Topological Objects in Ordered Electronic Systems

Most of correlated electronic systems possess ground states with broken crystal symmetries. These include superconductivity, spin orderings, a vast family of electronic crystals (including charge-/spin-density waves, Wigner crystals, arrays of stripes, charge ordering, and electronic ferroelectrics), and other translationally periodic states like super-structures in spin systems, spin-polarized density waves, and superconductors. Ground-state degeneracy allows for topologically nontrivial configurations connecting equivalent, whilst different, states. These “topological defects” include extended objects such as plane domain walls, lines of dislocations or phase vortices, various solitons or skyrmions as local microscopic objects, and transient processes—referred to as instantons—such as phase slips as space–time vortices. Embedded or transient topologically nontrivial configurations are readily induced by doping or optical pumping, by electric or magnetic fields, and under stresses or sliding. The Special Issue of the MDPI journal Symmetry and this corresponding e-book address these phenomena, as well as other topology-related electronic properties. This editorial article offers a schematic introduction to the field, placing emphasis on microscopic solitons and density waves, and provides a short summary of the content of this Special Issue.

1. Topologically Nontrivial Configurations: From Condensed Matter via the Solid State to Electronic Realizations

Embedded or transient topologically nontrivial configurations are common among symmetry-broken ground states [1]. State degeneracy tolerates the formation of various configurations, commonly called topological defects, connecting equivalent but different ground states.

The importance of topology-related phenomena in condensed matter physics has been recognized in the last decade. For example, in 2014, the Lars Onsager Prize was awarded to V.P. Mineev and G.E. Volovik “for or their contribution to a comprehensive classification of topological defects in condensed matter phases with broken symmetry”. Then, in 2016, the Nobel Prize in Physics was awarded to D.J. Thouless, F.D. Haldane, and J.M Kosterlitz, more generally “for theoretical discoveries of topological phase transitions and topological phases of matter”.

The classification of topological defects and questions on their allowance, stability, and protection primarily depend on the dimensions of the degeneracy manifold of their order parameters with respect to the dimensions of the space in which they are embedded [2,3]. Topological defects include macroscopic extended objects (domain walls, dislocation lines, and vorticity lines of phases or directors), and microscopic ones, including various solitons [4] and instantons for related transient processes.

The numerous existing or discussed realizations of topological defects involve particle physics [5], cosmology [6,7], quantum liquids [8,9] and gases, optical condensates [10], cold atoms [11], liquid crystals [12], conjugated polymers [13], and other quasi-one-dimensional conductors and biological macro-molecules, etc.

Solid states reveal topological defects, such as as domain walls or discommensurations in superstructures ([14] in this Special Issue), current vortices and phase slips in superconductors [15], displacement vortices as dislocations [16], vortices [17,18], and phase slips [19] in sliding superstructures, walls, and skyrmions in magnetic media [20,21,22].

Topological defects appear in the frame of strongly correlated electronic systems which typically show various types of symmetry breaking, giving rise to degenerate ground states. Among them, the vast family of electronic crystals includes charge- and spin-density waves; Wigner crystals; arrays of stripes [23,24,25], which will be discussed in this Special Issue; charge ordering; electronic ferroelectrics; and spin superstructures under magnetic fields. The latter include the spin-Peierls state [26], spin-polarized charge-density waves, and the FFLO state [27,28,29] in superconductors. Some of these superlattices, particularly incommensurate density waves [23,30,31,32] and Wigner crystals [23], are able to maintain the collective electric current by means of sliding (the so called Frölich conductivity [23,30,33]). For these incommensurate electronic crystals, their number of unit cells is not fixed which allows to absorb excess electrons into the extended ground state. This exchange among normal and condensed charge carriers requires for steps involving topologically nontrivial objects, like amplitude and phase solitons; phase vortices – dislocations, proceeding via transient processes of phase slips); and space–time vortices [15,17,19]. All of these result in a rich complex of nonlinear and nonstationary behavior with vast experimental observations. Here, topological defects are crucially necessary for the conversion between normal and collective currents, and they also appear in depinning processes to initiate sliding in the presence of host defects and constraints.

