#  Classification of symmetry-enriched topological quantum spin liquids

Weicheng Ye  Perimeter Institute for Theoretical Physics, Waterloo, Ontario,
Canada N2L 2Y5  Liujun Zou  Perimeter Institute for Theoretical Physics,
Waterloo, Ontario, Canada N2L 2Y5

###### Abstract

We present a systematic framework to classify symmetry-enriched topological
quantum spin liquids in two spatial dimensions. This framework can deal with
all topological quantum spin liquids, which may be either Abelian or non-
Abelian, chiral or non-chiral. It can systematically treat a general symmetry,
which may include both lattice symmetry and internal symmetry, may contain
anti-unitary symmetry, and may permute anyons. The framework applies to all
types of lattices, and can systematically distinguish different lattice
systems with the same symmetry group using their Lieb-Schultz-Mattis
anomalies. We apply this framework to classify $U(1)_{2N}$ chiral states and
non-Abelian Ising(ν) states enriched by a $p6\times SO(3)$ or $p4\times SO(3)$
symmetry, and $\mathbb{Z}_{N}$ topological orders and $U(1)_{2N}\times
U(1)_{-2N}$ topological orders enriched by a $p6m\times
SO(3)\times\mathbb{Z}_{2}^{T}$, $p4m\times SO(3)\times\mathbb{Z}_{2}^{T}$,
$p6m\times\mathbb{Z}_{2}^{T}$ or $p4m\times\mathbb{Z}_{2}^{T}$ symmetry, where
$p6$, $p4$, $p6m$ and $p4m$ are lattice symmetries, while $SO(3)$ and
$\mathbb{Z}_{2}^{T}$ are spin rotation and time reversal symmetries,
respectively. In particular, we identify symmetry-enriched topological quantum
spin liquids that are not easily captured by the usual parton-mean-field
approach, including examples with the familiar $\mathbb{Z}_{2}$ topological
order.

###### Contents

  1. I Introduction
  2. II Outline and summary
  3. III Universal characterization of symmetry-enriched topological quantum spin liquids
    1. III.1 Topological order
    2. III.2 Global symmetry
    3. III.3 Anomalies of symmetry-enriched topological orders
    4. III.4 Crystalline equivalence principle
  4. IV Symmetry properties and quantum anomalies of lattice systems
  5. V Framework of classification
  6. VI $\mathrm{U}(1)_{2N}$ topological orders: Generalized Abelian chiral spin liquids
    1. VI.1 Example: $\mathbb{Z}_{2}\times SO(3)$
      1. VI.1.1 No anyon permutation
      2. VI.1.2 $C_{2}$ acts as charge conjugation
    2. VI.2 $p6\times SO(3)$
    3. VI.3 $p4\times SO(3)$
  7. VII Ising(ν) topological orders: Kitaev’s non-Abelian chiral spin liquids
  8. VIII $Z_{N}$ topological orders: Generalized toric codes
  9. IX $\mathrm{U}(1)_{2N}\times\mathrm{U}(1)_{-2N}$ topological orders: Generalizations of the double-semion state
  10. X Discussion
  11. A Translation between the characterization of reflection symmetry and time-reversal symmetry
  12. B Wallpaper group symmetries: group structure and $\mathbb{Z}_{2}$ cohomology
    1. B.1 $p6$
    2. B.2 $p6m$
    3. B.3 $p4$
    4. B.4 $p4m$
  13. C Details of realizations: Anyon permutation patterns and symmetry fractionalization classes
    1. C.1 $\mathrm{U}(1)_{2N}$
    2. C.2 Ising(ν)
    3. C.3 $\mathbb{Z}_{2}$ topological order
    4. C.4 $\mathbb{Z}_{N}$ topological order ($N\geqslant 3$)
    5. C.5 $\mathrm{U}(1)_{2}\times\mathrm{U}(1)_{-2}$ (Double Semion)
    6. C.6 $\mathrm{U}(1)_{4}\times\mathrm{U}(1)_{-4}$
  14. D Anomaly indicators
  15. E Symmetry fractionalization classes of the “beyond-parton” $\mathbb{Z}_{2}$ topological quantum spin liquids

##  I Introduction

Topological quantum spin liquids, which are more formally referred to as
bosonic topological orders, are exotic gapped quantum phases of matter with
long-range entanglement beyond the phenomenon of spontaneous symmetry
breaking.111All symmetries discussed in this paper are 0-form invertible
symmetries, unless otherwise stated. In two space dimensions, they can host
anyons, i.e., point-like excitations that are neither bosons nor fermions Wen
(2004). Besides being fundamentally interesting, they are also potential
platforms for fault-tolerant quantum computation Kitaev (2003); Nayak  _et
al._ (2008).

Roughly speaking, in the absence of symmetries, the universal properties of a
topological quantum spin liquid are encoded in the properties of its anyons,
such as their fusion rules and statistics. We refer to these properties as the
topological properties of a topological quantum spin liquid. In the presence
of symmetries, there can be interesting interplay between these topological
properties and the symmetries. In particular, even with fixed topological
properties, there can be sharply distinct topological quantum spin liquids
that cannot evolve into each other without breaking the symmetries or
encountering a quantum phase transition. These are known as different
symmetry-enriched topological quantum spin liquids.

The goal of this paper is to classify symmetry-enriched topological quantum
spin liquids in two space dimensions. That is, given the topological and
symmetry properties, we would like to understand which symmetry-enriched
topological quantum spin liquids are possible. This problem is first of
fundamental conceptual interest, as understanding different types of quantum
matter is one of the central goals of condensed matter physics. Moreover,
although topological quantum spin liquids have been identified in certain
lattice models and small-sized quantum simulators, its realization and
detection in macroscopic quantum materials remains elusive Broholm  _et al._
(2020). The knowledge of which symmetry-enriched topological quantum spin
liquids are possible is helpful for identifying the correct observable
signatures of them, thus paving the way to the realization and detection of
these interesting quantum phases in the future.

In the previous condensed matter literature, there are two widely used
approaches to classify symmetry-enriched topological quantum spin liquids, and
most of the other approaches can be viewed as variations or generalizations of
them. The first approach is based on parton mean fields and projective
symmetry groups, which starts by representing the microscopic degrees of
freedom (such as spins) via certain fractional degrees of freedom, and then
studies the possible projective representations of the symmetries carried by
these fractional degrees of freedom Wen (2002). The advantages of this
approach include its simplicity and its intimate connection to the microscopic
degrees of freedom of the system. It has been applied to classify various
symmetry-enriched quantum spin liquids in many different lattice systems, and
it treats internal symmetries and lattice symmetries on equal footing.

However, this approach is not perfect. One of the main disadvantages of this
approach is that it is not easily applicable to general topological quantum
spin liquids. For example, it is inconvenient to apply this approach to
classify symmetry-enriched $\mathbb{Z}_{N}$ topological orders with $N>2$.
This is because such topological orders are most naturally described by
partons coupled to a dynamical $\mathbb{Z}_{N}$ gauge field. When $N>2$, the
partons themselves are interacting, so a non-interacting parton mean field
description is inadequate. Also, it is often challenging to use this approach
to study topological quantum spin liquids where anyons are permuted in a
complicated manner by symmetries. Moreover, even for quantum spin liquids that
are often assumed to be captured by parton mean fields, some of their symmetry
enrichment patterns may be beyond parton mean fields. Ref. Ye  _et al._ (2022)
presented such phenomenon for the gapless Dirac spin liquid, and in Sec. VIII
we find that this phenomenon can occur even for the familiar $\mathbb{Z}_{2}$
topological quantum spin liquid. Another disadvantage of this approach is that
the projective representations carried by the fractional degrees of freedom
may not be the same as the symmetry properties of the physical anyons, which
sometimes require one more nontrivial step to obtain. For examples, see Refs.
Cincio and Qi (2015); Zou and He (2020).

The second approach is based on the categorical theoretic description of the
topological quantum spin liquids, and studies how the category theory
corresponding to a topological quantum spin liquid can be consistently
extended to include a symmetry group Barkeshli  _et al._ (2019a); Tarantino
_et al._ (2016). The advantage of this approach is that it is fully general
and can be applied to situations where the parton-based approach is
inconvenient, since it is believed that the topological properties of all
topological quantum spin liquids can be described by category theory. Also, it
yields the symmetry properties of the anyons directly, without relying on some
artificial fractional degrees of freedom as in the first approach. Overall,
this approach is very systematic and powerful in studying topological quantum
spin liquids with only internal symmetries.

However, this approach also has its disadvantages: Unlike the first approach,
it has a relatively weaker connection with the microscopic properties of the
physical system, and it is particularly inconvenient when applied to systems
with lattice symmetries, which are often physically important. Specifically,
within this approach all symmetry properties of the physical system are
assumed to be captured by its symmetry group. However, such a description of
the symmetries is inadequate for many purposes. For example, spin-1/2 systems
defined on a triangular lattice, kagome lattice and honeycomb lattice can all
have the same symmetry group, say, the $SO(3)$ spin rotational symmetry and
the $p6m$ lattice symmetry. But they are physically distinct, and any
symmetry-enriched topological quantum spin liquid that can emerge in one of
these three systems cannot emerge in the other two Po  _et al._ (2017); Ye
_et al._ (2022). A physically relevant classification of symmetry-enriched
topological quantum spin liquids should take into account this distinction,
which reflects some microscopic symmetry properties beyond the symmetry group.
In the above example, these properties can be viewed as the locations of the
spin-1/2’s in those three lattices.

In the present paper, we develop a framework to classify symmetry-enriched
topological quantum spin liquids, which combines the advantages of the above
two approaches while avoiding their disadvantages. Specifically, we will use
the language of category theory to directly describe the topological
properties of a topological quantum spin liquid and the symmetry properties of
the anyons, and we will also keep track of the robust microscopic symmetry-
related information of the physical system. Roughly speaking, this information
includes:

  * •

The symmetry group. For example, the $SO(3)$ spin rotational symmetry and
$p6m$ lattice symmetry.

  * •

The nature of the microscopic degrees of freedom, or more precisely, how the
microscopic degrees of freedom transform under the symmetry group. For
example, whether the microscopic degrees of freedom are spin-1/2’s or
spin-1’s.

  * •

The locations of the microscopic degrees of freedom. For example, whether the
microscopic degrees of freedom reside on the sites of a triangular, kagome or
honeycomb lattice.

All three pieces of information above are taken into account in the
aforementioned first approach, but the second approach usually only considers
the first piece (see exceptions in Refs. Cheng  _et al._ (2016); Qi  _et al._
(2015); Qi and Cheng (2018); Ning  _et al._ (2020); Manjunath and Barkeshli
(2021, 2020); Song and Senthil (2022) studying certain specific symmetry-
enriched quantum spin liquids, but the methods therein have not been
generalized to the generic case before). The reason why the other two pieces
of information are physically relevant is because they are also robust
properties of a system as long as the symmetries are preserved. In fact, if
two systems have the same symmetry group but their microscopic degrees of
freedom are of different nature or at different locations, they may not be
able to evolve into each other without breaking the symmetry (even if going
across quantum phase transitions is allowed) Po  _et al._ (2017). Moreover,
just like the symmetry group, these two pieces of information are often
relatively easy to determine experimentally, and they are the basic aspects of
any theoretical model. We will make these concepts more precise in Sec. IV.

Ultimately, our framework takes as input 1) the topological properties of a
topological quantum spin liquid and 2) a collection of the above symmetry-
related properties of a microscopic system, and outputs which symmetry-
enriched topological quantum spin liquids are possible. Each of these
symmetry-enriched topological quantum spin liquids is characterized by the
symmetry properties of its anyons.

This framework relies on the state-of-the-art in the studies of the quantum
anomalies of lattice systems and topological quantum spin liquids. In
particular, in Ref. Ye  _et al._ (2022) we and our coauthors obtained the
precise characterizations of the full set of the quantum anomalies of a large
class of lattice systems, which exactly encode the aforementioned robust
microscopic symmetry-related information of the lattice system. These quantum
anomalies are referred to as Lieb-Schultz-Mattis anomalies. Moreover, in Ref.
Ye and Zou (2023) we developed a systematic framework to calculate the quantum
anomaly of a generic symmetry-enriched topological quantum spin liquid. With
these previous developments, in Sec. V we will present the general framework
to classify any two dimensional symmetry-enriched topological quantum spin
liquids in any lattice systems with any symmetries, by matching the anomaly of
the lattice system and the anomaly of the symmetry-enriched topological
quantum spin liquid. Here the topological quantum spin liquid can be described
by a topological order that is either chiral or non-chiral, Abelian or non-
Abelian. The symmetry can include both internal and lattice symmetries,
unitary and anti-unitary symmetries, and symmetries that permute anyons. In
the rest of the paper, we will apply this framework to classify various
symmetry-enriched topological quantum spin liquids. We will focus on examples
with symmetry group $G=G_{s}\times G_{int}$, where $G_{s}$ is certain lattice
symmetry, and $G_{int}$ is an on-site internal symmetry.

We remark that our philosophy of encoding the robust microscopic symmetry-
related information of a physical system into its quantum anomaly and using
anomaly-matching to classify quantum many-body states is very general. Besides
classifying symmetry-enriched topological quantum spin liquids, it can also be
used to classify other quantum states. In fact, we and our coauthors have
applied it to classify certain symmetry-enriched gapless quantum spin liquids
in Ref. Ye  _et al._ (2022). Our anomaly-based philosophy can be viewed as a
generalization of the theories of symmetry indicators or topological quantum
chemistry, which classify band structures Bradlyn  _et al._ (2017); Po (2020).
In fact, the robust microscopic symmetry-related information of a physical
system we take is identical to what those theories take, but those theories
only apply to weakly correlated systems that can be described by band theory,
while ours applies to generic strongly correlated systems.

##  II Outline and summary

The outline of the rest of the paper and summary of the main results are as
follows.

  * •

In Sec. III, we briefly review how to describe a symmetry-enriched topological
quantum spin liquid and its quantum anomaly. The description is based on the
category theory, but no previous knowledge of category theory is required.

  * •

In Sec. IV, we discuss the symmetry properties of a lattice system and how to
encode them into the quantum anomaly of a lattice system.

  * •

In Sec. V, we present our general framework to classify symmetry-enriched
topological quantum spin liquid, which may be Abelian or non-Abelian, and
chiral or non-chiral. This framework applies to topological quantum spin
liquids with any symmetry, which may contain both lattice symmetry and
internal symmetry, may contain anti-unitary symmetry and may permute anyons.

  * •

In Sec. VI, we apply the framework in Sec. V to classify the $U(1)_{2N}$
topological quantum spin liquid enriched by a $p6\times SO(3)$ or $p4\times
SO(3)$ symmetry, where $p6$ and $p4$ are lattice symmetries and $SO(3)$ is the
spin rotational symmetry. In this section we spell out the details of the
analysis, including the symmetry fractionalization class of each symmetry-
enriched state and the calculation of the quantum anomalies. The results are
summarized in Table 7.

  * •

In Sec. VII, we apply our framework in Sec. V to classify the non-Abelian
Ising(ν) topological quantum spin liquid enriched by a $p6\times SO(3)$ or
$p4\times SO(3)$ symmetry.

  * •

In Sec. VIII, we apply our framework in Sec. V to classify the
$\mathbb{Z}_{N}$ topological quantum spin liquids enriched by one of these
four symmetries: $p6m\times SO(3)\times\mathbb{Z}_{2}^{T}$, $p4m\times
SO(3)\times\mathbb{Z}_{2}^{T}$, $p6m\times\mathbb{Z}_{2}^{T}$ and
$p4m\times\mathbb{Z}_{2}^{T}$, where the $p6m$ and $p4m$ are lattice
symmetries, and $SO(3)$ and $\mathbb{Z}_{2}^{T}$ are on-site spin rotational
symmetry and time reversal symmetry, respectively. The results are summarized
in Table 16. In particular, even for the simple case with $N=2$, we find many
states beyond the description based on the usual parton mean field.

  * •

In Sec. IX, we apply our framework in Sec. V to classify the $U(1)_{2N}\times
U(1)_{-2N}$ topological quantum spin liquids enriched by one of these four
symmetries: $p6m\times SO(3)\times\mathbb{Z}_{2}^{T}$, $p4m\times
SO(3)\times\mathbb{Z}_{2}^{T}$, $p6m\times\mathbb{Z}_{2}^{T}$ and
$p4m\times\mathbb{Z}_{2}^{T}$. The resutls are summarized in Table 16.

  * •

We close our paper in Sec. X.

  * •

Various appendices contain further details. Appendix A presents a formal
treatment of the connection between the data characterizing a topological
order enriched by reflection symmetry and the data characterizing a
topological order enriched by time reversal symmetry. Appendix B reviews the
properties of the lattice symmetries involved in this paper. Appendix C
details the information of all symmetry fractionalization classes of all
states studied in this paper. Appendix D presents the anomaly indicators for
various symmetries, and their values in the physically relevant cases
considered in this paper. Appendix E presents the details of the symmetry
fractionalization classes of the symmetry-enriched $\mathbb{Z}_{2}$
topological orders that are beyond the usual parton mean fields.

##  III Universal characterization of symmetry-enriched topological quantum
spin liquids

In this section, we review the universal characterization of symmetry-enriched
topological quantum spin liquids. This characterization is divided into two
parts. We first specify the topological order corresponding to the topological
quantum spin liquid, which is reviewed in Sec. III.1. After this, assuming the
symmetry is an internal symmetry, we will specify the global symmetry of the
topological quantum spin liquid and how it acts on the topological order,
which is reviewed in Sec. III.2. We review the anomaly of a symmetry-enriched
topological order in Sec. III.3. Finally, we review the crystalline
equivalence principle in Sec. III.4, which allows us to connect a topological
order with lattice symmetry to one with internal symmetry.

###  III.1 Topological order

The characteristic feature of a topological order is the presence of anyons,
point-like excitations that can have self-statistics other than bosonic or
fermionic statistics, and nontrivial mutual braiding statistics. When multiple
anyons are present, there may also be a protected degenerate state space. It
is believed that the category theory can universally characterize all
topological orders or the anyons therein. In this subsection we briefly review
the concepts relevant to this paper. Our review will be minimal and does not
assume any knowledge of the category theory itself. For a more comprehensive
review, see e.g., Refs. Barkeshli  _et al._ (2019a); Kitaev (2006); Kong and
Zhang (2022) for a more physics oriented introduction, or Refs. Bakalov and
Kirillov (2001); Etingof  _et al._ (2016); Turaev (1994); Selinger (2011) for
a more mathematical treatment.

