# Dihedral twist liquid models in the orthogonal family

Jeffrey C. Y. Teo  Department of Physics, University of Virginia,
Charlottesville, VA22904, USA  Yichen Hu  The Rudolf Peierls Centre for
Theoretical Physics, University of Oxford, Oxford OX1 3PU, UK

###### Abstract

We present a family of electron-based coupled-wire models of bosonic orbifold
topological phases, referred to as twist liquids, in two spatial dimensions.
All local fermion degrees of freedom are gapped and removed from the
topological order by many-body interactions. Bosonic chiral spin liquids and
anyonic superconductors are constructed on an array of interacting wires, each
supports emergent massless Majorana fermions that are non-local (fractional)
and constitute the $SO(N)$ Kac-Moody Wess-Zumino-Witten algebra at level 1. We
focus on the dihedral $D_{k}$ symmetry of $SO(2n)_{1}$, and its promotion to a
gauge symmetry by manipulating the locality of fermion pairs. Gauging the
symmetry (sub)group generates the $\mathcal{C}/G$ twist liquids, where
$G=\mathbb{Z}_{2}$ for $\mathcal{C}=U(1)_{l}$, $SU(n)_{1}$, and
$G=\mathbb{Z}_{2}$, $\mathbb{Z}_{k}$, $D_{k}$ for $\mathcal{C}=SO(2n)_{1}$. We
construct exactly solvable models for all of these topological states. We
prove the presence of a bulk excitation energy gap and demonstrate the
appearance of edge orbifold conformal field theories corresponding to the
twist liquid topological orders. We analyze the statistical properties of the
anyon excitations, including the non-Abelian metaplectic anyons and a new
class of quasiparticles referred to as Ising-fluxons. We show an eight-fold
periodic gauging pattern in $SO(2n)_{1}/G$ by identifying the non-chiral
components of the twist liquids with discrete gauge theories.

###### Contents

  1. I Introduction
    1. I.1 Summary of results
  2. II Boson wires from interacting electrons
  3. III Topological phases in the orthogonal and unitary families
    1. III.1 The $SO(N)_{1}$ family
    2. III.2 The $U(1)_{l}$ and $SU(n)_{1}$ family
    3. III.3 Gauging the $\mathbb{Z}_{k}$ or $\mathbb{Z}_{2}$ symmetries in $SO(2n)_{1}$
      1. III.3.1 The $SO(2n)_{1}/\mathbb{Z}_{2}$ twist liquid
      2. III.3.2 The $SO(2n)_{1}/\mathbb{Z}_{k}$ twist liquid
  4. IV Dihedral twist liquids
    1. IV.1 The $SU(n)_{1}/\mathbb{Z}_{2}$ twist liquid
    2. IV.2 The $U(1)_{l}/\mathbb{Z}_{2}$ twist liquid
    3. IV.3 The $SO(2n)_{1}/D_{k}$ twist liquid
  5. V Conclusion and discussion
  6. A The $SO(N)_{1}$ current algebra and its primary fields
  7. B The anyon content of $SO(n)_{2}$ and $U(1)_{l}/\mathbb{Z}_{2}$
  8. C Discrete gauge theories with cyclic, dihedral or dicyclic gauge groups
    1. C.1 The $\mathbb{Z}_{k}$ gauge theory
    2. C.2 The $D_{k}$ gauge theory
    3. C.3 The $Q_{4k}$ gauge theory
  9. D Useful group cohomology identities

##  I Introduction

Topological phases in two spatial dimensions are long-range entangled states
of quantum matter that support fractional quasiparticle excitations, known as
anyons (a review of these ideas can be found in ref. Wilczek, 1990; Wen, 2004;
Fradkin, 2013; Wen, 2017). The topological order of a topological phase is the
characterization of the fusion, exchange, and braiding properties of its
anyonic excitations. When symmetry is present, these phases are further
distinguished by their symmetry-protected or symmetry-enriched topological
(SPT/SET) orders. Chen  _et al._ (2012); Lu and Vishwanath (2012); Mesaros and
Ran (2013); Essin and Hermele (2013); Kapustin (2014); Bi  _et al._ (2015);
Else and Nayak (2014); Wang  _et al._ (2015); Lu and Vishwanath (2016);
Zaletel  _et al._ (2017); Chen (2017) Symmetries in topological phases can be
analyzed theoretically by studying the statistical properties of fluxes,
referred to as twist defects Kitaev (2006); Etingof  _et al._ (2010);
Barkeshli and Wen (2010a); Bombin (2010, 2011); Kitaev and Kong (2012); Kong
(2014a); You and Wen (2012); You  _et al._ (2013); Petrova  _et al._ (2014);
Barkeshli and Qi (2012, 2014); Barkeshli  _et al._ (2013); Teo  _et al._
(2014a, b); Khan  _et al._ (2014); Teo  _et al._ (2015); Bais and Haaker
(2015); Tarantino  _et al._ (2016); Teo  _et al._ (2016); Teo (2016);
Barkeshli  _et al._ (2019); Bridgeman  _et al._ (2020). These topological
point defects “rotate” the local winding low-energy degrees of freedom
according to the symmetry actions. For example, a twist defect associated to
an anyonic symmetry Khan  _et al._ (2014) (or outer automorphism) permutes the
anyon type of an orbiting quasiparticle excitation. Such defects must exhibit
non-Abelian statistical behaviors even when the underlying topological order
is Abelian, and may be useful in the construction of a topological quantum
computer Preskill (1997); Freedman (1998); Kitaev (2003); Walter Ogburn and
Preskill (1999); Preskill (2004); Freedman  _et al._ (2002a, b); Nayak  _et
al._ (2008); Wang (2010); Stern and Lindner (2013).

Twist liquids Teo  _et al._ (2015) are topological phases where (1) the
symmetries are “elevated” into local gauge symmetries, and (2) the gauge
fluxes are themselves deconfined quantum excitations. The promotion of a
(parent) SPT/SET phase to a (descendant) twist liquid phase is known as
gauging. Teo  _et al._ (2015); Barkeshli  _et al._ (2019) The fundamental
examples are discrete gauge theories Bais  _et al._ (1992); de Wild Propitius
(1995); de Wild Propitius and Bais (1995); Preskill (2004) $D^{[\omega]}(G)$
in 2+1D. These are twist liquids promoted from a short-range entangled (non-
fractional) SPT phase equipped with the symmetry group $G$. The cohomological
classification Chen  _et al._ (2011, 2013) $[\omega]$ of the parent SPT phase
determines the Dijkgraaf-Witten deformation Dijkgraaf and Witten (1990);
Dijkgraaf  _et al._ (1991); Altschuler and Coste (1992); Bais  _et al._
(1993); de Wild Propitius (1995) of the discrete gauge theory that dictates
the anyon braiding statistics Levin and Gu (2012) after gauging. Despite their
structural simplicity (excitations have integral quantum dimensions and their
braid groups have finite image Etingof  _et al._ (2008)), the exact evaluation
of link invariants in some gauge groups, such as the alternating group
$G=A_{m}$ for $m\geq 5$, are in the #P-complete computational complexity class
Krovi and Russell (2015). Anyons are capable of universal quantum computing
when the finite gauge group $G$ is non-solvable Mochon (2003) (e.g. $S_{3}$)
or non-nilpotent Mochon (2004) (e.g. $A_{5}$) if the gate-set involves
measurements Preskill (1997); Walter Ogburn and Preskill (1999); Bonderson
_et al._ (2008).

In general, a twist liquid $\mathcal{C}/G$ is the resulting phase when a
symmetry group $G$ of a long-range entangled (fractional) SET phase
$\mathcal{C}$ is gauged. In this paper, we focus on a family of chiral twist
liquids. In a chiral topological phase, the 1+1D conformal field theory
Ginsparg (1988a); Di Francesco  _et al._ (1999); Blumenhagen (2009) (CFT) that
describes the gapless edge excitations is identified with the unitary modular
tensor category Walker (1991); Turaev (1992); Bakalov and Kirillov (2001);
Fuchs  _et al._ (2002); Rowell (2005); Kitaev (2006); Bonderson (2007); Rowell
_et al._ (2009); Turaev (2010); Delaney (2019) (UMTC) that describes the 2+1D
gapped topological order by the bulk-edge correspondence Fröhlich and Gabbiani
(1990); Moore and Read (1991); Wen (1995); Read and Rezayi (1999); Kitaev
(2006). In a chiral twist liquid, symmetry gauging acts as an orbifolding
procedure Dixon  _et al._ (1985, 1986); Ginsparg (1988b); Dijkgraaf  _et al._
(1988); Moore and Seiberg (1989) on the edge CFT so that the bulk-edge
correspondence is preserved Chen  _et al._ (2017). The deconfined gauge fluxes
in the bulk associate to twist fields that change the boundary conditions on
the edge. Examples include orbifold fractional quantum Hall states Barkeshli
and Wen (2010b, c, 2011, 2012); Möller  _et al._ (2014); Kane and Stern
(2018); Tam  _et al._ (2020) and quantum spin liquids Levin and Gu (2012); Lu
and Vishwanath (2016). In this paper, with the interest on the topological and
modular properties, we do not distinguish between the twist liquid bulk UMTC
and its edge orbifold CFT, and we will refer to both as $\mathcal{C}/G$.

While non-chiral twist liquids can be constructed microscopically from
exactly-solvable string-net models Teo  _et al._ (2015); Barkeshli  _et al._
(2019); Levin and Wen (2005) and this provides a proof of principle that the
orbifold topological order can be supported by a gapped local Hamiltonian, the
method is not designed to analyze chiral electronic twist liquid phases.
Moreover, the fundamental relationship between symmetry gauging and the change
of locality is obscure. On one hand, the gauge charges in a twist liquid are
non-local and fractional because they carry non-trivial mutual braiding
statistics with the gauge fluxes. On the other hand, they are derived from
field operators in the parent globally-symmetric SET phase that are local
integral combinations of electrons and “condense” in the anyon condensation
Bais and Slingerland (2009); Kong (2014b); Neupert  _et al._ (2016); Burnell
(2018) picture. In other words, gauging is the reversal of gauge charge
condensation. In this paper, we seek a microscopic model description based on
electrons that explicitly demonstrates such reconfiguration of gauge charge
locality. At the same time, we explore models that display the bulk-edge
correspondence between chiral twist liquid bulk topological order and edge
orbifold CFT.

The coupled-wire construction Kane  _et al._ (2002); Teo and Kane (2014)
provides a strategy to obtain exactly-solvable model Hamiltonians for these
purposes. The theoretical technique was inspired by sliding Lutthinger liquids
O’Hern  _et al._ (1999); Emery  _et al._ (2000); Vishwanath and Carpentier
(2001); Sondhi and Yang (2001); Mukhopadhyay  _et al._ (2001) and was first
implemented by Kane, Mukhopadhyay, and Lubensky Kane  _et al._ (2002) to
describe the Abelian Laughlin Laughlin (1983) and Haldane-Halperin hierarchy
Haldane (1983); Halperin (1984) fractional quantum Hall states. The method was
later generalized to other (Abelian or non-Abelian, integer or fractional)
quantum Hall states Teo and Kane (2014); Klinovaja and Loss (2014); Meng  _et
al._ (2014); Sagi  _et al._ (2015); Fuji  _et al._ (2016); Kane  _et al._
(2017); Fuji and Lecheminant (2017); Kane and Stern (2018); Fuji and Furusaki
(2019a); Sirota  _et al._ (2019); Fontana  _et al._ (2019); Lopes  _et al._
(2019); Imamura  _et al._ (2019); Tam  _et al._ (2020); Tam and Kane (2021),
anyon models Oreg  _et al._ (2014); Stoudenmire  _et al._ (2015); Iadecola
_et al._ (2019); Tam  _et al._ (2021), spin liquids Meng  _et al._ (2015);
Gorohovsky  _et al._ (2015); Huang  _et al._ (2016); Patel and Chowdhury
(2016), (fractional) topological insulators Neupert  _et al._ (2014);
Klinovaja and Tserkovnyak (2014); Sagi and Oreg (2014); Mross  _et al._
(2015); Santos  _et al._ (2015); Raza  _et al._ (2019); Han and Teo (2019) and
superconductors Mong  _et al._ (2014); Seroussi  _et al._ (2014); Sahoo  _et
al._ (2016); Hu and Kane (2018); Park  _et al._ (2018); Cheng (2018); Yang
_et al._ (2020), fracton models Sullivan  _et al._ (2021a, b) and higher-
dimensional fractional phases Iadecola  _et al._ (2016); Fuji and Furusaki
(2019b); Sagi and Oreg (2015); Meng (2015); Meng  _et al._ (2016), as well as
the exploration of dualities Mross  _et al._ (2016, 2017) and quantum
entanglement Cano  _et al._ (2015); Sohal  _et al._ (2020); Kau Lim  _et al._
(2021). The coupled-wire construction is a highly anisotropic microscopic
description where the low-energy electronic degrees of freedom are confined on
a 2D array of one-dimensional wires. Local many-body intra- and inter-wire
electron backscattering interactions are responsible for the finite excitation
energy gap in the bulk but leave behind gapless modes on the edge.