Among electronic crystals, the most frequently studied ordered states are the charge- and spin-density waves, which are ubiquitous in quasi-one-dimensional systems [30,31,32]. They demonstrate spectacular nonlinear conduction through collective sliding and lability to the electric field and the current injection. Static and transient topological defects emerge necessarily to maintain these phenomena; dislocations as space vortices and space–time vortices are known as instantons or phase-slip centers. Dislocations [18] are built-in statically under a transverse electric field [34]; their sweeping provides conversion among the normal carriers and the condensate [35,36], which ensures the onset of collective sliding. A special realization in a high magnetic field [37], when the density wave is driven by the Hall voltage originated by the current of quantized normal carriers, reveals the dynamical vorticity that serves to annihilate compensating normal and collective currents.

In charge-density waves, topological defects appear under stresses coming from various sources: surface mismatches of periodicity [38] or injection ([14,39] in this Special Issue ), proximity to commensurability ([40] in this Special Issue), a constraint geometry [41]; and an imbalance of normal and collective currents near junctions in the sliding regime [35,36]. The stress can easily exceed a plastic threshold, leading to the appearance of topological defects. For commensurate or close-to-commensurability density waves, the related strains can leave particular fingerprints, like a solitonic lattice or a system of random solitons, which allows for the identification of underlying structures via the STM [42,43] or by space-resolved X-ray scattering ([35,36,40] in this Special Issue).

Finally, spin-density waves, with their rich multiplicative order parameter, reveal complex objects with half-integer topologically bound vorticities in phases and directional degrees of freedom ([44] in this Special Issue).

2. Microscopic Solitons in (Quasi)-One-Dimensional Electronic Systems

The popular notion of “topological solitons” is a shortening for “topologically nontrivial solitons” which still need to be more precise, such as “topologically stable …” or “topologically protected …”. The “topological stability” is based on conservation of “topological charges” and the related “irreducibility of trajectories” connecting different states of a systems, see reviews e.g., [2,3].

Here, in applications to electronic systems, following only the minimal feature of a broadly spanning trajectory with no respect to its reducibility, we imply a broader, while less precise, definition of a “topologically nontrivial object”, as a local configuration exploring manifold degenerate ground states, connecting different equivalent ones. Their configurations may not be topologically stable, thus allowing for their transmutations with trivial electronic excitations. Such solitons appear or are preserved because they are energetically preferable, and/or because the total electric charge or spin, which these solitons carry, is preserved or monitored externally.

In one-dimensional systems, topological solitons become truly microscopic objects, carrying energy, charge, and other quantum numbers at a single-electron scale. As microscopic quasi-particles, they may become the lowest energy excitations and take over the role of conventional electrons in transport or optical properties (see short reviews [45,46]). Since the solitons possess similar quantum numbers such as the charge or the spin, their stable ensembles can be controllably created, maintained, and observed. Such ensembles can experience a sequence of phase transitions, accompanied by the formation of structures at an increasing scale: from individual solitons, via their microscopic complexes and growing aggregates, to macroscopic domain walls and stripes.

The role of solitons in electronic properties was anticipated in theories since the mid 1970s (see short reviews [45,46]). The common double degeneracy of systems with dimerization, like (sin-)Peierls states or the Mott state with a charge ordering), gives rise to solitons as kinks of the scalar order parameter A. The continuous degeneracy of the complex order parameter $A \exp \left(i\theta \right)$ (superconductors or charge-density waves) gives rise to phase vortices, amplitudes solitons, and their combinations. These degrees of freedom can be controlled or accessed independently via either the spin polarization or the charge doping.

