We propose Majorana-assisted nonlocal spin correlation as a manifestation of Majorana nonlocality in quasi-one-dimensional (1D) Kitaev spin liquids. Focusing on the flux-free sector of the Kitaev honeycomb model in a quasi-1D geometry, we uncover its topological nature and show that it hosts Majorana zero modes localized at both ends, which are stabilized by finite-size-induced topology. We further show that the nonlocal Majorana fermion parity operator, P_MF=iγ_Lγ_R, is mapped to a nonlocal spin-string operator, producing an end-to-end spin correlation proportional to the product of P_MF and total fermion parity operators when local perturbations remove redundant ground-state degeneracies while preserving the Majorana and total fermion parities in the flux-free sector. Numerical calculations confirm a finite nonlocal spin correlation generated by these Majorana zero modes without any local magnetization. Our results establish a concrete signature of intrinsic Majorana nonlocality in quantum spin liquids.

We study the quantum correction to the conductivity due to disorder in two-dimensional fragile topological bands with nonzero Euler class. Contrary to graphene where two Dirac nodes have opposite vorticities, two bands with a unit Euler number possess two Dirac points with the same vorticity, which may affect the Anderson localization. Most notably, we report an anomalous localization behavior in spinful magnetic Euler bands based on symmetry analysis and diagrammatic calculations. Despite the presence of spin-orbit coupling and an in-plane magnetization that explicitly breaks physical time-reversal symmetry, the system exhibits weak localization behavior characteristic of the orthogonal symmetry class. We demonstrate that this counter-intuitive phenomenon originates from an emergent effective time-reversal symmetry composed of crystalline and spacetime inversion symmetries, allowing it to supersede the standard localization behavior. Our findings reveal that the effective crystalline symmetries can fundamentally alter the universality class of disordered systems, rendering the localization behavior independent of the specific vorticity configuration of Dirac nodes.

Real band topology often appears in systems with space-time inversion symmetry and is characterized by invariants such as the Euler and second Stiefel-Whitney classes. Here, we examine the generic band topology of Bogoliubov-de Gennes (BdG) Hamiltonians with C2zT symmetry, where C2z and T are twofold rotation about the z axis and time-reversal symmetries, respectively. We discuss the Euler band topology of superfluids and superconductors in the DIII and CI Altland-Zirnbauer symmetry classes, where the Euler class serves as an integer-valued topological invariant of the 4 x 4 BdG Hamiltonian. Using expressions for the Euler class under n-fold rotational symmetry, we derive formulas relating the Euler class to previously known topological invariants of class DIII and CI systems. We demonstrate that three-dimensional class DIII topological phases with an odd winding number, including the B phase of superfluid helium-3, are topological superconductors or superfluids with a nontrivial Euler class. We refer to these as Euler superconductors or superfluids. Specifically, the 3He-B superfluid in a magnetic field is identified as an Euler superfluid. Three-dimensional class CI topological phases with twice an odd winding number are also Euler superconductors or superfluids. When spatial inversion symmetry is present, class CI superconductors with a nontrivial Euler class exhibit superconducting nodal lines with a linking structure. This phenomenon is demonstrated using a model of a three-dimensional s +/--wave superconductor. These findings provide a unified framework for exploring Euler band topology in superfluids and superconductors, connecting various phenomena associated with T-breaking perturbations, including Majorana Ising susceptibility and higher-order topology.

We study the one-dimensional S=1 XXZ spin model with single-ion anisotropy. It is known that at the transition points between the Haldane and large-D phases, the model exhibits a quantum criticality described by the Gaussian theory, i.e., a conformal field theory with the central charge c=1. Using the bosonization approach, we investigate various correlation functions at the phase transition and derive their asymptotic forms. This allows us to clarify their peculiar behavior: the longitudinal (transverse) two-point spin correlation function has components that decay algebraically only in the uniform (staggered) sector. These theoretical predictions are verified by the numerical calculations using the density-matrix renormalization group method. The effect of weak bond alternation on the critical ground state at the phase transition is also discussed. It is shown that the bond alternation induces the missing power-law components in the correlation functions.

The anisotropic quantum spin-1/2 XY model on a linear chain was solved by Lieb, Schultz, and Mattis in 1961 and shown to display a continuous quantum phase transition at the O(2) symmetric point separating two gapped phases with competing Ising long-range order. For the square lattice, the following is known. The two competing Ising ordered phases extend to finite temperatures, up to a boundary where a transition to the paramagnetic phase occurs, and meet at the O(2) symmetric critical line along the temperature axis that ends at a tricritical point at the Berezinskii-Kosterlitz-Thouless transition temperature where the two competing phases meet the paramagnetic phase. We show that the first-order zero-temperature (quantum) phase transition that separates the competing phases as a function of the anisotropy parameter is smoothed by thermal fluctuations into deconfined classical criticality.