A universal procedure of constructing a coupled-wire model for a general UMTC
has not been established, and it is out of the scope of this paper to
construct electronic models for general twist liquids. Among orbifold phases
$\mathcal{C}/G$, there is a sub-collection where the parent $G$-symmetric SET
phase $\mathcal{C}$ corresponds to an edge CFT described by an affine Kac-
Moody Kac (1968); Moody (1968) (KM) Wess-Zumino-Witten Wess and Zumino (1971);
Witten (1983, 1984) (WZW) current Lie algebra. These WZW phases are of
significant interest in the coupled-wire construction because (1) some of the
many-body electron interactions can be introduced as backscattering of the
local KM currents (such as the Gross-Neveu interactions Gross and Neveu
(1974); Zamolodchikov and Zamolodchikov (1978); Witten (1978); Shankar and
Witten (1978)), and (2) some non-local KM current operators can be associated
with the gauge charges in the twist liquid. In particular, when $\mathcal{C}$
is an affine simply-laced KM WZW Lie algebra at level 1 (i.e. the
$A_{r}=SU(r+1)$, $D_{r}=SO(2r)$ series and the exceptional $E_{6}$, $E_{7}$,
$E_{8}$), it has an Abelian topological order. $N$-dimensional real (complex)
point groups are discrete finite subgroups $G$ in $SO(N)$ ($SU(N)$), and they
correspond to the orbifold phases $SO(N)_{k}/G$ ($SU(N)_{k}/G$), where $k$ is
the level of the WZW algebra. 3-dimensional point group orbifold CFTs
$SU(2)_{1}/G$ are known Ginsparg (1988b); Dijkgraaf  _et al._ (1988). In this
paper, we focus on gauging a dihedral symmetry group $D_{k}$ in $SO(2n)_{1}$.
In doing so, the construction also encompasses the anyon relabeling
$\mathbb{Z}_{2}$ conjugation symmetry in $U(1)_{l}$ and $SU(n)_{1}$.

We summarize the main results of this paper below. We discuss the implications
and possible future directions in section V.

###  I.1 Summary of results

In this paper, we investigate various aspects and consequences of gauging a
non-Abelian dihedral symmetry group $D_{k}$ of topological phases of matter in
the orthogonal family. The results we achieved are threefold.

First, we present two different ways of constructing a one-dimensional quantum
wire of gapless bosons with an effective $SO(N)$ symmetry in low energy. We
begin with a one-dimensional quantum wire of interacting spinful electrons. By
taking advantage of either an umklapp scattering term or a superconducting
pairing term, we lift all charge modes and local fermion excitations above a
finite energy gap. The residual low-energy degrees of freedom below the energy
gap are gapless bosons, which can be effectively described by the $SO(N)_{1}$
WZW CFT. These boson wires serve as the basic building blocks for the
subsequent coupled wire constructions of the globally symmetric “parent”
topological order phases ($\mathcal{C}$) and their gauged and locally
symmetric “descendant” counterparts (twist liquid / orbifold phases
$\mathcal{C}/G$) in the orthogonal and unitary family. They include (1)
$\mathcal{C}=SO(2n)_{1}$ and its $G=\mathbb{Z}_{2}$, $\mathbb{Z}_{k}$, or the
non-Abelian $D_{k}=\mathbb{Z}_{2}\ltimes\mathbb{Z}_{k}$ twist liquid phases,
(2) $\mathcal{C}=U(1)_{l}$, $SU(n)_{1}$, or $U(1)_{l}\times SU(n)_{1}$
(conformally embedded as WZW subalgebras of $SO(2n)_{1}$) and their
$G=\mathbb{Z}_{2}$ twist liquid phases. The degree, rank, and level are
related by $k=n$ and $l=4n$ (or $k=n/2$ and $l=n$) when $n$ is odd (resp.
even). The level $l$ of $U(1)_{l}$ corresponds to the radius $R=\sqrt{l}/2$ of
the compactified free boson CFT at central charge $c=1$. The coupled-wire
constructions based on the umklapp (pairing) approach lead to a topologically
ordered spin liquid (superconductor).

Figure 1: Coupled-wire models for (a) $\mathcal{C}=U(1)_{l}$ or $SU(n)_{1}$
Abelian topological states, and (b)
$\mathcal{C}/\mathbb{Z}_{2}=U(1)_{l}/\mathbb{Z}_{2}$ or
$SU(n)_{1}/\mathbb{Z}_{2}$ twist liquid topological states from bosonic
$SO(N)_{1}$ wires. Intra-/inter-wire interactions $\mathcal{U}^{y}$,
$\mathcal{U}^{y+1/2}$ introduce a finite bulk excitation energy gap and leave
behind chiral gapless boundaries described by CFTs that correspond to the bulk
topological orders. Yellow boxes represent splitting of channels by (a) the
conformal embedding $SO(2n)_{1}=U(1)_{l}\times SU(n)_{1}$, and (b) the coset
decomposition $SO(n)_{1}^{2}=SO(n)_{2}\times[SO(n)_{1}^{2}/SO(n)_{2}]$.

Since any intra- and inter-wire coupling term in a coupled-wire Hamiltonian
has to be made out of integral combinations of local electrons, it is crucial
to keep track of the locality of field operators and their physical origin.
This is our second key result. Starting from our wires of bosons, upon re-
fermionization, each wire carries $N$ pairs of counter-propagating Majorana
fermions. Different from the local electronic Majorana fermions based on the
Bogoliubov–de Gennes (BdG) formalism in a conventional
Bardeen–Cooper–Schrieffer (BCS) superconductor, our Majorana fermions here are
emergent and fractional. Only even fermion products are integral combinations
of local electrons. More importantly, we explicitly connect symmetry gauging
and the modification of the locality of field operators. For a general twist
liquid phase $\mathcal{C}/G$, local fields in $\mathcal{C}/G$ are exclusively
$G$-invariant local fields in $\mathcal{C}$. Taking $\mathcal{C}=SO(2n)_{1}$
as an example, for its Majorana fermions $\psi^{i=1,\cdots,2n}$, the
$\mathbb{Z}_{2}$ symmetry sends
$\psi_{A}^{i=1,\cdots,n}\equiv\psi^{i}\rightarrow\psi_{A}^{i}$ and
$\psi_{B}^{i=1,\cdots,n}\equiv\psi^{i+n}\rightarrow-\psi_{B}^{i}$. At the same
time, the $\mathbb{Z}_{k}$ symmetry adds a phase to the Dirac fermion
$d^{i}\sim\psi_{A}^{i}+i\psi_{B}^{i}\rightarrow e^{i2\pi/k}d^{i}$. While
$\psi_{A}^{i}\psi_{B}^{i}$ are integral in $SO(2n)_{1}$, they become
fractional in $SO(2n)_{1}/\mathbb{Z}_{2}$ because they are not invariant under
the internal $\mathbb{Z}_{2}$ symmetry. Similarly, the Dirac pairs
$d^{i}d^{j}$ are not local fields in $SO(2n)_{1}/\mathbb{Z}_{k}$ because their
phases are rotated by the internal $\mathbb{Z}_{k}$ symmetry. Moving on to the
non-Abelian $D_{k}$ symmetry, we find that the

$\displaystyle\frac{SU(n)_{1}}{\mathbb{Z}_{2}}=SO(n)_{2}\quad\mbox{and}\quad\frac{U(1)_{l}}{\mathbb{Z}_{2}}=\frac{SO(n)_{1}\times
SO(n)_{1}}{SO(n)_{2}}$ (1)

topological orders are based on local fields of $SO(2n)_{1}$ that are
invariant under the $D_{k}$ symmetry. Inside these topological phases, we
highlight the existence of metaplectic anyons in the form of $\mathbb{Z}_{2}$
fluxes, when $n$ is odd, together with a microscopic electronic Hamiltonian
from the coupled-wire construction. (See figure 1 for a diagrammatic outline
of the coupled-wire models and figure 2 for a summary of the gauging relations
between different twist liquids constructed in this paper.) Moreover, we
discover a novel type of anyon with quantum dimension $\sqrt{2}n$ in

$\displaystyle\frac{SO(2n)_{1}}{D_{k}}=\frac{U(1)_{l}\times
SU(n)_{1}}{\mathbb{Z}_{2}}$ (2)

when $n$ is odd. It is a composite quasiparticle that combines twist fields
from both the $SO(n)_{2}$ and $U(1)_{l}/\mathbb{Z}_{2}$ theories. We dubbed it
an “Ising-fluxon”.

Figure 2: Gauging flowchart of $SO(2n)_{1}$ and twist liquid orbifold phases
constructed in this paper.

Third, we provide a systematic characterization of the $G$-twist liquids
($G=\mathbb{Z}_{2}$, $\mathbb{Z}_{k}$, or $D_{k})$ at the level of unitary
modular tensor categories (UMTC). Loosely speaking Teo  _et al._ (2015),
anyons in $\mathcal{C}/G$ are super-selection combinations of anyons in
$\mathcal{C}$ attached with deconfined $G$-gauge fluxes and gauge charges. If
$G$ contains only inner automorphisms that do not permute anyon types, such as
the case when $\mathcal{C}=SO(2n)_{1}$ and (i) $G=\mathbb{Z}_{k}$ for all $n$,
and (ii) $G=\mathbb{Z}_{2}$, or $D_{k}$ for even $n$, we demonstrate the UMTC
equivalence

$\displaystyle SO(2n)_{1}/G=SO(2n)_{1}\boxtimes D^{[\omega]}(\widehat{G}).$
(3)

Here, the quantum symmetry group $\widehat{G}$ is a subgroup of an extension
of $G$ by the Abelian fusion group $\mathcal{A}$ of anyons in $\mathcal{C}$.
In general, the group extension can be projective Wen (2002); Etingof  _et
al._ (2010); Barkeshli  _et al._ (2019) (i.e. non-symmorphic Teo  _et al._
(2015)) and is classified by the group cohomology Brown (1982)
$H^{2}(G,\mathcal{A})$. $D^{[\omega]}(\widehat{G})$ denotes the discrete
$\widehat{G}$ gauge theory Bais  _et al._ (1992); de Wild Propitius (1995); de
Wild Propitius and Bais (1995); Preskill (2004) deformed by the Dijkgraaf-
Witten invariant Dijkgraaf and Witten (1990); Dijkgraaf  _et al._ (1991);
Altschuler and Coste (1992); Bais  _et al._ (1993); de Wild Propitius (1995)
$[\omega]$ in $H^{3}(\widehat{G},U(1))$. The invariant $[\omega]$ specifies
the quantum basis transformations (i.e. associations) between flux
combinations with different orders,
$\hat{g}_{1}(\hat{g}_{2}\hat{g}_{3})\cong(\hat{g}_{1}\hat{g}_{2})\hat{g}_{3}$,
and in turn, dictates the spin-exchange statistics of gauge fluxes. The tensor
product $\boxtimes$ is relative to a set of bosonic pairs of anyons from both
sides that are in fact local and are “condensed” in the anyon condensation
picture Bais and Slingerland (2009); Kong (2014b); Neupert  _et al._ (2016);
Burnell (2018). (The decoupled product $\times$ is a special case when there
is no anyon pair other than the vacuum is “condensed”.) Projective quantum
symmetry groups $\widehat{G}$ relevant to this paper include (i)
$\widehat{\mathbb{Z}_{2}}=\mathbb{Z}_{4}$, $\widehat{D_{k}}=Q_{4k}$ when
$n=2k\equiv 2$ modulo 4, and (ii) $\widehat{\mathbb{Z}_{k}}=\mathbb{Z}_{2k}$,
$\widehat{D_{k}}=D_{2k}$ when $n=2k\equiv 0$ modulo 4. In particular, the
dicyclic group $Q_{4k}$ is the double cover of the dihedral group $D_{k}$ that
contains the perpendicular axes of $k$-fold and 2-fold rotations in a half-
integral spin representation.

