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arXiv:1511.02226

Classification and detection of symmetry fractionalization in chiral spin liquids

Łukasz Cincio, Yang Qi

Published Nov 6, 2015• 13 citations
arXivHTMLPDF

Abstract

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-$\frac12$ 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-$\frac12$ kagome lattice model. These fractional symmetry quantum numbers are confirmed numerically using ground state wave functions obtained with the density matrix renormalization group method.

DOI
10.48550/arxiv.1511.02226

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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.
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.
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 Z[subscript 2] 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 Z[subscript 2] spin liquid states on the kagome lattice.
The anyonic excitations of a spin liquid can feature fractional quantum numbers under space group symmetries. Detecting these fractional quantum numbers, which are analogs of the fractional charge of Laughlin quasiparticles, may prove easier than the direct observation of anyonic braiding and statistics. Motivated by the recent numerical discovery of spin-liquid phases in the kagome Heisenberg antiferromagnet, we theoretically predict the pattern of space group symmetry fractionalization in the kagome lattice SO(3)-symmetric chiral spin liquid. We provide a method to detect these fractional quantum numbers in finite-size numerics which is simple to implement in the density matrix renormalization group. Applying these developments to the chiral spin liquid phase of a kagome Heisenberg model, we find perfect agreement between our theoretical prediction and numerical observations.
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.
Recent years have seen the discovery of a chiral spin liquid state - a bosonic analogue of a fractional Quantum Hall state first put forward by Kalmeyer and Laughlin in 1987 - in several deformations of the Heisenberg model on the Kagome lattice. Here, we apply state-of-the-art numerical techniques to one such model, where breaking of the time-reversal symmetry drives the system into the chiral phase. Our methods allow us to obtain explicit matrix-product state representations of the low-lying excitations of the chiral spin liquid state, including the topologically non-trivial semionic excitation. We characterize these excitations and study their energetics as the model is tuned towards a topological phase transition.