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arXiv:1606.04544 · Physical review. B./Physical review. B

Classification of symmetry fractionalization in gapped Z2 spin liquids

Yang Qi, Meng Cheng

Published Mar 19, 2018• 23 citations
arXivHTMLPDF

Abstract

In quantum spin liquids, fractional spinon excitations carry half-integer spins and other fractional quantum numbers of lattice and time-reversal symmetries. Different patterns of symmetry fractionalization distinguish different spin liquid phases. In this work, we derive a general constraint on the symmetry fractionalization of spinons in a gapped spin liquid, realized in a system with an odd number of spin-$\frac{1}{2}$ per unit cell. In particular, when applied to kagome/triangular lattices, we obtain a complete classification of symmetric gapped ${\mathbb{Z}}_{2}$ spin liquids.

DOI
10.1103/physrevb.97.115138

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In quantum spin liquids, fractional spinon excitations carry half-integer spins and other fractional quantum numbers of lattice and time-reversal symmetries. Different patterns of symmetry fractionalization distinguish different spin liquid phases. In this work, we derive a general constraint on the symmetry fractionalization of spinons in a gapped spin liquid, realized in a system with an odd number of spin-12\frac{1}{2}21​ per unit cell. In particular, when applied to kagome/triangular lattices, we obtain a complete classification of symmetric gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquids.
Despite rapid progress in understanding gapped topological states, much less is known about gapless topological phases of matter, especially in strongly correlated electrons. In this work, we discuss a large class of robust gapless quantum spin liquids in frustrated magnets made of half-integer spins, which are described by gapless fermionic spinons coupled to dynamical Z2{\mathbb{Z}}_{2}Z2​ gauge fields. Requiring U(1)\text{U}(1)U(1) spin conservation, time-reversal, and certain space-group symmetries, we show that certain spinon symmetry fractionalization class necessarily leads to a gapless spectrum. These gapless excitations are stable against any perturbations, as long as the required symmetries are preserved. Applying these gapless criteria to spin-12\frac{1}{2}21​ systems on square, triangular, and kagome lattices, we show that all gapped symmetric Z2{\mathbb{Z}}_{2}Z2​ spin liquids in Abrikosov-fermion representation can also be realized in Schwinger-boson representation. This leads to 64 gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquids on square lattice, and 8 gapped states on both kagome and triangular lattices.
In this work we study the crystal symmetry fractionalization in chiral spin liquids with the chiral-semion topological order. We show that if such a chiral spin liquid is realized in a two-dimensional lattice model with odd number of spin-12\frac1221​ per unit cell and the state preserves spin rotation symmetry and translation symmetries, the semion excitation must carry both half-integer spin and fractional crystal quantum numbers. As a result, only a unique symmetry enriched topological phase can be realized in chiral spin liquids in a spin-12\frac1221​ kagome lattice model. These fractional symmetry quantum numbers are confirmed numerically using ground state wave functions obtained with the density matrix renormalization group method.
In quantum spin liquid states, the fractionalized spinon excitations can carry fractional crystal symmetry quantum numbers, and this symmetry fractionalization distinguishes different symmetry-enriched spin liquid states with identical intrinsic topological order. In this work we propose a simple way to detect signatures of such crystal symmetry fractionalizations from the crystal symmetry representations of the ground state wave function. We demonstrate our method on projected Z2{\mathbb{Z}}_{2}Z2​ spin liquid wave functions on the kagome lattice, and show that it can be used to classify generic wave functions. Particularly our method can be used to distinguish several proposed candidates of Z2{\mathbb{Z}}_{2}Z2​ spin liquid states on the kagome lattice.
