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Classification and Properties of Symmetry Enriched Topological Phases: A Chern-Simons approach with applications to Z2 spin liquids

Yuan-Ming Lu, Ashvin Vishwanath

TL;DR

The paper develops a K-matrix Chern-Simons framework to classify symmetry enriched topological ($SET$) phases in $2+1$-D with on-site symmetries, focusing on Abelian topological orders. It introduces a bulk-edge consistent edge sewing criterion to distinguish distinct SETs and shows how symmetry acts on edge fields via ${\bf W}^{\boldsymbol{g}}$ and $\delta\vec{\phi}^{\boldsymbol{g}}$, linking this to bulk quasiparticle content. Applying the method to $Z_2$ spin liquids (time-reversal and onsite $Z_2$), double semion, and Laughlin states reveals both conventional and unconventional SETs, including phases with symmetry-protected Majorana edge modes and non-Abelian orders arising upon gauging. A key result is that gauging a finite unitary symmetry preserves the total quantum dimension across distinct SETs, while distinct Dijkgraaf-Witten terms can yield the same intrinsic topological order, illustrating subtle interplays between symmetry, edge physics, and bulk topology. The framework connects edge state structure, symmetry realization, and gauging to guide lattice realizations and experimental probes of SET physics.

Abstract

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 K-matrix Chern-Simons approach to identify all distinct phases with Abelian topological order, in the presence of unitary or anti-unitary global symmetries . A key step is the identification of an edge sewing condition that is used to check if two putative phases are indeed distinct. We illustrate this method for the case of $Z_2$ topological order ($Z_2$ spin liquids), in the presence of an internal Z$_2$ global symmetry. We find 6 distinct phases. The well known quantum number fractionalization patterns account for half of these states. Phases also differ due to the addition of a symmetry protected topological (SPT) phase. Also, we allow for the unconventional possibility that anyons are exchanged by the symmetry. This leads to 2 additional phases with 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. Two surprising lessons that emerge are: (i) gauging the global symmetry of distinct SET phases always lead to different topological orders with the same total quantum dimension, (ii) gauge theories with distinct Dijkgraaf-Witten topological terms may have the same topological order.

Classification and Properties of Symmetry Enriched Topological Phases: A Chern-Simons approach with applications to Z2 spin liquids

TL;DR

The paper develops a K-matrix Chern-Simons framework to classify symmetry enriched topological () phases in -D with on-site symmetries, focusing on Abelian topological orders. It introduces a bulk-edge consistent edge sewing criterion to distinguish distinct SETs and shows how symmetry acts on edge fields via and , linking this to bulk quasiparticle content. Applying the method to spin liquids (time-reversal and onsite ), double semion, and Laughlin states reveals both conventional and unconventional SETs, including phases with symmetry-protected Majorana edge modes and non-Abelian orders arising upon gauging. A key result is that gauging a finite unitary symmetry preserves the total quantum dimension across distinct SETs, while distinct Dijkgraaf-Witten terms can yield the same intrinsic topological order, illustrating subtle interplays between symmetry, edge physics, and bulk topology. The framework connects edge state structure, symmetry realization, and gauging to guide lattice realizations and experimental probes of SET physics.

Abstract

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 K-matrix Chern-Simons approach to identify all distinct phases with Abelian topological order, in the presence of unitary or anti-unitary global symmetries . A key step is the identification of an edge sewing condition that is used to check if two putative phases are indeed distinct. We illustrate this method for the case of topological order ( spin liquids), in the presence of an internal Z global symmetry. We find 6 distinct phases. The well known quantum number fractionalization patterns account for half of these states. Phases also differ due to the addition of a symmetry protected topological (SPT) phase. Also, we allow for the unconventional possibility that anyons are exchanged by the symmetry. This leads to 2 additional phases with 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. Two surprising lessons that emerge are: (i) gauging the global symmetry of distinct SET phases always lead to different topological orders with the same total quantum dimension, (ii) gauge theories with distinct Dijkgraaf-Witten topological terms may have the same topological order.