Anyons in a topological order are denoted by $a,b,c,\cdots$. A single anyon
cannot be converted into a different single anyon via any local process. There
is always a trivial anyon in all topological orders, obtained by performing
some local operation on the ground state. Roughly speaking, in a many-body
system where a topological order emerges at low energies, the quantum state is
specified by two pieces of data: the global data characterizing the anyons,
which cannot be changed by any local process, and the local data independent
of the anyons, which can change by local operations.

For the purpose of this paper, two important properties of an anyon $a$ will
be used, its quantum dimension $d_{a}$ and topological spin $\theta_{a}$.
Suppose $n$ anyons $a$ are created, the dimension of the degenerate state
space scales as $d_{a}^{n}$ when $n$ is large. So the quantum dimension
$d_{a}$ effectively measures how rich the internal degree of freedom the anyon
$a$ carries. If $d_{a}=1$, then $a$ is said to be an Abelian anyon. Otherwise
it is a non-Abelian anyon. The topological spin $\theta_{a}$ measures the
self-statistics of $a$. For bosons and fermions, the topological spin is $\pm
1$. For a generic anyon, the topological spin can take other values.

Two anyons $a$ and $b$ can fuse into other anyons, expressed using the
equation $a\times b\cong\sum_{c}N^{c}_{ab}c$, where $N^{c}_{ab}$ are positive
integers and $c$ is said to be a fusion outcome of $a$ and $b$. Generically,
$a$ and $b$ may have multiple fusion outcomes, and there may also be multiple
different ways to get each fusion outcome, which is why the right hand side of
the fusion equation involves a summation and $N^{c}_{ab}$ can be larger than
1. Physically, fusion means that if we only perform measurements far away from
the anyons $a$ and $b$, we will think they look like the fusion outcome $c$.
Diagramatically, we can use

to represent a fusion process, or

to represent a splitting process, which can be viewed as the reversed process
of fusion. In the above, $\mu\in\\{1,2,\cdots,N^{c}_{ab}\\}$, and the arrows
can be viewed as the world lines of the anyons. If the trivial anyon is in the
fusion product of two anyons, we say that these two anyons are anti-particles
of each other, and we denote the anti-particle of $a$ by $\bar{a}$.

When multiple anyons are present, we may imagine fusing them in different
orders. Similarly, when a given anyon splits into multiple other anyons, it
may also do so in different orders. For example, the two sides of the
following equation represent two different orders of splitting an anyon $d$
into three anyons, $a$, $b$ and $c$. Generically, the state obtained after
these two processes are not identical, but they can be related by the so-
called $F$-symbol, which is a unitary matrix acting on the degenerate state
space.

$\raisebox{-0.5pt}{\begin{pspicture}[missing](0.0pt,-12.80373pt)(51.21494pt,51.21494pt)\small\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}
{\psline(5.69054pt,42.67911pt)(28.45274pt,14.22636pt)}{\psline(28.45274pt,14.22636pt)(28.45274pt,0.0pt)}{\psline(51.21494pt,42.67911pt)(28.45274pt,14.22636pt)}{\psline(17.07164pt,28.45274pt)(28.45274pt,42.67911pt)}{\psline{->}(17.07164pt,28.45274pt)(8.53581pt,39.12251pt)}{\psline{->}(17.07164pt,28.45274pt)(25.60747pt,39.12251pt)}{\psline{->}(28.45274pt,14.22636pt)(48.36966pt,39.12251pt)}{\psline{->}(28.45274pt,14.22636pt)(19.91692pt,24.89615pt)}{\psline{->}(28.45274pt,0.0pt)(28.45274pt,10.66977pt)}\rput[bl]{0.0}(1.42262pt,45.52438pt){$a$}
\rput[bl]{0.0}(27.0301pt,45.52438pt){$b$}
\rput[bl]{0.0}(49.7923pt,45.52438pt){${c}$}
\rput[bl]{0.0}(14.22636pt,14.22636pt){$e$}
\rput[bl]{0.0}(25.60747pt,-8.53581pt){$d$}
\scriptsize\rput[bl]{0.0}(8.53581pt,22.76219pt){$\alpha$}
\rput[bl]{0.0}(19.91692pt,7.11317pt){$\beta$}
\end{pspicture}}=\sum_{f,\mu,\nu}\left[F_{d}^{abc}\right]_{(e,\alpha,\beta),(f,\mu,\nu)}\raisebox{-0.5pt}{\begin{pspicture}[missing](0.0pt,-12.80373pt)(51.21494pt,51.21494pt)\small\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}
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\rput[bl]{0.0}(27.0301pt,45.52438pt){$b$}
\rput[bl]{0.0}(49.7923pt,45.52438pt){${c}$}
\rput[bl]{0.0}(35.56592pt,12.80373pt){$f$}
\rput[bl]{0.0}(25.60747pt,-8.53581pt){$d$}
\scriptsize\rput[bl]{0.0}(42.67911pt,22.76219pt){$\mu$}
\rput[bl]{0.0}(19.91692pt,7.11317pt){$\nu$} \end{pspicture}}$

In this equation, $e$ is a fusion out come of $a$ and $b$, where
$\alpha\in\\{1,\cdots N^{e}_{ab}\\}$, $d$ is a fusion outcome of $c$ and $e$,
where $\beta\in\\{1,\cdots,N^{d}_{ec}\\}$, $f$ is a fusion outcome of $b$ and
$c$, where $\mu\in\\{1,\cdots,N^{f}_{bc}\\}$, and $f$ and $a$ also fuse into
$d$, where $\nu\in\\{1,\cdots,N^{d}_{af}\\}$.

When anyons move around each other, the state may acquire some nontrivial
braiding phase factor, and it may even be acted by a unitary matrix in
general. The statistics and braiding properties of the anyons are encoded in
the $R$-symbol, which is also a unitary matrix acting in the degenerate state
space and is defined via the the following diagram:

$\raisebox{-0.5pt}{\begin{pspicture}[missing](-2.84526pt,-5.69054pt)(42.67911pt,34.14328pt)\small\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}
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{\psarc(17.07164pt,19.15453pt){5.69054pt}{-60.0}{35.0}}
{\psline(17.45291pt,25.50531pt)(7.59688pt,31.19586pt)}{\psline{->}(17.45291pt,25.50531pt)(10.0606pt,29.77322pt)}{\psline(19.91692pt,24.08267pt)(32.23695pt,31.19586pt)}{\psline{->}(19.91692pt,24.08267pt)(29.77322pt,29.77322pt)}\rput[bl]{0.0}(11.38109pt,0.0pt){$c$}
\rput[br]{0.0}(38.4112pt,24.18483pt){$b$}
\rput[bl]{0.0}(1.42262pt,24.18483pt){$a$}
\scriptsize\rput[bl]{0.0}(23.33124pt,9.95845pt){$\mu$}
\end{pspicture}}=\sum\limits_{\nu}\left[R_{c}^{ab}\right]_{\mu\nu}\raisebox{-0.5pt}{\begin{pspicture}[missing](-2.84526pt,-5.69054pt)(42.67911pt,34.14328pt)\small\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}
{\psline{->}(19.91692pt,0.0pt)(19.91692pt,12.80373pt)}{\psline(19.91692pt,0.0pt)(19.91692pt,15.649pt)}{\psline(19.91692pt,15.649pt)(7.11317pt,28.45274pt)}{\psline{->}(19.91692pt,15.649pt)(8.53581pt,27.0301pt)}{\psline(19.91692pt,15.649pt)(32.72064pt,28.45274pt)}{\psline{->}(19.91692pt,15.649pt)(31.298pt,27.0301pt)}\rput[bl]{0.0}(11.38109pt,0.0pt){$c$}
\rput[br]{0.0}(39.83383pt,22.76219pt){$b$}
\rput[bl]{0.0}(0.0pt,22.76219pt){$a$}
\scriptsize\rput[bl]{0.0}(23.33124pt,10.52751pt){$\nu$} \end{pspicture}}.$

The topological spin can be expressed using the $R$-symbol via

$\theta_{a}=\theta_{\overline{a}}=\sum\limits_{c,\mu}\frac{d_{c}}{d_{a}}\left[R_{c}^{aa}\right]_{\mu\mu}=\frac{1}{d_{a}}\raisebox{-0.5pt}{\begin{pspicture}[missing](-36.98856pt,-17.07164pt)(36.98856pt,17.07164pt)\small\psset{linewidth=0.9pt,linecolor=black,arrowscale=1.5,arrowinset=0.15}
{\psset{linewidth=0.9pt,linecolor=black}\psarc(20.11893pt,0.0pt){14.22636pt}{-135.0}{135.0}}
{\psset{linewidth=0.9pt,linecolor=black}\psarc(-20.11893pt,0.0pt){14.22636pt}{45.0}{315.0}}
{\psline(-10.06088pt,10.06088pt)(10.06088pt,-10.06088pt)}{\psset{border=2.3pt}\psline(-10.06088pt,-10.06088pt)(10.06088pt,10.06088pt)}{\psset{border=2.3pt}\psline{->}(-10.06088pt,-10.06088pt)(0.0pt,0.0pt)}\rput[bl]{0.0}(-5.69054pt,-14.22636pt){$a$}
\end{pspicture}}$

The $F$\- and $R$-symbols satisfy strong constraints and have some “gauge”
freedom i.e., two sets of $\\{F,R\\}$ data related by certain gauge
transformations are physically equivalent. We remark that the $F$\- and
$R$-symbols can all be defined microscopically Kawagoe and Levin (2020).

###  III.2 Global symmetry

In the presence of global symmetry, anyons can display interesting phenomena
including (1) anyon permutation and (2) symmetry fractionalization. We will
review the concepts relevant to this paper below, and more comprehensive
reviews can be found in, e.g., Refs. Barkeshli  _et al._ (2019a); Etingof  _et
al._ (2016, 2010). In this subsection and the next, we will assume that all
symmetries are internal symmetries, i.e., they do not move the locations of
the degrees of freedom. We will comment on how to deal with the case with
lattice symmetry in Sec. III.4.

Before specifying any microscopic symmetry of interest, it is useful to first
discuss the abstract topological symmetry group of a topological order, whose
elements can be viewed as invertible maps that take one anyon into another, so
that the fusion properties of the topological order are invariant. For unitary
(anti-unitary) topological symmetry action, the braiding properties, encoded
in the $F$\- and $R$-symbols, is preserved (conjugated). Later in the paper we
will see many examples of topological symmetries of various topological
orders.

Now we consider a microscopic symmetry described by a group $G$. Suppose
$R_{\bf g}$ is a symmetry action, where ${\bf g}$ labels an element of $G$.
This action can change anyons into other types, for example, it change an
anyon $a$ into another anyon ${}^{\bf g}a$. We use

$\displaystyle\rho_{\bf g}:a\rightarrow\ ^{\bf g}a$ (1)

to represent the anyon permutation pattern. Mathematically, we may say that
$\rho_{\rm g}$ describes a group homomorphism from $G$ to the topological
symmetry group of a topological order. From here we see why the notion of
topological symmetry is important: It encodes all possible anyon permutation
patterns.

However, $\rho_{\rm g}$ by itself is insufficient to fully characterize a
symmetry-enriched topological order. Consider creating three anyons $a_{1}$,
$a_{2}$ and $a_{3}$ from the ground states, such that $a_{1}\times
a_{2}\rightarrow\bar{a}_{3}$, i.e., these three anyons can fuse into the
trivial anyon. After separating these anyons far away from each other, there
are generically $N^{\bar{a}_{3}}_{a_{1}a_{2}}$ degenerate such states. What
effect does $R_{\bf g}$ have when it acts on a state in this degenerate space,
denoted by $|\Psi_{a_{1},a_{2},a_{3}}\rangle$?

Since the state of a topological order is specified by the two pieces of
information, the global one and the local one, the symmetry localization
ansatz states that Essin and Hermele (2013)

$\displaystyle\begin{aligned} &R_{\bf g}|\Psi_{a_{1},a_{2},a_{3}}\rangle\\\
&=V_{\bf g}^{(1)}V_{\bf g}^{(2)}V_{\bf g}^{(3)}U_{\bf g}(^{\bf g}a_{1},^{\bf
g}a_{2};^{\bf g}\bar{a}_{3})|\Psi_{{}^{\bf g}a_{1},^{\bf g}a_{2},^{\bf
g}a_{3}}\rangle\end{aligned}$ (2)

where $V_{\bf g}^{(i)}$ is a local unitary operation supported only around the
anyon $a_{i}$, for $i=1,2,3$, and $U_{\rm g}(^{\bf g}a_{1},^{\bf g}a_{2};^{\bf
g}\bar{a}_{3})$ is a unitary matrix with rank $N^{\bar{a}_{3}}_{a_{1}a_{2}}$,
which acts on the degenerate state space and describes the symmetry action on
the global part of the information contained in the state. Notice that the
state appearing on the right hand side is $|\Psi_{{}^{\bf g}a_{1},^{\bf
g}a_{2},^{\bf g}a_{3}}\rangle$, i.e., generically the anyons are permuted by
the symmetry.

It can be shown that the local operations $V$ satisfy

$\displaystyle\eta_{a_{i}}({\bf g},{\bf h})V_{\bf
gh}^{(i)}|\Psi_{a_{1},a_{2},a_{3}}\rangle=R_{\bf g}V_{\bf h}^{(i)}R_{\bf
g}^{-1}V_{\bf g}^{(i)}|\Psi_{a_{1},a_{2},a_{3}}\rangle$ (3)

for a pair of group elements ${\bf g}$ and ${\bf h}$. Here $\eta_{a_{i}}({\bf
g},{\bf h})$ are generically nontrivial phase factors, implying that the
anyons may carry fractional charge or projective quantum number under the
symmetry, i.e., there can be symmetry fractionalization.222In the context of
symmetry fractionalization, it is well known that the notion of fractional
charge or projective quantum number is generically not the same as the
projective representations more commonly discussed in mathematics, which are
classified by $H^{2}(G,U(1))$.

For a given topological order and a symmetry group $G$, it turns out that the
data $\\{\rho_{\bf g};U_{\bf g}(a,b;c),\eta_{a}({\bf g},{\bf h})\\}$
completely characterizes how this topological order is enriched by the
symmetry $G$. We will often call $U_{\bf g}(a,b;c)$ the $U$-symbol, and
$\eta_{a}({\bf g},{\bf h})$ the $\eta$-symbol. This data $\\{\rho_{\bf
g};U_{\bf g}(a,b;c),\eta_{a}({\bf g},{\bf h})\\}$ also satisfies strong
constraints and have some “gauge” freedom, i.e., two sets of $\\{\rho_{\bf
g};U_{\bf g}(a,b;c),\eta_{a}({\bf g},{\bf h})\\}$ data related by certain
gauge transformations are physically equivalent. These gauge transformations
will not be explicitly used in this paper, but they are summarized in, e.g.,
Ref. Ye and Zou (2023). Moreover, even if two sets of data $\\{\rho_{\bf
g};U_{\bf g}(a,b;c),\eta_{a}({\bf g},{\bf h})\\}$ are not related by a gauge
transformation, they still correspond to the physical state if they are
related to each other by anyon relabeling that preserves the fusion and
braiding properties Essin and Hermele (2013); Barkeshli  _et al._ (2019a),
i.e., such relabeling is precisely a unitary element in the topological
symmetry group.333Namely, for any unitary element ${\bf k}_{0}$ in the
topological symmetry group (which does not have to be a microscopic symmetry),
we can relabel each anyon $a$ by ${}^{{\bf k}_{0}}a$ (under this relabeling
the data $\\{F^{abc}_{d},R^{ab}_{c}\\}$ is the same as $\\{F^{{}^{{\bf
k}_{0}}a^{{\bf k}_{0}}b^{{\bf k}_{0}}c}_{{}^{{\bf k}_{0}}d},R^{{}^{{\bf
k}_{0}}a^{{\bf k}_{0}}b}_{{}^{{\bf k}_{0}}c}\\}$ up to gauge transformations).
Then the symmetry-enriched topological order with data $\\{\rho_{\bf
g}:a\rightarrow\,^{\bf g}a;U_{\bf g}(a,b;c),\eta_{a}({\bf g},{\bf h})\\}$ is
the same as the one with data $\\{\rho^{\prime}_{\bf g}:a\rightarrow\,^{{\bf
k}_{0}}\left[{}^{\bf g}\left({}^{\overline{{\bf
k}_{0}}}a\right)\right];U^{\prime}_{\bf g}(a,b;c)=U_{\bf g}(^{\overline{{\bf
k}_{0}}}a,^{\overline{{\bf k}_{0}}}b;^{\overline{{\bf
k}_{0}}}c),\eta^{\prime}_{a}({\bf g},{\bf h})=\eta_{{}^{\overline{{\bf
k}_{0}}}a}({\bf g},{\bf h})\\}$.

For a given anyon permutation pattern, one can show that distinct symmetry
fractionalization classes form a torsor over $H^{2}_{\rho}(G,\mathcal{A})$.
Namely, different possible symmetry fractionalization classes can be related
to each other by elements of $H^{2}_{\rho}(G,\mathcal{A})$, where
$\mathcal{A}$ is an Abelian group whose group elements correspond to the
Abelian anyons in this topological order, and the group multiplication
corresponds to the fusion of these Abelian anyons. The subscript $\rho$
represents the permutation action of $G$ on these Abelian anyons. In
particular, given an element $[t]\in H^{2}_{\rho}(G,\mathcal{A})$, we can go
from one symmetry fractionalization class with data $\eta_{a}({\bf g},{\bf
h})$ to another with data $\tilde{\eta}_{a}({\bf g},{\bf h})$ given by

$\displaystyle\tilde{\eta}_{a}({\bf g},{\bf h})=\eta_{a}({\bf g},{\bf
h})M_{a,t({\bf g},{\bf h})}$ (4)

where $t({\bf g},{\bf h})\in\mathcal{A}$ is a representative 2-cocyle for the
cohomology class $[t]$ and $M_{a,t({\bf g},{\bf h})}=\frac{\theta_{a\times
t({\bf g},{\bf h})}}{\theta_{a}\theta_{t({\bf g},{\bf h})}}$ is the braiding
statistics between $a$ and $t({\bf g},{\bf h})$ Bonderson  _et al._ (2008).