We propose a general necessary condition for a spin chain with SO(3) spin-rotation symmetry to be gapped. Specifically, we prove that the ground state(s) of an SO(3)-symmetric gapped spin chain must be spin singlet(s), and the expectation value of a twisting operator asymptotically approaches unity in the thermodynamic limit, where finite-size corrections are inversely proportional to the system size. This theorem provides (i) supporting evidence for various conjectured gapped phases, and (ii) a sufficient criterion for identifying gapless spin chains. We verify our theorem by numerical simulations for a variety of spin models and show that it offers a novel efficient way to identify gapless phases in spin chains with spin-rotation symmetry.

The Euler class is a $\mathbb{Z}$-valued topological invariant that characterizes a pair of real bands in a two-dimensional Brillouin zone. One of the symmetries that permits its definition is $C_{2z}T$, where $C_{2z}$ denotes a twofold rotation about the $z$ axis and $T$ denotes time-reversal symmetry. Here, we study three-dimensional spinless insulators characterized by the Euler class, focusing on the case where additional $C_{4z}$ or $C_{6z}$ rotational symmetry is present, and investigate the relationship between the Euler class of the occupied bands and their rotation eigenvalues. We first consider two-dimensional systems and clarify the transformation rules for the real Berry connection and curvature under point group operations, using the corresponding sewing matrices. Applying these rules to $C_{4z}$ and $C_{6z}$ operations, we obtain explicit formulas that relate the Euler class to the rotation eigenvalues at high-symmetry points. We then extend our analysis to three-dimensional systems, focusing on the difference in the Euler class between the two $C_{2z}T$-invariant planes. We derive analytic expressions that relate the difference in the Euler class to two types of representation-protected invariants and analyze their phase transitions. We further construct tight-binding models and perform numerical calculations to support our analysis and elucidate the bulk-boundary correspondence.

We demonstrate that topological insulators exhibit an intrinsic interfacial instability that amplifies small interface fluctuations, resulting in chaotic behavior during interface growth. This mechanism is different from conventional interfacial instabilities in crystal growth that are driven by external non-uniformities such as surface diffusion, and instead arises from intrinsic electronic properties of topological materials. We find that the boundary states of topological insulators have a pronounced impact on the surface stiffness, which quantifies how strongly a surface resists changes in its shape or orientation. While trivial insulators possess positive stiffness that smooths out surface roughness, topological insulators exhibit negative stiffness that amplifies small shape fluctuations. We derive an effective equation of the interface growth with this negative stiffness and demonstrate that the interface dynamics is governed by the Kuramoto–Sivashinsky equation, a prototypical nonlinear equation exhibiting spatiotemporal chaos.

The anisotropic quantum spin-12 XY model on a linear chain was solved by Lieb, Schultz, and Mattis [] and shown to display a continuous quantum phase transition at the O(2) symmetric point separating two gapped phases with competing Ising long-range order. For the square lattice, the following is known. The two competing Ising ordered phases extend to finite temperatures, up to a boundary where a transition to the paramagnetic phase occurs, and meet at the O(2) symmetric critical line along the temperature axis that ends at a tricritical point at the Berezinskii-Kosterlitz-Thouless transition temperature where the two competing phases meet the paramagnetic phase. We show that the first-order zero-temperature (quantum) phase transition that separates the competing phases as a function of the anisotropy parameter is smoothed by thermal fluctuations into deconfined classical criticality.

Lieb-Schultz-Mattis (LSM) theorems impose non-perturbative constraints on the zero-temperature phase diagrams of quantum lattice Hamiltonians (always assumed to be local in this paper). LSM theorems have recently been interpreted as the lattice counterparts to mixed ’t Hooft anomalies in quantum field theories that arise from a combination of crystalline and global internal symmetry groups. Accordingly, LSM theorems have been reinterpreted as LSM anomalies. In this work, we provide a systematic diagnostic for LSM anomalies in one spatial dimension. We show that gauging subgroups of the global internal symmetry group of a quantum lattice model obeying an LSM anomaly delivers a dual quantum lattice Hamiltonian such that its internal and crystalline symmetries mix non-trivially through a group extension. This mixing of crystalline and internal symmetries after gauging is a direct consequence of the LSM anomaly, i.e., it can be used as a diagnostic of an LSM anomaly. We exemplify this procedure for a quantum spin-1/2 chain obeying an LSM anomaly resulting from combining a global internal \mathbb{Z}^{\,}_{2}×\mathbb{Z}^{\,}_{2} ℤ2×ℤ2 symmetry with translation or reflection symmetry. We establish a triality of models by gauging a \mathbb{Z}^{\,}_{2}\subset\mathbb{Z}^{\,}_{2}×\mathbb{Z}^{\,}_{2} ℤ2⊂ℤ2×ℤ2 symmetry in two ways, one of which amounts to performing a Kramers-Wannier duality, while the other implements a Jordan-Wigner duality. We discuss the mapping of the phase diagram of the quantum spin-1/2 XYZ XYZ chains under such a triality. We show that the deconfined quantum critical transitions between Neel and dimer orders are mapped to either topological or conventional Landau-Ginzburg transitions. Finally, we extend our results to \mathbb{Z}^{\,}_{n} ℤn clock models with \mathbb{Z}^{\,}_{n}×\mathbb{Z}^{\,}_{n} ℤn×ℤn global internal symmetry, and provide a reinterpretation of the dual internal symmetries in terms of \mathbb{Z}^{\,}_{n} ℤn charge and dipole symmetries.

