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arXiv:1509.02927

Symmetry fractionalization of visons in $\mathbb Z_2$ spin liquids

Yang Qi, Meng Cheng, Chen Fang

Published Sep 9, 2015• 15 citations
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Abstract

In this work we study symmetry fractionalization of vison excitations in topological $\mathbb{Z}_2$ spin liquids. We show that in the presence of the full $\mathrm{SO}(3)$ spin rotational symmetry and if there is an odd number of spin-$\frac12$ per unit cell, the symmetry fractionalization of visons is completely fixed. On the other hand, visons can have different classes of symmetry fractionalization if the spin rotational symmetry is reduced. As a concrete example, we show that visons in the Balents-Fisher-Girvin $\mathbb{Z}_2$ spin liquid have crystal symmetry fractionalization classes which are not allowed in $\mathrm{SO}(3)$ symmetric spin liquids, due to the reduced spin rotational symmetry.

DOI
10.48550/arxiv.1509.02927

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Symmetry fractionalization of visons in Z2\mathbb Z_2Z2​ spin liquids
In this work we study symmetry fractionalization of vison excitations in topological Z2\mathbb{Z}_2Z2​ spin liquids. We show that in the presence of the full SO(3)\mathrm{SO}(3)SO(3) spin rotational symmetry and if there is an odd number of spin-12\frac1221​ per unit cell, the symmetry fractionalization of visons is completely fixed. On the other hand, visons can have different classes of symmetry fractionalization if the spin rotational symmetry is reduced. As a concrete example, we show that visons in the Balents-Fisher-Girvin Z2\mathbb{Z}_2Z2​ spin liquid have crystal symmetry fractionalization classes which are not allowed in SO(3)\mathrm{SO}(3)SO(3) symmetric spin liquids, due to the reduced spin rotational symmetry.
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.
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.
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 study the fractionalization of space group symmetries in two-dimensional topologically ordered phases. Specifically, we focus on Z2{Z}_{2}Z2​-fractionalized phases in two dimensions whose deconfined topological excitations transform trivially under translational symmetries but projectively under glide reflections, whose quantum numbers are hence fractionalized. We accomplish this by generalizing the dichotomy between even and odd gauge theories to incorporate additional symmetries inherent to nonsymmorphic crystals. We show that the resulting fractionalization of point group quantum numbers can be detected in numerical studies of ground state wave functions. We illustrate these ideas using a microscopic model of a system of bosons at integer unit cell filling on a lattice with space group p4gp4gp4g that can be mapped to a half-magnetization plateau for an S=1/2S=1/2S=1/2 spin system on the Shastry-Sutherland lattice.
Symmetry fractionalization on topological excitations is one of the most remarkable quantum phenomena in topological orders with symmetry, i.e., symmetry-enriched topological phases. While much progress has been theoretically and experimentally made in two dimensions (2D), the understanding on symmetry fractionalization in 3D is far from complete. A long-standing challenge is to understand symmetry fractionalization on looplike topological excitations, which are spatially extended objects. In this paper, we construct a powerful topological-field-theoretical framework approach for 3D topological orders, which leads to a systematic characterization and classification of symmetry fractionalization. For systems with Abelian gauge groups (Gg{G}_{g}Gg​) and Abelian symmetry groups (Gs{G}_{s}Gs​), we successfully establish equivalence classes that lead to a finite number of patterns of symmetry fractionalization, although there are notoriously infinite number of Lagrangian-descriptions of the system. Based on this, we compute topologically distinct types of fractional symmetry charges carried by particles. Then, for each type, we compute topologically distinct statistical phases of braiding processes among loop excitations and external symmetry fluxes. As a result, we are able to unambiguously list all physical observables for each pattern of symmetry fractionalization. We present detailed calculations on many concrete examples. As an example, we find that the symmetry fractionalization in an untwisted ${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ topological order with Z2{\mathbb{Z}}_{2}Z2​ symmetry is classified by ${({\mathbb{Z}}_{2})}^{6}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}\ensuremath{\bigoplus}{({\mathbb{Z}}_{2})}^{2}$. If the topological order is twisted, the classification reduces to (Z2)6{({\mathbb{Z}}_{2})}^{6}(Z2​)6 in which particle excitations always carry integer charge. Inspired by the field-theoretical analysis, we find that our classification of symmetry fractionalization with twist $\ensuremath{\omega}$ can be formally organized into an algebraic formalism: ${⨁}_{{\ensuremath{\nu}}_{i}}{\mathcal{H}}^{4}({G}_{g}{\ensuremath{\leftthreetimes}}_{{\ensuremath{\nu}}_{i}}{G}_{s},U(1))/{\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i}),$ where anomaly-free symmetry fractionalization on particles ${\ensuremath{\nu}}_{i}\ensuremath{\in}{\mathcal{H}}_{non}^{2}({G}_{s},{G}_{g})$. Despite the lack of detailed dependence of ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i})$ on $\ensuremath{\omega}$ and ${\ensuremath{\nu}}_{i}$, the present field-theoretical approach allows us to efficiently calculate and understand ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\omega}}({\ensuremath{\nu}}_{i})$. Several future directions are presented at the end of this paper.