In the case where no anyon is permuted by any symmetry, there is always a
canonical notion of a trivial symmetry fractionalization class, where
$\eta_{a}({\bf g},{\bf h})=1$ for all anyon $a$ and all ${\bf g},{\bf h}\in
G$. In this case, an element of $H^{2}(G,\mathcal{A})$ is sufficient to
completely characterize the symmetry fractionalization class.

Later in the paper, for a symmetry-enriched topological order we will often
just specify $\rho_{\bf g}$, without explicitly specifying the $U$\- and
$\eta$-symbols. However, we will specify the $U$\- and $\eta$-symbols of the
topological symmetry group of this topological order, which allows us to
determine the $U$\- and $\eta$-symbols of the microscopic symmetry as follows.

Denote the microscopic symmetry group by $G$ and the topological symmetry
group by $G_{0}$, then $\rho_{\bf g}$ defines a group homomorphism
$\varphi:G\rightarrow G_{0}$. Denote the $U$\- and a set of $\eta$-symbols of
$G_{0}$ by $U^{(0)}_{\bf g_{0}}(a,b;c)$ and $\eta_{a}^{(0)}({\bf g}_{0},{\bf
h}_{0})$, for any ${\bf g}_{0},{\bf h}_{0}\in G_{0}$. These $U$\- and
$\eta$-symbols are some data intrinsic to the topological order, independent
of the microscopic symmetry $G$, just like the $F$\- and $R$-symbols. The
$U$\- and a set of $\eta$-symbols of the microscopic symmetry $G$ can be
written as

$\displaystyle\begin{aligned} &U_{\bf g}(a,b;c)=U_{\varphi(\bf g)}(a,b;c),\\\
&\eta_{a}({\bf g},{\bf h})=\eta_{a}(\varphi({\bf g}),\varphi({\bf
h}))\end{aligned}$ (5)

for any ${\bf g},{\bf h}\in G$. Other symmetry fractionalization classes
correponding to other sets of $\eta$-symbols can be related to this one via
Eq. (4).

###  III.3 Anomalies of symmetry-enriched topological orders

A symmetry-enriched topological order can have a quantum anomaly. Roughly
speaking, the anomaly characterizes the interplay between locality and the
symmetry of the system. There are a few different definitions of quantum
anomaly that are believed to be equivalent. In general, the anomaly can be
viewed as an obstruction to gauging the symmetry, an obstruction to having a
symmetric short-range entangled ground state, an obstruction to describing the
system using a Hilbert space with a tensor product structure and on-site
symmetry actions, or as the boundary manifestation of a higher dimensional
bulk.

For a symmetry-enriched topological order, we can characterize its anomaly via
anomaly indicators, a set of quantities expressed in terms of the data like
$\\{F^{abc}_{d},R^{ab}_{c},\rho_{\bf g},U_{\bf g}(a,b;c),\eta_{a}({\bf g},{\bf
h})\\}$. For example, consider a $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
symmetry. The anomalies of $(2+1)$-dimensional bosonic systems with a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry are classified by
$(\mathbb{Z}_{2})^{2}$. Suppose the two generators of
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ are $C_{1}$ and $C_{2}$. The two anomaly
indicators can be given by $\mathcal{I}_{3}(C_{1},C_{2})$ and
$\mathcal{I}_{3}(C_{2},C_{1})$, where

$\displaystyle\begin{aligned}
\mathcal{I}_{3}\left(C_{1},C_{2}\right)=\frac{1}{D^{2}}\sum_{\begin{subarray}{c}a,b,x,u\\\
\mu\nu\tilde{\mu}\tilde{\nu}\rho\sigma\alpha\\\ {}^{C_{1}}a=a\\\ a\times
b\times^{C_{1}}b\rightarrow^{C_{2}}a\end{subarray}}&{d_{b}}\frac{\theta_{x}}{\theta_{a}}\left(R_{u}^{b,^{C_{1}}b}\right)_{\rho\sigma}\left(F_{{{}^{C_{2}}}a}^{a,b,{{}^{C_{1}}}b}\right)^{*}_{(x,\tilde{\mu},\tilde{\nu})(u,\sigma,\alpha)}\left(F_{{{}^{C_{2}}}a}^{a,{{}^{C_{1}}}b,b}\right)_{({{}^{C_{1}}}x,\mu,\nu)(u,\rho,\alpha)}\\\
&\times
U_{C_{1}}^{-1}(a,b;x)_{\tilde{\mu}\mu}U_{C_{1}}^{-1}(x,^{C_{1}}b;^{C_{2}}a)_{\tilde{\nu}\nu}\times\frac{1}{\eta_{b}(C_{1},C_{1})}\frac{\eta_{a}(C_{2},C_{1})}{\eta_{a}(C_{1},C_{2})}\end{aligned}$
(6)

and $\mathcal{I}_{3}(C_{2},C_{1})$ is obtained from the above equation by
swapping $C_{1}\leftrightarrow C_{2}$ Ye and Zou (2023). The reason for the
subscript of $\mathcal{I}_{3}$ can be seen in Appendix D. The summation is
over all anyon types $a$ and $b$ satisfying ${}^{C_{1}}a=a$ and that
${}^{C_{2}}a$ is a fusion outcome of $a\times b\times^{C_{1}}b$, $x$ also
denotes anyon types, and the Greek letters index different ways of getting a
particular fusion outcome (e.g.,
$\mu=1,2,\cdots,N^{{}^{C_{2}}a}_{a\,^{C_{1}}b}$). These two anomaly indicators
take values in $\pm 1$, and each set of values of these anomaly indicators
specifies an element in the $(\mathbb{Z}_{2})^{2}$ group, which classifies
these anomalies.

###  III.4 Crystalline equivalence principle

In the above two subsections our focus is topological orders with purely
internal symmetries. But as mentioned in the Introduction, one of the main
goals of this paper is to classify topological quantum spin liquids enriched
by a general symmetry, which may contain both lattice symmetry and internal
symmetry. The crystalline equivalence principle Song  _et al._ (2017a);
Thorngren and Else (2018) provides a convenient way to describe a topological
order with lattice symmetry (and possibly also internal symmetry) using a
topological order with a purely internal symmetry.

More concretely, the crystalline equivalence principle asserts that for each
topological phase with a symmetry group $G$, where $G$ may contain both
lattice symmetry and internal symmetry, there is a corresponding topological
phase with only internal symmetries, where the symmetry group is still $G$,
and all orientation reversing symmetries in the original topological phase
should be viewed as anti-unitary symmetries in the corresponding topological
phase. For example, Appendix A explains how to translate the data
characterizing a symmetry-enriched topological order with reflection symmetry
into the data for a time reversal symmetry-enriched topological order.

Strictly speaking, the above statement only applies to bosonic systems, which
is the focus of the present paper. The fermionic version of this statement is
still under development (see Refs. Cheng and Wang (2022); Debray (2021); Zhang
_et al._ (2022); Manjunath  _et al._ (2023) for recent progress).

##  IV Symmetry properties and quantum anomalies of lattice systems

In the Introduction we mentioned that the three pieces of robust symetry-
related information of a lattice system can be encoded in its quantum anomaly.
In this section, we make this notion more precise. Although this idea is
general, for concreteness, we focus on lattice spin systems in two spatial
dimensions with one of these six symmetry groups: $p6\times SO(3)$, $p4\times
SO(3)$, $p6m\times\mathbb{Z}_{2}^{T}$, $p4m\times\mathbb{Z}_{2}^{T}$,
$p6m\times SO(3)\times\mathbb{Z}_{2}^{T}$, and $p4m\times
SO(3)\times\mathbb{Z}_{2}^{T}$. Here $p6$, $p4$, $p6m$ and $p4m$ are lattice
symmetry groups, whose definitions are explained in Figs. 1 and 2. These
lattice symmetries are assumed to only move the locations of the microscopic
degrees of freedom, without acting on their internal states, i.e., there is no
spin-orbit coupling. The $SO(3)$ and $\mathbb{Z}_{2}^{T}$ are on-site spin
rotational symmetry and time reversal symmetry, respectively. These symmetry
settings are relevant to many theoretical, experimental and numerical studies,
and the examples we consider in the later part of the paper will also be based
on these symmetry settings.

Figure 1: Panel (a) shows the generators of the $p6m$ group, including
translations $T_{1}$ and $T_{2}$, a 6-fold rotation $C_{6}$ and a mirror
reflection $M$. The two translation vectors have the same length, and their
angle is $2\pi/3$. The reflection axis of $M$ bisects these two translation
vectors. The $p6$ symmetry is generated by $T_{1}$, $T_{2}$ and $C_{6}$.
Namely, $p6$ has no $M$ compared to $p6m$. In panel (b), the hexagon is a
translation unit cell of either $p6m$ or $p6$ lattice symmetry. There are
three types of high symmetry points, labelled by $a$, $b$ and $c$, and they
form the sites of the triangular, honeycomb and kagome lattices, respectively.
The $C_{6}$ rotation center in panel (a) is at a type-$a$ point. Figure 2:
Panel (a) shows the generators of the $p4m$ group, including translations
$T_{1}$ and $T_{2}$, a 4-fold rotation $C_{4}$ and a mirror reflection $M$.
The two translation vectors have the same length, and their angle is $\pi/2$.
The reflection axis of $M$ is parallel to the translation vector of $T_{2}$.
The $p4$ symmetry is generated by $T_{1}$, $T_{2}$ and $C_{4}$. Namely, $p4$
has no $M$ compared with $p4m$. In panel (b), the square is a translation unit
cell of either $p4m$ or $p4$ lattice symmetry. There are three high symmetry
points, labelled by $a$, $b$ and $c$. Both type-$a$ and type-$b$ points form a
square lattice, and type-$c$ points form a checkerboard lattice. The $C_{4}$
rotation center in panel (a) is taken to be at a type-$a$ point.

Given such a symmetry group, different lattice systems can be organized into
the so-called lattice homotpy classes Po  _et al._ (2017); Else and Thorngren
(2020). Two lattice systems are in the same lattice homotopy class if and only
if they can be deformed into each other by these operations: 1) moving the
microscopic degrees of freedom while preserving the lattice symmetry, 2) at
each location, identifying degrees of freedom with the same type of projective
representation under the on-site symmetry, and 3) adding or removing degrees
of freedom with linear representation (i.e., trivial projective
representation) under the on-site symmetry, in a way preserving the lattice
symmetry. Lattice systems within the same class share the same robust
symmetry-related properties, while those in different classes have distinct
symmetry properties and cannot be smoothly connected without breaking the
symmetry. So the robust symmetry-related information of a lattice system is
the lattice homotopy class it belongs to, while colloquially it is the three
pieces of information mentioned in the Introduction.

To make the above discussion less abstract, consider systems with $p6\times
SO(3)$ symmetry. From Fig. 1, there are three types of high symmetry points,
forming a triangular, honeycomb and kagome lattice, respectively. The on-site
symmetry $SO(3)$ has two types of projective representations: half-odd-integer
spins and integer spins. According to the above discussion, although spin-1/2
systems defined on triangular, honeycomb and kagome lattices have the same
symmetry group, they are in different lattice homotopy classes and have
sharply distinct symmetry properties, because they cannot be deformed into
each other via the above operations.

Below we enumerate all lattice hotomopy classes in our symmetry settings. To
this end, we first specify the types of projective representations under the
internal symmetries we consider. As mentioned above, there are two types of
projective representations for the $SO(3)$ symmetry. For time reversal
symmetry $\mathbb{Z}_{2}^{T}$, there are also two types of projective
representations: Kramers singlet and Kramers doublet. For symmetry
$SO(3)\times\mathbb{Z}_{2}^{T}$, there are actually four types of projective
representations: integer spin under $SO(3)$ while Kramers singlet under
$\mathbb{Z}_{2}^{T}$, half-odd-integer spin under $SO(3)$ while Kramers
doublet under $\mathbb{Z}_{2}^{T}$, half-odd-integer spin under $SO(3)$ while
Kramers singlet under $\mathbb{Z}_{2}^{T}$, and integer spin under $SO(3)$
while Kramers doublet under $\mathbb{Z}_{2}^{T}$. The first two types of
projective representations are more common in physical systems and theoretical
models than the last two, so below we will only consider the first two.
Therefore, for all three types of internal symmetries we consider, i.e.,
$SO(3)$, $\mathbb{Z}_{2}^{T}$ and $SO(3)\times\mathbb{Z}_{2}^{T}$, there is a
trivial projective representation and a nontrivial one under consideration.

Then for lattice systems with symmetry group being either $p6\times SO(3)$,
$p6m\times\mathbb{Z}_{2}^{T}$ or $p6m\times SO(3)\times\mathbb{Z}_{2}^{T}$,
there are 4 different lattice homotopy classes Po  _et al._ (2017):

  1. 1.

Class “0”. A representative configuration: A system with degrees of freedom
only carrying the trivial projective representation under the internal
symmetry.

  2. 2.

Class “a”. A representative configuration: A system with degrees of freedom
carrying the nontrivial projective representation under the internal symmetry,
which locate at the triangular lattice sites (type-$a$ high symmetry points in
Fig. 1).

  3. 3.

Class “c”. A representative configuration: A system with degrees of freedom
carrying the nontrivial projective representation under the internal symmetry,
which locate at the kagome lattice sites (type-$c$ high symmetry points in
Fig. 1).

  4. 4.

Class “a+c”. A representative configuration: A system with degrees of freedom
carrying the nontrivial projective representation under the internal symmetry,
which locate at both the triangular and kagome lattice sites (both type-$a$
and type-$c$ high symmetry points in Fig. 1).

Note that a system with degrees of freedom carrying nontrivial projective
representation that locate at the honeycomb lattice sites (type-$b$ high
symmetry points) is in class 0 Po  _et al._ (2017).

For lattice systems with symmetry group being either $p4\times SO(3)$,
$p4m\times\mathbb{Z}_{2}^{T}$ or $p4m\times SO(3)\times\mathbb{Z}_{2}^{T}$,
there are 8 different lattice homotopy classes. Using labels similar to the
above, these are classes “0”, “a”, “b”, “c”, “a+b”, “a+c”, “b+c” and “a+b+c”,
respectively, where each label represents the type of the high symmetry points
at which the degrees of freedom carrying the nontrivial projective
representation locate (“0” means all degrees of freedom carry the regular
representation, i.e., the trivial projective representation). Note that
type-$a$ and type-$c$ high symmetry points are physically distinct once the
$C_{4}$ rotation center is specified, although they may look identical at the
first glance.

To turn the above picture into useful mathematical formulations, Ref. Ye  _et
al._ (2022) shows how to characterize each lattice homotopy class using its
quantum anomaly, i.e., Lieb-Schultz-Mattis anomaly. Different lattice homotopy
classes have different anomalies, and the lattice homotopy class 0 has a
trivial anomaly. In the context of topological orders, the anomalies can be
expressed via the anomaly indicators. For topological quantum spin liquids
with $p6\times SO(3)$ symmetry, the anomaly indicators are given in Appendix
D:

$\displaystyle\begin{aligned}
\mathsf{I}_{1}&=\mathcal{I}_{3}(C_{2}U_{\pi},C_{2}U^{\prime}_{\pi})\,,\\\
\mathsf{I}_{2}&=\mathcal{I}_{3}(T_{1}C_{2}U_{\pi},T_{1}C_{2}U^{\prime}_{\pi})\,.\\\
\end{aligned}$ (7)

where the expression of $\mathcal{I}_{3}$ is given by Eq. (6), $C_{2}$ is a
2-fold rotation symmetry (i.e., $C_{2}\equiv C_{6}^{3}$), while $U_{\pi}$ and
$U^{\prime}_{\pi}$ are $\pi$ spin rotations around two orthogonal axes. We can
think of $\mathsf{I}_{1}$ and $\mathsf{I}_{2}$ as respectively detecting half-
odd-integer spins at type-$a$ and type-$c$ high symmetry points, which are
respectively the 2-fold rotation centers of the $C_{2}$ and $T_{1}C_{2}$
symmetries. More generally, the values of these anomaly indicators for the 4
lattice homotopy classes enumerated above are shown in Table 1.

| 0 | a | c | a+c
---|---|---|---|---
$\mathsf{I}_{1}$ | 1 | $-1$ | 1 | $-1$
$\mathsf{I}_{2}$ | 1 | $1$ | $-1$ | $-1$
Table 1: Values of the anomaly indicators for the 4 lattice homotopy classes
with symmetry group $p6\times SO(3)$.

For topological quantum spin liquids with $p4\times SO(3)$ symmetry, the
anomaly indicators are

$\displaystyle\begin{aligned}
\mathsf{I}_{1}&=\mathcal{I}_{3}(C_{2}U_{\pi},C_{2}U^{\prime}_{\pi})\,,\\\
\mathsf{I}_{2}&=\mathcal{I}_{3}(T_{1}T_{2}C_{2}U_{\pi},T_{1}T_{2}C_{2}U^{\prime}_{\pi})\,,\\\
\mathsf{I}_{3}&=\mathcal{I}_{3}(T_{1}C_{2}U_{\pi},T_{1}C_{2}U^{\prime}_{\pi})\,,\\\
\end{aligned}$ (8)

where $C_{2}$ is still a 2-fold rotational symmetry (but $C_{2}=C_{4}^{2}$ in
this case), while $U_{\pi}$ and $U^{\prime}_{\pi}$ are still $\pi$ spin
rotations around two orthogonal axes. We can think of $\mathsf{I}_{1}$,
$\mathsf{I}_{2}$ and $\mathsf{I}_{3}$ as respectively detecting half-odd-
integer spins at type-$a$, type-$b$ and type-$c$ high symmetry point, which
are respectively the 2-fold rotation centers of the $C_{2}$, $T_{1}T_{2}C_{2}$
and $T_{1}C_{2}$ symmetries. More generally, the values of these anomaly
indicators for the 8 lattice homotopy classes enumerated above are shown in
Table 2.

| 0 | a | b | c | a+b | a+c | b+c | a+b+c
---|---|---|---|---|---|---|---|---
$\mathsf{I}_{1}$ | 1 | $-1$ | 1 | $1$ | $-1$ | $-1$ | 1 | $-1$
$\mathsf{I}_{2}$ | 1 | $1$ | $-1$ | $1$ | $-1$ | 1 | $-1$ | $-1$
$\mathsf{I}_{3}$ | 1 | $1$ | $1$ | $-1$ | 1 | $-1$ | $-1$ | $-1$
Table 2: Values of the anomaly indicators for the 8 lattice homotopy classes
with symmetry group $p4\times SO(3)$.