There is a vast amount of experimental evidence on the existence of microscopic solitons and their determining role in electronic processes of quasi-one-dimensional electronic crystals. Different types of solitons appear in experiments. These include the amplitude king which can be charged spinless or neutral spin carrying, and the “holon” and the “spinon”, which are general terms used for strongly correlated electronic systems. There are also the polarization kinks carrying a fractional charge ([47] in this Special Issue), and the topologically bound charge–spin soliton. Solitons were firstly accessed in experiments on conducting polymers in the early 1980s [48]. They received renewed attention in the early 2000s from discoveries of ferroelectric charge ordering (see a review [49]) in organic conductors, from access via nano-scale tunneling experiments [41] in materials with charge-density waves, and from the optics of conducting polymers [50]. Today, various solitons appear in conductivity, tunneling spectroscopy, and optical absorption. Instantons, the corresponding dynamical processes, are responsible for subgap transitions leading to a pseudogap formation. The ferroelectric charge ordering in organic conductors provides access to several types of solitons ([47,49,51] in this Special Issue) observed in conductivity (holons), NMR (spinons), permittivity (polar kinks), as bound pairs in optics, and also compound charge–spin solitons in cases of combined symmetry breaking in polymers [50] and in charge-transfer molecular crystals ([47,51] in this Special Issue). In charge-density waves, individual solitons as the amplitude kinks, namely the spinons, have been visually captured in STM experiments [42,43]; notably, they have also been captured in diverse local probes in magnetic systems ([52] in this Special Issue). The subgap spectra of the coherent internal tunneling [41] recover the solitons as instantons.

Beyond the exceptional case of a truly one-dimensional (1D) system [43], experimentally, solitons have been observed or looked for in quasi-one-dimensional systems within low-temperature phases with long-range orders. Here, commuting between degenerate minima at only one chain would lead to a loss of the inter-chain-ordering energy proportional to the distance along the chain until the next soliton or a boundary. This energy dominates at long distances, even if it can be unimportant locally for a weak inter-chain interaction, which gives rise to the confinement of solitons (see [45,53] in this Special Issue). These interactions can appear already in some specific 1D systems where the ground-state degeneracy is not exact, so the soliton connects the true and false vacuums, losing the confinement energy.

The effect of confinement is omnipresent at higher dimensions where the inter-chain interactions, which are responsible for establishing the long-range 2D or 3D ordering, lift the degeneracy locally. In cases of discrete symmetries, the solitons are bound in topologically trivial pairs with an option for a subsequent phase transition to form cross-sample domain walls. In cases of continuous symmetries, the gapless mode can cure solve the interruption from the amplitude kink, which allows for individual solitons to exist in the low-temperature phases with long-range ordered states. The solitons adapt by forming topologically bound combined complexes with half-integer vortices of gapless modes: $\pi$-rotons [45,46]. In cases of repulsing and attracting electronic interactions correspondingly, this results in spin- or charge-roton configurations with charge- or spin-amplitude solitons localized in the core.

For the ensemble with a finite density of solitons, the confinement forces lead to a sequence of phase transitions ([53] in this Special Issue). The higher-temperature transition forces the confinement of solitons into topologically bound complexes: pairs of kinks or the amplitude solitons dressed by exotic half-integer vortices. At a second, lower temperature transition, the solitons aggregate into rods of bi-kinks or into walls of amplitude solitons terminated by rings of half-integer vortices [29,45,46,53]. As temperature lowers, the walls multiply, passing sequentially across the sample.

3. Special Issue Summary

This Special Issue features nine articles (including this introduction) which largely, though not exclusively, cover contemporary studies of topological objects in electronic, and related, systems.

The article by J. Tranquada [25] describes the concept of topological doping realized in high-Tc cuprate superconductors. Hole doping into a correlated antiferromagnet leads to charge stripes that separate antiferromagnetic spin stripes from opposite phases. The anti-phase Josephson coupling across the spin stripes can lead to a pair-density-wave order in which the broken translation symmetry of the superconducting wave function is accommodated by pairs with finite momentum.

The article by A. Kranjec et al. [14] describes studies in the dynamics of electronic dislocations and discommensurations in a Wigner crystal state. The studies exploit the scanning probe microscopy of an ensemble of structurally ordered polarons perturbed by optical pulses or a local charge injection. The experiments were performed on a layered compound which exhibits the Wigner crystal, displaying discommensurations and domain patterns when an additional charge is injected, either through contact or by photoexcitation. The domain walls and their crossings display topologically metastable entangled structures. The studies demonstrate the significance of topological protection at the microscopic level: the topologically trivial defects are rapidly annihilated pair-density-wave order with respect to the (meta-)stable non-trivial defects.

