Antiunitary Symmetry Research Articles

Duality and stacking of bosonic and fermionic SPT phases

We study the interplay of duality and stacking of bosonic and fermionic symmetry-protected topological phases in one spatial dimension. In general the classifications of bosonic and fermionic phases have different group structures under the operation of stacking, but we argue that they are often isomorphic and give an explicit isomorphism when it exists. This occurs for all unitary symmetry groups and many groups with antiunitary symmetries, which we characterize. We find that this isomorphism is typically not implemented by the Jordan-Wigner transformation, nor is it a consequence of any other duality transformation that falls within the framework of topological holography. Along the way to this conclusion, we recover the fermionic stacking rule in terms of G\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{G} $$\\end{document}-pin partition functions, give a gauge-invariant characterization of the twisted group cohomology invariant, and state a procedure for stacking gapped phases in the formalism of symmetry topological field theory.

Lieb-Schultz-Mattis Theorem for 1D Quantum Magnets with Antiunitary Translation and Inversion Symmetries.

We study quantum many-body systems in the presence of an exotic antiunitary translation or inversion symmetry involving time reversal. Based on a symmetry-twisting method and spectrum robustness, we propose that a half-integer spin chain that respects any of these two antiunitary crystalline symmetries in addition to the discrete Z_{2}×Z_{2} global spin-rotation symmetry must either be gapless or possess degenerate ground states. This explains the gaplessness of a class of chiral spin models not indicated by the Lieb-Schultz-Mattis theorem and its known extensions. Moreover, we present symmetry classes with minimal sets of generators that give nontrivial Lieb-Schultz-Mattis-type constraints, argued by the bulk-boundary correspondence in 2D symmetry-protected topological phases as well as lattice homotopy. Our results for detecting the ingappability of 1D quantum magnets from the interplay between spin-rotation symmetries and magnetic space groups are applicable to systems with a broader class of spin interactions, including Dzyaloshinskii-Moriya and triple-product interactions.

The Floquet Baxterisation

Quantum integrability has proven to be a useful tool to study quantum many-body systems out of equilibrium. In this paper we construct a generic framework for integrable quantum circuits through the procedure of Floquet Baxterisation. The integrability is guaranteed by establishing a connection between Floquet evolution operators and inhomogeneous transfer matrices obtained from the Yang-Baxter relations. This allows us to construct integrable Floquet evolution operators with arbitrary depths and various boundary conditions. Furthermore, we focus on the example related to the staggered 6-vertex model. In the scaling limit we establish a connection of this Floquet protocol with a non-rational conformal field theory. Employing the properties of the underlying affine Temperley-Lieb algebraic structure, we demonstrate the dynamical anti-unitary symmetry breaking in the easy-plane regime. We also give an overview of integrability-related quantum circuits, highlighting future research directions.

Robust Fermi-Liquid Instabilities in Sign Problem-Free Models.

Determinant quantum MonteCarlo (DQMC) is a powerful numerical technique to study many-body fermionic systems. In recent years, several classes of sign-free (SF) models have been discovered, where the notorious sign problem can be circumvented. However, it is not clear what the inherent physical characteristics and limitations of SF models are. In particular, which zero-temperature quantum phases of matter are accessible within such models, and which are fundamentally inaccessible? Here, we show that a model belonging to any of the known SF classes within DQMC cannot have a stable Fermi-liquid ground state in spatial dimension d≥2, unless the antiunitary symmetry that prevents the sign problem is spontaneously broken (for which there are currently no known examples in SF models). For SF models belonging to one of the symmetry classes (where the absence of the sign problem follows from a combination of nonunitary symmetries of the fermionic action), any putative Fermi liquid fixed point generically includes an attractive Cooper-like interaction that destabilizes it. In the recently discovered lower-symmetry classes of SF models, the Fermi surface (FS) is generically unstable even at the level of the quadratic action. Our results suggest a fundamental link between Fermi liquids and the fermion sign problem. Interestingly, our results do not rule out a non-Fermi-liquid ground state with a FS in a sign-free model.