The anomaly indicators for the other symmetry groups we consider (i.e.,
$p6m\times SO(3)\times\mathbb{Z}_{2}^{T}$, $p4m\times
SO(3)\times\mathbb{Z}_{2}^{T}$, $p6m\times\mathbb{Z}_{2}^{T}$ and
$p4m\times\mathbb{Z}_{2}^{T}$) and their values in each lattice homotopy
classes are more complicated, and they are presented in Appendix D.

We remark that, strictly speaking, Eqs. (7) and (8) are anomaly indicators of
topological quantum spin liquids with purely internal $p6\times SO(3)$ or
$p4\times SO(4)$ symmetry, but the crystalline equivalence principle discussed
in Sec. V still allows us to use them to classify symmetry-enriched
topological quantum spin liquids.

##  V Framework of classification

Now we are ready to present our framework to classify symmetry-enriched
topological quantum spin liquids.

Our framework is based on the hypothesis of emergibility Ye  _et al._ (2022);
Zou  _et al._ (2021). Namely, suppose the anomaly of the lattice system is
$\omega$, then, by tuning the parameters of this lattice system, a quantum
many-body state (or its low-energy effective theory) with anomaly $\Omega$ can
emerge at low energies if and only if the anomaly-matching condition holds:
$\omega=\Omega$.

The “only if” part of this statement is established and well known Hooft
(1980). The “if” part is hypothetical, but there is no known counterexample to
it and it is supported by multiple nontrivial examples Kimchi  _et al._
(2013); Jian and Zaletel (2016); Kim  _et al._ (2016); Latimer and Wang
(2021). So we will assume this hypothesis to be true and use it as our basis
of analysis.

With this hypothesis, the framework to classify symmetry-enriched topological
quantum spin liquids, or equivalently, to obtain the possible symmetry-
enriched topological quantum spin liquids that can emerge in the lattice
system of interest, is as follows.

  1. 1.

Given the symmetry group, which may contain both lattice symmetry and internal
symmetry, we first use the crystalline equivalence principle in Sec. III.4 to
translate it into a purely internal symmetry.

  2. 2.

Based on the above internal symmetry and the topological quantum spin liquid,
we use the method in Ref. Barkeshli  _et al._ (2019a) to obtain the
classification of the internal symmetry enriched topological quantum spin
liquids.

  3. 3.

For each of the internal symmetry enriched topological quantum spin liquids,
we use the method in Ref. Ye and Zou (2023) to obtain its anomaly, $\Omega$.

  4. 4.

As discussed in Sec. IV, the original lattice system has its own quantum
anomaly, $\omega$. We check if the anomaly-matching condition,
$\omega=\Omega$, holds. If it does (doesn’t), then the corresponding symmetry-
enriched topological quantum spin liquid can (cannot) emerge in this lattice
system, according to the hypothesis of emergibility.

If there is only an internal symmetry but no lattice symmetry, then step 1 in
the framework can be ignored. In this case, sometimes one is only interested
in anomaly-free states, then $\omega$ in step 4 should be the trivial anomaly.
For steps 3 and 4, in our context $\Omega$ ($\omega$) can be represented by
the values of the anomaly indicators of the topological quantum spin liquid
(lattice system), so checking whether $\omega=\Omega$ becomes checking whether
the values of these two sets of anomaly indicators match.

We reiterate that the above framework can be straightforwardly generalized to
classify quantum states other than symmetry-enriched topological quantum spin
liquids. For example, it has been used to classify some gapless quantum spin
liquids in Ref. Ye  _et al._ (2022).

In the following sections, we will apply the above framework to obtain the
classification of some representative two dimensional symmetry-enriched
topological quantum spin liquids on various lattice systems.

##  VI $\mathrm{U}(1)_{2N}$ topological orders: Generalized Abelian chiral
spin liquids

Our first class of examples are topological quantum spin liquids with
$U(1)_{2N}$ topological orders. These are Abelian chiral states, and the $N=1$
case is the well-known Kalmeyer-Laughlin state Kalmeyer and Laughlin (1987,
1989). We will classify the $U(1)_{2N}$ topological quantum spin liquid
enriched by $p6\times SO(3)$ or $p4\times SO(4)$ symmetry. As discussed in
Sec. III.4, these symmetries can be viewed as purely internal symmetries
according to the crystalline equivalence principle. Our results are summarized
in Table 7.

The topological properties of the $U(1)_{2N}$ topological order can be
described by either the Laughlin-$(1/{2N})$ wave function or a Chern-Simons
theory with Lagrangian
$\mathcal{L}=-\frac{2N}{4\pi}\epsilon^{\mu\nu\lambda}A_{\mu}\partial_{\nu}A_{\lambda}$,
with $A$ a dynamical $U(1)$ gauge field. These states also allow a description
using parton mean field. Specifically, one can consider $2N$ species of
fermionic partons with an $SU(2N)$ gauge structure. When all species are in a
Chern band with a unit Chern number, the resulting state is the $U(1)_{2N}$
topological order.444The special case with $N=1$ allows another parton mean
field description, described by fermionic partons with a $U(1)$ gauge
structure. When these fermionic partons are in a Chern band with Chern number
2, the resulting state is the $U(1)_{2}$ topological order. The special case
with $N=2$ also allows another parton mean field description, described by
fermionic partons with a $\mathbb{Z}_{2}$ gauge structure. When these
fermionic partons form a $d+id$ superconductor, the resulting state is the
$U(1)_{4}$ topological order Kitaev (2006).

The above descriptions of these topological quantum spin liquids all suffer
from some disadvantages. Concretely, the Laughlin wave function is a single
specific state, and it cannot describe different symmetry-enriched states. To
capture the symmetry actions in the Chern-Simons theory, one needs to invoke
the concept of 2-group symmetries Benini  _et al._ (2019), which are not exact
symmetries of the physical system. Also, in the $SU(2N)$ parton mean field
description, the projective quantum number of the fermionic partons are not
exactly the same as the symmetry fractionalization class of the anyons.

Below we will discuss the topological properties of these states in the
language of Sec. III, which does not suffer from the above disadvantages,
since it can describe general symmetry-enriched $U(1)_{2N}$ topological
quantum spin liquids directly in terms of the symmetry properties of the
anyons.

We label anyons in $\mathrm{U}(1)_{2N}$ by $(a)$, where $a$ is an element in
$\\{0,\dots,2N-1\\}$. These are all Abelian anyons with $d_{(a)}=1$. The
fusion rule is given by addition modulo $2N$, i.e.,

$\displaystyle(a)\otimes(b)=([a+b]_{2N})\,$ (9)

In this paper, we use the notation $[x]_{y}$ to denote $x$ modulo $y$ for any
integer $x$ and positive integer $y$, and $[x]_{y}$ takes values in
$\\{0,\dots,y-1\\}$. The $F$-symbols can be written as

$F^{(a)(b)(c)}=e^{\frac{i\pi}{2N}a(b+c-[b+c]_{2N})},$ (10)

the $R$-symbols are

$R^{(a)(b)}=e^{\frac{i\pi}{2N}ab}$ (11)

which yield the topological spins:

$\displaystyle\theta_{(a)}=e^{\frac{i\pi}{2N}a^{2}}$ (12)

The topological symmetry of $\mathrm{U}(1)_{2N}$ is complicated for general
$N$ Delmastro and Gomis (2021) 555However, as far as we understand, the
statement in Ref. Delmastro and Gomis (2021) about the topological symmetry of
$\mathrm{U}(1)_{2N}$ is not true. The authors there claim that the topological
symmetry is isomorphic to the automorphism group $\text{Aut}(\mathbb{Z}_{2N})$
of $\mathbb{Z}_{2N}$, i.e., it contains all actions labeled by $q$ where
$(q,2N)=1$, such that anyons are transformed according to
$(a)\rightarrow([qa]_{2N})$. We believe that this statement cannot be true:
the topological spins of anyon $(1)$ and anyon $(q)$ are
$\exp(\frac{\pi}{2N}i)$ and $\exp(\frac{q^{2}\pi}{2N}i)$, respectively, and
for generic values of $q$ relatively prime to $2N$ these two topological spins
are not the same. An explicit example is $q=3$ and $N=5$.. For $N=1$, there is
no nontrivial topological symmetry. For $N\geqslant 2$, there is always a
$\mathbb{Z}_{2}$ topological symmetry generated by the charge conjugation
symmetry $C$, such that anyon $(a)\rightarrow([-a]_{2N})$ under $C$. For this
topological symmetry, we can take the $U$-syombols as

$\displaystyle U_{C}((a),(b);([a+b]_{2N}))=\begin{cases}(-1)^{a},&b>0\\\
1,&b=0\end{cases}$ (13)

and a set of $\eta$-symbols all equal to 1. From this set of $\eta$-symbols,
we can obtain all other possible $\eta$-symbols via Eq. (4). When $2\leqslant
N\leqslant 5$, this $\mathbb{Z}_{2}$ is the full topological symmetry group.
When $N\geqslant 6$, there can be other topological symmetries.666For example,
when $N=6$, the action that takes $(a)\rightarrow([5a]_{12})$ is a unitary
$\mathbb{Z}_{2}$ topological symmetry. One can easily check that this map
preserves the fusion and braiding properties of the $U(1)_{12}$ state.
Furthermore, a consistent set of $U$\- and $\eta$-symbols can indeed be
constructed for this symmetry. In the later discussion we consider general
$N\geqslant 2$, but limit to the cases where the microscopic symmetry can
permute anyons only as charge conjugation (i.e., we ignore anyon permutation
patterns other than $C$, if any). To read off the results for $N=1$ from those
for $N\geqslant 2$, we just need to ignore the cases where the microscopic
symmetry permutes anyons.

As mentioned in Sec. III, the $U$\- and $\eta$-symbols contain redundant
“gauge” freedom, and only their gauge invariant combinations detect different
symmetry fractionalization classes. Below we construct three types of such
invariants for the $\mathrm{U}(1)_{2N}$ topological order.

For the first type, consider two group elements ${\bf g}$ and ${\bf h}$ which
commute with each other and do not permute anyons, we can define the following
topological invariant,

$\displaystyle\lambda_{1}({\bf g},{\bf h})=\frac{\eta_{(1)}({\bf g},{\bf
h})}{\eta_{(1)}({\bf h},{\bf g})}\,,$ (14)

which can be thought of as the phase we get after acting on anyon $(1)$ by
$\bf{gh}\bf{g}^{-1}\bf{h}^{-1}$. For the second type, consider a group element
${\bf g}$ such that ${\bf g}^{k}={\bf 1}$ and does not permute anyons, we can
define the following topological invariant,

$\displaystyle\lambda_{2}({\bf g})=\left(\prod_{i=1}^{k-1}\eta_{(1)}({\bf
g}^{i},{\bf g})\right)^{\frac{2N}{(2N,k)}}\left(-\prod_{a=1}^{2N-1}U_{{\bf
g}}((1),(a);([1+a]_{2N}))\right)^{\frac{k}{(2N,k)}}\,,$ (15)

where $(2N,k)$ denotes the greatest common divisor of $2N$ and $k$. Finally,
for the third type, consider a group element ${\bf g}$ such that ${\bf
g}^{2k}={\bf 1}$ and acts on anyons by charge conjugation, we can define the
following topological invariant,

$\displaystyle\lambda_{3}({\bf g})=\left(\prod_{i=1}^{2k-1}\eta_{(1)}({\bf
g}^{i},{\bf g})\right)\left(U_{{\bf
g}}((1),(2N-1);(0))R^{(2N-1),(1)}\theta_{(1)}\right)^{k}\,$ (16)

It turns out that, for our purpose, $\lambda_{1}$, $\lambda_{2}$ and
$\lambda_{3}$ are enough to specify all symmetry fractionalization classes for
$\mathrm{U}(1)_{2N}$.

###  VI.1 Example: $\mathbb{Z}_{2}\times SO(3)$

To illustrate our calculation of the anomaly of the $U(1)_{2N}$ topological
order with $p6\times SO(3)$ or $p4\times SO(3)$ symmetry, let us first discuss
the example where the symmetry is $\mathbb{Z}_{2}\times SO(3)$ in detail. It
turns out that the calculation of the anomaly when the symmetry is $p6\times
SO(3)$ or $p4\times SO(3)$ can be reduced to this example, by restricting $p6$
or $p4$ to its various $\mathbb{Z}_{2}$ subgroups.

The anomalies associated with the $\mathbb{Z}_{2}\times SO(3)$ symmetry are
classified by

$\displaystyle H^{4}(\mathbb{Z}_{2}\times
SO(3),\mathrm{U}(1))\cong\mathbb{Z}_{2}\,.$ (17)

Hence there is only one type of nontrivial anomaly, which can be detected by
the anomaly indicator
$\mathsf{I}=\mathcal{I}_{3}(C_{2}U_{\pi},C_{2}U^{\prime}_{\pi})$, where
$C_{2}$ is the generator of $\mathbb{Z}_{2}$, while $U_{\pi}$ and
$U^{\prime}_{\pi}$ are elements of $SO(3)$, representing the $\pi$-rotations
about two orthogonal axes.

The $SO(3)$ symmetry cannot permute anyons because all elements of $SO(3)$ are
continuously connected to the identity element. Hence, only the generator of
$\mathbb{Z}_{2}$, denoted by $C_{2}$ here, can permute anyons by charge
conjugation, and there are two possibilities.

####  VI.1.1 No anyon permutation

The first possibility is that the action of $C_{2}$ is trivial and there is no
anyon permutation. For the case with $N=1$, this is the only possibility to be
considered. Then the symmetry fractionalization classes are classified by

$\displaystyle\begin{aligned} &H^{2}_{(1)}(\mathbb{Z}_{2}\times
SO(3),\mathbb{Z}_{2N})\\\ =&H^{2}(\mathbb{Z}_{2},\mathbb{Z}_{2N})\oplus
H^{2}(SO(3),\mathbb{Z}_{2N})\\\ =&(\mathbb{Z}_{2})^{2}\,.\end{aligned}$ (18)

Namely, there are 2 generators that generate 4 different symmetry
fractionalization classes. To understand these symmetry fractionalization
classes, we can directly write down representative cochains of them. A
representative cochain of the first generator, which we denote by
$\widetilde{\beta}(x)$ and comes from $H^{2}(\mathbb{Z}_{2},\mathbb{Z}_{2N})$,
is:

$\displaystyle\widetilde{\beta}(x)(C_{2}^{i},C_{2}^{j})=\frac{i+j-[i+j]_{2}}{2}=ij\mod{2N}\,,$
(19)

with $i,j\in\\{0,1\\}$. The reason for the name of this generator is explained
in Appendix C. Physically, this generator detects whether the anyon $(1)$
carries a fractional charge under the $\mathbb{Z}_{2}$ symmetry.

The second generator, which comes from $H^{2}(SO(3),\mathbb{Z}_{2N})$, detects
whether the anyon $(1)$ carries a half-odd-integer spin under the $SO(3)$
symmetry, and we denote it by $Nw_{2}$, for reasons explained in Appendix C.
To have a representative cochain of $Nw_{2}$, it is convenient to consider a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ subgroup of $SO(3)$ generated by
$U_{\pi}$ and $U^{\prime}_{\pi}$, and an element in this subgroup can be
written as $U_{\pi}^{i}{U^{\prime}}_{\pi}^{j}$, with $i,j\in\\{0,1\\}$. Then,
restricting $SO(3)$ to this $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ subgroup, the
representative cochain of $Nw_{2}$ is

$\displaystyle\begin{aligned}
&\left(Nw_{2}\right)(U_{\pi}^{i_{1}}{U^{\prime}}_{\pi}^{i_{2}},U_{\pi}^{j_{1}}{U^{\prime}}_{\pi}^{j_{2}})\\\
=&N(i_{1}j_{1}+i_{2}j_{2}+i_{1}j_{2})\mod{2N}\,.\end{aligned}$ (20)

So the symmetry fractionalization classes can be written as

$\displaystyle w=n_{1}\cdot\widetilde{\beta}(x)+n_{2}\cdot Nw_{2}\,,$ (21)

and labeled as $\\{n_{1},n_{2}\\}$ with $n_{1,2}\in\\{0,1\\}$. When the
$SO(3)$ symmetry is restricted to its $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
subgroup generated by $U_{\pi}$ and $U^{\prime}_{\pi}$, a representative
cochain can be taken as

$\displaystyle\begin{aligned}
&w(C_{2}^{i_{1}}U_{\pi}^{i_{2}}{U^{\prime}}_{\pi}^{i_{3}},C_{2}^{j_{1}}U_{\pi}^{j_{2}}{U^{\prime}}_{\pi}^{j_{3}})\\\
=&n_{1}i_{1}j_{1}+n_{2}N(i_{2}j_{2}+i_{3}j_{3}+i_{2}j_{3})\mod{2N}\end{aligned}$
(22)

Combining the above equation and Eq. (4), we get

$\displaystyle\begin{aligned}
&\eta_{(a)}(C_{2}U_{\pi},C_{2}U^{\prime}_{\pi})=\eta_{(a)}(C_{2}U_{\pi},C_{2}U_{\pi})\\\
&=\exp\left(\frac{i\pi}{N}a(n_{1}+Nn_{2})\right),\\\
&\eta_{(a)}(C_{2}U^{\prime}_{\pi},C_{2}U_{\pi})=\exp\left(\frac{i\pi}{N}an_{1}\right)\end{aligned}$
(23)

Now we plug Eqs. (10), (11), (12) and (23) into Eq. (6) (the $U$-symbols
therein can all be taken as 1 since there is no anyon permutation), and get
the anomaly indicator of the state in symmetry fractionalization class
$\\{n_{1},n_{2}\\}$:

$\displaystyle\mathcal{I}_{3}(C_{2}U_{\pi},C_{2}U^{\prime}_{\pi})=(-1)^{n_{1}n_{2}}\,.$
(24)

####  VI.1.2 $C_{2}$ acts as charge conjugation

The second possibility is that $C_{2}$ acts by charge conjugation. This
possibility occurs only if $N\geqslant 2$. Then the symmetry fractionalization
classes are classified by

$\displaystyle H^{2}_{(2)}(\mathbb{Z}_{2}\times
SO(3),\mathbb{Z}_{2N})=(\mathbb{Z}_{2})^{2}\,.$ (25)

There are also 2 generators that generate 4 different symmetry
fractionalization classes, but these symmetry fractionalization classes are
different from those in Sec. VI.1.1. Explicitly, a representative cochain of
the first generator, which we denote by $Nx^{2}$, is:

$\displaystyle(Nx^{2})(C_{2}^{i},C_{2}^{j})=Nij\mod{2N}\,,$ (26)

with $i,j\in\\{0,1\\}$. The second generator also detects whether the anyon
$(1)$ carries a half-odd-integer spin under the $SO(3)$ symmetry, and we also
denote it by $Nw_{2}$. The representative cochain restricted to the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ subgroup generated by $U_{\pi}$ and
$U^{\prime}_{\pi}$ is still given by Eq. (20).