The article by J. Seidel [52] presents a scanning probe microscopy investigation of topological defects in magnetically and/or ferroelectrically ordered media. The article offers a comprehensive comparison of various versions of the techniques, from the more common STM, AFM, and SQUID to less known ones. The examples of nano-scale topological defects in the study span from straight domain walls to nontrivial monopoles and skyrmions.

The article by L. Vigliotti et al. [54] investigates bound states in a one-dimensional topological superconductor, addressing the dilemma of competition between the cases of Majorana- and Tamm-type edge states. This question is important in view of envisaged applications in topological quantum computation, which require the engineering of non-Abelian Majorana zero modes, the presence of which can be misleading due to the appearance of Tamm- or Andreev-bound states. The authors designed a model in which both Majorana- and Tamm-bound states can be present and compete. The model is a finite-size, one-dimensional topological superconductor in the presence of a competing normal gapping mechanism arising from a position-dependent potential, akin to a CDW, in which the phase can be monitored.

The article by K. Sunami [47] et al. is devoted to studies of solitons appearing in a quasi-one-dimensional ferroelectric-conducting material, composed of stacks of alternating donor and acceptor molecules. The dimerizational symmetry breaking which accompanies the neutral-–ionic phase transition gives rise to solitonic topological excitations as mobile boundaries between alternating ferroelectric domains. These solitons are expected to carry fractional charges, and may also carry the electronic spins that lead to the anomalous charge transport and the spin response. The article reviews related properties, which are studied by using a combination of NMR, NQR, and electrical resistivity measurements, and recalls the underlying theoretical concepts.

The article by D. Le Bolloc’h et al. [40] describes the application of space-resolved coherent X-ray diffraction in studies of solitonic lattices in sliding charge-density waves. The main features emerging from the local probe experiments include the influence of charge-density wave pinning on the sample surfaces and the propagation of periodic phase defects, such as charge solitons, across the entire sample. Corresponding numerical modeling is also presented.

The article by P. Karpov et al. [53] presents numerical simulations of the pattern formation and aggregation across phase transition in ensembles of solitons in a quasi-one-dimensional system. The long-range ordering enforced by the inter-chain coupling imposes super-long-range confinement forces upon the solitons, leading to a sequence of phase transitions in their ensembles. The higher-temperature transition enforces the confinement of solitons into topologically bound complexes, comprising pairs of kinks or the amplitude solitons dressed by half-integer vortices. At a second, lower-temperature transition, the solitons aggregate into rods of bi-kinks or into walls of amplitude solitons terminated by rings of half-integer vortices. The efficient Monte Carlo algorithm was employed, allowing to extend them to the three-dimensional case and to include the long-range Coulomb interactions.

The article by N. Kirova et al. [44] presents the phenomenological theory and numerical simulations of dynamical electronic vortices in charge- and spin-density waves. Their collective sliding requires the emergence of static and transient topological defects: there are dislocations as space vortices and space–time vortices known as phase-slip centers, which are a kind of instantons. The rich order parameter of spin-density waves reveals complex objects with half-integer topologically bound vorticities in charge- and spin degrees of freedom. The presented modeling is based upon numerical solutions to partial differential equations for the dissipative dynamics. It takes into account the complex order parameter, the self-consistent electric field, and the normal carriers. The traditional time-dependent Ginzburg–Landau approach, which is shown to be contradictory with respect to the charge conservation, was generalized, allowing the authors to treat the intrinsic normal carriers consistently.

In conclusion, we believe that this Special Issue and the subsequent e-book will illustrate the vitality of the field and the diversity of its subjects. We appreciate the efforts of the authors and help from referees, all of whose contributions have made this Special Issue a true success. We acknowledge the Editorial Office for supporting this Special Issue and the edited e-book.