On the other hand, the $\mathbb{Z}_{2}$ symmetry is an outer automorphism for
$U(1)_{l}$ and $SU(n)_{1}$. The twofold symmetry non-trivially transposes
anyon classes by conjugation. Gauging these symmetries then takes us beyond
conventional discrete gauge theories. This, in general, enables non-Abelian
quasiparticles with non-integral quantum dimensions, such as the
$\mathbb{Z}_{2}$ twist fields in the two prototypical $\mathbb{Z}_{2}$ twist
liquid phases in (1). For $SO(2n)_{1}$ with odd rank $n=k$, the
$\mathbb{Z}_{2}$, and subsequently, the $D_{k}$ symmetries contain outer
automorphisms (also known as anyon relabeling symmetries Khan  _et al._
(2014); Teo (2016)). Upon gauging, we show

$\displaystyle SO(2n)_{1}/G=SO(2n)_{1}\boxtimes\mathcal{Z}_{n}(G),$ (4)

where $\mathcal{Z}_{n}(G)$ is a non-chiral quantum double Kassel (1995);
Bakalov and Kirillov (2001); Levin and Wen (2005) carrying a $G$-gauge
symmetry. $\mathcal{Z}_{n}(\mathbb{Z}_{2})$ has a non-chiral Ising topological
order and is the twist liquid phase from gauging the twofold electric-magnetic
symmetry of the $\mathbb{Z}_{2}$ discrete gauge theory. Barkeshli and Wen
(2012); Teo  _et al._ (2015); Chen  _et al._ (2017) $\mathcal{Z}_{n}(D_{k})$
is the twist liquid phase of gauging a mixed $\mathbb{Z}_{2}$ symmetry of the
$\mathbb{Z}_{k}$ discrete gauge theory. It supports the Ising-fluxon
excitations and is referred to as the Ising-fluxon topological phase in this
article. Combining both even and odd $n$ cases in (3) and (4), we observe an
eight-fold periodic pattern of the topological orders of the twist liquids
$SO(2n)_{1}/G$.

The paper is organized as the following. In section II, we present the
superconductor and spin liquid constructions of 1+1D gapless boson wires from
interacting spinful electrons. The $SO(N)_{1}$ WZW symmetry, locality and
internal $\mathbb{Z}_{2}$ symmetry of field operators within a single wire are
also discussed. In section III, we proceed to describe the coupled-wire
constructions for 2+1D topological phases of the orthogonal and unitary
families. The $SO(N)_{1}$ models are presented in section III.1 while the
$U(1)_{l}$ and $SU(n)_{1}$ models are presented in section III.2. The global
$D_{k}$ symmetry of the “parent” $SO(2n)_{1}$ phase is described in section
III.3. Its $\mathbb{Z}_{2}$ and $\mathbb{Z}_{k}$ twist liquid “descendants”
$SO(2n)_{1}/\mathbb{Z}_{2}=SO(n)_{1}^{2}$ and
$SO(2n)_{1}/\mathbb{Z}_{k}=U(1)_{l}\times SU(n)_{1}$ are introduced in section
III.3.1 and III.3.2 respectively. In section IV, we develop the dihedral twist
liquid phase $SO(2n)_{1}/D_{k}=\left[U(1)_{l}\times
SU(n)_{1}\right]/\mathbb{Z}_{2}$. Its construction relies on the simpler
$SU(n)_{1}/\mathbb{Z}_{2}=SO(n)_{2}$ and
$U(1)_{l}/\mathbb{Z}_{2}=SO(n)_{1}^{2}/SO(n)_{2}$ orbifold phases, which are
presented in section IV.1 and IV.2. For each case, the relation between
locality and internal symmetries of field operators is carefully examined at
the beginning. Then, we explicitly present the exactly solvable intra- and
inter-wire interaction terms from local electrons. The interactions introduce
a finite 2D bulk excitation energy gap but leave behind gapless edge CFT that
corresponds to the bulk topological order. The novel eight-fold periodic
topological order of $SO(2n)_{1}/D_{k}$ is fully narrated in section IV.3.

While the new and essential content is presented in the main sections, we
include comprehensive revisions on the relevant background in four appendices.
The WZW KM algebras and their primary field modular contents of $SO(N)_{1}$,
$SO(n)_{2}$ and the $U(1)_{l}/\mathbb{Z}_{2}$ orbifold CFT are reviewed in
appendix A and B. Discrete gauge theories with cyclic, dihedral, and dicyclic
gauge groups, $\mathbb{Z}_{k}$, $D_{k}$ and $Q_{4k}$, appear as non-chiral
components of the various $SO(2n)_{1}/G$ twist liquids introduced in the main
sections. They are reviewed in appendix C. The group extensions and Dijkgraaf-
Witten invariants relevant to the discrete gauge theories are classified by
the group cohomologies $H^{2}(G,\mathcal{A})$ and $H^{3}(G,U(1))$. We present
these cohomology classifications and the explicit cycle representations in
appendix D.

##  II Boson wires from interacting electrons

The building blocks of the coupled-wire models are gapless bosons in one
spatial dimension that carry an effective $SO(N)$ symmetry in low energy.
These wires can be realized by 1D electrons under strong interactions that
lift all charge modes and local fermion excitations above a finite energy gap.
The remaining electrically neutral gapless modes below the gap are described
by a $SO(N)$ Wess-Zumino-Witten Wess and Zumino (1971); Witten (1983, 1984)
(WZW) Kac-Moody Kac (1968); Moody (1968) (KM) conformal field theory Ginsparg
(1988a); Di Francesco  _et al._ (1999); Blumenhagen (2009) (CFT) at level 1.
We illustrate this using $\mathcal{N}$ copies of Rashba spin-orbit coupled
electrons. The electron operators $c_{sj}(x)\approx
c_{sj}^{L}(x)e^{i(k_{f}+sk_{SO})x}+c_{sj}^{R}(x)e^{i(-k_{f}+sk_{SO})x}$ near
the Fermi level can be represented by vertex operators of bosonized variables
$c_{sj}^{\sigma}(x)\sim e^{i\Phi_{sj}^{\sigma}(x)}$, where
$s=\uparrow(\downarrow)=+(-)$ labels spins, $\sigma=L(R)=+(-)$ specifies the
left/right propagation directions, $j=1,\ldots,\mathcal{N}$ designates the
electron channels, and the Fermi momenta of the channels are
$k_{s}^{\sigma}=\sigma k_{f}+sk_{SO}$. The low-energy kinetic behavior of the
electrons can be described by the Luttinger liquid Tomonaga (1950); Luttinger
(1963) Lagrangian density

$\displaystyle\begin{split}\mathcal{L}_{0}&=\frac{1}{4\pi}\sigma\delta_{\sigma\sigma^{\prime}}\delta^{ss^{\prime}}\delta^{jj^{\prime}}\partial_{x}\Phi_{sj}^{\sigma}\partial_{t}\Phi_{s^{\prime}j^{\prime}}^{\sigma^{\prime}}-\mathcal{H}_{0},\\\
\mathcal{H}_{0}&=v^{sj,s^{\prime}j^{\prime}}_{\sigma\sigma^{\prime}}\partial_{x}\Phi_{sj}^{\sigma}\partial_{x}\Phi_{s^{\prime}j^{\prime}}^{\sigma^{\prime}}.\end{split}$
(5)

The Hamiltonian density
$\mathcal{H}_{0}=\mathcal{H}_{\mathrm{Dirac}}+\mathcal{H}_{\mathrm{int}}$
includes the single-body massless Dirac theory
$\mathcal{H}_{\mathrm{Dirac}}=iv\sigma{c_{sj}^{\sigma}}^{\dagger}\partial_{x}c_{sj}^{\sigma}=\frac{v}{4\pi}(\partial_{x}\Phi^{\sigma}_{sj})^{2}$
and density-density interactions
$\mathcal{H}_{\mathrm{int}}=u^{sj,s^{\prime}j^{\prime}}_{\sigma\sigma^{\prime}}n_{sj}^{\sigma}n_{s^{\prime}j^{\prime}}^{\sigma^{\prime}}$,
where
$n_{sj}^{\sigma}={c_{sj}^{\sigma}}^{\dagger}c_{sj}^{\sigma}=\sigma\partial_{x}\Phi_{sj}^{\sigma}/(2\pi)$.
The “$p\dot{q}$” term of the Lagrangian sets the equal-time commutation
relation

$\displaystyle\left[\Phi^{\sigma}_{sj},\Phi^{\sigma^{\prime}}_{s^{\prime}j^{\prime}}\right]=2\pi
i\sigma\delta^{\sigma\sigma^{\prime}}\delta_{ss^{\prime}}\delta_{jj^{\prime}}\delta(x-x^{\prime}).$
(6)

The charged $U(1)$ sector can be gapped by either a $\mathcal{N}$-body umklapp
scattering or a superconducting pairing

$\displaystyle\mathcal{U}_{\mathrm{umklapp}}=u\cos\theta_{\rho}\sim
u\prod_{j=1}^{\mathcal{N}}\prod_{s=\uparrow,\downarrow}{c_{sj}^{L}}^{\dagger}c_{sj}^{R}+h.c.,$
$\displaystyle\mathcal{U}_{\mathrm{SC}}=\Delta\cos\varphi_{\rho}\sim\Delta\prod_{j=1}^{\mathcal{N}}\prod_{s=\uparrow,\downarrow}c_{sj}^{L}c_{sj}^{R}+h.c.,$
(7)
$\displaystyle\varphi_{\rho}=\sum_{sj}\left(\Phi^{L}_{sj}+\Phi^{R}_{sj}\right),\quad\theta_{\rho}=\sum_{sj}\left(\Phi^{L}_{sj}-\Phi^{R}_{sj}\right).$

This can be shown by re-expressing the kinetic Lagrangian density (5) in terms
of new variables

$\displaystyle\mathcal{L}_{0}$
$\displaystyle=\frac{1}{4\pi}\sum_{\sigma=+,-}\sigma\left[2\partial_{x}\phi_{\rho}^{\sigma}\partial_{t}\phi_{\rho}^{\sigma}+\sum_{a=1}^{2\mathcal{N}-1}\partial_{x}\phi_{a}^{\sigma}\partial_{t}\phi_{a}^{\sigma}\right]-\mathcal{H}_{0}.$
(8)

This can be achieved by using the basis transformation

$\displaystyle
2\phi_{\rho}^{\sigma}=\varphi_{0}+\sigma\theta_{\rho},\quad\phi^{\sigma}_{a}=\varphi_{0}-\Phi^{\sigma}_{s_{a}j_{a}},$
(9)

for the umklapp scattering case, or

$\displaystyle
2\phi_{\rho}^{\sigma}=\varphi_{\rho}+\sigma\theta_{0},\quad\phi^{\sigma}_{a}=\theta_{0}-\sigma\Phi^{\sigma}_{s_{a}j_{a}},$
(10)

for the superconducting case. Here $a=1,\ldots,2\mathcal{N}-1$ enumerate the
spins $s=\uparrow,\downarrow$ and channels $j=1,\ldots,\mathcal{N}$ but
exclude $(s,j)=(\uparrow,1)$ for the $\sigma=L$ sector and $(\downarrow,1)$
for the $\sigma=R$ sector, and

$\displaystyle\varphi_{0}=\frac{\Phi^{L}_{\uparrow 1}+\Phi^{R}_{\downarrow
1}}{2},\quad\theta_{0}=\frac{\Phi^{L}_{\uparrow 1}-\Phi^{R}_{\downarrow
1}}{2}.$ (11)

With a particular set of electron density interactions, the Hamiltonian
density takes a diagonal form

$\displaystyle\begin{split}\mathcal{H}_{0}&=\frac{v}{4\pi}\sum_{\sigma=L,R}\sum_{a=1}^{2\mathcal{N}-1}\left(\partial_{x}\phi_{a}^{\sigma}\right)^{2}+v_{\rho}\left[g_{\rho}\left(\partial_{x}\varphi\right)^{2}+\frac{1}{g_{\rho}}\left(\partial_{x}\theta\right)^{2}\right]\end{split}$
(12)

where $(\varphi,\theta)=(\varphi_{0},\theta_{\rho})$ for the umklapp case or
$(\varphi_{\rho},\theta_{0})$ for the superconducting case.