So the symmetry fractionalization classes can be written as

$\displaystyle w=n_{1}\cdot Nx^{2}+n_{2}\cdot Nw_{2}\,,$ (27)

and also labeled as $\\{n_{1},n_{2}\\}$ with $n_{1,2}\in\\{0,1\\}$. When the
$SO(3)$ symmetry is restricted to its $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
subgroup generated by $U_{\pi}$ and $U^{\prime}_{\pi}$, a representative
cochain can be taken as

$\displaystyle\begin{aligned}
&w(C_{2}^{i_{1}}U_{\pi}^{i_{2}}{U^{\prime}}_{\pi}^{i_{3}},C_{2}^{j_{1}}U_{\pi}^{j_{2}}{U^{\prime}}_{\pi}^{j_{3}})\\\
=&n_{1}Ni_{1}j_{1}+n_{2}N(i_{2}j_{2}+i_{3}j_{3}+i_{2}j_{3})\mod{2N}\end{aligned}$
(28)

Combining the above equation and Eq. (4), we get

$\displaystyle\begin{aligned}
&\eta_{(a)}(C_{2}U_{\pi},C_{2}U^{\prime}_{\pi})=\eta_{(a)}(C_{2}U_{\pi},C_{2}U_{\pi})\\\
&=\exp\left(i\pi a(n_{1}+n_{2})\right),\\\
&\eta_{(a)}(C_{2}U^{\prime}_{\pi},C_{2}U_{\pi})=\exp\left(i\pi
an_{1}\right)\end{aligned}$ (29)

Now we plug Eqs. (10), (11), (12), (13) and (29) into Eq. (6), and get the
anomaly indicator of the state in symmetry fractionalization class
$\\{n_{1},n_{2}\\}$:

$\displaystyle\mathcal{I}_{3}(C_{2}U_{\pi},C_{2}U^{\prime}_{\pi})=(-1)^{(n_{1}+1)n_{2}N}\,.$
(30)

Hence when $N$ is even the anomaly is always absent, and when $N$ is odd,
$n_{1}=0,n_{2}=1$ gives nonzero anomaly and otherwise the anomaly is absent.

With this warm-up, we are ready to classify $U(1)_{2N}$ topological quantum
spin liquids enriched by $p6\times SO(3)$ or $p4\times SO(3)$ symmetry using
the framework in Sec. V. The results are summarized in Table 7.

###  VI.2 $p6\times SO(3)$

The generator $T_{1,2}$ and $SO(3)$ cannot permute anyons777The generator
$T_{1,2}$ cannot permute anyons because $C_{6}T_{2}C_{6}^{-1}=T_{1}^{-1}$,
implying that $T_{1}$ and $T_{2}$ should permute anyons in opposite ways,
which, combined with $C_{6}T_{1}C_{6}^{-1}=T_{1}T_{2}$, implies that neither
$T_{1}$ nor $T_{2}$ can permute anyons., and only the generator $C_{6}$ can
permute anyons by charge conjugation. Hence, there are two possibilities
regarding how $p6\times SO(3)$ can permute anyons:

  1. 1.

Trivial $C_{6}$ action: no anyon permutation.

In this case, the possible symmetry fractionalization classes are classified
by

$\displaystyle H^{2}_{(1)}(p6\times
SO(3),\mathbb{Z}_{2N})=\mathbb{Z}_{2N}\oplus\mathbb{Z}_{(2N,6)}\oplus\mathbb{Z}_{2}\,,$
(31)

whose elements can be written as

$\displaystyle
w=n_{1}\cdot\widetilde{\mathscr{B}}^{(1)}_{xy}+n_{2}\cdot\widetilde{\mathscr{B}}^{(1)}_{c^{2}}+n_{3}\cdot
Nw_{2}\,,$ (32)

and labeled by $\\{n_{1},n_{2},n_{3}\\}$, with $n_{1}\in\\{0,\dots,2N-1\\}$,
$n_{2}\in\\{0,\dots,(2N,6)-1\\}$, $n_{3}\in\\{0,1\\}$. Here
$\widetilde{\mathscr{B}}^{(1)}_{xy}$, $\widetilde{\mathscr{B}}^{(1)}_{c^{2}}$
and $Nw_{2}$ are generators of $\mathbb{Z}_{2N}$, $\mathbb{Z}_{(2N,6)}$ and
$\mathbb{Z}_{2}$, respectively (the representative cochains and the reason for
the names of these generators are given in Appendix C). We can identify these
generators in terms of the following three topological invariants defined in
Eqs. (14) and (15): $\lambda_{1}(T_{1},T_{2})$, $\lambda_{2}(C_{6})$ and
$\lambda_{1}(U_{\pi},U^{\prime}_{\pi})$, and the values of these topological
invariants for the generators are listed in Table 3. Physically, we can think
of $\widetilde{\mathscr{B}}^{(1)}_{xy}$,
$\widetilde{\mathscr{B}}^{(1)}_{c^{2}}$ and $Nw_{2}$ as detecting whether the
anyon $(1)$ carries projective representation under translation symmetries,
$C_{6}$ and $SO(3)$, respectively.888In fact, the generator $A_{c}^{2}$
actually detects whether the anyon $(1)$ simutaneously carries projective
representation under four different symmetries, generated by $C_{6}$,
$T_{1}C_{6}^{3}$, $T_{2}C_{6}^{3}$ and $T_{1}T_{2}C_{6}^{3}$, respectively.
For each symmetry fractionalization class, the $U$\- and $\eta$-symbols can be
obtained via Eqs. (4), (5) and (13).

SF class | $\lambda_{1}(T_{1},T_{2})$ | $\lambda_{2}(C_{6})$ | $\lambda_{1}(U_{\pi},U^{\prime}_{\pi})$
---|---|---|---
$\widetilde{\mathscr{B}}^{(1)}_{xy}$ | $e^{\frac{2\pi i}{2N}}$ | 1 | 1
$\widetilde{\mathscr{B}}^{(1)}_{c^{2}}$ | 1 | $e^{\frac{2\pi i}{(2N,6)}}$ | 1
$Nw_{2}$ | 1 | 1 | $-1$
Table 3: Values of the topological invariants, given $p6\times SO(3)$ symmetry
with trivial action.

Without considering anomaly matching, $p6\times SO(3)$ symmetric $U(1)_{2N}$
topological quantum spin liquids are classified by $\\{n_{1},n_{2},n_{3}\\}$,
if no symmetry permutes anyons. Recall that two symmetry fractionalization
classes related to each other by relabeling anyons are physically identical,
so $\\{n_{1},n_{2},n_{3}\\}$ and $\\{[-n_{1}]_{2N},[-n_{2}]_{(2N,6)},n_{3}\\}$
are identified.

As argued in the Introduction and Sec. IV, in systems with lattice symmetry it
is important to consider anomaly matching for the classification of symmetry-
enriched topological quantum spin liquids. The values of the anomaly
indicators for different lattice homotopy classes with $p6\times SO(3)$
symmetry are given in Table 1. The anomaly indicators for the $U(1)_{2N}$
state in the symmetry fractionalization class $\\{n_{1},n_{2},n_{3}\\}$ can be
calculated in a way similar to Sec. VI.1, which yields 999Another way to
calculate these anomaly indicators is as follows. We can restrict the
$p6\times SO(3)$ symmetry into two of its $\mathbb{Z}_{2}\times SO(3)$
subgroups, where the $\mathbb{Z}_{2}$ in the first subgroup is generated by
$C_{2}\equiv C_{6}^{3}$, and in the second it is generated by $T_{1}C_{2}$. By
comparing the representative cochains of $B_{xy}$, $A_{c}^{2}$ and $Nw_{2}$ in
Appendix C with Eq. (22), we see that, after the restriction to the first
$\mathbb{Z}_{2}\times SO(3)$ subgroup, the $p6\times SO(3)$ symmetry
fractionalization class labeled by $\\{n_{1},n_{2},n_{3}\\}$ becomes the class
labeled by $\\{n_{2},n_{3}\\}$ in Sec. VI.1.1, while after the restriction to
the second $\mathbb{Z}_{2}\times SO(3)$ subgroup, the class labeled by
$\\{n_{1},n_{2},n_{3}\\}$ becomes the class labeled by
$\\{n_{1}+n_{2},n_{3}\\}$ in Sec. VI.1.1. According to Eq. (24), the anomaly
indicators in Eq. (7) for $\mathrm{U}(1)_{2N}$ with the symmetry
fractionalization class labeled by $\\{n_{1},n_{2},n_{3}\\}$ is Eq. (33).

$\displaystyle\mathsf{I}_{1}=(-1)^{n_{2}n_{3}}\,,\quad\mathsf{I}_{2}=(-1)^{(n_{1}+n_{2})n_{3}}\,.$
(33)

Therefore, by matching these anomaly indicators with Table 1, we arrive at the
classification in Table 4.

Lattice homotopy class |  Symmetry fractionalization class
---|---
0 |  $\\{n_{1},n_{2},0\\}$, or $\\{n_{1},n_{2},1\\}$ with $n_{1,2}$ even
a |  $\\{n_{1},n_{2},1\\}$ with $n_{1,2}$ odd
c |  $\\{n_{1},n_{2},1\\}$ with $n_{1}$ odd, $n_{2}$ even
a+c |  $\\{n_{1},n_{2},1\\}$ with $n_{1}$ even, $n_{2}$ odd
Table 4: Classification of symmetry-enriched $U(1)_{2N}$ topological quantum
spin liquids in lattice systems with $p6\times SO(3)$ symmetry, if no symmetry
permutes anyons.

  2. 2.

Nontrivial $C_{6}$ action: $C_{6}$ as charge conjugation.

In this case, the possible symmetry fractionalization classes are given by

$\displaystyle H^{2}_{(2)}(p6\times
SO(3),\mathbb{Z}_{2N})=(\mathbb{Z}_{2})^{3}$ (34)

whose elements can be written as

$\displaystyle w=n_{1}\cdot NB_{xy}+n_{2}\cdot NA_{c}^{2}+n_{3}\cdot
Nw_{2}\,,$ (35)

and labeled by $\\{n_{1},n_{2},n_{3}\\}$ with $n_{1,2,3}\in\\{0,1\\}$. Here
$NB_{xy}$, $NA_{c}^{2}$ and $Nw_{2}$ are generators of the three
$\mathbb{Z}_{2}$ pieces respectively (the representative cochains and the
reason for the names of these generators are given in Appendix C). We can
identify these generators in terms of the following three topological
invariants defined in Eqs. (14) and (16): $\lambda_{3}(C_{2})$,
$\lambda_{3}(T_{1}C_{2})$ and $\lambda_{1}(U_{\pi},U^{\prime}_{\pi})$, and the
values of these topological invariants for the generators are listed in Table
5. For each symmetry fractionalization class, the $U$\- and $\eta$-symbols can
be obtained via Eqs. (4), (5) and (13).

SF class | $\lambda_{3}(C_{2})$ | $\lambda_{3}(T_{1}C_{2})$ | $\lambda_{1}(U_{\pi},U^{\prime}_{\pi})$
---|---|---|---
$NB_{xy}$ | 1 | $-1$ | 1
$NA_{c}^{2}$ | $-1$ | $-1$ | 1
$Nw_{2}$ | 1 | 1 | $-1$
Table 5: Values of the topological invariants, given $p6\times SO(3)$ symmetry
with nontrivial $C_{6}$ action.

Similar to the previous case, without considering anomaly matching, $p6\times
SO(3)$ symmetric $U(1)_{2N}$ topological quantum spin liquids are classified
by the above $\\{n_{1},n_{2},n_{3}\\}$ if $C_{6}$ acts as charge conjugation.
Calculating the anomaly indicators for $U(1)_{2N}$ with symmetry
fractionalization class labeled by $\\{n_{1},n_{2},n_{3}\\}$ as before, we get

$\displaystyle\mathsf{I}_{1}=(-1)^{(n_{2}+1)n_{3}N}\,,\quad\mathsf{I}_{2}=(-1)^{(n_{1}+n_{2}+1)n_{3}N}\,.$
(36)

Therefore, by matching these anomaly indicators with Table 1, we arrive at the
classification in Table 6.

Lattice homotopy class |  Symmetry fractionalization class
---|---
0 |  |  $\\{n_{1},n_{2},n_{3}\\}$ for $N$ even,
---
$\\{n_{1},n_{2},0\\}$ or $\\{0,1,1\\}$ for $N$ odd
a |  $\\{1,0,1\\}$ for $N$ odd
c |  $\\{1,1,1\\}$ for $N$ odd
a+c |  $\\{0,0,1\\}$ for $N$ odd
Table 6: Classification of symmetry-enriched $U(1)_{2N}$ topological quantum
spin liquids in lattice systems with $p6\times SO(3)$ symmetry, if $C_{6}$
acts as charge conjugation.

Summarizing all cases, the total number of different $p6\times SO(3)$
symmetry-enriched $U(1)_{2N}$ topological quantum spin liquids is summarized
in Table 7. Note that this classification is complete for $N\leqslant 5$, but
incomplete for $N\geqslant 6$, because we have assumed that the only way the
symmetry can permute anyons is via charge conjugation, while for $N\geqslant
6$ a symmetry can in principle permute anyons in other manners.

Symmetry group | Lattice homotopy class | $N=1$ | odd $N>1$ | even $N$
---|---|---|---|---
$p6\times SO(3)$ | 0 | 5 | $\left(\frac{5(N(N,3)+1)}{2}\right)+(5)$ | $\left(\frac{5N(N,3)}{2}+3\right)+(8)$
a | 1 | $\left(\frac{N(N,3)+1}{2}\right)+(1)$ | $\left(\frac{N(N,3)}{2}\right)+(0)$
c | 1 | $\left(\frac{N(N,3)+1}{2}\right)+(1)$ | $\left(\frac{N(N,3)}{2}\right)+(0)$
a+c | 1 | $\left(\frac{N(N,3)+1}{2}\right)+(1)$ | $\left(\frac{N(N,3)}{2}+1\right)+(0)$
$p4\times SO(3)$ | 0 | 9 | $\left(\frac{9(N+1)}{2}\right)+(9)+(9)+(9)$ | $(9N+6)+(12)+(20)+(20)$
a | 1 | $\left(\frac{N+1}{2}\right)+(1)+(1)+(1)$ | $(N)+(0)+(4)+(0)$
b | 1 | $\left(\frac{N+1}{2}\right)+(1)+(1)+(1)$ | $(N)+(0)+(0)+(4)$
c | 1 | $\left(\frac{N+1}{2}\right)+(1)+(1)+(1)$ | $(N+2)+(4)+(0)+(0)$
a+b | 1 | $\left(\frac{N+1}{2}\right)+(1)+(1)+(1)$ | $(N)+(0)+(0)+(0)$
a+c | 1 | $\left(\frac{N+1}{2}\right)+(1)+(1)+(1)$ | $(N)+(0)+(0)+(0)$
b+c | 1 | $\left(\frac{N+1}{2}\right)+(1)+(1)+(1)$ | $(N)+(0)+(0)+(0)$
a+b+c | 1 | $\left(\frac{N+1}{2}\right)+(1)+(1)+(1)$ | $(N)+(0)+(0)+(0)$
Table 7: Number of $p6\times SO(3)$ and $p4\times SO(3)$ symmetry-enriched
$U(1)_{2N}$ topological quantum spin liquids. For the case with a $p6\times
SO(3)$ symmetry, each of the last two columns is written as a sum of two
terms, representing the number of states where no anyon is permuted by
symmetries and where $C_{6}$ acts as charge conjugation, respectively. For the
case with $p4\times SO(3)$ symmetry, each of the last two columns is written
as a sum of four terms, representing the number of states where no anyon is
permuted by symmetries, where $C_{4}$ acts as charge conjugation while
$T_{1,2}$ do not, where $T_{1,2}$ act as charge conjugation while $C_{4}$ does
not, and where $C_{4}$ and $T_{1,2}$ all act as charge conjugation,
respectively. The details of the symmetry fractionalization class of each
state can be found in Tables 4, 6, 9, 11, 13 and 15.

###  VI.3 $p4\times SO(3)$

$SO(3)$ cannot permute anyons, and both the generators $T_{1,2}$ and the
generator $C_{4}$ can permute anyons by charge conjugation. Hence, there are
four possibilities regarding how $p4\times SO(3)$ can permute anyons:

  1. 1.

Trivial $T_{1,2}$ and $C_{4}$ action.

In this case, the possible symmetry fractionalization classes are given by

$\displaystyle H^{2}_{(1)}(p4\times
SO(3),\mathbb{Z}_{2N})=\mathbb{Z}_{2N}\oplus\mathbb{Z}_{(2N,4)}\oplus(\mathbb{Z}_{2})^{2}\,.$
(37)

whose elements can be labeled as

$\displaystyle\begin{aligned}
&w=n_{1}\cdot\widetilde{\mathscr{B}}^{(1)}_{xy}+n_{2}\cdot\widetilde{\mathscr{B}}^{(1)}_{c^{2}}\\\
&+n_{3}\cdot\widetilde{\beta}(A_{x+y})+n_{4}\cdot Nw_{2}\,,\end{aligned}$ (38)

with $n_{1}\in\\{0,\dots,2N-1\\}$, $n_{2}\in\\{0,\dots,(2N,4)-1\\}$,
$n_{3,4}\in\\{0,1\\}$. Here $\widetilde{\mathscr{B}}^{(1)}_{xy}$,
$\widetilde{\mathscr{B}}^{(1)}_{c^{2}}$, $\widetilde{\beta}(A_{x+y})$ and
$Nw_{2}$ are generators of $\mathbb{Z}_{2N}$, $\mathbb{Z}_{(2N,4)}$ and two
$\mathbb{Z}_{2}$ pieces, respectively (the representative cochains and the
reason for the names of these generators are given in Appendix C). We can
identify them in terms of the following four topological invariants defined in
Eqs. (14) and (15): $\lambda_{1}(T_{1},T_{2})$, $\lambda_{2}(C_{4})$,
$\lambda_{2}(T_{1}C_{2})$ and $\lambda_{1}(U_{\pi},U^{\prime}_{\pi})$, and we
list the value of the topological invariants for the generators of symmetry
fractionalization classes in Table 8. For each symmetry fractionalization
class, the $U$\- and $\eta$-symbols can be obtained via Eqs. (4), (5) and
(13).