Paper Structure

This paper contains 26 sections, 158 equations, 5 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Chern-Simons approach to symmetry enriched Abelian topological orders in 2+1-D
  3. Chern-Simons theory description of 2+1-D Abelian topological orders
  4. Edge excitations of an Abelian topological order
  5. Different Chern-Simons theories can describe the same topological order
  6. Implementing symmetries in Abelian topological orders
  7. Criteria for different symmetry enriched topological orders
  8. Examples
  9. Classifying $Z_2$ spin liquids with time reversal symmetry
  10. Classifying $Z_2$ spin liquids with onsite $Z_2$ symmetry
  11. "Conventional" $Z_2$-symmetry-enriched $Z_2$ spin liquids
  12. "Unconventional" $Z_2$-symmetry-enriched $Z_2$ spin liquids
  13. Measurable effects of "unconventional" $Z_2$ symmetry realizations
  14. Gauging symmetry in "unconventional" SET phases
  15. Classifying bosonic Laughlin state at $\nu=\frac{1}{2k},~(k\in{\mathbb{Z}})$ with onsite $Z_2$ symmetry
  16. ...and 11 more sections

Figures (5)

  • Figure 1: (color online) Edge Sewing Criterion to distinguish symmetry enriched topological (SET) phases. Only the microscopic degrees of freedom i.e. "electrons" (and not gauge charged objects such as anyons/fractionalized quasiparticles) can tunnel between the two edges of a pair of semi-infinite cylinders. If two SET phases can be continuously tuned into one another without a phase transition (while preserving symmetry), there is a "smooth" sewing between the two cylinders of SET phases $\#1$ and $\#2$. This implies that all edge excitations are gapped by a few symmetry-allowed terms that tunnel "electrons" between the two edges. In the thermodynamic limit these tunneling terms lead to $M$ degenerate ground states, corresponding exactly to the $M$-fold torus degeneracy of the topological order. On the other hand, if the two SET phases are different, there is no such "smooth" boundary condition to sew the two edges. A precise version of this statement is formulated in Criterion I in Section \ref{['CRITERIA']}.
  • Figure 2: (color online) A fermion mode ($f$) localized at the boundary between two subsystem $A$ and $B$ which from a bipartition of the on a sphere, where the "unconventional" Ising-symmetry-enriched $Z_2$ spin liquid resides. Under the Ising symmetry operation, an electric charge $e$ will transform into a magnetic vortex $m$. Consider one electric charge is created in each subsystem. If we perform Ising ($Z_2$) symmetry only on subsystem $A$, a fermion mode will emerge on the boundary, as the electric $e$ charge turns into a magnetic vortex $m$ in $A$.
  • Figure 3: (color online) The Ising symmetry eigenstates are linear combinations of minimal entropy states (MESs) for a "unconventional" Ising-symmetry-enriched $Z_2$ spin liquid, since one MES $|e\rangle$ transforms into another MES $|m\rangle$ under Ising symmetry operation.
  • Figure 4: (color online) Domain wall bound state on the edge of "unconventional" Ising-symmetry-enriched $Z_2$ spin liquids (see TABLE \ref{['tab:Z2SL:z2:unconventional']}). In these SET phases, under $Z_2$ symmetry operation, one electric charge will transform into a magnetic vortex and vice versa. The on-site unitary $Z_2$ (Ising) symmetry can be, e.g. a spin-flip symmetry. On the two sides of the Ising-symmetry domain wall, two different backscattering "mass" terms related by spin-flip Ising symmetry are added to gap out the edge states. These two mass terms break $Z_2$ symmetry in opposite ways. A non-Abelian bound state with quantum dimension $d_{q_{\boldsymbol{g}}}=\sqrt2$ is localized at each Ising domain wall. For a "conventional" $Z_2$-SET phase, such a Ising mass domain wall will trap an Abelian bound state with quantum dimension $1$.
  • Figure 5: (color online) The process in which a symmetry flux $q_{\boldsymbol{g}}$ and its antiparticle $\bar{q}_{\boldsymbol{g}}$ are created out of the vaccum, dragged around a quasiparticle $a$ and then annihilated. The dashed line denotes the trajectory of $q_{\boldsymbol{g}}$ and $\bar{q}_{\boldsymbol{g}}$ from their creation to annihilation. After this process, symmetry ${\boldsymbol{g}}$ is implemented in the region inside the closed loop (dashed line), and hence quasiparticle $a$ is transformed into $b$ under ${\boldsymbol{g}}$ symmetry operation.