The umklapp scattering (7) requires translation symmetry breaking and can
arise in a half-filled lattice when the Fermi momentum is commensurate,
$4k_{f}=2\pi/a$, where $a$ is a microscopic lattice translation. The process
is relevant in the renormalization group Giamarchi (2003); Sénéchal (2004);
Tsvelik (2007); Fradkin (2013) (RG) sense when $g_{\rho}<4$ and leads to a
spin liquid. The charge-violating pairing (7) can be induced by proximity with
a bulk $s$-wave superconductor and is relevant when $g_{\rho}>1/4$. The
unaffected low-energy theory is generated by $2\mathcal{N}-1$ counter-
propagating pairs of Dirac fermions $d_{a}^{\sigma}\sim
e^{i\phi_{a}^{\sigma}}$, which are non-local and must come in pairs. Their
fractional nature is revealed by their charge and spin numbers
$Q(e^{ia^{sj}_{\sigma}\Phi^{\sigma}_{sj}})=e\sum_{sj\sigma}a^{sj}_{\sigma}$,
$S(e^{ia^{sj}_{\sigma}\Phi^{\sigma}_{sj}})=\sum_{sj\sigma}sa^{sj}_{\sigma}/2$.
Contrary to an electron/hole which carries charge $Q=\pm e$ and angular spin
$S=\pm 1/2$, the Dirac fermion $d_{a}$ has $Q=0$ and $S=\pm 1/2$ ($Q=e$ modulo
$2e$ and $S=0,\pm 1$) in the spin liquid (resp. superconductor). In the spin
liquid, the momentum of these Dirac fermions becomes 0 or $\pi/a$ when the
electron Fermi level is tuned near the band crossing of the two spins so that
$k_{f}=k_{SO}$. In the superconductor, the momentum is zero when spin-orbit
coupling is negligible $k_{SO}=0$. The Dirac fermions split into Majorana
components $\psi=\psi^{\dagger}$ by the real and imaginary decomposition

$\displaystyle
d_{a}^{\sigma}=\frac{1}{\sqrt{2}}\left(\psi_{2a-1}^{\sigma}+i\psi_{2a}^{\sigma}\right)\sim
e^{i\phi^{\sigma}_{a}}.$ (13)

A subset of counter-propagating pairs can turn massive and be removed from
low-energy by introducing the fermion backscattering

$\displaystyle\mathcal{U}_{\mathrm{mass}}$
$\displaystyle=im\sum_{j=N+1}^{4\mathcal{N}-2}\psi_{j}^{L}\psi_{j}^{R},$ (14)

where $\psi^{L}\psi^{R}$ are a local integral combination of electrons
operators. In energy much smaller than $m$ and $u$ or $\Delta$, the
Hamiltonian (12) is effectively truncated to a theory with $N$ counter-
propagating pairs of massless Majorana fermions

$\displaystyle\mathcal{H}_{0}=\frac{iv}{2}\sum_{\sigma=+,-}\sum_{j=1}^{N}\sigma\psi_{j}^{\sigma}\partial_{x}\psi_{j}^{\sigma}.$
(15)

The low-energy theory (15) has an emergent $SO(N)$ symmetry and is invariant
under an orthogonal transformation $\psi_{j}\to\mathcal{O}_{j}^{i}\psi_{i}$ of
the fermion vector. Each chiral sector $\sigma=L,R$ carries a $SO(N)$ WZW KM
algebra at level 1 generated by the current operators
$J^{\sigma}_{jk}=i\psi_{j}^{\sigma}\psi_{k}^{\sigma}$, for $1\leq j<k\leq N$
(see appendix A for a review). The conformal primary fields are vector and
spinor representations of $SO(N)$. From (9) or (10), the vertex operator
representation of the Dirac fermion $d_{a}^{\sigma}\sim
e^{i\phi^{\sigma}_{a}}=e^{ia^{sj}_{\sigma\sigma^{\prime}}\Phi^{\sigma^{\prime}}_{sj}}$
in terms of the electronic bosonized variables $\Phi^{\sigma}_{sj}$ contains
the half-integral coefficients $a^{\uparrow 1}_{L},a^{\downarrow 1}_{R}=\pm
1/2$. The electronic origin dictates that the Dirac fermions $d_{a}^{\sigma}$
and Majorana fermions $\psi_{j}^{\sigma}$ are non-local fractional operators
and cannot exist individually. On the other hand, all even products of
fermions, such as the $SO(N)_{1}$ current operators and the fermion
backscattering $\psi_{j}^{L}\psi_{k}^{R}$, are linear combination of integral
products of electron operators, and are therefore local. A spinor field can be
represented by a vertex operator

$\displaystyle
s_{\boldsymbol{\varepsilon}}^{\sigma}=e^{i\varepsilon^{a}\phi^{\sigma}_{a}/2},$
(16)

where
$\boldsymbol{\varepsilon}=(\varepsilon^{1},\ldots,\varepsilon^{2\mathcal{N}-1})$
has unit entries $\varepsilon^{a}=\pm 1$. The spinor fields separate into even
and odd primary sectors $[s^{\sigma}_{\pm}]$ distinguished by the parity
$\prod_{a}\varepsilon^{a}=\pm 1$. A spinor pair of the same chirality
$s^{\sigma}_{\boldsymbol{\epsilon}}s^{\sigma}_{-\boldsymbol{\epsilon}^{\prime}}$
is a local integral combination of electrons when
$\prod_{a}\varepsilon_{a}\varepsilon^{\prime}_{a}=1$, or is equivalent to a
non-local odd fermion combination when
$\prod_{a}\varepsilon_{a}\varepsilon^{\prime}_{a}=-1$. Moreover, when
projected onto the ground state Hilbert space of the umklapp (or pairing)
potential in (7), the corresponding sine-Gordon variable
$\langle\theta_{\rho}\rangle$ (resp. $\langle\varphi_{\rho}\rangle$) admits a
ground state expectation value. Consequently, all pair of spinor fields with
the same parity but opposite chirality
$s^{L}_{\boldsymbol{\epsilon}}s^{R}_{\boldsymbol{\epsilon}^{\prime}}$ is
effectively local in low-energy because it is equivalent to an integral
combination of electrons up to the vertex operator $e^{i\theta_{\rho}/2}$
($e^{i\varphi_{\rho}/2}$), which becomes a $U(1)$ phase when projected onto
the ground state. In addition, when projecting to the ground states of (14),
the local spinor pairs
$s^{\sigma}_{\boldsymbol{\varepsilon}}s^{\sigma}_{-\boldsymbol{\varepsilon}^{\prime}}$
and
$s^{L}_{\boldsymbol{\varepsilon}}s^{R}_{-\boldsymbol{\varepsilon}^{\prime}}$
operate as pairs of spinor twist fields in the low-energy $SO(N)_{1}$ sector
if the ground state projection is non-vanishing. In general, an operator is
effectively integral and local if and only if it is invariant under the
internal $\mathbb{Z}_{2}$ gauge symmetry at any given wire $y$

$\displaystyle\mathbb{Z}_{2}(y):\quad\psi_{y^{\prime}j}^{\sigma}\to(-1)^{\delta_{yy^{\prime}}}\psi_{y^{\prime}j}^{\sigma},\quad\phi^{\sigma}_{y^{\prime}a}\to\phi^{\sigma}_{y^{\prime}a}+\sigma\pi\delta_{yy^{\prime}}.$
(17)

The bosonic $SO(N)_{1}$ wire in (15) will be the building blocks of the 2D
coupled-wire models described in the following sections. After an energy gap
has been established by inter-wire interactions, the topological phase will be
stable against perturbations smaller than the energy gap. In particular, the
fine-tuning of the electron density interactions
$u^{sjs^{\prime}j^{\prime}}_{\sigma\sigma^{\prime}}$ and the Fermi level that
enable the bosonic $SO(N)_{1}$ theory can be relaxed in the upcoming 2D
topological model.

##  III Topological phases in the orthogonal and unitary families

We construct exactly-solvable coupled-wire models of topological phases in two
spatial dimensions with $SO(N)_{1}$, $U(1)_{l}$, and $SU(n)_{1}$ topological
orders. The models are built on a 2D array of boson wires, each being
effectively described by a $SO(N)_{1}$ WZW KM CFT (15) in low-energy and
originated from strongly correlated electrons. In a closed torus geometry, a
finite excitation energy gap is created by backscattering interactions between
neighboring wires. In an open geometry, the 1D boundary hosts chiral gapless
degrees of freedom that propagate in a single forward direction and are
effectively described by a CFT corresponding to the topological order Fröhlich
and Gabbiani (1990); Moore and Read (1991); Wen (1995); Read and Rezayi
(1999); Kitaev (2006). We show the topological phases in the orthogonal and
unitary families are related by gauging Teo  _et al._ (2015); Barkeshli  _et
al._ (2019) (also referred to as orbifolding Dixon  _et al._ (1985, 1986);
Ginsparg (1988b); Dijkgraaf  _et al._ (1988); Moore and Seiberg (1989)) the
discrete $\mathbb{Z}_{2}$ and $\mathbb{Z}_{k}$ symmetries

$\displaystyle\begin{split}\frac{SO(2n)_{1}}{\mathbb{Z}_{2}}&=SO(n)_{1}\times
SO(n)_{1}\\\ \frac{SO(2n)_{1}}{\mathbb{Z}_{k}}&=U(1)_{l}\times
SU(n)_{1}\end{split}$ (18)

where $k=n$ ($k=n/2$) and $l=4n$ ($l=n$) for odd (even) $n$.

###  III.1 The $SO(N)_{1}$ family

The model begin with a 2D array of bosonic $SO(N)_{1}$ wires. All the wires
are parallel to the horizontal $x$ direction and each one is labeled by its
vertical $y$ position, for $y=1,2,\ldots,\mathsf{L}$. The array can occupy an
open plane with top (bottom) edges at $y=\mathsf{L}$ ($y=1$), or it can wrap
around a periodic cylinder geometry when $y\equiv y+\mathsf{L}$. The boson
wires can emerge from strongly interacting 1D electrons in a spin liquid or a
superconducting setting presented in the previous section. In both cases, all
fermionic local excitations are gapped and the remaining gapless boson modes
are represented by even combinations of non-local emergent Majorana fermions
$\psi_{yj}^{\sigma}(x)=\psi_{j}^{\sigma}(x,y)$, where $j=1,\ldots,N$ and
$\sigma=L,R=+,-$ labels the left and right propagating directions. Unlike BdG
fermions which are integral combinations of electrons and holes in a
conventional BCS superconductor, these Majorana fermions are fractional and
cannot appear individually. Only pairs of them on the same wire are
combinations of integral products of electrons and holes. This includes the
current generators $J^{\sigma}_{y,jk}=i\psi^{\sigma}_{yj}\psi^{\sigma}_{yk}$
of the $SO(N)_{1}$ WZW KM algebra and the intra-wire backscattering
$\psi^{L}_{yj}\psi^{R}_{yk}$. The effective low-energy kinetic Hamiltonian
density is $\mathsf{L}$ copies of (15)

$\displaystyle\mathcal{H}_{0}=\frac{iv}{2}\sum_{y=1}^{\mathsf{L}}\sum_{j=1}^{N}\sum_{\sigma=+,-}\sigma\psi_{yj}^{\sigma}\partial_{x}\psi_{yj}^{\sigma}.$
(19)

Interwire single-fermion tunnelings
$\psi^{\sigma}_{yj}\psi^{\sigma^{\prime}}_{y^{\prime}j^{\prime}}$, for $y\neq
y^{\prime}$, are forbidden by locality, but two-fermion processes are allowed
and can be represented by local integral combinations of electrons. In
particular, the interwire Gross-Neveu interaction Gross and Neveu (1974);
Zamolodchikov and Zamolodchikov (1978); Witten (1978); Shankar and Witten
(1978) that backscatters the $SO(N)_{1}$ currents ${\bf
J}_{y}^{\sigma}=(J_{y,jk}^{\sigma})$

$\displaystyle\mathcal{U}_{SO(N)_{1}}$ $\displaystyle=u\sum_{y}{\bf
J}_{y+1}^{L}\cdot{\bf J}_{y}^{R}$ (20) $\displaystyle=-u\sum_{y}\sum_{1\leq
j<k\leq N}\psi_{y+1,j}^{L}\psi_{y+1,k}^{L}\psi_{y,j}^{R}\psi_{y,k}^{R}$

gaps all bulk degrees of freedom and leads to the $SO(N)_{1}$ topological
phase Kitaev (2006). The interaction involves interwire backscattering of the
$SO(N)_{1}$ WZW KM currents
$J_{y,jk}^{\sigma}=i\psi_{y,j}^{\sigma}\psi_{y,k}^{\sigma}$. The model
$\mathcal{H}_{0}+\mathcal{U}_{SO(N)_{1}}$ admits a mean-field approximation
with real order parameters $\hat{\mathcal{O}}_{y+1/2,j}=\int
dxi\psi^{L}_{y+1,j}\psi^{R}_{y,j}$, whose ground state expectation values
$\mathcal{O}_{y+1/2}=\langle\hat{\mathcal{O}}_{y+1/2,j}\rangle$ are non-
vanishing and independent from $j$ due to the $SO(N)$ symmetry. The mean-field
approximation of the the Gross-Neveu potential is quadratic in $\psi$. It
introduces a finite excitation energy gap and turns the Majorana fermions
massive.