SF class | $\lambda_{1}(T_{1},T_{2})$ | $\lambda_{2}(C_{4})$ | $\lambda_{2}(T_{1}C_{2})$ | $\lambda_{1}(U_{\pi},U^{\prime}_{\pi})$
---|---|---|---|---
$\widetilde{\mathscr{B}}^{(1)}_{xy}$ | $e^{\frac{2\pi i}{2N}}$ | 1 | 1 | 1
$\widetilde{\mathscr{B}}^{(1)}_{c^{2}}$ | 1 | $e^{\frac{2\pi i}{(2N,4)}}$ | $-1$ | 1
$\widetilde{\beta}(A_{x+y})$ | 1 | 1 | $-1$ | 1
$Nw_{2}$ | 1 | 1 | 1 | $-1$
Table 8: Values of the topological invariants, given $p4\times SO(3)$ symmetry
with trivial action.

Again, because symmetry fractionalization classes related by relabeling anyons
are physically identical, different symmetry realizations on $U(1)_{2N}$ in
this case are specified by $\\{n_{1},n_{2},n_{3},n_{4}\\}$, where
$\\{n_{1},n_{2},n_{3},n_{4}\\}$ is identified with
$\\{[-n_{1}]_{2N},[-n_{2}]_{(2N,4)},n_{3},n_{4}\\}$. Calculating the anomaly
indicators for $U(1)_{2N}$ state with symmetry fractionalization class labeled
by $\\{n_{1},n_{2},n_{3},n_{4}\\}$, we get

$\displaystyle\begin{aligned}
&\mathsf{I}_{1}=(-1)^{n_{2}n_{4}}\,,\quad\mathsf{I}_{2}=(-1)^{(n_{1}+n_{2})n_{4}}\,,\\\
&\mathsf{I}_{3}=(-1)^{(n_{2}+n_{3})n_{4}}\,.\\\ \end{aligned}$ (39)

Therefore, by matching these anomaly indicators with Table 2, we arrive at the
classification in Table 9.

Lattice homotopy class |  Symmetry fractionalization class
---|---
0 |  $\\{n_{1},n_{2},n_{3},0\\}$, or $\\{n_{1},n_{2},0,1\\}$ with $n_{1,2}$ even
a |  $\\{n_{1},n_{2},1,1\\}$ with $n_{1,2}$ odd
b |  $\\{n_{1},n_{2},0,1\\}$ with $n_{1}$ odd, $n_{2}$ even
c |  $\\{n_{1},n_{2},1,1\\}$ with $n_{1,2}$ even
a+b |  $\\{n_{1},n_{2},1,1\\}$ with $n_{1}$ even, $n_{2}$ odd
a+c |  $\\{n_{1},n_{2},0,1\\}$ with $n_{1,2}$ odd
b+c |  $\\{n_{1},n_{2},1,1\\}$ with $n_{1}$ odd, $n_{2}$ even
a+b+c |  $\\{n_{1},n_{2},0,1\\}$ with $n_{1}$ even, $n_{2}$ odd
Table 9: Classification of symmetry-enriched $U(1)_{2N}$ topological quantum
spin liquids in lattice systems with $p4\times SO(3)$ symmetry, if no symmetry
permutes anyons.

  2. 2.

Nontrivial $C_{4}$ action, trivial $T_{1,2}$ action.

In this case, the possible symmetry fractionalization classes are given by

$\displaystyle H^{2}_{(2)}(p4\times
SO(3),\mathbb{Z}_{2N})=(\mathbb{Z}_{2})^{4}\,.$ (40)

We can write these elements as

$\displaystyle\begin{aligned} &w=n_{1}\cdot NB_{xy}+n_{2}\cdot NB_{c^{2}}\\\
&+n_{3}\cdot\widetilde{\beta}(A_{x+y})+n_{4}\cdot Nw_{2}\,,\end{aligned}$ (41)

with $n_{1,2,3,4}\in\\{0,1\\}$. Here $NB_{xy}$, $NB_{c^{2}}$,
$\widetilde{\beta}(A_{x+y})$ and $Nw_{2}$ are generators of the four
$\mathbb{Z}_{2}$ pieces respectively (the representative cochains and the
reason for the names of these generators are given in Appendix C). We can
identify them in terms of the following four topological invariants defined in
Eqs. (14), (15) and (16): $\lambda_{1}(T_{1},T_{2})$, $\lambda_{3}(C_{4})$,
$\lambda_{2}(T_{1}C_{2})$ and $\lambda_{1}(U_{\pi},U^{\prime}_{\pi})$, and we
list the value of the topological invariants for the generators of symmetry
fractionalization classes in Table 10. For each symmetry fractionalization
class, the $U$\- and $\eta$-symbols can be obtained via Eqs. (4), (5) and
(13).

SF class | $\lambda_{1}(T_{1},T_{2})$ | $\lambda_{3}(C_{4})$ | $\lambda_{2}(T_{1}C_{2})$ | $\lambda_{1}(U_{\pi},U^{\prime}_{\pi})$
---|---|---|---|---
$NB_{xy}$ | $-1$ | 1 | 1 | 1
$NB_{c^{2}}$ | 1 | $-1$ | $(-1)^{N}$ | 1
$\widetilde{\beta}(A_{x+y})$ | 1 | 1 | $-1$ | 1
$Nw_{2}$ | 1 | 1 | 1 | -1
Table 10: Values of the topological invariants, given $p4\times SO(3)$
symmetry with nontrivial $C_{4}$ action.

Calculating the anomaly indicators for $U(1)_{2N}$ with symmetry
fractionalization class labeled by $(n_{1},n_{2},n_{3},n_{4})$ as before, we
get

$\displaystyle\begin{aligned}
&\mathsf{I}_{1}=(-1)^{n_{2}n_{4}N}\,,\quad\mathsf{I}_{2}=(-1)^{(n_{1}+n_{2})n_{4}N}\,,\\\
&\mathsf{I}_{3}=(-1)^{(n_{2}N+n_{3})n_{4}}\,.\end{aligned}$ (42)

Therefore, by matching these anomaly indicators with Table 2, we arrive at the
classification in Table 11.

Lattice homotopy class |  Symmetry fractionalization class
---|---
0 |  |  $\\{n_{1},n_{2},n_{3},0\\}$ or $\\{n_{1},n_{2},0,1\\}$ for $N$ even,
---
$\\{n_{1},n_{2},n_{3},0\\}$ or $\\{0,0,0,1\\}$ for $N$ odd
a |  $\\{1,1,1,1\\}$ for $N$ odd
b |  $\\{1,0,0,1\\}$ for $N$ odd
c |  |  $\\{n_{1},n_{2},1,1\\}$ for $N$ even,
---
$\\{0,0,1,1\\}$ for $N$ odd
a+b |  $\\{0,1,1,1\\}$ for $N$ odd
a+c |  $\\{1,1,0,1\\}$ for $N$ odd
b+c |  $\\{1,0,1,1\\}$ for $N$ odd
a+b+c |  $\\{0,1,0,1\\}$ for $N$ odd
Table 11: Classification of symmetry-enriched $U(1)_{2N}$ topological quantum
spin liquids in lattice systems with $p4\times SO(3)$ symmetry, if $C_{4}$
acts as charge conjugation.

  3. 3.

Nontrivial $T_{1,2}$ action, trivial $C_{4}$ action.

In this case, the possible symmetry fractionalization classes are given by

$\displaystyle H^{2}_{(3)}(p4\times
SO(3),\mathbb{Z}_{2N})=\mathbb{Z}_{(2N,4)}\oplus(\mathbb{Z}_{2})^{3}\,.$ (43)

We can write these elements as

$\displaystyle\begin{aligned}
w&=n_{1}\cdot\widetilde{\mathscr{B}}^{(3)}_{xy}+n_{2}\cdot NB_{c^{2}}\\\
&+n_{3}\cdot NA_{x+y}^{2}+n_{4}\cdot Nw_{2}\,,\end{aligned}$ (44)

with $n_{1}\in\\{0,\dots,(2N,4)-1\\}$, $n_{2,3,4}\in\\{0,1\\}$. Here
$\widetilde{\mathscr{B}}^{(3)}_{xy}$, $NB_{c^{2}}$, $NA_{x+y}^{2}$ and
$Nw_{2}$ are generators of the $\mathbb{Z}_{(2N,4)}$ and the three
$\mathbb{Z}_{2}$ pieces, respectively (the representative cochains and the
reason for the names of these generators are given in Appendix C)We can
identify them in terms of the following four topological invariants defined in
Eqs. (14), (15) and (16): $\lambda_{2}(C_{4})$,
$\lambda_{2}(T_{1}T_{2}C_{2})$, $\lambda_{3}(T_{1}C_{2})$ and
$\lambda_{1}(U_{\pi},U^{\prime}_{\pi})$, and we list the value of the
topological invariants for the generators of symmetry fractionalization
classes in Table 12. For each symmetry fractionalization class, the $U$\- and
$\eta$-symbols can be obtained via Eqs. (4), (5) and (13).

SF class | $\lambda_{2}(C_{4})$ | $\lambda_{3}(T_{1}C_{4})$ | $\lambda_{3}(T_{1}C_{2})$ | $\lambda_{1}(U_{\pi},U^{\prime}_{\pi})$
---|---|---|---|---
$\widetilde{\mathscr{B}}^{(3)}_{xy}$ | $e^{\frac{2\pi}{(2N,4)}i}$ | 1 | 1 | 1
$NB_{c^{2}}$ | $(-1)^{N}$ | $-1$ | $-1$ | 1
$NA_{x+y}^{2}$ | 1 | 1 | $-1$ | 1
$Nw_{2}$ | 1 | 1 | 1 | -1

Table 12: Values of the topological invariants, given $p4\times SO(3)$
symmetry with nontrivial $T_{1,2}$ action.

Again, because symmetry fractionalization classes related by relabeling anyons
are physically identical, different symmetry realizations on $U(1)_{2N}$ in
this case are specified by $\\{n_{1},n_{2},n_{3},n_{4}\\}$, where
$\\{n_{1},n_{2},n_{3},n_{4}\\}$ is identified with
$\\{[-n_{1}]_{(2N,4)},n_{2},n_{3},n_{4}\\}$. Calculating the anomaly
indicators for $U(1)_{2N}$ with symmetry fractionalization class labeled by
$(n_{1},n_{2},n_{3},n_{4})$ as before, we get

$\displaystyle\begin{aligned}
&\mathsf{I}_{1}=(-1)^{(n_{1}+n_{2}N)n_{4}}\,,\quad\mathsf{I}_{2}=(-1)^{n_{2}n_{4}N}\,,\\\
&\mathsf{I}_{3}=(-1)^{(n_{2}+n_{3}+1)n_{4}N}\,.\end{aligned}$ (45)

Therefore, by matching these anomaly indicators with Table 2, we arrive at the
classification in Table 13.

Lattice homotopy class |  Symmetry fractionalization class
---|---
0 |  |  $\\{n_{1},n_{2},n_{3},0\\}$, $\\{0,n_{2},n_{3},1\\}$,
---
or $\\{2,n_{2},n_{3},1\\}$ for $N$ even,
$\\{n_{1},n_{2},n_{3},0\\}$ or $\\{0,0,1,1\\}$ for $N$ odd
a |  |  $\\{1,n_{2},n_{3},1\\}$ for $N$ even,
---
$\\{1,0,1,1\\}$ for $N$ odd
b |  $\\{1,1,0,1\\}$ for $N$ odd
c |  $\\{0,0,0,1\\}$ for $N$ odd
a+b |  $\\{0,1,0,1\\}$ for $N$ odd
a+c |  $\\{1,0,0,1\\}$ for $N$ odd
b+c |  $\\{1,1,1,1\\}$ for $N$ odd
a+b+c |  $\\{0,1,1,1\\}$ for $N$ odd
Table 13: Classification of symmetry-enriched $U(1)_{2N}$ topological quantum
spin liquids in lattice systems with $p4\times SO(3)$ symmetry, if
translations act as charge conjugation.

  4. 4.

Nontrivial $T_{1,2}$ and $C_{4}$ action.

In this case, the possible symmetry fractionalization classes are given by

$\displaystyle H^{2}_{(4)}(p4\times
SO(3),\mathbb{Z}_{2N})=\mathbb{Z}_{(2N,4)}\oplus(\mathbb{Z}_{2})^{3}\,.$ (46)

We can label these elements as

$\displaystyle\begin{aligned}
w=&n_{1}\cdot\widetilde{\mathscr{B}}^{(4)}_{xy}+n_{2}\cdot NB_{c^{2}}\\\
&+n_{3}\cdot NA_{x+y}^{2}+n_{4}\cdot Nw_{2}\,,\end{aligned}$ (47)

with $n_{1}\in\\{0,\dots,(2N,4)-1\\}$, $n_{2,3,4}\in\\{0,1\\}$. Here
$\widetilde{\mathscr{B}}^{(4)}_{xy}$, $NB_{c^{2}}$, $\widetilde{A_{x+y}^{2}}$
and $Nw_{2}$ are generators of the $\mathbb{Z}_{(2N,4)}$ and the three
$\mathbb{Z}_{2}$ pieces, respectively (the representative cochains and the
reason for the names of these generators are given in Appendix C). We can
identify them in terms of the following four topological invariants defined in
Eqs. (14), (15) and (16): $\lambda_{3}(C_{4})$, $\lambda_{2}(T_{1}C_{4})$,
$\lambda_{3}(T_{1}C_{2})$ and $\lambda_{1}(U_{\pi},U^{\prime}_{\pi})$, and we
list the value of the topological invariants for the generators of symmetry
fractionalization classes in Table 14. For each symmetry fractionalization
class, the $U$\- and $\eta$-symbols can be obtained via Eqs. (4), (5) and
(13).

SF class | $\lambda_{3}(C_{4})$ | $\lambda_{2}(T_{1}C_{4})$ | $\lambda_{3}(T_{1}C_{2})$ | $\lambda_{1}(U_{\pi},U^{\prime}_{\pi})$
---|---|---|---|---
$\widetilde{\mathscr{B}}^{(4)}_{xy}$ | 1 | $e^{\frac{2\pi}{(2N,4)}i}$ | 1 | 1
$NB_{c^{2}}$ | $-1$ | $(-1)^{N}$ | $-1$ | 1
$NA_{x+y}^{2}$ | 1 | 1 | $-1$ | 1
$Nw_{2}$ | 1 | 1 | 1 | -1
Table 14: Values of the topological invariants, given $p4\times SO(3)$
symmetry with nontrivial $T_{1,2}$ and $C_{4}$ actions.

Again, because symmetry fractionalization classes related by relabeling anyons
are physically identical, different symmetry realizations on $U(1)_{2N}$ in
this case are specified by $\\{n_{1},n_{2},n_{3},n_{4}\\}$, where
$\\{n_{1},n_{2},n_{3},n_{4}\\}$ is identified with
$\\{[-n_{1}]_{(2N,4)},n_{2},n_{3},n_{4}\\}$. Calculating the anomaly
indicators for $U(1)_{2N}$ with symmetry fractionalization class labeled by
$(n_{1},n_{2},n_{3},n_{4})$ as before, we get

$\displaystyle\begin{aligned}
&\mathsf{I}_{1}=(-1)^{n_{2}n_{4}N}\,,\quad\mathsf{I}_{2}=(-1)^{(n_{1}+n_{2}N)n_{4}}\,,\\\
&\mathsf{I}_{3}=(-1)^{(n_{2}+n_{3}+1)n_{4}N}\,.\end{aligned}$ (48)

Therefore, by matching these anomaly indicators with Table 2, we arrive at the
classification in Table 15.

Lattice homotopy class | Symmetry fractionalization class
---|---
0 |  |  $\\{n_{1},n_{2},n_{3},0\\}$, $\\{0,n_{2},n_{3},1\\}$ ,
---
or $\\{2,n_{2},n_{3},1\\}$ for $N$ even,
$\\{n_{1},n_{2},n_{3},0\\}$ or $\\{0,0,1,1\\}$ for $N$ odd
a |  $\\{1,1,0,1\\}$ for $N$ odd
b |  |  $\\{1,n_{2},n_{3},1\\}$ for $N$ even,
---
$\\{1,0,1,1\\}$ for $N$ odd
c |  $\\{0,0,0,1\\}$ for $N$ odd
a+b |  $\\{0,1,0,1\\}$ for $N$ odd
a+c |  $\\{1,1,1,1\\}$ for $N$ odd
b+c |  $\\{1,0,0,1\\}$ for $N$ odd
a+b+c |  $\\{0,1,1,1\\}$ for $N$ odd
Table 15: Classification of symmetry-enriched $U(1)_{2N}$ topological quantum
spin liquids in lattice systems with $p4\times SO(3)$ symmetry, if
translations and $C_{4}$ both act as charge conjugation.

Summarizing all cases, the total number of different $p4\times SO(3)$
symmetry-enriched $U(1)_{2N}$ topological quantum spin liquids is summarized
in Table 7. Note that this classification is complete for $N\leqslant 5$, but
incomplete for $N\geqslant 6$, because we have assumed that the only way the
symmetry can permute anyons is via charge conjugation, while for $N\geqslant
6$ a symmetry can in principle permute anyons in other manners.

##  VII Ising(ν) topological orders: Kitaev’s non-Abelian chiral spin liquids

Our next class of examples are non-Abelian chiral spin liquid states, which we
dub the “Ising(ν) states”, with $\nu$ an odd integer. Their topological
properties are discussed in detail by Kitaev Kitaev (2006) and will be
reviewed below. The exactly solvable model in Ref. Kitaev (2006) has triggered
enormous interest in realizing the Ising(1) state in real materials Rau  _et
al._ (2016); Trebst (2017); Winter  _et al._ (2017). We remark that usually
these Kitaev quantum spin liquids are discussed in the context of spin-orbit
coupled systems, but here we consider them in systems without spin-orbit
coupling for simplicity. In particular, we will classify Ising(ν) states in
lattice systems with $p6\times SO(3)$ or $p4\times SO(3)$ symmetry.