Nonperturbative constraints from symmetry and chirality on Majorana zero modes and defect quantum numbers in (2+1) dimensions

In (1+1)D topological phases, unpaired Majorana zero modes (MZMs) can arise only if the internal symmetry group $G_f$ of the ground state splits as $G_f = G_b \times \mathbb{Z}_2^f$, where $\mathbb{Z}_2^f$ is generated by fermion parity, $(-1)^F$. In contrast, (2+1)D topological superconductors (TSC) can host unpaired MZMs at defects even when $G_f$ is not of the form $G_b \times \mathbb{Z}_2^f$. In this paper we study how $G_f$ together with the chiral central charge $c_-$ strongly constrain the existence of unpaired MZMs and the quantum numbers of symmetry defects. Our results utilize a recent algebraic characterization of (2+1)D invertible fermionic topological states, which provides a non-perturbative approach based on topological quantum field theory, beyond free fermions. We study physically relevant groups such as $\mathrm{U}(1)^f\rtimes H,\mathrm{SU}(2)^f \times H, \mathrm{U}(2)^f\rtimes H $, generic Abelian groups, as well as more general compact Lie groups, antiunitary symmetries and crystalline symmetries. We present an algebraic formula for the fermionic crystalline equivalence principle, which gives an equivalence between states with crystalline and internal symmetries. In light of our theory, we discuss several previously proposed realizations of unpaired MZMs in TSC materials such as Sr$_2$RuO$_4$, transition metal dichalcogenides and iron superconductors, in which crystalline symmetries are often important; in some cases we present additional predictions for the properties of these models.

Signatures of spectral crossovers in the short- and long-range spectral correlations of a disordered spin-chain with Kramers degeneracy

We investigate several distinct spectral crossovers amongst various integrable (Poissonian) and quantum-chaotic (Wigner-Dyson) limits of a $1\mathrm{D}$ disordered quantum spin ($\mathrm{S}=1/2$) model, by tuning the relative amplitudes of various Hamiltonian parameters to retain or break relevant unitary and antiunitary symmetries. The spin model consists of an isotropic and deterministic Heisenberg term, a random Ising term, an anisotropic and antisymmetric, but deterministic Dzyaloshinskii-Moriya (DM) term, and a Zeeman coupling to a random, inhomogeneous magnetic field. Since we are specially interested in crossovers involving a Gaussian symplectic ensemble (GSE) limit, we carry out all our calculations with an odd number of lattice sites (spins) that naturally results in eigenspectra with Kramers degeneracies (KD's). The various crossovers (viz., the reentrant Poissonian-to-GSE-to-Poissonian, Poissonian-to-GUE, GSE-to-GUE and the reentrant Poissonian-to-GOE-to-Poissonian crossovers) are investigated via detailed studies of both short-range (nearest-neighbour-spacings distribution, NNSD) and long-range [spectral rigidity ${\mathrm{\ensuremath{\Delta}}}_{3}(L)$ and number variance ${\mathrm{\ensuremath{\Sigma}}}^{2}(L)$ ] spectral correlations, where $L$ is the spectral interval over which the long-range statistic is examined. The short-range studies show excellent agreement with RMT predictions. One of the highlights of this study is the systematic investigation of the consequences of retaining both eigenvalues corresponding to every Kramers doublet, in a crossover involving the GSE limit, and see how it evolves to a limit where the KD is naturally lifted. This is seen most clearly in the NNSD study of the GSE-to-GUE (Gaussian unitary ensemble) transition, achieved by gradually lifting the KD, using the random magnetic field. The NNSD plot in the GSE limit here exhibits a Dirac delta peak at zero splitting and a renormalized GSE hump at finite splitting, whose general analytical form and its asymptotic limit are derived. With an increasing symmetry breaking magnetic field the NNSD shows an interesting, dynamic two-peaked structure that finally converges to the standard GUE lineshape. We explain this trend in terms of a competition between the splittings amongst distinct Kramers doublets (related to unitary symmetries) and the Zeeman-like splittings induced by a breaking of the antiunitary time-reversal symmetry (TRS). This is investigated via the NNSD, the marginal spectral density (MSD) and the densities of states (DOS) for both spin models and RMT crossover matrix models. The first and the final short-range studies involve reentrant Poissonian-to-GOE(GSE)-to-Poissonian crossovers, where the final Poissonian is obtained by a many-body localization of states in the strongly disordered limit, whereas the initial Poissonian regime involve much more delocalized eigenstates. In the long-range spectral correlation studies, we shed light on the extent of agreement between our physical spin systems and RMT predictions. We find that the spin systems depart from the ideal RMT predictions for relatively finite $L\ensuremath{\sim}10\ensuremath{-}15$ at least, for the spectral rigidity and a much smaller $L\ensuremath{\sim}2\ensuremath{-}4$ for the number variance. It is further seen that the departure is usually sooner at the uncorrelated (Poissonian) upper end compared to the correlated (Wigner-Dyson) lower end. We carry out a detailed comparison between the local and the global crossover points, associated with the short-range and the long-range statistics respectively, and find that in most cases they seem to agree reasonably well, but for a few exceptions. Our studies also show that the long-range correlations may serve to distinguish between the two Poissonian limits (nonlocalized and localized) in the reentrant crossovers, which the short-range correlations fail to distinguish.