Bosonizing Di Francesco  _et al._ (1999); Giamarchi (2003); Gogolin  _et al._
(2004); Sénéchal (2004); Fradkin (2013) according to

$\displaystyle
d_{y,a}^{\sigma}=\frac{1}{\sqrt{2}}\left(\psi_{y,a}^{\sigma}+i\psi_{y,n+a}^{\sigma}\right)\sim
e^{i\phi^{\sigma}_{y,a}},$ (21)

the $SO(N)_{1}$ Gross-Neveu interaction is equivalent to the sine-Gordon
potential

$\displaystyle\mathcal{U}_{SO(2n)_{1}}$
$\displaystyle=-\tilde{u}\sum_{y}\sum_{a=1}^{n}\partial_{x}\phi^{L}_{y+1,a}\partial_{x}\phi^{R}_{y,a}$
$\displaystyle-\Delta\sum_{y}\sum_{1\leq a<b\leq
n}\sum_{\epsilon=\pm}\cos\left(\Theta_{y+1/2}^{a}+\epsilon\Theta_{y+1/2}^{b}\right),$
$\displaystyle\mathcal{U}_{SO(2n+1)_{1}}$
$\displaystyle=\mathcal{U}_{SO(2n)_{1}}$ (22)
$\displaystyle-\tilde{\Delta}\sum_{y}\left(\sum_{a=1}^{n}\cos\Theta_{y+1/2}^{a}\right)i\psi^{L}_{y+1,2n+1}\psi^{R}_{y,2n+1},$

where the angle variables $\Theta^{a}_{y+1/2}=\phi^{L}_{y+1,a}-\phi^{R}_{y,a}$
are non-chiral combinations of the bosonized variables. The potentials
$\Delta$ and $\tilde{\Delta}$ are relevant in the RG sense under repulsive
density interactions when $\tilde{u}>0$. They introduces a finite energy gap
above the ground state, where the order parameter quantum fluctuate about the
expectation values

$\displaystyle\left\langle\Theta^{a}_{y+1/2}(x)\right\rangle$
$\displaystyle=m^{a}_{y+1/2}\pi,$ (23)
$\displaystyle\mathrm{sgn}\mathcal{O}_{y+1/2}$
$\displaystyle=\mathrm{sgn}\left\langle\int
dxi\psi^{L}_{y+1,j}\psi^{R}_{y,j}\right\rangle=(-1)^{m^{a}_{y+1/2}},$

where ${\bf m}_{y+1/2}=(m^{1}_{y+1/2},\ldots,m^{n}_{y+1/2})$ either have all
even or all odd integral entries.

In a periodic geometry where $y\equiv y+\mathsf{L}$, all excitations are
separated away from the ground state by a finite energy gap. In an open
geometry where the system terminates at wire $y=1$ and $\mathsf{L}$, the
Gross-Neveu interaction (20) leaves behind the gapless left (right) moving
$SO(N)_{1}$ WZW CFT on the boundary at $y=1$ ($y=\mathsf{L}$), where the
emergent Majorana fermions $\psi^{L}_{y=1,j}$ and $\psi^{R}_{y=\mathsf{L},j}$
remain massless. The primary field excitations on the boundaries and bulk
anyon quasiparticle excitations are correlated due to the bulk topology and
boson locality. They are created and non-locally connected by Wilson string
operators, which are integral combination of electrons. The edge primary
fields and bulk anyons are classified by the super-selection sectors
$X=1,\psi,s_{+},s_{-}$ when $N=2n$ is even, or $X=1,\psi,\sigma$ when $N=2n+1$
is odd. Fields and anyons in the same class differ from each other by local
electronic combinations. The vacuum sector 1 contains all integral fields.
$\psi$ contains all odd combinations of the non-local Majorana fermions.
$s_{\pm}$ and $\sigma$ are non-local spinor fields with spin (or conformal
scaling dimension) $h=N/16$. A full braid between a spinor and a fermion
associates a $\pi$ monodromy phase. Therefore a spinor field carries a
$\mathbb{Z}_{2}$ flux component with respect to the internal fermion parity
symmetry (17). The $SO(2n)_{1}$ topological order is Abelian and the anyons
obey the single-channel fusion rules

$\displaystyle\psi\times\psi=1,\quad\psi\times s_{\pm}=s_{\mp},$
$\displaystyle s_{\pm}\times
s_{\pm}=\left\\{\begin{array}[]{*{20}l}1,&\mbox{for even $n$}\\\
\psi,&\mbox{for odd $n$}\end{array}\right..$ (26)

The $SO(2n+1)_{1}$ topological order is non-Abelian. The Ising twist field
$\sigma$ obeys the fusion rules

$\displaystyle\psi\times\sigma=\sigma,\quad\sigma\times\sigma=1+\psi.$ (27)

Since all excitations are created by strings of electron operators, the edge
primary fields and bulk anyons must have balancing topological charges
$X_{\mathrm{top-edge}}\times X_{\mathrm{bottom-edge}}\times
X_{\mathrm{bulk}}=1$.

The $SO(N)_{1}$ topological orders are sixteen-fold periodic Kitaev (2006).
The anyon fusion and braiding structures of $SO(N)_{1}$ and $SO(N+16)_{1}$ are
identical. Therefore, as UMTCs, $SO(N)_{1}=SO(N+16)_{1}$. As CFTs, their
chiral charge charges differ by 8. This can be compensated by subtracting from
$SO(N+16)_{1}$ the $E_{8}$ WZW CFT at level 1. Sahoo  _et al._ (2016) The
subtraction does not alter the topological order because $(E_{8})_{1}$ does
not support fractional excitations.

###  III.2 The $U(1)_{l}$ and $SU(n)_{1}$ family

The coupled-wire models in the unitary family $U(1)_{l}$ and $SU(n)_{1}$ are
summarized in figure 1(a). The level of $U(1)$ is $l=4n$ when $n$ is odd or
$l=n$ when $n$ is even. It corresponds to the compactification radius
$R=\sqrt{l}/2$ of the free boson on a circle. The models are constructed based
on the conformal embedding $U(1)_{l}\times SU(n)_{1}\subseteq SO(2n)_{1}$.
Among the $SO(2n)$ rotations of the Majorana fermions $\psi_{j}$, unitary
rotations $d_{a}\to U_{a}^{b}d_{b}$ of the Dirac fermions
$d_{a}=(\psi_{a}+i\psi_{n+a})/\sqrt{2}\sim e^{i\phi_{a}}$ form a unitary
subgroup $U(n)$ that commutes with the artificial $U(1)$ symmetry
$\phi_{a}\to\phi_{a}+\vartheta$. The corresponding $U(1)$ invariant WZW KM
subalgebra is generated by the current operators ${\bf J}_{U(1)_{l}\times
SU(n)_{1}}$ consisting of the Cartan-Weyl generators $\partial_{x}\phi_{a}\sim
d_{a}^{\dagger}d_{a}$ and roots $e^{i\left(\phi_{a}-\phi_{b}\right)}\sim
d_{b}^{\dagger}d_{a}$ for $a\neq b$. The subalgebra decomposes into a diagonal
$U(1)_{l}$ sector generated by $\partial_{x}\phi_{\perp}$, where

$\displaystyle\phi_{\perp}=\phi_{1}+\ldots+\phi_{n},$ (28)

and an off-diagonal $SU(n)_{1}$ sector. The Cartan-Weyl generators of
$SU(n)_{1}$ can be chosen to be $\partial_{x}\tilde{\phi}_{p}$, for
$p=1,\ldots,n-1$, using an orthogonal transformation

$\displaystyle\tilde{\phi}_{p}=\frac{1}{\sqrt{p(p+1)}}\left(-p\phi_{p+1}+\sum_{a=1}^{p}\phi_{a}\right),$
(29)

all of which being orthogonal to (28). Together with the roots
$e^{i\left(\phi_{a}-\phi_{b}\right)}$, they generate the $SU(n)$ WZW KM
algebra at level 1.

The kinetic Hamiltonian density (19) of the array of bosonic $SO(2n)_{1}$
wires can be re-expressed in the bosonized form

$\displaystyle\mathcal{H}_{0}$
$\displaystyle=\frac{v}{4\pi}\sum_{y=1}^{\mathsf{L}}\sum_{\sigma=L,R}\sum_{a=1}^{n}\left(\partial_{x}\phi_{ya}^{\sigma}\right)^{2}=\mathcal{H}_{0}^{U(1)}+\mathcal{H}_{0}^{SU(n)},$
$\displaystyle\mathcal{H}_{0}^{U(1)}$
$\displaystyle=\frac{v/n}{4\pi}\sum_{y=1}^{\mathsf{L}}\sum_{\sigma=L,R}\left(\partial_{x}\phi_{y\perp}^{\sigma}\right)^{2},$
(30) $\displaystyle\mathcal{H}_{0}^{SU(n)}$
$\displaystyle=\frac{v}{4\pi}\sum_{y=1}^{\mathsf{L}}\sum_{\sigma=L,R}\sum_{p=1}^{n-1}\left(\partial_{x}\tilde{\phi}_{yp}^{\sigma}\right)^{2}.$

We consider backscattering interactions that are composed of inter-wire or
intra-wire sine-Gordon potentials. The interactions among the $U(1)_{l}$
sectors are either the intra-wire or inter-wire backscattering

$\displaystyle\mathcal{U}^{y}_{U(1)_{l}}$
$\displaystyle=-u_{\perp}^{\mathrm{intra}}\partial_{x}\phi_{y\perp}^{L}\partial_{x}\phi_{y\perp}^{R}-\Delta_{\perp}^{\mathrm{intra}}\cos\Theta_{y}^{\perp},$
(31) $\displaystyle\mathcal{U}^{y+1/2}_{U(1)_{l}}$
$\displaystyle=-u_{\perp}^{\mathrm{inter}}\partial_{x}\phi_{y+1,\perp}^{L}\partial_{x}\phi_{y\perp}^{R}-\Delta_{\perp}^{\mathrm{inter}}\cos\Theta_{y+1/2}^{\perp},$

where $\Theta_{y}^{\perp}=\phi_{y\perp}^{L}-\phi_{y\perp}^{R}$ and

$\displaystyle\Theta_{y+1/2}^{\perp}=q\left(\phi_{y+1,\perp}^{L}-\phi_{y,\perp}^{R}\right),\quad
q=\left\\{\begin{array}[]{*{20}l}1,&\mbox{for even $n$}\\\ 2,&\mbox{for odd
$n$}\end{array}\right..$ (34)

The extra factor of 2 for the odd $n$ case is to ensure an even number of
Dirac fermions are backscattered from one wire to the next so that the
interaction can be constructed by an integral combination of electrons. No
such distinction is necessary for the intra-wire interaction because
$\cos\Theta_{y}^{\perp}$ is already an even product of Dirac fermions on the
same wire. The interactions within the $SU(n)_{1}$ sector consist of
backscattering processes of the $SU(n)_{1}$ currents ${\bf J}_{SU(n)_{1}}$
inside a wire or in-between wires

$\displaystyle\mathcal{U}_{SU(n)_{1}}^{y}$
$\displaystyle=u^{\mathrm{intra}}{\bf J}_{y,SU(n)_{1}}^{L}\cdot{\bf
J}_{y,SU(n)_{1}}^{R}$
$\displaystyle=-\tilde{u}^{\mathrm{intra}}\sum_{p=1}^{n-1}\partial_{x}\tilde{\phi}_{yp}^{L}\partial_{x}\tilde{\phi}_{yp}^{R}$
$\displaystyle\;\;\;-\Delta^{\mathrm{intra}}\sum_{1\leq a<b\leq
n}\cos\left(\Theta_{y}^{a}-\Theta_{y}^{b}\right),$ (35)
$\displaystyle\mathcal{U}_{SU(n)_{1}}^{y+1/2}$
$\displaystyle=u^{\mathrm{inter}}{\bf J}_{y+1,SU(n)_{1}}^{L}\cdot{\bf
J}_{y,SU(n)_{1}}^{R}$
$\displaystyle=-\tilde{u}^{\mathrm{inter}}\sum_{p=1}^{n-1}\partial_{x}\tilde{\phi}_{y+1,p}^{L}\partial_{x}\tilde{\phi}_{yp}^{R}$
$\displaystyle\;\;\;-\Delta^{\mathrm{inter}}\sum_{1\leq a<b\leq
n}\cos\left(\Theta_{y+1/2}^{a}-\Theta_{y+1/2}^{b}\right),$

where $\Theta_{y}^{a}=\phi_{ya}^{L}-\phi_{ya}^{R}$ and
$\Theta_{y+1/2}^{a}=\phi_{y+1,a}^{L}-\phi_{y,a}^{R}$.