AbstractTopological semimetals, known for their intriguing properties arising from band degeneracies, have garnered significant attention. However, the discovery of a material realization and the detailed characterization of spinless Dirac semimetals have not yet been accomplished. Here, we propose from first-principles calculations that the RE8CoX3 group (RE = rare earth elements, X = Al, Ga, or In) contains ideal spinless Dirac semimetals whose Fermi surfaces are fourfold degenerate band-crossing points (without including spin degeneracy). Despite the lack of space inversion symmetry in these materials, Dirac points are formed on the rotation-symmetry axis due to accidental degeneracies of two bands corresponding to different 2-dimensional irreducible representations of the C6v group. We also investigate, through first-principles calculations and effective model analysis, various phase transitions caused by lattice distortion or elemental substitutions from the Dirac semimetal phase to distinct topological semimetallic phases such as nonmagnetic linked-nodal-line and Weyl semimetals (characterized by the second Stiefel–Whitney class) and ferromagnetic Weyl semimetals.

We discuss the topology of the parameter space of invertible phases with an onsite symmetry G , i.e., quantum many-body ground states that have neither fractionalization nor spontaneous breaking of the symmetry. The classification of invertible phases is known to be obtained by counting the connected components in the parameter space of the invertible phases. We consider its generalization - the deformation classes of the mappings from n-dimensional spheres S-n to this parameter space for arbitrary integer n . We argue a direct one-to-one correspondence in the framework of lattice models between the noncontractibility of S-n and (i) the classification of invertible phases in d dimensions when d n ; or (ii) zero-dimensional invertible Hamiltonians parametrized by S( n - d )when d < n , using an isotropic modulating Hamiltonian approach. Explicitly, we construct the noncontractible spheres of two-dimensional invertible phases, i.e., n = 2 and d = 2. We also propose a large class of crystalline topological phases protected by a generalized magnetic translations.

We show that the ground-state expectation value of twisting operator is a topological order parameter for U(1)- and Z_{N}-symmetric symmetry-protected topological (SPT) phases in one-dimensional "spin" systems-it is quantized in the thermodynamic limit and can be used to identify different SPT phases and to diagnose phase transitions among them. We prove that this (nonlocal) order parameter must take values in Nth roots of unity, and its value can be changed by a generalized lattice translation acting as an N-ality transformation connecting distinct phases. This result also implies the Lieb-Schultz-Mattis (LSM) ingappability for SU(N) spins if we further impose a general translation symmetry. Furthermore, our exact result for the order parameter of SPT phases can predict a large number of LSM ingappabilities by the general lattice translation. We also apply the N-ality property to provide an efficient way to construct possible multicritical phase transitions starting from a single Hamiltonian with a unique gapped ground state.

We show that spin-singlet s-wave multiband superconductors have a topological phase protected by rotation symmetry and time-reversal symmetry without spin-orbit coupling in two and three dimensions. This topological phase, an example of a representation-protected topological phase, has a Z(2) topological index and is stable as long as the bands at the Fermi energy are formed by a doublet of orbital states with finite angular momenta. In the limit of weak superconducting pair potential, the Z(2) index gives a Fermi-surface formula and is related to the winding number of three-dimensional strong topological superconductors of class CI. We present a model of a topological s f-wave superconductor that has gapless surface states with a quadratic dispersion and suggest a connection with iron-based superconductors.