Non-local order parameters for fermion chains via the partial transpose

In the last two decades, a vast variety of topological phases have been described, predicted, classified, proposed, and measured. While there is a certain unity in method and philosophy, the phenomenology differs wildly. This work deals with the simplest such case: fermions in one spatial dimension, in the presence of a symmetry group G, which contains anti-unitary symmetries. A complete classification of topological phases, in this case, is available. Nevertheless, these methods are to some extent lacking as they generally do not allow to determine the class of a given system easily. This paper will take up proposals for non-local order parameters defined through anti-unitary symmetries. They are shown to be homotopy invariants on a suitable set of ground states. For matrix product states, an interpretation of these invariants is provided: in particular, for a particle–hole symmetry, the invariant determines a real division super algebra D such that the bond algebra is a matrix algebra over D.

Protection of all nondefective twofold degeneracies by antiunitary symmetries in non-Hermitian systems

Non-Hermitian degeneracies are classified as defective and nondefective exceptional points~(EP). While in defective EPs, both eigenvalues and eigenenergies coalesce, nondefective EPs are characterized merely by the occurrence of degeneracies in their eigenvalues. It is also known that all degeneracies are either symmetry-protected or accidental. In this paper, I prove that anti-unitary symmetries protect all nondefective twofold EPs. By developing a 2D non-Hermitian tight-binding model, I have demonstrated that these symmetries are composite of various symmetry operations such as discrete or spatial point-group symmetries and Wick's rotation in the non-Hermitian parameter space. Introducing these composite symmetries, I present the protection of nondefective EPs in various parameter regimes of my model. This work paves the way to stabilizing nondefective EPs and offers a new perspective on understanding non-Hermitian band crossings.

Construction of maximally-localized Wannier functions using crystal symmetry

In this study, we show two methods concerning constructions of Maximally localized Wannier functions (MLWFs) using crystal symmetry. The first one is to compute input matrices for generating MLWFs only using wavefunctions in the irreducible Brillouin zone (IBZ). This approach reduces the computational costs and the size of intermediate files in proportion to the number of k points in the IBZ. The second method is to calculate the symmetry-adapted Wannier functions that are compatible with so-called frozen window technique, that is, the procedure to disentangle band structures by forcing Bloch states in a certain energy range to be included. We demonstrate how these work in the case of Fe, Co3Sn2S2, Cu, and Nb. These methods are implemented in PW2WANNIER90 code, which interfaces Quantum-ESPRESSO with the Wannier90, and the open-source python library, SymWannier. The code supports both unitary and anti-unitary symmetry operations as well as nonsymmorphic group operations, spin-orbit couplings, and ultrasoft/PAW pseudopotentials. All of these codes are available on GitHub (https://github.com/wannier-utils-dev/symWannier).

Majorana fermions in multiband quantum wires: A semiclassical analysis

In this paper, we calculate the Andreev bound state spectrum in a half-infinite superconducting wire with an arbitrary number of bands crossing the chemical potential. Electrons in the wire are affected by the asymmetric (Rashba) spin–orbit coupling due to the presence of a substrate. The normal state of the wire is assumed to have an antiunitary symmetry (time reversal or its combination with a point group operation), which may be broken by the superconducting order parameter and/or the boundary. The bound state spectrum is calculated without relying on any particular microscopic model, using the semiclassical approach with the boundary conditions described by a phenomenological scattering matrix.

Magnetic band representations, Fu-Kane-like symmetry indicators, and magnetic topological materials

To realize novel topological phases and to pursue potential applications in low-energy consumption spintronics, the study of magnetic topological materials is of great interest. Starting from the theory of nonmagnetic topological quantum chemistry [Bradlyn et al., Nature 547, 298 (2017)], we have obtained irreducible (co)representations and compatibility relations (CRs) in momentum space, and we constructed a complete list of magnetic band (co)representations (MBRs) in real space for other magnetic space groups (MSGs) with antiunitary symmetries (i.e.,type-III and type-IV MSGs). The results are consistent with the magnetic topological quantum chemistry [Elcoro et al., Nat. Commun. 12, 5965 (2021)]. Using the CRs and MBRs, we reproduce the symmetry-based classifications for MSGs, and we obtain a set of Fu-Kane-like formulas of symmetry indicators (SIs) in both spinless (bosonic) and spinful (fermionic) systems, which are implemented in an automatic code---TopMat---to diagnose topological magnetic materials. The magnetic topological materials, whose occupied states cannot be decomposed into a sum of MBRs, are consistent with nonzero SIs. Lastly, using our online code, we have performed spin-polarized calculations for magnetic compounds in the materials database, and we find many magnetic topological candidates.