The $U(1)_{l}$ topological phase is constructed by combining inter-wire
interactions (31) in the $U(1)_{l}$ sector and intra-wire interactions (35) in
the $SU(n)_{1}$ sector

$\displaystyle\mathcal{H}_{U(1)_{l}}=\mathcal{H}_{0}+\sum_{y=1}^{\mathsf{L}}\left(\mathcal{U}^{y+1/2}_{U(1)_{l}}+\mathcal{U}^{y}_{SU(n)_{1}}\right).$
(36)

This gaps all excitations in the bulk. For
$\Delta^{\mathrm{inter}}_{\perp},\Delta^{\mathrm{intra}}>0$, the sine-Gordon
potentials pin the finite ground state expectation values of the angle
variables $\left\langle\Theta^{\perp}_{y+1/2}\right\rangle$ and
$\left\langle\Theta^{a}_{y}-\Theta^{b}_{y}\right\rangle$ to integer multiples
of $2\pi$. The $SU(n)_{1}$ potential $\Delta^{\mathrm{intra}}$ is relevant in
the RG sense when the density interaction is “repulsive”, i.e.
$\tilde{u}^{\mathrm{intra}}>0$. The $U(1)_{l}$ sine-Gordon potential
$\Delta^{\mathrm{inter}}_{\perp}$ is relevant when
$u^{\mathrm{inter}}_{\perp}>v(q^{4}n^{2}-4)/[2\pi n(q^{4}n^{2}+4)]$. In an
open geometry, the model (36) leaves behind the gapless chiral $U(1)_{l}$ KM
CFT on the edges generated by $\phi^{R}_{y=\mathsf{L},\perp}$ and
$\phi^{L}_{y=1,\perp}$.

The $U(1)_{l}$ topological phase supports $l$ Abelian anyon types
$1,[e],\left[e^{2}\right],\ldots,\left[e^{l-1}\right]$. They carry spins
$h_{e^{m}}=m^{2}/(2l)$ and obey the fusion rules $e^{m}\times
e^{m^{\prime}}=e^{m+m^{\prime}}$ and
$\left[e^{l}\right]=\left[e^{0}\right]=1$. The anyon $e^{m}$ corresponds to
the primary field

$\displaystyle
e^{m}=\left\\{\begin{array}[]{*{20}l}e^{im\phi_{\perp}/n}&\mbox{if $n$
even}\\\ e^{im\phi_{\perp}/(2n)}&\mbox{if $n$ odd}\end{array}\right.$ (39)

on the edge CFT. The level $l$ is determined by the smallest non-trivial local
boson $e^{l}$. When $n$ is even, $e^{i\phi_{\perp}}\sim d_{1}\ldots d_{n}$ is
an even product of Dirac fermions and hence it is a local integral combination
of electrons. This sets $l=n$ by equating $e^{l}=e^{i\phi_{\perp}}$. When $n$
is odd, $e^{i\phi_{\perp}}$ is not local because it is an odd product of Dirac
fermions and has $\pi$ monodromy with $e^{m}$ for odd $m$. Instead, the
primitive local boson is $e^{l}=e^{i2\phi_{\perp}}\sim(d_{1}\partial
d_{1})\ldots(d_{n}\partial d_{n})$. This sets $l=4n$. The $U(1)_{l}$
topological order can be described by the $2+1$D Chern-Simons field theory
Witten (1989); Frohlich and Zee (1991); Lopez and Fradkin (1991); Wen and Zee
(1992)

$\displaystyle\mathcal{S}_{\mathrm{CS}}\left[U(1)_{l}\right]=\frac{l}{4\pi}\int_{2+1}a\wedge
da.$ (40)

On the other hand, the $SU(n)_{1}$ topological phase – which is the “particle-
hole” conjugate of the $U(1)_{l}$ – is constructed by combining inter-wire
interactions in the $SU(n)_{1}$ sector and intra-wire interactions in the
$U(1)_{l}$ sector

$\displaystyle\mathcal{H}_{SU(n)_{1}}=\mathcal{H}_{0}+\sum_{y=1}^{\mathsf{L}}\left(\mathcal{U}^{y}_{U(1)_{l}}+\mathcal{U}^{y+1/2}_{SU(n)_{1}}\right).$
(41)

Similar to the $U(1)_{l}$ case, it creates a finite excitation energy gap in
the bulk, but leaves behind the $SU(n)_{1}$ WZW CFT on the edges. The
$SU(n)_{1}$ topological phase supports $n$ Abelian anyon types
$1,[\Psi],\left[\Psi^{2}\right],\ldots,\left[\Psi^{n-1}\right]$. They carry
spins $h_{\Psi^{m}}=m(n-m)/(2n)$ and obey the fusion rules
$[\Psi^{m}]\times[\Psi^{m^{\prime}}]=\left[\Psi^{m+m^{\prime}}\right]$ and
$\left[\Psi^{n}\right]=\left[\Psi^{0}\right]=1$. $\left[\Psi^{m}\right]$
corresponds not to a single field, but a super-selection sector of fields that
rotate irreducibly under the $SU(n)_{1}$ WZW KM algebra. On the edge
$SU(n)_{1}$ CFT, $\left[\Psi^{m}\right]$ is spanned by
$C^{n}_{m}=n!/[m!(n-m)!]$ primary fields

$\displaystyle\left[\Psi^{m}\right]=\mathrm{span}\left\\{e^{i\left(\phi_{a_{1}}+\ldots+\phi_{a_{m}}-m\phi_{\perp}/n\right)}\right\\}_{1\leq
a_{1}<\ldots<a_{m}\leq n}.$ (42)

The $SU(n)_{1}$ topological order can be described by the $2+1$D Chern-Simons
field theory

$\displaystyle\mathcal{S}_{\mathrm{CS}}\left[SU(n)_{1}\right]=\frac{K_{IJ}}{4\pi}\int_{2+1}a^{I}\wedge
da^{J}$ (43)

with $n-1$ components $a^{1},\ldots,a^{n-1}$, where
$K_{IJ}=2\delta_{IJ}-\delta_{I,J+1}-\delta_{I,J-1}$ is the Cartan matrix of
$SU(n)$.

The trivial topological phase with a finite excitation energy gap can be
constructed using solely intra-wire backscattering interactions

$\displaystyle\mathcal{H}_{\mathrm{trivial}}=\mathcal{H}_{0}+\sum_{y=1}^{\mathsf{L}}\left(\mathcal{U}^{y}_{U(1)_{l}}+\mathcal{U}^{y}_{SU(n)_{1}}\right).$
(44)

The model is a stack of decoupled 1D gapped boson wires. The edge of an open
system does not carry gapless modes, and the ground state of any system,
regardless of being open or closed, is non-degenerate. On the contrary, a non-
trivial topological phase can be constructed using solely inter-wire
backscattering interactions

$\displaystyle\mathcal{H}_{SO(2n)_{1}}=\mathcal{H}_{0}+\sum_{y=1}^{\mathsf{L}}\left(\mathcal{U}^{y+1/2}_{U(1)_{l}}+\mathcal{U}^{y+1/2}_{SU(n)_{1}}\right).$
(45)

It is worth observing that the topological order is not $U(1)_{l}\times
SU(n)_{1}$, but instead $SO(2n)_{1}$, which was presented in the previous
subsection. This is because the bosonic anyon pair

$\displaystyle
z=\left\\{\begin{array}[]{*{20}l}\left[e^{2}\right]\times\left[\Psi^{2}\right]&\mbox{if
$n$ even}\\\ \left[e^{4}\right]\times\left[\Psi^{2}\right]&\mbox{if $n$
odd}\end{array}\right.$ (48)

along with its higher powers $z^{p}$ are even products of Dirac fermions
($z\sim d_{a}d_{b}$) and are therefore actually integral combinations of local
electrons. They should belong in the vacuum sector, and are condensed in the
context of anyon condensation Bais and Slingerland (2009); Kong (2014b);
Neupert  _et al._ (2016); Burnell (2018). The topological phase only carries 4
distinct anyon types – the trivial class 1 of local fields, the fermion
$\psi=[e]\times[\Psi]$ if $n$ even or $\psi=[e^{2}]\times[\Psi]$ if $n$ odd,
the even spinor $s_{+}=[e^{n/2}]$ if $n$ even or $s_{+}=[e^{n}]$ if $n$ odd,
and the odd spinor $s_{-}=s_{+}\times\psi$ – matching the $SO(2n)_{1}$
topological order.

The topological phase with $U(1)_{l}\times SU(n)_{1}$ order can be constructed
by stacking the $U(1)_{l}$ and $SU(n)_{1}$ models

$\displaystyle\mathcal{H}_{U(1)_{l}\times
SU(n)_{1}}=\mathcal{H}_{U(1)_{l}}\oplus\mathcal{H}_{SU(n)_{1}}$ (49)

from (36) and (41). The two Hamiltonian components act independently on
decoupled Hilbert spaces. Here, the boson wires that constitute the $U(1)_{l}$
layer are completely distinct from those that constitute the $SU(n)_{1}$
layer. Consequently, the anyon pair $z$ in (48) is not a local integral
combination of electrons in this construction. The $U(1)_{l}\times SU(n)_{1}$
topological phase is a promotion from $SO(2n)_{1}$ by gauging a
$\mathbb{Z}_{k}$ symmetry as explained in the subsection below.

###  III.3 Gauging the $\mathbb{Z}_{k}$ or $\mathbb{Z}_{2}$ symmetries in
$SO(2n)_{1}$

The $SO(2n)_{1}$ topological phase admits global $\mathbb{Z}_{k}$ and
$\mathbb{Z}_{2}$ symmetries, where $k=n$ when $n$ is odd or $k=n/2$ when $n$
is even. They originate from the rotation and conjugation of the Dirac
fermions
$d_{ya}^{\sigma}=(\psi_{y,a}^{\sigma}+i\psi_{y,n+a}^{\sigma})/\sqrt{2}\sim
e^{i\phi^{\sigma}_{ya}}$

$\displaystyle\begin{split}&\mathbb{Z}_{k}:\quad d^{\sigma}_{ya}\to e^{2\pi
i/k}d^{\sigma}_{ya},\quad\phi^{\sigma}_{ya}\to\phi^{\sigma}_{ya}+2\pi/k\\\
&\mathbb{Z}_{2}:\quad
d^{\sigma}_{ya}\to{d^{\sigma}_{ya}}^{\dagger},\quad\phi^{\sigma}_{ya}\to-\phi^{\sigma}_{ya}\end{split}$
(50)

where the symmetries apply to all species $a=1,\ldots,n$, propagation
directions $\sigma=L,R$ on all wires $y=1,\ldots,\mathsf{L}$. Together, they
form the dihedral group $D_{k}=\mathbb{Z}_{2}\ltimes\mathbb{Z}_{k}$. The
symmetry group is non-Abelian since the $\mathbb{Z}_{k}$ and $\mathbb{Z}_{2}$
actions do not mutually commute. Unlike the internal $\mathbb{Z}_{2}$ symmetry
in (17), which is a local gauge symmetry of $SO(2n)_{1}$ dictated by locality,
the $\mathbb{Z}_{k}$ and $\mathbb{Z}_{2}$ symmetries in (50) are global
symmetries of the $SO(2n)_{1}$ Hamiltonian
$\mathcal{H}_{0}+\mathcal{U}_{SO(2n)_{1}}$ in (19) and (22). While all local
operators are necessarily invariant under the internal $\mathbb{Z}_{2}$
symmetry (17), they may change under the global symmetries. For example, the
fermion pair $\psi^{\sigma}_{yj}\psi^{\sigma^{\prime}}_{yj^{\prime}}$ on a
given wire is an integral combination of electrons and is invariant under
(17), but in general is not fixed by the global $\mathbb{Z}_{k}$ or
$\mathbb{Z}_{2}$ symmetries. In the following, we discuss topological phases
that promote the $\mathbb{Z}_{k}$ or $\mathbb{Z}_{2}$ symmetries into local
gauge symmetries.