The Ising(ν) state has three anyons $\\{I,\sigma,\psi\\}$, where the trivial
anyon here is denoted $I$, and the nontrivial fusion rules are given by

$\psi\times\psi=I,\quad\sigma\times\psi=\sigma,\quad\sigma\times\sigma=I+\psi.$
(49)

The nontrivial $F$-symbols are

$\begin{gathered}F^{\psi\sigma\psi}_{\sigma}=F^{\sigma\psi\sigma}_{\psi}=-1\\\
\left[F^{\sigma\sigma\sigma}_{\sigma}\right]_{ab}=\frac{\varkappa_{\sigma}}{\sqrt{2}}\left[\begin{matrix}1&1\\\
1&-1\end{matrix}\right]_{ab}.\end{gathered}$ (50)

Here, the column and row labels of the matrix take values $I$ and $\psi$ (in
this order). All other $F$-symbols are 1 if it is compatible with the fusion
rule and 0 if it is not. $\varkappa_{\sigma}=(-1)^{\frac{\nu^{2}-1}{8}}$ is
the Frobenius-Schur indicator of $\sigma$.

The nontrivial $R$-symbols are

$\begin{gathered}R^{\psi\psi}=-1,\quad
R^{\psi\sigma}_{\sigma}=R^{\sigma\psi}_{\sigma}=(-i)^{\nu}\\\
R^{\sigma\sigma}_{I}=\varkappa_{\sigma}e^{-i\frac{\pi}{8}\nu},\quad
R^{\sigma\sigma}_{\psi}=\varkappa_{\sigma}e^{i\frac{3\pi}{8}\nu}.\end{gathered}$
(51)

The topological spins are $\theta_{\psi}=-1$,
$\theta_{\sigma}=e^{i\frac{\pi}{8}\nu}$, and the chiral central charge
$c_{-}=\frac{\nu}{2}$.

The topological symmetry of Ising(ν) is trivial and no symmetry of Ising(ν)
can permute anyons. The $U$-symbol and a set of $\eta$-symbol can all be
chosen to be 1.

The symmetry fractionalization classes of $p6\times SO(3)$ are classified by

$\displaystyle H^{2}(p6\times
SO(3),\mathbb{Z}_{2})\cong(\mathbb{Z}_{2})^{3}\,.$ (52)

We can label these elements as

$\displaystyle w=n_{1}\cdot B_{xy}+n_{2}\cdot A_{c}^{2}+n_{3}\cdot w_{2}\,,$
(53)

with $n_{1,2,3}\in\\{0,1\\}$. The symmetry fractionalization classes of
$p4\times SO(3)$ are given by

$\displaystyle H^{2}(p4\times
SO(3),\mathbb{Z}_{2})\cong(\mathbb{Z}_{2})^{4}\,.$ (54)

We can label these elements as

$\displaystyle w=n_{1}\cdot B_{xy}+n_{2}\cdot B_{c^{2}}+n_{3}\cdot
A_{x+y}^{2}+n_{4}\cdot w_{2}\,,$ (55)

with $n_{1,2,3,4}\in\\{0,1\\}$. The representative cochains of these elements
are presented in Appendix C. Physically, these generators can be viewed as
detecting whether the non-Abelian anyon $\sigma$ carries a projective quantum
number under these global symmetries. For each symmetry fractionalization
class, the $U$\- and $\eta$-symbols can be obtained via Eqs. (4) and (5).

The above discussion implies that, without considering anomaly matching, there
are in total $2^{3}=8$ different $p6\times SO(3)$ symmetric Ising(ν) states,
and $2^{4}=16$ different $p4\times SO(3)$ symmetric Ising(ν) states.
Calculating the anomaly indicators for the Ising(ν) state in a way similar to
the the calculation for the $U(1)_{2N}$ state, we find that for any symmetry
fractionalization class of either $p6\times SO(3)$ or $p4\times SO(3)$
symmetry, all anomaly indicators always evaluate to 1 and hence the anomaly is
always absent. Therefore, all $p6\times SO(3)$ or $p4\times SO(3)$ symmetry-
enriched Ising(ν) topological quantum spin liquids can emerge in lattice
systems within lattice homotopy class 0 (including, for example, honeycomb
lattice spin-1/2 system or spin-1 system on any lattice), but not other
lattice homotopy class (including, for example, spin-1/2 system on triangular,
kagome, square and checkerboard lattices). We notice that in most previous
discussions of the Ising(1) state in spin-orbit coupled systems, the
underlying lattice systems indeed have a trivial anomaly, since they can be
obtained from the lattice homotopy class 0 here by breaking certain
symmetries.

##  VIII $Z_{N}$ topological orders: Generalized toric codes

In this section, we consider the $\mathbb{Z}_{N}$ topological order, which is
the $\mathbb{Z}_{N}$ generalization of the famous $\mathbb{Z}_{2}$ topological
order Kitaev (2003); Barkeshli  _et al._ (2019a); Kivelson  _et al._ (1987);
Rokhsar and Kivelson (1988); Read and Sachdev (1991); Wen (1991); Senthil and
Fisher (2000). The case with $N=2$ has been studied extensively in many
different types of lattice systems. However, as mentioned in the Introduction,
when $N>2$ these states do not allow a description in terms of a simple parton
mean field (instead, the partons have to be strongly interacting), and they
are much less explored (see examples in Refs. Motrunich (2003); Myers and
Herdman (2017); Devakul (2018); Kurečić  _et al._ (2019); Giudice  _et al._
(2022)). Our framework in Sec. V allows us to classify a general
$\mathbb{Z}_{N}$ topological quantum spin liquid enriched by a general
symmetry. For concreteness, the symmetry we will consider below are one of
these four: $p6m\times SO(3)\times\mathbb{Z}_{2}^{T}$, $p4m\times
SO(3)\times\mathbb{Z}_{2}^{T}$, $p6m\times\mathbb{Z}_{2}^{T}$ and
$p4m\times\mathbb{Z}_{2}^{T}$, where $p6m$ and $p4m$ are lattice symmetries,
while $SO(3)$ and $\mathbb{Z}_{2}^{T}$ are on-site spin rotational symmetry
and time reversal symmetry, respectively.

In the $\mathbb{Z}_{N}$ topological order, there are $N^{2}$ anyons in total,
which can be labeled by two integers as $a=(a_{e},a_{m})$, with
$a_{e},a_{m}\in\\{0,\dots,N-1\\}$. Following the convention in the
$\mathbb{Z}_{2}$ toric code, we will call the anyon labeled by $(1,0)$ as $e$,
and the anyon labeled by $(0,1)$ as $m$. The fusion rules are element-wise
addition modulo $N$, i.e.,

$(a_{e},a_{m})\times(b_{e},b_{m})=([a_{e}+b_{e}]_{N},[a_{m}+b_{m}]_{N})\,,$
(56)

In a choice of gauge, the $F$-symbols of this topological order are all 1 and
the $R$-symbols are given by

$R^{ab}=e^{i\frac{2\pi}{N}a_{m}b_{e}}\,.$ (57)

The topological symmetry group is complicated to determine for general $N$
Delmastro and Gomis (2021); Geiko and Moore (2023). For $N=2$, the topological
symmetry is $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{T}$, generated by the unitary
electric-magnetic duality symmetry $S$ that exchanges $e$ and $m$, i.e.,
$(a_{e},a_{m})\rightarrow(a_{m},a_{e})$, and an anti-unitary symmetry $T$
which exchanges $e$ and $m$ the same way as $S$. For this
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{T}$ symmetry, we can choose the
$U$-symbols as

$U_{\bf g}(a,b;c)=\begin{cases}(-1)^{a_{m}b_{e}},&\text{${\bf g}$ permutes
anyons,}\\\ 1,&\text{otherwise,}\end{cases}$ (58)

and a set of $\eta$-symbols as

$\displaystyle\eta_{a}({\bf g},{\bf
h})=\begin{cases}(-1)^{a_{e}a_{m}},&\text{${\bf g}$, ${\bf h}$ permute
anyons,}\\\ 1,&\text{otherwise.}\end{cases}$ (59)

with $(a_{e},a_{m}),(b_{e},b_{m})$ the anyon labels of anyons $a,b$.

For $N\geqslant 3$, there is always a
$\mathbb{Z}_{4}^{T}\rtimes\mathbb{Z}_{2}$ topological symmetry. The anti-
unitary $\mathbb{Z}_{4}^{T}$ is generated by an action
$T:(a_{e},a_{m})\rightarrow(a_{m},[-a_{e}]_{N})$, and the unitary
$\mathbb{Z}_{2}$ is generated by an action
$S:(a_{e},a_{m})\rightarrow(a_{m},a_{e})$. The two generators satisfy the
relation

$\displaystyle S^{2}=\textbf{1}\,,\quad T^{4}=\textbf{1}\,,\quad
STS=T^{-1}\,.$ (60)

For this $\mathbb{Z}_{4}^{T}\rtimes\mathbb{Z}_{2}$ symmetry, writing a group
element as ${\bf g}=T^{g_{1}}S^{g_{2}}$, with $g_{1}\in\\{0,1,2,3\\}$ and
$g_{2}\in\\{0,1\\}$, we can choose the $U$-symbols as

$U_{\bf g}(a,b;c)=\begin{cases}e^{\frac{2\pi
i}{N}a_{m}b_{e}},&\text{$g_{1}+g_{2}$ is odd,}\\\
1,&\text{otherwise}\end{cases},$ (61)

and a set of $\eta$-symbols as

$\displaystyle\eta_{a}({\bf g},{\bf h})=\begin{cases}e^{\frac{2\pi
i}{N}a_{e}a_{m}},&\text{$g_{1}+g_{2}$, $h_{1}+h_{2}$ are odd,}\\\
1,&\text{otherwise}\end{cases}$ (62)

For certain $N\geqslant 3$ there can be other topological symmetries, in
addition to the above $\mathbb{Z}_{4}\rtimes\mathbb{Z}_{2}^{T}$ symmetry. For
example, when $N=5$, the action
$(a_{e},a_{m})\rightarrow([3a_{e}]_{5},[3a_{m}]_{5})$ is an anti-unitary
topological symmetry. For simplicity, below we will focus on the cases where
$N=2,3,4$.

The analysis of the classification is similar to the previous cases. In the
present case, we need to understand the anomaly indicators of the $p6m\times
SO(3)\times\mathbb{Z}_{2}^{T}$, $p4m\times SO(3)\times\mathbb{Z}_{2}^{T}$,
$p6m\times\mathbb{Z}_{2}^{T}$ and $p4m\times\mathbb{Z}_{2}^{T}$ symmetries.
These anomaly indicators and their values for different lattice homotopy
classes can be found in Appendix D. Carrying out the procedure listed in Sec.
V, we can obtain the classification. In Tables 16, we list the number of
different symmetry-enriched $\mathbb{Z}_{N}$ topological quantum spin liquids
in different lattice homotopy classes under these symmetries. The precise
symmetry fractionalization classes in each case can be found in Appendix C. We
also upload codes using which one can 1) see all symmetry fractionalization
classes of the symmetry-enriched states within each lattice homotopy class,
and 2) check which lattice homotopy class a given symmetry-enriched state
belongs to cod . Below we comment on some of these results.

For the case with $N=2$ and the $p6m\times SO(3)\times\mathbb{Z}_{2}^{T}$
symmetry, the classification was carried out for spin-1/2 systems on a
triangular, kagome and honeycomb lattice Qi  _et al._ (2015); Qi and Cheng
(2018); Lu and Ran (2011), which belongs to the lattice homotopy class $a$,
$c$ and 0, respectively. For lattice homotopy classes $a$ and $c$, our results
agree with those in Refs. Qi  _et al._ (2015); Qi and Cheng (2018). For the
lattice homotopy class 0, using the parton-mean-field approach and assuming
that one of $e$ and $m$ carries spin-1/2 under the $SO(3)$ symmetry, Ref. Lu
and Ran (2011) found 128 different states. We find 336 states in total, where
128 of them have one of $e$ and $m$ carrying half-odd-integer spin, and in the
other 208 states both $e$ and $m$ carry integer spin, 9 of which also have
symmetries permuting $e$ and $m$. For the case with $N=2$ and the $p4m\times
SO(3)\times\mathbb{Z}_{2}^{T}$ symmetry, Ref. Lu (2018) found 64 states on
square lattice spin-1/2 system, which belongs to our lattice homotopy class
$a$, agreeing with our results.

For the case with $N=2$ and the $p4m\times\mathbb{Z}_{2}^{T}$ symmetry, using
the parton-mean-field approach, Ref. Yang and Wang (2016) found 64 states on
square lattice system with Kramers doublet spins, which can all be obtained
from the $p4m\times SO(3)\times\mathbb{Z}_{2}^{T}$ symmetric $\mathbb{Z}_{2}$
topological quantum spin liquids by breaking the $SO(3)$ symmetry. Suppose in
the $p4m\times SO(3)\times\mathbb{Z}_{2}^{T}$ symmetric version of these
states, the anyon $e$ carries half-odd-integer spin under $SO(3)$, then
projective quantum numbers of $m$ are fixed for all these 64 states Qi  _et
al._ (2015). In particular, $m$ experiences no nontrivial symmetry
fractionalization pattern that simultaneously involves the time reversal and
lattice symmetries. The absence of such symmetry fractionalization pattern
still holds in the 64 “within parton” $p4m\times\mathbb{Z}_{2}^{T}$ symmetric
states obtained by breaking $SO(3)$. However, in addition to these 64 states,
we have found $117-64=53$ other states, with their symmetry fractionalization
classes presented in Appendix E (anyons are not permuted by symmetries in all
these 117 states). A common property of these 53 states is the presence of
nontrivial symmetry fractionalization involving both the lattice symmetry and
time reversal symmetry for the anyon $m$, e.g., translation and time reversal
may not commute for $m$. Furthermore, for all 117 states, the $C_{2}\equiv
C_{4}^{2}$ symmetry fractionalizes on the $m$ anyon, i.e., effectively
$C_{2}^{2}=-1$ for $m$. Usually, the interpretation of this phenomenon is that
there is a background $e$ anyon at each square lattice site (the $C_{4}$
center), and the mutual braiding statistics between $e$ and $m$ yields
$C_{2}^{2}=-1$. However, for 16 of the 53 “beyond-parton” states,
$(T_{1}C_{2})^{2}=(T_{2}C_{2})^{2}=-1$ for $m$, which seems to suggest that
there are also background $e$ anyons at the 2-fold rotation centers of
$T_{1}C_{2}$ and $T_{2}C_{2}$, although microscopically there is no spin at
those positions. So the analysis based on anomaly matching suggests that the
simple picture where the fractionalization of rotational symmetries purely
comes from background anyons is actually incomplete.

The above example shows that even for simple states like the $\mathbb{Z}_{2}$
topological order, the parton-mean-field approach may miss some of their
symmetry enrichment patterns, and our framework in Sec. V is more general.
Note that here by “parton mean field”, we are referring to the ususal parton
mean fields where the partons are non-interacting. If the partons are allowed
to interact strongly, say, if they form nontrivial interacting symmetry-
protected topological states under the projective symmetry group of the
partons, symmetry-enriched states not captured by Ref. Yang and Wang (2016)
may arise, but it is technically complicated to study them. Also, by using
parton constructions other than the one in Ref. Yang and Wang (2016), one may
also obtain states beyond those in Ref. Yang and Wang (2016), but it is
challenging to make this approach systematic.

We also notice that the number of $\mathbb{Z}_{3}$ topological quantum spin
liquids is nonzero only in the lattice homotopy class 0. This phenomenon is
actually true for general odd $N$. To see it, first notice all lattice
homotopy classes except 0 have some mixed anomalies between the $SO(3)$
symmetry and the lattice symmetry Ye  _et al._ (2022). In order to match this
anomaly, it is impossible for both $e$ and $m$ to carry integer spin. Suppose
that $e$ carries half-odd-integer spin, and consider threading an $SO(3)$
monopole through the system. The monopole will be viewed as a $\pi$ flux from
the perspective of $e$. Then the local nature of the monopole implies that it
must trap an anyon that has $\pi$ braiding statistics with $e$. For odd $N$,
no such anyon exists, which leads to a contradiction. So $\mathbb{Z}_{N}$
topological quantum spin liquids with $N$ odd cannot possibly arise in lattice
homotopy class other than 0. Note that the above argument does not rely on the
time reversal symmetry, and it is valid no matter how the symmetries permute
anyons.

For $\mathbb{Z}_{N}$ topological quantum spin liquids in systems belonging to
a lattice homotopy class other than 0, which requires $N$ to be even, anyons
$(1,0)$ and $(0,N/2)$ cannot simutaneously carry half-odd-integer spin,
otherwise there would be a mixed anomaly between the $SO(3)$ and time reversal
symmetries Zou  _et al._ (2018).

