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arXiv:0803.2300 · Physical Review B

Mutual Chern-Simons theory for Z2 topological order

Su-Peng Kou, Michael Levin, Xiao-Gang Wen

Published Oct 31, 2008• 90 citations
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Abstract

We study several different ${Z}_{2}$ topological ordered states in frustrated spin systems. The effective theories for those different ${Z}_{2}$ topological orders all have the same form---a ${Z}_{2}$ gauge theory which can also be written as a mutual $U(1)\ifmmode\times\else\texttimes\fi{}U(1)$ Chern-Simons theory. However, we find that the different ${Z}_{2}$ topological orders are reflected in different projective realizations of lattice symmetry in the same effective mutual Chern-Simons theory. This result is obtained by comparing the ground-state degeneracy, the ground-state quantum numbers, the gapless edge state, and the projective symmetry group of quasiparticles calculated from the slave-particle theory and from the effective mutual Chern-Simons theories. Our study reveals intricate relations between topological order and symmetry.

DOI
10.1103/physrevb.78.155134

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We study several different Z2{Z}_{2}Z2​ topological ordered states in frustrated spin systems. The effective theories for those different Z2{Z}_{2}Z2​ topological orders all have the same form---a Z2{Z}_{2}Z2​ gauge theory which can also be written as a mutual U(1)\ifmmode×\else\texttimes\fiU(1)U(1)\ifmmode\times\else\texttimes\fi{}U(1)U(1)\ifmmode×\else\texttimes\fiU(1) Chern-Simons theory. However, we find that the different Z2{Z}_{2}Z2​ topological orders are reflected in different projective realizations of lattice symmetry in the same effective mutual Chern-Simons theory. This result is obtained by comparing the ground-state degeneracy, the ground-state quantum numbers, the gapless edge state, and the projective symmetry group of quasiparticles calculated from the slave-particle theory and from the effective mutual Chern-Simons theories. Our study reveals intricate relations between topological order and symmetry.
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 this paper, we develop an effective mutual Chern-Simons Landau-Ginzburg (MCSLG) theory to describe the continuous topological quantum phase transition (TQPT). In particular, we consider the TQPT between a spin-polarized phase (a state without topological order) and a Z2{Z}_{2}Z2​ topologically ordered state. The TQPT is not induced by spontaneous symmetry breaking. Instead the Z2{Z}_{2}Z2​ topological order is broken down by the condensation of Z2{Z}_{2}Z2​ charged quasiparticles. By generalizing the hierarchy theory of fractional quantum Hall effect to Z2{Z}_{2}Z2​ topological order, we show that the TQPT belongs to the universal class of three-dimensional Ising phase transition. In the end, we applied the MCSLG theory to the toric code model.
Topological phase transitions involving intrinsic topological orders are usually characterized by qualitative changes of ground state quantum entanglement, which cannot be described by conventional mean-field theories with local order parameters. Here, we apply the lattice Chern-Simons theory to study frustrated quantum magnets and show that the conventional concepts, such as the order parameter and symmetry breaking, can still play a crucial role in certain topological phase transitions. The lattice Chern-Simons representation establishes a nonlocal mapping from quantum spin models to interacting spinless Dirac fermions. We show that breaking certain emergent symmetries of the fermionic theory could provide a unified approach to describing both magnetic and topological orders, as well as the topological phase transitions between them. We apply this method to the perturbed spin-1/2 ${J}_{1}\ensuremath{-}{J}_{2}$ XY model on the honeycomb lattice and predict a nonuniform chiral spin liquid ground state in the strong frustration region. This is further verified by our high-precision tensor network calculations. These results suggest that the lattice Chern-Simons theory can simplify the complicated topological phase transitions to effective mean-field theories in terms of fermionic degrees of freedom, which lead to different understandings that help to understand the frustrated quantum magnets.
In this paper we systematically study a simple class of translation-symmetry protected topological orders in quantum spin systems using slave-particle approach. The lattice spin systems are translation invariant, but may break any other symmetries. We consider topologically ordered ground states that do not spontaneously break any symmetry. Those states can be described by Z2AZ2AZ2A or Z2BZ2BZ2B projective symmetry group. We find that the Z2AZ2AZ2A translation symmetric topological orders can still be divided into 16 subclasses corresponding to 16 translation-symmetry protected topological orders. We introduced four Z2{Z}_{2}Z2​ topological indices ${\ensuremath{\zeta}}_{\mathbf{k}}=0,1$ at k=(0,0)\mathbf{k}=(0,0)k=(0,0), $(0,\ensuremath{\pi})$, $(\ensuremath{\pi},0)$, and $(\ensuremath{\pi},\ensuremath{\pi})$ to characterize those 16 topological orders on square lattice. We calculated the topological degeneracies and crystal momenta for those 16 topological phases on even-by-even, even-by-odd, odd-by-even, and odd-by-odd lattices, which allows us to physically measure such topological orders. We predict the appearance of gapless fermionic excitations at the quantum phase transitions between those symmetry protected topological orders. Our result can be generalized to any dimensions. We find 256 translation-symmetry protected Z2AZ2AZ2A topological orders for a system on three-dimensional cubic lattice.
We present a general approach to understanding the quantum phases and phase transitions of quantum antiferromagnets in two spatial dimensions. We begin with the simplest spin-liquid state, the Z2{Z}_{2}Z2​ spin liquid, whose elementary excitations are spinons and visons, carrying Z2{Z}_{2}Z2​ electric and magnetic charges, respectively. Their dynamics is expressed in terms of a doubled U(1) Chern-Simons theory, which correctly captures the ``topological'' order of the Z2{Z}_{2}Z2​ spin-liquid state. We show that the same theory also yields a description of the variety of ordered phases obtained when one or more of the elementary excitations condense. Field theories for the transitions and multicritical points between these phases are obtained. We survey experimental results on antiferromagnets on the anisotropic triangular lattice, and make connections between their phase diagrams and our results.
In this paper, we develop an effective mutual Chern-Simons Landau-Ginzburg (MCSLG) theory to describe the continuous topological quantum phase transition (TQPT). In particular, we consider the TQPT between a spin-polarized phase (a state without topological order) and a Z[subscript 2] topologically ordered state. The TQPT is not induced by spontaneous symmetry breaking. Instead the [subscript 2]topological order is broken down by the condensation of [subscript 2] charged quasiparticles. By generalizing the hierarchy theory of fractional quantum Hall effect to [subscript 2] topological order, we show that the TQPT belongs to the universal class of three-dimensional Ising phase transition. In the end, we applied the MCSLG theory to the toric code model.