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arXiv:1312.3093 · Physical Review B

Detection of symmetry-enriched topological phases

Ching-Yu Huang, Xie Chen, Frank Pollmann

Published Jul 30, 2014• 40 citations
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Abstract

Topologically ordered systems in the presence of symmetries can exhibit new structures which are referred to as symmetry-enriched topological (SET) phases. We introduce simple methods to detect certain SET orders directly from a complete set of topologically degenerate ground-state wave functions. In particular, we first show how to directly determine the characteristic symmetry fractionalization of the quasiparticles from the reduced density matrix of the minimally entangled states. Second, we show how a simple generalization of a nonlocal order parameter can be measured to detect SET phases. The usefulness of the proposed approach is demonstrated by examining two concrete model states which exhibit SET phases: (i) a spin-1 model on the honeycomb lattice and (ii) the resonating valence bond (RVB) state on a kagome lattice. We conclude that the spin-1 model and the RVB state are in the same SET phases.

DOI
10.1103/physrevb.90.045142

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Topologically ordered systems in the presence of symmetries can exhibit new structures which are referred to as symmetry-enriched topological (SET) phases. We introduce simple methods to detect certain SET orders directly from a complete set of topologically degenerate ground-state wave functions. In particular, we first show how to directly determine the characteristic symmetry fractionalization of the quasiparticles from the reduced density matrix of the minimally entangled states. Second, we show how a simple generalization of a nonlocal order parameter can be measured to detect SET phases. The usefulness of the proposed approach is demonstrated by examining two concrete model states which exhibit SET phases: (i) a spin-1 model on the honeycomb lattice and (ii) the resonating valence bond (RVB) state on a kagome lattice. We conclude that the spin-1 model and the RVB state are in the same SET phases.
Abstract How do we detect topological phases from a single ground state? Entanglement entropy and spectrum have long been the standard tools — but the reduced density matrix (RDM) itself contains far more information. We show that the RDM of a 2D topologically ordered system, expressed at the entanglement cut, realizes a 1D mixed-state phase. For the Z 2 toric code phase, it is a 1D Z 2 strong-to-weak spontaneous symmetry breaking (SW-SSB) phase, where deconfinement of anyons manifests as the short-range correlation of both Z 2 charge and Z 2 domain-wall in the RDM. The bulk e - m duality translates into a Kramers–Wannier self-duality of the SW-SSB phase. Extending the framework to gapped Z 2 spin liquids, the global spin-rotation symmetry manifests as an additional weak symmetry for the 1D RDM. Spin-1/2 spinons result in a cusp on the disorder parameter of spin-rotation at θ = π , providing a direct, ground-state signature of symmetry fractionalization. We verify this prediction analytically using the matrix product density operator formalism and numerically for the kagome-lattice resonating valence bond state. The proposed observable requires only a single ground-state wavefunction, making it amenable to quantum simulation platforms.
We study (2+1)-dimensional phases with topological order, such as fractional quantum Hall states and gapped spin liquids, in the presence of global symmetries. Phases that share the same topological order can then differ depending on the action of symmetry, leading to symmetry-enriched topological (SET) phases. Here, we present a KKK-matrix Chern-Simons approach to identify distinct phases with Abelian topological order, in the presence of unitary or antiunitary global symmetries. A key step is the identification of a smooth edge sewing condition that is used to check if two putative phases are indeed distinct. We illustrate this method by classifying Z2{Z}_{2}Z2​ topological order (Z2{Z}_{2}Z2​ spin liquids) in the presence of an internal Z2{Z}_{2}Z2​ global symmetry for which we find six distinct phases. These include two phases with an unconventional action of symmetry that permutes anyons leading to symmetry-protected Majorana edge modes. Other routes to realizing protected edge states in SET phases are identified. Symmetry-enriched Laughlin states and double-semion theories are also discussed. Somewhat surprisingly, we observe that (i) gauging the global symmetry of distinct SET phases leads to topological orders with the same total quantum dimension, and (ii) a pair of distinct SET phases can yield the same topological order on gauging the symmetry.
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 interplay of symmetry and topological order leads to a variety of distinct phases of matter, the symmetry enriched topological (SET) phases. Here we discuss physical observables that distinguish different SETs in the context of Z2{\mathbb{Z}}_{2}Z2​ quantum spin liquids with SU(2) spin rotation invariance. We focus on the cylinder geometry, and show that ground-state quantum numbers for different topological sectors are robust invariants which can be used to identify the SET phase. More generally, these invariants are related to 1D symmetry protected topological phases when viewing the cylinder geometry as a 1D spin chain. In particular, we show that the kagome spin liquid SET can be determined by measurements on one ground state, by wrapping the kagome in a few different ways on the cylinder. In addition to guiding numerical studies, this approach provides a transparent way to connect bosonic and fermionic mean-field theories of spin liquids. When fusing quasiparticles, it correctly predicts nontrivial phase factors for combining their space group quantum numbers.
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.