Symmetry group | Lattice homotopy class | $\mathbb{Z}_{2}$ | $\mathbb{Z}_{3}$ | $\mathbb{Z}_{4}$ | $\mathrm{U}(1)_{2}\times\mathrm{U}(1)_{-2}$ | $\mathrm{U}(1)_{4}\times\mathrm{U}(1)_{-4}$
---|---|---|---|---|---|---
$p6m\times SO(3)\times\mathbb{Z}_{2}^{T}$ | 0 | 336 | 8 | 16453 | 32 | 144
a | 8 | 0 | 70 | 0 | 0
c | 8 | 0 | 70 | 0 | 0
a+c | 4 | 0 | 82 | 0 | 0
$p6m\times\mathbb{Z}_{2}^{T}$ | 0 | 208 | 8 | 4725 | 16 | 72
a | 13 | 0 | 61 | 0 | 0
c | 13 | 0 | 61 | 0 | 0
a+c | 12 | 0 | 167 | 0 | 0
$p4m\times SO(3)\times\mathbb{Z}_{2}^{T}$ | 0 | 3653 | 9 | 886740 | 128 | 1344
a | 64 | 0 | 5008 | 0 | 0
b | 64 | 0 | 5008 | 0 | 0
c | 64 | 0 | 8872 | 0 | 0
a+b | 16 | 0 | 636 | 0 | 0
a+c | 16 | 0 | 656 | 0 | 0
b+c | 16 | 0 | 656 | 0 | 0
a+b+c | 8 | 0 | 318 | 0 | 0
$p4m\times\mathbb{Z}_{2}^{T}$ | 0 | 2629 | 9 | 280852 | 64 | 672
a | 117 | 0 | 3491 | 0 | 0
b | 117 | 0 | 3491 | 0 | 0
c | 193 | 0 | 12449 | 0 | 0
a+b | 33 | 0 | 513 | 0 | 0
a+c | 34 | 0 | 610 | 0 | 0
b+c | 34 | 0 | 610 | 0 | 0
a+b+c | 21 | 0 | 309 | 0 | 0
Table 16: Number of various topological quantum spin liquids enriched by
$p6m\times SO(3)\times\mathbb{Z}_{2}^{T}$, $p6m\times\mathbb{Z}_{2}^{T}$,
$p4m\times SO(3)\times\mathbb{Z}_{2}^{T}$or $p4m\times\mathbb{Z}_{2}^{T}$
symmetry, where the third, fourth and fifth columns represent
$\mathbb{Z}_{2}$, $\mathbb{Z}_{3}$ and $\mathbb{Z}_{4}$ topological orders,
while the last two colomns represent the $U(1)_{2}\times U(1)_{-2}$ and
$U(1)_{4}\times U(1)_{-4}$ topological orders, respectively. The details of
the symmetry fractionalization classes of each state can be found in Appendix
C. For $\mathbb{Z}_{2}$ and $\mathbb{Z}_{4}$ topological orders, we also
upload codes containing the symmetry fractionalization class for each state in
each lattice homotopy class cod . For $\mathbb{Z}_{3}$, $U(1)_{2}\times
U(1)_{-2}$, and $U(1)_{4}\times U(1)_{-4}$ topological orders, all symmetry-
enriched states are anomaly-free.

##  IX $\mathrm{U}(1)_{2N}\times\mathrm{U}(1)_{-2N}$ topological orders:
Generalizations of the double-semion state

In this section, we consider the $\mathrm{U}(1)_{2N}\times\mathrm{U}(1)_{-2N}$
topological order, which is the generalization of the double-semion state,
i.e., the case with $N=1$. Effectively, this state can be obtained by stacking
a $U(1)_{2N}$ state, which is discussed in Sec. VI, on its time reversal
partner, the $U(1)_{-2N}$ state. We would like to classify the
$\mathrm{U}(1)_{2N}\times\mathrm{U}(1)_{-2N}$ topological order enriched by
one of these four symmetries: $p6m\times SO(3)\times\mathbb{Z}_{2}^{T}$,
$p4m\times SO(3)\times\mathbb{Z}_{2}^{T}$, $p6m\times\mathbb{Z}_{2}^{T}$ and
$p4m\times\mathbb{Z}_{2}^{T}$.

In a $U(1)_{2N}\times U(1)_{-2N}$ topological quantum spin liquid, there are
$4N^{2}$ anyons in total, which can be labeled by two integers as
$a=(a_{s},a_{\bar{s}})$, with $a_{s},a_{\bar{s}}\in\\{0,\dots,2N-1\\}$.
Following the convention in the double-semion state, we will call the anyon
labeled by $(1,0)$ as $s$, and the anyon labeled by $(0,1)$ as $\bar{s}$ (note
in this convention $s$ and $\bar{s}$ are not anti-particles of each other).
The fusion rules are element-wise addition modulo $2N$, i.e.,

$(a_{s},a_{\bar{s}})\times(b_{s},b_{\bar{s}})=([a_{s}+b_{s}]_{2N},[a_{\bar{s}}+b_{\bar{s}}]_{2N})\,,$
(63)

In a choice of gauge, the $F$-symbols of the theory are

$F^{abc}=\exp\left(i\frac{2\pi}{N}\left(a_{s}(b_{s}+c_{s}-[b_{s}+c_{s}]_{2N})-a_{\bar{s}}(b_{\bar{s}}+c_{\bar{s}}-[b_{\bar{s}}+c_{\bar{s}}]_{2N})\right)\right)\,$
(64)

and the $R$-symbols are

$R^{ab}=\exp\left(i\frac{2\pi}{N}(a_{s}b_{s}-a_{\bar{s}}b_{\bar{s}})\right)\,.$
(65)

The topological symmetry group is complicated to determine for general $N$,
just like the $U(1)_{2N}$ state Delmastro and Gomis (2021); Geiko and Moore
(2023). Here we list the topological symmetry group for $N=1,2$ here. For
$N=1$, the topological symmetry is $\mathbb{Z}_{2}^{T}$, generated by
$\tilde{S}$ exchanging $s$ and $\bar{s}$, i.e.,

$\displaystyle(a_{s},a_{\bar{s}})\rightarrow(a_{\bar{s}},a_{s})\,.$ (66)

We can choose the $U$-symbols and a set of $\eta$-symbols all equal to 1.

For $N=2$, the topological symmetry is
$\mathbb{Z}_{4}^{T}\rtimes\mathbb{Z}_{2}^{T}$, generated by an order 2 anti-
unitary symmetry $\tilde{S}$ which exchanges $s$ and $\bar{s}$,
$\tilde{S}\colon(a_{s},a_{\bar{s}})\rightarrow(a_{\bar{s}},a_{s})$, and
another order 4 anti-unitary symmetry $T$, which permutes anyons in the
following way
$T\colon(a_{s},a_{\bar{s}})\rightarrow(a_{\bar{s}},[-a_{s}]_{2N})$. The two
generators satisfy the relation

$\displaystyle\tilde{S}^{2}=\textbf{1}\,,\quad
T^{4}=\textbf{1}\,,\quad\tilde{S}T\tilde{S}=T^{-1}\,.$ (67)

An element in $\mathbb{Z}_{4}^{T}\rtimes\mathbb{Z}_{2}^{T}$ can be written as
$T^{g_{1}}\tilde{S}^{g_{2}}$, with $g_{1}\in\\{0,\dots,3\\}$ and
$g_{2}\in\\{0,1\\}$. To define the $U$-symbols, first we define the following
function

$\displaystyle\tilde{U}(a_{s},b_{s})=\begin{cases}(-1)^{a_{s}}&b_{s}\neq 0\\\
1&b_{s}=0\end{cases}$ (68)

Given an element ${\bf g}\in\mathbb{Z}_{4}^{T}\rtimes\mathbb{Z}_{2}^{T}$, the
$U$-symbols can be chosen such that

$\displaystyle U_{\bf g}(a,b;c)=\begin{cases}1&g_{1}=0\\\
\tilde{U}(a_{\bar{s}},b_{\bar{s}})&g_{1}=1\\\
\tilde{U}(a_{s},b_{s})\tilde{U}(a_{\bar{s}},b_{\bar{s}})&g_{1}=2\\\
\tilde{U}(a_{s},b_{s})&g_{1}=3\\\ \end{cases}$ (69)

And a set of $\eta$-symbols can be chosen to be all identity.

Carrying out the procedure in Sec. V in a manner similar to the previous
examples, we can obtain the classification of $U(1)_{2}\times U(1)_{-2}$ and
$U(1)_{4}\times U(1)_{-4}$ topological quantum spin liquids enriched by
$p6m\times SO(3)\times\mathbb{Z}_{2}^{T}$, $p4m\times
SO(3)\times\mathbb{Z}_{2}^{T}$, $p6m\times\mathbb{Z}_{2}^{T}$ or
$p4m\times\mathbb{Z}_{2}^{T}$ symmetry. The results are summarized in Table
16.

We notice that in all symmetry groups considered here, $U(1)_{2}\times
U(1)_{-2}$ and $U(1)_{4}\times U(1)_{-4}$ can only arise in the lattice
homotopy class 0. Ref. Zaletel and Vishwanath (2015) presented a physical
reason for this phenomenon. If we only consider the symmetry group $p6m\times
SO(3)\times\mathbb{Z}_{2}^{T}$ and $p4m\times SO(3)\times\mathbb{Z}_{2}^{T}$,
the following simpler argument can explain it. To be concrete, suppose the
symmetry group is $p6m\times SO(3)\times\mathbb{Z}_{2}^{T}$, and a similar
argument can be made if the symmetry group is $p4m\times
SO(3)\times\mathbb{Z}_{2}^{T}$. Now suppose breaking the symmetry to $p6\times
SO(3)$. Then the system can be viewed as a $p6\times SO(3)$ symmetric
$U(1)_{2N}$ state on top of a $p6\times SO(3)$ symmetric $U(1)_{-2N}$ state,
and these two states must have the opposite anomalies under the $p6\times
SO(3)$ symmetry, otherwise they cannot be connected by time reversal to form
the original $p6m\times SO(3)\times\mathbb{Z}_{2}^{T}$ symmetric state.
Namely, after breaking $p6m\times SO(3)\times\mathbb{Z}_{2}^{T}$ to $p6\times
SO(3)$, there is no remaining anomaly and the state is in lattice homotopy
class 0. Now we ask which lattice homotopy class with a $p6m\times
SO(3)\times\mathbb{Z}_{2}^{T}$ symmetry becomes the lattice homotopy class 0
with a $p6\times SO(3)$ symmetry after this symmetry breaking. From the
representative configurations of all lattice homotopy classes in Sec. IV,
clearly only the lattice homotopy class 0 does.

##  X Discussion

In this paper, we have presented a general framework in Sec. V to classify
symmetry-enriched topological quantum spin liquids in two spatial dimensions.
This framework applies to all topological quantum spin liquids, which may be
Abelian or non-Abelian, and chiral or non-chiral. The symmetry we consider may
include both lattice symmetry and internal symmetry, may contain anti-unitary
symmetry, and may permute anyons. We then apply this framework to various
examples in Secs. VI, VII, VIII and IX. As argued in the Introduction, our
framework combines the advantages of the previous approaches in the
literature, while avoiding their disadvantages. Indeed, we are able to
identify symmetry-enriched topological quantum spin liquids that are not
easily captured by the usual parton-mean-field approach (see examples in Sec.
VIII), and we can systematically distinguish different lattice systems with
the same symmetry group using their quantum anomalies.

We finish this paper by discussing some open questions.

  * •

In this paper, we characterize a topological quantum spin liquid with a
lattice symmetry by one with an internal symmetry via the crystalline
equivalence principle in Sec. III.4. However, it is more ideal to have a
theory that directly describes topological quantum spin liquids with lattice
symmetries.

Such a theory should be able to tell how an arbitrary symmetry acts on a state
obtained by creating some anyons from the ground state and putting them at
arbitrary positions. The symmetry action should be some analog of Eq. (2), but
it is subtle to understand what constraints the analogs of $U_{\bf g}(a,b;c)$
and $\eta_{a}({\bf g},{\bf h})$ should satisfy. So far this question has been
answered if the lattice symmetry only contains translation symmetry Cheng  _et
al._ (2016), but for cases with point group symmetries it is answered in a
very specific case, where the lattice symmetry is reflection, and the state
only contains two anyons that are 1) anti-particles of each other, 2)
transformed into each other under the reflection symmetry, and 3) located at
two reflection related positions Barkeshli  _et al._ (2019b). It is useful to
have a complete theory that can answer this question in full generality. Such
a theory is also helpful for the purpose of identifying observable signatures
of different symmetry-enriched topological quantum spin liquids.

  * •

Strictly speaking, our classification is a classification of different
patterns of how symmetries permute anyons and the symmetry fractionalization
patterns. In principle, one should further consider how the classification is
modified upon stacking an invertible state on the topological quantum spin
liquid with the same symmetry. This question is subtle because some nontrivial
invertible states can be trivialized in the presence of a long-range entangled
state Barkeshli  _et al._ (2019a); Wang and Senthil (2016); Zou  _et al._
(2018). We leave this problem for future study.

  * •

In this paper, we focus on how symmetry permutes anyons and the symmetry
fractionalization classes, which can be viewed as the bulk properties of
different symmetry-enriched topological quantum spin liquids. It is also
interesting to explore their boundary properties in the future. In particular,
sometimes the symmetry enrichment pattern may enforce the boundary of the
topological quantum spin liquid to be gapless, even if it is non-chiral Wang
_et al._ (2017); Wen and Potter (2023). Similarly, it is intriguing to study
the properties of defects in different symmetry-enriched topological quantum
spin liquids, and examine their potential to perform quantum computation
Barkeshli  _et al._ (2013, 2019a).

  * •

It is useful to find numerical algorithms to identify the symmetry enrichement
pattern of a topological quantum spin liquid that emerges in a lattice model
that is not fine-tuned, and find experimental methods to detect the symmetry
enrichment pattern in experiments. Some previous proposals for various
specific cases include Refs. Essin and Hermele (2014); Qi and Fu (2015);
Zaletel  _et al._ (2015); Sun  _et al._ (2018); Lu and Lu (2022), but it is
useful to find algorithms and methods applicable to the general setting.

  * •

After classifying different symmetry-enriched topological quantum spin liquids
and finding methods to detect them numerically and experimentally, it is
important to construct explicit models that realize these topological orders.
For many topological orders enriched by internal symmetries, Refs. Heinrich
_et al._ (2016); Cheng  _et al._ (2017); Bulmash and Barkeshli (2020); Wang
and Cheng (2022) construct their exactly solvable models with explicit
Hamiltonians and ground-state wavefunctions. Moreover, there are many
proposals for realizing symmetry-protected and symmetry-enriched topological
states with lattice symmetries in the literature, including Refs. Song  _et
al._ (2017b); Huang  _et al._ (2017); Song  _et al._ (2018, 2019); Zhang  _et
al._ (2022, 2020); Qi  _et al._ (2019); Else and Thorngren (2020). We
anticipate that we can combine the above constructions to obtain exactly
solvable models with concrete Hamiltonians that realize the symmetry-enriched
topological quantum spin liquids discussed in this paper.

It will be also interesting to find quantum materials and develop quantum
simulators to realize these different phases, and explore interesting
continuous quantum phase transitions out of them, which are beyond the
conventional Landau-Ginzburg-Wilson-Fisher paradigm.

  * •

In this paper, our focus is topological quantum spin liquids in two spatial
dimensions. It is interesting to generalize our work to other systems, such as
fermionic systems, systems in higher dimensions, gapless systems and fractonic
systems. In particular, there are many experimental candidates of
$(3+1)$-dimensional symmetry-enriched gapless $U(1)$ quantum spin liquids in
pyrochlores Gingras and McClarty (2014), and their classification has been
discussed within the framework of projective symmetry groups Chern and Kim
(2021); Liu  _et al._ (2021); Desrochers  _et al._ (2022); Chern  _et al._
(2022); Desrochers  _et al._ (2023); Chauhan  _et al._ (2023). As discussed in
the present paper, the classification based on projective symmetry groups may
be incomplete. Using more general approaches, $U(1)$ quantum spin liquids with
only internal symmetries have been classified Wang and Senthil (2016); Zou
_et al._ (2018); Ning  _et al._ (2020); Hsin and Turzillo (2020), and some
examples of their lattice symmetry enriched versions have been constructed Zou
(2018). However, a systematic classification of $(3+1)$-dimensional $U(1)$
quantum spin liquids enriched by both lattice and internal symmetries is
lacking, and it is interesting to apply the idea in the present paper to those
settings in the future.

Note:  Codes for checking anomaly matching and details of realizations for
$\mathbb{Z}_{2}$ and $\mathbb{Z}_{4}$ topological order available at
https://github.com/Weicheng-Ye/Classification-of-QSL.git.

###### Acknowledgements.

We thank Maissam Barkeshli, Dominic Else, Meng Guo, Yin-Chen He, T. Senthil,
Chong Wang, Fa Wang and Qing-Rui Wang for helpful discussion. Research at
Perimeter Institute is supported in part by the Government of Canada through
the Department of Innovation, Science and Industry Canada and by the Province
of Ontario through the Ministry of Colleges and Universities.

##  Appendix A Translation between the characterization of reflection symmetry
and time-reversal symmetry

It is a common folklore that “reflection symmetry $=$ time-reversal symmetry
$\times$ charge-conjugation”. However, one precise formulation of the
statement is based on the CPT theorem, which is formulated in relativistic
quantum field theory and requires Lorentz symmetry as premise Weinberg (2005).
In the context of topological order, even though Lorentz symmetry is not
explicitly present, it is also widely believed that the statement also holds
true. However, the precise correspondence between reflection symmetry and
time-reversal symmetry, especially the matching between the data $\\{\rho_{\bf
g};U_{\bf g}(a,b;c),\eta_{a}({\bf g},{\bf h})\\}$ for these two symmetries, is
little known in the literature. We summarize this correspondence in this
appendix.

For this purpose, we need more formal treatments of the topological order in
terms of a unitary modular tensor category (UMTC), which go beyond what is
reviewed in Sec. III, and we refer the interested readers to Sec. II of Ref.
Ye and Zou (2023) for the basics. Following the convention of Refs. Barkeshli
_et al._ (2019a, b); Walker (2017), we model the time-reversal symmetry action
on the UMTC as a $\mathbb{C}$-anti-linear functor and the (unitary) reflection
symmetry action as an anti-monoidal functor (also see Ref. Kong and Zhang
(2022) for slightly different treatments). Therefore, mathematically speaking,
in this appendix we establish a precise correspondence between the data of a
$\mathbb{C}$-anti-linear functor and the data of an anti-monoidal functor. We
believe that such correspondence will imply the correspondence of the data
$\\{\rho_{\bf g};U_{\rm g}(a,b;c),\eta_{a}({\bf g},{\bf h})\\}$ for these two
symmetries on the explicit wavefunctions of the topological order, and we
defer it to future study.

Throughout the appendix, we assume that the reflection symmetry is unitary. We
can also consider anti-unitary reflection symmetry, which in the crystalline
equivalence principle should correspond to a unitary symmetry which does not
reflect spacetime. Following the treatment in this appendix, we can similarly
establish a correspondence between a $\mathbb{C}$-linear functor for the
unitary symmetry and a $\mathbb{C}-$anti-linear anti-monoidal functor for the
anti-unitary reflection symmetry. The details can be worked out by following
closely the treatment in this appendix and we omit them.

Recall that anyon lines may be “bent” using the $A$ and $B$ symbols, given
diagrammatically by

$\raisebox{-0.5pt}{