Vortex lines in superconducting Dirac semimetals realize crystal-symmetry-protected gapless vortex-line phases in which gapless excitations propagate inside a vortex line, in the presence of appropriate crystal symmetry, spin-orbit coupling, and multi-band structures. Here we present a general scheme to classify possible gapless vortex-line phases in superconducting Dirac semimetals with rotation (or screw) symmetry and inversion symmetry, assuming that the rotation (screw) axis is parallel to the vortex line. The rotation (screw)-symmetry-protected gapless modes are stable as long as they have different rotation (screw) eigenvalues. The underlying mechanism for the formation of gapless vortex bound states depends on irreducible representations of rotation (screw) symmetry subject to a vortex field and is classified into three types: (i) accidental band crossing of two vortex bound-state modes under rotation symmetry; (ii) accidental and (iii) enforced band crossing of four vortex bound-state modes under screw symmetry. We present a tight-binding model of screw-symmetry-protected Dirac semimetal with an $s$-wave pair potential, demonstrating a gapless vortex-line phase of type (ii). We obtain four gapless modes of vortex bound states whose gapless points (Majorana zero modes) pinned at a time-reversal invariant momentum (TRIM) when the Fermi energy is close to the Dirac points. As the Fermi energy is moved away from the Dirac points, the four gapless modes are split into a pair of two gapless modes with vanishing excitation energy at non-TRIMs. In closing, we discuss Nb$_3$Pt as a candidate material with the four-fold screw-symmetry-protected Dirac cones that can host a gapless vortex-line phase.

Three-dimensional (3D) gapped topological phases with fractional excitations are divided into two subclasses: one has topological order with point-like and loop-like excitations fully mobile in the 3D space, and the other has fracton order with point-like excitations constrained in lower-dimensional subspaces. These exotic phases are often studied by exactly solvable Hamiltonians made of commuting projectors, which, however, are not capable of describing those exhibiting surface states with gapless chiral dispersion. Here we introduce a systematic way, based on cellular construction recently proposed for 3D topological phases, to construct another type of exactly solvable models in terms of coupled quantum wires with given inputs of cellular structure, two-dimensional Abelian topological order, and their gapped interfaces. We show that our models can describe both 3D topological and fracton orders (and even their hybrid) and study their universal properties such as quasiparticle statistics and topological ground-state degeneracy. We also apply this construction to two-dimensional coupled-wire models with ordinary topological orders and translation-symmetry-enriched topological orders. Our results pave the way for effective quantum field theory descriptions or microscopic model realizations of fracton orders with chiral gapless surface states.

Received 16 January 2022DOI:https://doi.org/10.1103/PhysRevB.105.039903©2022 American Physical SocietyPhysics Subject Headings (PhySH)Research AreasSymmetry protected topological statesTopological insulatorsTopological phases of matterCondensed Matter, Materials & Applied Physics

We propose an index I_{G} which characterizes the degree of gappability, namely the difficulty to induce a unique ground state with a nonvanishing excitation gap, in the presence of a symmetry G. I_{G} represents the dimension of the subspace of ambient uniquely gapped theories in the entire G-invariant "theory space." The celebrated Lieb-Schultz-Mattis theorem corresponds, in our formulation, to the case I_{G}=0 (completely ingappable) for the symmetry G including the lattice translation symmetry. We illustrate the usefulness of the index by discussing the phase diagram of spin-1/2 antiferromagnets in various dimensions, which do not necessarily have the translation symmetry.

We propose a geometric {approach to Lieb-Schultz-Mattis theorem for} quantum many-body systems with discrete spin-rotation symmetries and lattice inversion or rotation symmetry, but without translation symmetry assumed. Under symmetry-twisting on a $(d-1)$-dimensional plane, we find that any $d$-dimensional inversion-symmetric spin system possesses a doubly degenerate spectrum when it hosts a half-integer spin at the inversion-symmetric point. We also show that any rotation-symmetric generalized spin model with a projective representation at the rotation center has a similar degeneracy under symmetry-twisting. We argue that these degeneracies imply that {a unique symmetric gapped ground state that is smoothly connected to product states} is forbidden in the original untwisted systems -- generalized inversional/rotational Lieb-Schultz-Mattis theorems without lattice translation symmetry imposed. The traditional Lieb-Schultz-Mattis theorems with translations also fit in the proposed framework.

The fractional Josephson effect is known to be a characteristic phenomenon of topological Josephson junctions hosting Majorana zero modes (MZMs), where the Josephson current has a 4π (rather than a 2π) periodicity in the phase difference between the two topological superconductors. We introduce a one-dimensional model of a topological superconductor/normal-metal/superconductor (SNS) junction with the normal-metal (N) region of finite length, which is intermediate regime between the shortand long-junction limits. Assuming weak tunneling at the SN interfaces, we investigate resonance and finite-size effects on the fractional Josephson effect due to the existence of several discrete energy levels in the N region in which wavefunctions have oscillating nodal structure. Through careful analysis of the sign change in the transmission amplitudes through the junction and the fermion parity of the two MZMs, we find that the fractional Josephson current is proportional to the parity of total fermion numbers including both filled normal levels and two MZMs. Furthermore, we elucidate drastic enhancement of the Josephson current due to the resonance between a discrete level in the N region and MZMs.