Realizing exceptional points of any order in the presence of symmetry

Exceptional points~(EPs) appear as degeneracies in the spectrum of non-Hermitian matrices at which the eigenvectors coalesce. In general, an EP of order $n$ may find room to emerge if $2(n-1)$ real constraints are imposed. Our results show that these constraints can be expressed in terms of the determinant and traces of the non-Hermitian matrix. Our findings further reveal that the total number of constraints may reduce in the presence of unitary and antiunitary symmetries. Additionally, we draw generic conclusions for the low-energy dispersion of the EPs. Based on our calculations, we show that in odd dimensions the presence of sublattice or pseudo-chiral symmetry enforces $n$th order EPs to disperse with the $(n-1)$th root. For two-, three- and four-band systems, we explicitly present the constraints needed for the occurrence of EPs in terms of system parameters and classify EPs based on their low-energy dispersion relations.

Spin-Group Symmetry in Magnetic Materials with Negligible Spin-Orbit Coupling

Symmetry formulated by group theory plays an essential role with respect to the laws of nature, from fundamental particles to condensed matter systems. Here, by combining symmetry analysis and tight-binding model calculations, we elucidate that the crystallographic symmetries of a vast number of magnetic materials with light elements, in which the neglect of relativistic spin-orbit coupling (SOC) is an appropriate approximation, are considerably larger than the conventional magnetic groups. Thus, a symmetry description that involves partially-decoupled spin and spatial rotations, dubbed as spin group, is required. Spin group permits more symmetry operations and thus more energy degeneracies that are disallowed by the magnetic groups. One consequence of the spin group is the new anti-unitary symmetries that protect SOC-free Z_2 topological phases with unprecedented surface node structures. Our work not only manifests the physical reality of materials with weak SOC, but also shed light on the understanding of all solids with and without SOC by a unified group theory.

Imaginarity-free quantum multiparameter estimation

Multiparameter quantum estimation is made difficult by the following three obstacles. First, incompatibility among different physical quantities poses a limit on the attainable precision. Second, the ultimate precision is not saturated until you discover the optimal measurement. Third, the optimal measurement may generally depend on the target values of parameters, and thus may be impossible to perform for unknown target states. We present a method to circumvent these three obstacles. A class of quantum statistical models, which utilizes antiunitary symmetries or, equivalently, real density matrices, offers compatible multiparameter estimations. The symmetries accompany the target-independent optimal measurements for pure-state models. Based on this finding, we propose methods to implement antiunitary symmetries for quantum metrology schemes. We further introduce a function which measures antiunitary asymmetry of quantum statistical models as a potential tool to characterize quantumness of phase transitions.

Spin-space groups and magnon band topology

Band topology is both constrained and enriched by the presence of symmetry. The importance of anti-unitary symmetries such as time reversal was recognized early on leading to the classification of topological band structures based on the ten-fold way. Since then, lattice point group and non-symmorphic symmetries have been seen to lead to a vast range of possible topologically nontrivial band structures many of which are realized in materials. In this paper we show that band topology is further enriched in many physically realizable instances where magnetic and lattice degrees of freedom are wholly or partially decoupled. The appropriate symmetry groups to describe general magnetic systems are the spin-space groups. Here we describe cases where spin-space groups are essential to understand the band topology in magnetic materials. We then focus on magnon band topology where the theory of spin-space groups has its simplest realization. We consider magnetic Hamiltonians with various types of coupling including Heisenberg and Kitaev couplings revealing a hierarchy of enhanced magnetic symmetry groups depending on the nature of the lattice and the couplings. We describe, in detail, the associated representation theory and compatibility relations thus characterizing symmetry-enforced constraints on the magnon bands revealing a proliferation of nodal points, lines, planes and volumes.

Symmetry-Protected Multifold Exceptional Points and Their Topological Characterization.

