virtually all superconductors.The prerequisite is the TRS breaking order parameter and a multiband one with non-zero interorbital/interband velocity matrices.
While the Kubo approach has been used in 14 , we 15 have calculated the Kerr effect from the definition of the optical dichroism 139,140 .In this formalism the conductivity tensor can The power has been calculated from the Fermi golden rule with the dipole matrix elements evaluated between the Bogolubov-de Gennes states.
The interorbital mechanism seem to be especially well suited for the understanding of the Kerr effect in strontium ruthenate, which is perhaps the cleanest superconductor, ever studied and the concurrent mechanism relying on higher order impurity scattering is not efficient.The novel mechanism of polar Kerr effect discovered during studies of the Kerr effect in this superconductor is of general importance and its validity for the semi-quantitative description of the effect in UBe 3 has been recently demonstrated 141 .

V. SUMMARY
We have reviewed aspects related to time reversal symmetry breaking in superconductors including Sr 2 RuO 4 , UPt 3 and other newly discovered systems.One has to note that the family of materials with this property is growing very fast.Moreover, many compounds belong to the non-centro-symmetric crystals.This calls for better understanding of the interplay between various symmetries in superconductors.
We briefly described standard techniques used to identify TRS breaking in superconductors, namely the µSR and the Kerr effect.We have discussed some of the many puzzling characteristics of strontium ruthenate (Sr 2 RuO 4 ), one of the cleanest and best studied superconductors with TRS breaking state below T c .The special attention has been paid to the discovery of the novel mechanism of the Kerr effect 14,15 and its application to Sr 2 RuO 4 .
However, many of the recent discoveries [142][143][144][145][146][147][148][149][150][151][152][153] have been left, partly due to lack of space in this short review.We only mention that the recent work 152 t 2g orbitals a = d xz , b = d yz and c = d xy ; i and j label the sites of a body centered tetragonal lattice.The hopping integrals t mm ′ (ij) and site energies ε m were fitted to reproduce the experimentally determined Fermi surface 106,107 .λ is the effective Ru 4d spin-orbit coupling parameter, and the effective Hubbard parameters U αβ,γδ mm ′ (ij) are generally spin as well as orbital dependent.In its simplest version the model uses just two effective Hubbard interaction parameters, both of which are completely determined by the requirement to fit the experimental T c .One of the parameters is responsible for the effective attraction of electrons in d xy orbitals of in plane Ru atoms, while the other is for attraction between electrons in out of plane Ru orbitals.
)d(k) = (sin k y , − sin k x , 0)it is lying in the (a,b)-plane of the strontium ruthenate.The early17 O Knight shift experiments performed in (ab) plane magnetic fields110 and neutron scattering experiments111 for m, m ′ = a, b or d xz and d yz orbitals forming α and β Fermi sheets.The experiments 118 be expressed in terms of the difference of the electromagnetic power absorption P (ω, ǫ) for light of left and right circular polarizations, ǫ L and ǫ R , respectively,Im[σ xy (ω)] = 1 V E 2 0 [P (ω, ǫ L ) − P (ω, ǫ R )] .
has established close relation between the Kerr rotation and odd-frequency superconductivity.Both are emerging from the same finite hybridization between different orbitals.Thus Sr 2 RuO 4 appears as one of the first bulk materials hosting odd-frequency superconductivity.The discussed multiorbital mechanism of the Kerr effect is universally valid in both clean and dirty materials.In view of the recent interests in the study of TRS breaking in superconductors it would be of great interest to evaluate the relative share of the impurity and inter-orbital contributions to the measured signal.This should be possible by means of the controlled disordering and the concomitant measurements of the Kerr angle of one of the many novel superconductors breaking the TRS.
and the Re 0.82 Nb 0.12 alloy (T c = 8.8K) without it.Both superconductors have been studied with the µSR technique and small but well defined magnetic fields were detected below T c .Such behaviour point out into the spontaneous time reversal symmetry breaking in both superconductors.The conclusion of the paper is that the lack of the inversion symmetry is not essential for the appearance of TRS breaking.This is also supported by the data presented in the Table, where many of the TRS breaking materials belong to C class.On the other hand the lack of TRS strongly constraints the allowed symmetry 45y La y Os 4 Sb 12 and Pr(Os 1−x Ru x ) 4 Sb 12 the magnetic field appearing below T c and revealed by µSR measurements has been found to initially decrease linearly with solute concentration44.It has been also established that Ru doping is considerably more efficient in decreasing magnetic field than La doping, with a 50% faster initial decrease.The data suggest that broken TRS state is suppressed for Ru concentration x ≥ 0.6 but persists for essentially all La concentrations.The detailed changes of the superconducting properties of the alloys are needed to understand this behaviour.It is visible from Table (I) that many superconductors with Re as one of the components belong to NC compounds and at the same time break TRS.This raises the question about the possible relation between both symmetries and the reasons while so many ReT M (T Mtransition metal) alloys feature such behaviour.To elucidate the issue, the authors45have performed the comparative studies of elemental Re superconductor (T c = 2.7K) with the center of inversion of the superconducting order parameter.This together with the temperature dependence of the specific heat and other thermodynamic and transport characteristics allows to judge the presence of the gaps in the order parameter.The details of the symmetry of the or-

TABLE I :
Time reversal symmetry breaking superconductors and their properties.We indicate the method of detection of TRS breaking, the structure of the material if it is centrosymmetric (C) or non-centrosymmetric (NC), the T c .The last entry provides the remarks concerning the structure of the order parameter.