####  III.3.1 The $SO(2n)_{1}/\mathbb{Z}_{2}$ twist liquid

The $SO(2n)_{1}$ WZW KM algebra is generated by current operators
$J_{jk}=i\psi_{j}\psi_{k}$, for $1\leq j<k\leq 2n$. The $\mathbb{Z}_{2}$
symmetry from (50) that sends $\psi_{a}\to\psi_{a}$ and
$\psi_{n+a}\to-\psi_{n+a}$, for $a=1,\ldots,n$, fixes the subalgebra
$SO(n)_{1}^{A}\times SO(n)_{1}^{B}$, where the first and second components are
generated by the $\mathbb{Z}_{2}$ invariant current operators
$J_{ab}^{A}=i\psi_{a}\psi_{b}$ and $J_{ab}^{B}=i\psi_{n+a}\psi_{n+b}$, for
$1\leq a<b\leq n$. The corresponding $2+1$D topological phase can be
constructed using the coupled-wire model consisting of two identical but
decoupled copies of $SO(n)_{1}$

$\displaystyle\mathcal{H}_{SO(n)_{1}^{A}\times
SO(n)_{1}^{B}}=\mathcal{H}_{SO(n)_{1}^{A}}\oplus\mathcal{H}_{SO(n)_{1}^{B}}$
(51)

where the Hamiltonian of each copy was presented in section III.1.

The $\mathbb{Z}_{2}$ symmetry fixes all fermions in the $A$ layer but flips
the sign of all fermions in the $B$ layer. Restricting to the $SO(n)_{1}^{B}$,
the symmetry is identical to (17), which is a local gauge symmetry. Overall,
the $SO(n)_{1}^{A}\times SO(n)_{1}^{B}$ double has two sets of local
$\mathbb{Z}_{2}$ gauge symmetries at any given wire $y$

$\displaystyle\begin{split}\mathbb{Z}_{2}^{A}(y):\quad\psi_{y^{\prime}a}^{\sigma}\to(-1)^{\delta_{yy^{\prime}}}\psi_{y^{\prime}a}^{\sigma},\quad\psi_{y^{\prime},n+a}^{\sigma}\to\psi_{y^{\prime},n+a}^{\sigma}\\\
\mathbb{Z}_{2}^{B}(y):\quad\psi_{y^{\prime}a}^{\sigma}\to\psi_{y^{\prime}a}^{\sigma},\quad\psi_{y^{\prime},n+a}^{\sigma}\to(-1)^{\delta_{yy^{\prime}}}\psi_{y^{\prime},n+a}^{\sigma}\end{split}$
(52)

for $\sigma=L,R$. An operator is a local integral combination of electrons (up
to intra-wire ground state expectation values) if and only if it is symmetric
under the local $\mathbb{Z}_{2}^{A}\times\mathbb{Z}_{2}^{B}$ symmetry.

The $SO(n)_{1}^{A}\times SO(n)_{1}^{B}$ topological phase is the
$\mathbb{Z}_{2}$ twist liquid of $SO(2n)_{1}$, where the symmetry is gauged.
On the $1+1$D edge, the $SO(n)_{1}^{A}\times SO(n)_{1}^{B}$ WZW edge CFT is
the $\mathbb{Z}_{2}$ orbifold of $SO(2n)_{1}$.

${SO(2n)_{1}}$${SO(2n)_{1}/\mathbb{Z}_{2}=SO(n)_{1}^{A}\times
SO(n)_{1}^{B}}$$\scriptstyle{\mathrm{gauging}}$$\scriptstyle{\mathrm{condensation}}$
(53)

The bosonic $\mathbb{Z}_{2}$ charge is the fermion pair
$\zeta=\psi^{A}\psi^{B}$, which consists of non-local products of fermions
$\psi_{a}\psi_{n+b}$ from both layers, and is odd under the $\mathbb{Z}_{2}$
symmetry. The spinor fields – $s^{A/B}_{\pm}$ for $n$ even and $\sigma^{A/B}$
for $n$ odd – from either the $A$ or the $B$ layer exhibit a $\pi$ monodromy
with $\zeta$ and therefore carry a $\mathbb{Z}_{2}$ flux component. During
anyon condensation, the $\mathbb{Z}_{2}$ charge $\zeta$ belongs to the vacuum
sector as it is part of the local currents in $SO(2n)_{1}$. The fermions in
the two layers are identified $\psi^{A}\equiv\psi^{B}$. The spinor fields
$s^{A/B}_{\pm}$ or $\sigma^{A/B}$ in individual layers are confined. The
spinor field pairs become the spinor fields $s_{\pm}$ in $SO(2n)_{1}$. When
$n$ is even, $s_{+}=s^{A}_{\pm}s^{B}_{\pm}$ and
$s_{-}=s^{A}_{\pm}s^{B}_{\mp}$. When $n$ is odd,
$\sigma^{A}\sigma^{B}=s_{+}+s_{-}$ decomposes.

When $n$ is even, the global $\mathbb{Z}_{2}$ symmetry (50) on $SO(2n)_{1}$ is
an inner automorphism and do not alter the anyon types. When $n\equiv 0$
modulo 8, the twist liquid $SO(2n)_{1}/\mathbb{Z}_{2}=[SO(n)_{1}]^{2}$, as an
unitary modular tensor category (UMTC), is equivalent to the product
$SO(2n)_{1}\times D^{[0]}(\mathbb{Z}_{2})$, where $D^{[0]}(\mathbb{Z}_{2})$ is
the $\mathbb{Z}_{2}$ discrete gauge theory Bais  _et al._ (1992); de Wild
Propitius (1995); de Wild Propitius and Bais (1995); Preskill (2004) in its
deconfined phase Jalabert and Sachdev (1991); Senthil and Fisher (2000);
Moessner  _et al._ (2001); Ardonne  _et al._ (2004) or equivalently the Kitaev
toric code Kitaev (2003). When $n\equiv 4$ modulo 8, the twist liquid (53) is
equivalent to $SO(2n)_{1}\times D^{[1]}(\mathbb{Z}_{2})$, where
$D^{[1]}(\mathbb{Z}_{2})$ is the double semion theory Levin and Wen (2005).
$D^{[0]}(\mathbb{Z}_{2})$ and $D^{[1]}(\mathbb{Z}_{2})$ can be described by a
2-component Chern-Simons field theory $\mathcal{S}=\int_{2+1}K_{IJ}a^{I}\wedge
da^{J}/(4\pi)$, where $K=2\sigma_{x}$ ($2\sigma_{z}$) for the toric code
(double semion) and $\sigma_{x,z}$ are $2\times 2$ Pauli matrices. They both
carry 4 Abelian anyon types $1,\zeta,\mu,\zeta\times\mu$, each being self-
conjugate, and the $\mathbb{Z}_{2}$ flux $\mu$ – can be identified as one of
$s^{A/B}_{\pm}$ – is bosonic with spin 0 (semionic with spin $1/4$) for the
toric code (double semion). The distinction between the two $\mathbb{Z}_{2}$
gauge theories $D^{[v]}(\mathbb{Z}_{2})$, for $v=0,1$, stems from the
$F$-symbol Levin and Wen (2005); Kitaev (2006) of the $\mathbb{Z}_{2}$ flux
that governs the fusion associativity
$\mu\times(\mu\times\mu)=(\mu\times\mu)\times\mu$.
$F^{\mu\mu\mu}_{\mu}=(-1)^{v}$ takes opposite signs for the toric code and
double semion theories. The two gauge inequivalent classes of $F$-symbols
represent distinct Dijkgraaf-Witten invariants $v=0,1$ modulo 2 in the group
cohomology Brown (1982) $H^{3}(\mathbb{Z}_{2},U(1))=\mathbb{Z}_{2}$. The
details of Dijkgraaf-Witten deformation Dijkgraaf and Witten (1990); Dijkgraaf
_et al._ (1991); Altschuler and Coste (1992); Bais  _et al._ (1993); de Wild
Propitius (1995) of discrete gauge theories can be found in appendix C.

When $n\equiv 2$ modulo 4, the twist liquid
$SO(2n)_{1}/\mathbb{Z}_{2}=SO(n)_{1}^{A}\times SO(n)_{1}^{B}$ cannot be
factorized by a $\mathbb{Z}_{2}$ gauge theory. This is because the
$\mathbb{Z}_{2}$ fluxes are no longer self-conjugate, but follow the fusion
rule $s^{A/B}_{\pm}\times s^{A/B}_{\pm}=\psi^{A/B}$ instead. The
$G=\mathbb{Z}_{2}$ symmetry group is extended by the anyon fusion group
$\mathcal{A}=\\{1,\psi,s_{+},s_{-}\\}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ of
$SO(2n)_{1}$ into a quantum symmetry group Wen (2002); Etingof  _et al._
(2010); Teo  _et al._ (2015); Barkeshli  _et al._ (2019) $\widehat{G}$, which
is analogous to a space-group consisting of “rotations” $G$ and “translations”
$\mathcal{A}$. The central extension, captured by the exact sequence

$\displaystyle 1\to\mathcal{A}\hookrightarrow\widehat{G}\to G\to 1,$ (54)

is classified by the group cohomology Brown (1982) $H^{2}(G,\mathcal{A})$ (see
appendix C), which in our current situation is
$H^{2}(\mathbb{Z}_{2},\mathcal{A})=\mathcal{A}$. The trivial cohomology class
corresponds to a symmorphic quantum symmetry group, which is a direct product
$\widehat{G}=G\times\mathcal{A}$, and applies in the previous cases when
$n\equiv 0$ modulo 4. Here, when $n\equiv 2$ modulo 4, the cohomology class is
non-trivial and corresponds to the non-symmorphic quantum symmetry group
$\widehat{\mathbb{Z}_{2}}=\mathbb{Z}_{4}\times\mathbb{Z}_{2}$. This is because
the $\mathbb{Z}_{2}$ fluxes are of order 4, $(s^{A/B}_{\pm})^{4}=1$. The
$G=\mathbb{Z}_{2}$ “rotation” becomes a twofold “screw rotation” and squares
to the $SO(2n)_{1}$ fermion $s^{2}_{\pm}=\psi$, which can be viewed as a
“translation” in the fusion group $\mathcal{A}$.

As an UMTC, the $SO(n)_{1}$ topological order is equivalent to the coset
$SO(2n)_{1}/SO(n)_{1}=SO(2n)_{1}\boxtimes\overline{SO(n)_{1}}$, where the
denominator group $SO(n)_{1}$ sits inside the numerator group $SO(2n)_{1}$
under the matrix embedding $O\to O\oplus\mathbb{I}_{n}$,
$\overline{SO(n)_{1}}$ is the time-reversal conjugate of $SO(n)_{1}$, and the
fermion pair $\psi\bar{\psi}$ is condensed in the relative tensor product
$\boxtimes$. Hence, the twist liquid
$SO(2n)_{1}/\mathbb{Z}_{2}=[SO(n)_{1}]^{2}$ is equivalent to

$\displaystyle
SO(2n)_{1}/\mathbb{Z}_{2}=SO(2n)_{1}\boxtimes\overline{SO(n)_{1}}\times
SO(n)_{1}.$ (55)

When $n\equiv 2$ modulo 4, the non-chiral product $\overline{SO(n)_{1}}\times
SO(n)_{1}$ is equivalent, as an UMTC, to the discrete gauge theory
$D^{[2]}(\mathbb{Z}_{4})$ deformed by the Dijkgraaf-Witten invariant $[2]$ in
the cohomology group $H^{3}(\mathbb{Z}_{4},U(1))=\mathbb{Z}_{4}$ (see appendix
C). $D^{[2]}(\mathbb{Z}_{4})$ can be described by a 2-component Chern-Simons
field theory $\mathcal{S}=\int_{2+1}K_{IJ}a^{I}\wedge da^{J}/(4\pi)$ with the
$K$-matrix $K=4\sigma_{z}$. This identifies the twist liquid
$SO(2n)_{1}/\mathbb{Z}_{2}$ with the relative tensor product
$SO(2n)_{1}\boxtimes D^{[2]}(\mathbb{Z}_{4})$, for $n\equiv 2$ modulo 4. Here,
the fermion $\psi$ in $SO(2n)_{1}$ is pair condensed with
$\bar{\psi}=\bar{s}^{2}_{\pm}$ (the square of the $\mathbb{Z}_{4}$ flux) in
$D^{[2]}(\mathbb{Z}_{4})$ in the product.