We investigate the occurrence of n-fold exceptional points (EPs) in non-Hermitian systems, and show that they are stable in n-1 dimensions in the presence of antiunitary symmetries that are local in parameter space, such as, e.g., parity-time (PT) or charge-conjugation parity (CP) symmetries. This implies in particular that threefold and fourfold symmetry-protected EPs are stable, respectively, in two and three dimensions. The stability of such multofold exceptional points (i.e., beyond the usual twofold EPs) is expressed in terms of the homotopy properties of a resultant vector that we introduce. Our framework also allows us to rephrase the previously proposed Z_{2} index of PT and CP symmetric gapped phases beyond the realm of two-band models. We apply this general formalism to a frictional shallow water model that is found to exhibit threefold exceptional points associated with topological numbers ±1. For this model, we also show different non-Hermitian topological transitions associated with these exceptional points, such as their merging and a transition to a regime where propagation is forbidden, but can counterintuitively be recovered when friction is increased furthermore.

Anomalies in bosonic symmetry-protected topological edge theories: Connection to F symbols and a method of calculation

We describe a systematic procedure for determining the identity of a 2D bosonic symmetry protected topological (SPT) phase from the properties of its edge excitations. Our approach applies to general bosonic SPT phases with either unitary or antiunitary symmetries, and with either continuous or discrete symmetry groups, with the only restriction being that the symmetries must be on-site. Concretely, our procedure takes a bosonic SPT edge theory as input, and produces an element $\omega$ of the cohomology group $H^3(G, U_T(1))$. This element $\omega \in H^3(G, U_T(1))$ can be interpreted as either a label for the bulk 2D SPT phase or a label for the anomaly carried by the SPT edge theory. The basic idea behind our approach is to compute the $F$-symbol associated with domain walls in a symmetry broken edge theory; this domain wall $F$-symbol is precisely the anomaly we wish to compute. We demonstrate our approach with several SPT edge theories including both lattice models and continuum field theories.

Fate of symmetry protected coherence in open quantum system

We investigate the fate of coherence in the dynamical evolution of symmetry protected quantum systems. Under the formalism of system plus bath for open quantum systems, antiunitary symmetries exhibit significant differences from the unitary ones in protecting initial coherence. Specifically, taking advantage of the Lindblad master equation, we find that a pure state in the symmetry protected degenerate subspace will be decoherent even though both the system Hamiltonian and system-environment interaction respect the same antiunitary symmetry. In contrast, the coherence will persist when the protecting symmetry is unitary. We provide an elaborate classification table to illustrate what kinds of symmetry combinations are able to preserve the coherence of the initial state, which is confirmed in several concrete models of single spin-$3/2$ systems. It can also be applied to the Haldane phase in interacting topological systems. Our results could be helpful in exploring possible experimental realization of stable time-reversal symmetric topological states.

The construction of Dirac operators on orientifolds

Motivated by Wigner's theorem, a canonical construction is described that produces an Atiyah-Singer Dirac operator with both unitary and anti-unitary symmetries. This Dirac operator includes the Dirac operator for KR-theory as a special case, filling a long-standing gap in the literature. In order to make the construction, orientifold Spinc-structures are defined and classified using semi-equivariant Dixmier-Douady theory, and analogues of several standard theorems on the existence of Spinc-structures are proved.

Antiferromagnetism-Induced Spin Splitting in Systems Described by Magnetic Layer Groups

In this paper, we report the classification of all the 528 magnetic layer groups into seven spin splitting prototypes, in analogy to the similar classification previously reported for the 1651 magnetic space groups [Yuan et al., Phys. Rev. Mat. 2021, 5, 014409]. According to this classification, induced (Pekar–Rashba) spin splitting is possible in the antiferromagnetic monolayers described by magnetic layer groups of type I (no anti-unitary operations) and III, both in the presence and in the absence of the space inversion operation. As one specific application, we have studied theoretically the (110) monolayer of antiferromagnetic (AFM) MnO2. Our density functional theory calculations for this structure predicted colossal (on the order of 1 eV) spin splitting in the entire Brillouin zone, similar to the case of the (110) monolayer of AFM MnF2 which was studied by us earlier. This splitting is much larger compared to the corresponding values for the bulk AFM1 structure of β-MnO2, which was rationalized by noting that the (110) monolayer is described by the type I magnetic layer group that does not contain anti-unitary symmetry operations, and therefore, spin splitting is not suppressed anywhere in the entire Brillouin zone, including high-symmetry points and lines.