We classify distinct types of quantum number fractionalization occurring in two-dimensional topologically ordered phases, focusing in particular on phases with Z2{\mathbb{Z}}_{2}Z2​ topological order, that is, on gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquids. We find that the fractionalization class of each anyon is an equivalence class of projective representations of the symmetry group, corresponding to elements of the cohomology group H2(G,Z2){H}^{2}(G,{\mathbb{Z}}_{2})H2(G,Z2​). This result leads us to a symmetry classification of gapped Z2{\mathbb{Z}}_{2}Z2​ spin liquids, such that two phases in different symmetry classes cannot be connected without breaking symmetry or crossing a phase transition. Symmetry classes are defined by specifying a fractionalization class for each type of anyon. The fusion rules of anyons play a crucial role in determining the symmetry classes. For translation and internal symmetries, braiding statistics plays no role, but can affect the classification when point group symmetries are present. For square lattice space group, time-reversal, and SO(3)\mathrm{SO}(3)SO(3) spin rotation symmetries, we find $2\phantom{\rule{0.16em}{0ex}}098\phantom{\rule{0.16em}{0ex}}176\ensuremath{\approx}{2}^{21}$ distinct symmetry classes. Our symmetry classification is not complete, as we exclude, by assumption, permutation of the different types of anyons by symmetry operations. We give an explicit construction of symmetry classes for square lattice space group symmetry in the toric code model. Via simple examples, we illustrate how information about fractionalization classes can, in principle, be obtained from the spectrum and quantum numbers of excited states. Moreover, the symmetry class can be partially determined from the quantum numbers of the four degenerate ground states on the torus. We also extend our results to arbitrary Abelian topological orders (limited, though, to translations and internal symmetries), and compare our classification with the related projective symmetry group classification of parton mean-field theories. Our results provide a framework for understanding and probing the sharp distinctions among symmetric Z2{\mathbb{Z}}_{2}Z2​ spin liquids and are a first step toward a full classification of symmetric topologically ordered phases.
Projective symmetry groups beyond spin-1/21/21/2: Classifying integer-spin quantum spin liquids
Wen's projective symmetry group (PSG), introduced a quarter of a century ago, reduced the classification of symmetric quantum spin liquids to an algebraic problem and has guided the study of spin-1/21/21/2 models and materials ever since. Many current candidate materials and quantum simulators, however, have spin-111 or larger moments, and no comparable classification exists for them. We formulate the PSG for fermionic partons of arbitrary spin SSS. Integer and half-odd-integer spins already differ in their gauge structure. Half-odd-integer spins give the familiar SU(2)\mathrm{SU}(2)SU(2), U(1)\mathrm{U}(1)U(1), and Z2\mathbb{Z}_2Z2​ invariant gauge groups, while integer spins give a different hierarchy that includes U(1)⋊Z2\mathrm{U}(1)\rtimes\mathbb{Z}_2U(1)⋊Z2​ and Z2×Z2\mathbb{Z}_2\times\mathbb{Z}_2Z2​×Z2​; the latter has no spin-1/21/21/2 counterpart. Three lattice-independent results follow for integer spin. Singlet pairing of the partons is odd under site exchange. A lattice symmetry can exchange the particle and hole representations of the local constraint, which forces symmetry-related sites into opposite constraint sectors. And a time-reversal-symmetric U(1)\mathrm{U}(1)U(1) state with the same constraint on every site can be gapped only if the number of sites per unit cell is a multiple of 2S+12S+12S+1. We apply the formalism to the spin-111 diamond lattice, realized in frustrated A-site spinels, and find ten fully symmetric Ansätze up to second neighbors. In the J1J_1J1​-J2J_2J2​ Heisenberg model at the mean-field level, the ground state depends on how the constraint is imposed. With the particle and hole representations, the lowest states are fully gapped U(1)\mathrm{U}(1)U(1) states whose two sublattices sit in opposite constraint sectors, and one of them is gauge equivalent to a paired Ansatz. The classification provides a starting point for integer-spin quantum spin liquids on other lattices and in other materials.
Classification and Monte Carlo study of symmetric Z2Z_{2}Z2​ spin liquids on the triangular lattice
We study different ways of symmetry fractionalization in Z2Z_{2}Z2​ spin liquids on the triangular lattice. Our classification can be used to identify the symmetry fractionalization in the Z2Z_{2}Z2​ spin liquid reported in recent density-matrix-renormalization-group simulations for J1−J2J_{1}-J_{2}J1​−J2​ spin model on the triangular lattice. We find 64 types of symmetry enriched Z2Z_{2}Z2​ spin liquid states on triangular lattice. Besides 8 states constructed in Schwinger-boson parton wavefunctions, 12 more states can be realized in Abrikosov-fermion parton construction. Within a larger gauge group than SU(2), the rest 44 states are also found in a spin-3/2 system. Among 20 types of Abrikosov-fermion Gutzwillerprojected wavefunctions, No.B5 state is a promising candidate for the Z2Z_{2}Z2​ spin liquid for J1−J2J_{1}-J_{2}J1​−J2​ spin model on the triangular lattice. No.B5 lies close to Dirac spin liquid (DSL). However, variational Monte Carlo simulation find that DSL has a good variational energy and J1−J2J_{1}-J_{2}J1​−J2​ spin model cannot open a gap for spinons on top of DSL to stabilize No.B5 state.