Overall, we summarize the $\mathbb{Z}_{2}$ orbifold phases
$SO(2n)_{1}/\mathbb{Z}_{2}$ for a general even integer $n$ by the UMTC
equivalence

$\displaystyle\frac{SO(2n)_{1}}{\mathbb{Z}_{2}}=SO(n)_{1}\times SO(n)_{1}$
(56) $\displaystyle=\left\\{\begin{array}[]{*{20}l}SO(2n)_{1}\times
D^{[0]}(\mathbb{Z}_{2}),&\mbox{for $n\equiv 0$ mod 8}\\\ SO(2n)_{1}\times
D^{[1]}(\mathbb{Z}_{2}),&\mbox{for $n\equiv 4$ mod 8}\\\ SO(2n)_{1}\boxtimes
D^{[2]}(\mathbb{Z}_{4}),&\mbox{for $n\equiv 2$ mod 4}\end{array}\right..$ (60)

When $n$ is odd, the global $\mathbb{Z}_{2}$ symmetry (50) on $SO(2n)_{1}$ is
an outer automorphism. It corresponds to the mirror symmetry of the Dynkin
diagram of the simply-laced $D_{n}=SO(2n)$ Lie algebras Di Francesco  _et al._
(1999). It exchanges the even and odd spinor fields, $s_{+}\leftrightarrow
s_{-}$, in $SO(2n)_{1}$. Consequently, the $\mathbb{Z}_{2}$ fluxes
$\sigma^{A/B}$ in the twist liquid
$SO(2n)_{1}/\mathbb{Z}_{2}=SO(n)_{1}^{A}\times SO(n)_{1}^{B}$ are non-Abelian
and obey the Ising fusion rule $\sigma\times\sigma=1+\psi$. When applying (55)
for odd $n$, the non-chiral product
$\mathcal{Z}_{n}(\mathbb{Z}_{2})=\overline{SO(n)_{1}}\times SO(n)_{1}$ is
equivalent as UMTC to

$\displaystyle\begin{split}&Z(\mathrm{Ising})=\mathrm{Ising}\times\overline{\mathrm{Ising}},\quad\mbox{for
$n\equiv\pm 1$ modulo 8}\\\
&Z(SU(2)_{2})=SU(2)_{2}\times\overline{SU(2)_{2}},\quad\mbox{for $n\equiv\pm
3$ modulo 8}.\end{split}$ (61)

(Here $Z(\mathcal{C})$ stands for the Drinfeld center Kassel (1995); Bakalov
and Kirillov (2001); Levin and Wen (2005) of the fusion category
$\mathcal{C}$.) The $\mathrm{Ising}$ and $SU(2)_{2}$ topological orders have
identical anyon content $1,\psi,\sigma$ and fusion rules. They differ by the
Ising spin statistics $h_{\sigma}=1/16$ and Frobenius-Schur indicator Kitaev
(2006)
$\varkappa_{\sigma}=d_{\sigma}\left[F^{\sigma\sigma\sigma}_{\sigma}\right]^{1}_{1}=1$
for the former, and $h_{\sigma}=3/16$ and $\varkappa_{\sigma}=-1$ for the
latter. Both non-chiral products in (61) are $\mathbb{Z}_{2}$ twist liquids of
the Kitaev toric code after gauging an electric-magnetic symmetry Barkeshli
and Wen (2012); Teo  _et al._ (2015); Chen  _et al._ (2017). They are
differentiated from each other by the non-trivial Dijkgraff-Witten invariant
in $H^{3}(\mathbb{Z}_{2},U(1))$. The $\mathbb{Z}_{2}$ orbifold phases
$SO(2n)_{1}/\mathbb{Z}_{2}$ for odd $n$ can be summarized by the UMTC
equivalence

$\displaystyle\frac{SO(2n)_{1}}{\mathbb{Z}_{2}}=SO(n)_{1}\times
SO(n)_{1}=SO(2n)_{1}\boxtimes\mathcal{Z}_{n}(\mathbb{Z}_{2})$
$\displaystyle\mathcal{Z}_{n}(\mathbb{Z}_{2})=\left\\{\begin{array}[]{*{20}l}Z(\mathrm{Ising}),&\mbox{for
$n\equiv\pm 1$ mod 8}\\\ Z(SU(2)_{2}),&\mbox{for $n\equiv\pm 3$ mod
8}\end{array}\right.,$ (64)

where under the relative tensor product $\boxtimes$, the fermion in
$SO(2n)_{1}$ is pair condensed with the fermion in $\overline{\mathrm{Ising}}$
when $n\equiv 1$, in $\overline{SU(2)_{2}}$ when $n\equiv 3$, in $SU(2)_{2}$
when $n\equiv-3$, or in $\mathrm{Ising}$ when $n\equiv-1$ modulo 8.

####  III.3.2 The $SO(2n)_{1}/\mathbb{Z}_{k}$ twist liquid

The (complexified) $SO(2n)_{1}$ WZW KM algebra is spanned by the Cartan-Weyl
generators $d_{a}^{\dagger}d_{a}\sim\partial\phi_{a}$, the positive roots
$d_{a}d_{b}\sim e^{i(\phi_{a}+\phi_{b})}$ and $d_{a}d_{b}^{\dagger}\sim
e^{i(\phi_{a}-\phi_{b})}$, and the negative roots which are the hermitian
conjugate of the positive ones, where $1\leq a<b\leq n$. The $\mathbb{Z}_{k}$
symmetry in (50) (or the diagonal $U(1)$ symmetry) fixes $\partial\phi_{a}$
and $e^{\pm i(\phi_{a}-\phi_{b})}$, which generate the
$\mathbb{Z}_{k}$-invariant $U(1)_{l}\times SU(n)_{1}$ subalgebra. The
corresponding $2+1$D topological phase can be constructed using the coupled-
wire model consisting of the decoupled $U(1)_{l}$ and $SU(n)_{1}$ layers (see
(49) in section III.2).

The $U(1)_{l}\times SU(n)_{1}$ topological phase is the $\mathbb{Z}_{k}$ twist
liquid of $SO(2n)_{1}$, where the symmetry is gauged. On the $1+1$D edge, the
$U(1)_{l}\times SU(n)_{1}$ WZW edge CFT is the $\mathbb{Z}_{k}$ orbifold of
$SO(2n)_{1}$.

${SO(2n)_{1}}$${SO(2n)_{1}/\mathbb{Z}_{k}=U(1)_{l}\times
SU(n)_{1}}$$\scriptstyle{\mathrm{gauging}}$$\scriptstyle{\mathrm{condensation}}$
(65)

The $\mathbb{Z}_{k}$ symmetry is an inner autormophism of $SO(2n)$ and it does
not alter the anyon types of $SO(2n)_{1}$. After gauging, the $\mathbb{Z}_{k}$
gauge charge can be identified by the anyon pair

$\displaystyle
z=\left\\{\begin{array}[]{*{20}l}\left[e^{2}\right]\times\left[\Psi^{2}\right]&\mbox{if
$n$ even}\\\ \left[e^{4}\right]\times\left[\Psi^{2}\right]&\mbox{if $n$
odd}\end{array}\right.,$ (68)

where $e^{m}$ and $\Psi^{m}$ are the primary fields of $U(1)_{l}$ and
$SU(n)_{1}$ defined previously in (39) and (42). Unlike in model (45) where
the $\mathbb{Z}_{k}$ charge is local, here the $U(1)_{l}$ and $SU(n)_{1}$
states are constructed independently and therefore the $\mathbb{Z}_{k}$ charge
$z$ does not belong to the vacuum sector. The order of the gauge group
$\mathbb{Z}_{k}$ is set by the fusion rule $z^{k}=1$ for $k=n$ if $n$ is odd
or $n/2$ when $n$ is even.

When $k$ is odd (i.e. $n$ is not an integer multiple of 4), the
$U(1)_{l}\times SU(n)_{1}$ topological order factorizes into $SO(2n)_{1}\times
D^{[0]}(\mathbb{Z}_{k})$ as an UMTC, where $D^{[0]}(\mathbb{Z}_{k})$ is the
$\mathbb{Z}_{k}$ discrete gauge theory Bais  _et al._ (1992); de Wild
Propitius (1995); de Wild Propitius and Bais (1995); Preskill (2004) in its
deconfined phase or equivalently the $\mathbb{Z}_{k}$ Wen plaquette model Wen
(2003). The $D^{[0]}(\mathbb{Z}_{k})$ quantum double can be described by a
two-component Chern-Simons field theory
$\mathcal{S}=\int_{2+1}K_{IJ}a^{I}\wedge da^{J}/(4\pi)$, where
$K=k\sigma_{x}$. Its anyons are dyons $m^{a}z^{b}$ composed of fluxes $m$ and
charges $z$, where $a,b=0,1,\ldots,k-1$. The primitive gauge flux can be
identified with the bosonic anyon pair

$\displaystyle m=\left[e^{lr/k}\right]\times\left[\Psi^{-2r}\right].$ (69)

Here, $8r\equiv 1$ modulo $k=n$ when $n$ is odd, or $4r\equiv 1$ modulo
$k=n/2$ when $n$ is $2$ modulo 4, so that the monodromy braiding phase between
$z$ and $m$ is $e^{2\pi i/k}$. The $SO(2n)_{1}$ topological order can be
embedded in $U(1)_{l}\times SU(n)_{1}$ and decouples from
$D^{[0]}(\mathbb{Z}_{k})$. When $n$ is odd, $\left[e^{2n}\right]$ and
$\left[e^{\pm n}\right]$ – all of which being local with respect with the
$\mathbb{Z}_{k}$ flux and charge – take the roles of the fermion and the two
spinors of $SO(2n)_{1}$ respectively. When $n$ is 2 modulo 4,
$\left[e^{n/2}\right]$ and $\left[\Psi^{n/2}\right]$ can be treated as the two
spinor fields of $SO(2n)_{1}$, and the fermion is the product
$\left[e^{n/2}\right]\times\left[\Psi^{n/2}\right]$.

When $k$ is even (i.e. $n$ is an integer multiple of 4), the $\mathbb{Z}_{k}$
gauge part of $SO(2n)_{1}/\mathbb{Z}_{k}=U(1)_{l}\times SU(n)_{1}$ does not
decompose. This is because all anyon $m$ that carries a primitive gauge flux
component and exhibits a $e^{2\pi i/k}$ monodromy braiding phase with the
primitive gauge charge $z$ defined in (68) must be of order $2k$, i.e.
$m^{2k}=1$ but $m^{k}\neq 1$. For example, the generator
$\left[\Psi^{k-1}\right]$ of $SU(n)_{1}$ carries a unit of gauge flux and its
$k^{\mathrm{th}}$ power
$\left[\Psi^{k-1}\right]^{k}\equiv\left[\Psi^{k}\right]$ does not belong to
the vacuum sector but the even spinor sector $s_{+}$ in $SO(2n)_{1}$. The
quantum symmetry group
$\widehat{\mathbb{Z}_{k}}=\mathbb{Z}_{2k}\times\mathbb{Z}_{2}$ (see (54)) that
extends $G=\mathbb{Z}_{k}$ by $\mathcal{A}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
– the fusion group of Abelian anyons in $SO(2n)_{1}$ – is non-symmorphic and
is not the direct product $\mathbb{Z}_{k}\times\mathcal{A}$. The twist liquid
is represented by a non-trivial class in the group cohomology
$H^{2}(\mathbb{Z}_{k},\mathcal{A})=\mathcal{A}$ when $k$ is even (see appendix
C).

When $k=n/2\equiv 0$ modulo 4, $SO(2n)_{1}$ is equivalent as an UMTC to the
discrete $\mathbb{Z}_{2}$ gauge theory $D^{[0]}(\mathbb{Z}_{2})$ and carries
the same topological order as the Kitaev’s toric code Kitaev (2003). The
chiral central charge $c=n$ is a multiple of 8 and can be offset by copies of
the topological trivial $E_{8}$ at level 1. Kitaev (2006); Sahoo  _et al._
(2016); Lopes  _et al._ (2019) The even spinor $s_{+}$ in $SO(2n)_{1}$ can be
identified as the $\mathbb{Z}_{2}$ gauge flux. Gauging the $\mathbb{Z}_{k}$
symmetry extends the gauge group to $\mathbb{Z}_{2k}=\mathbb{Z}_{n}$. The
$\mathbb{Z}_{k}$ orbifold phase $SO(2n)_{1}/\mathbb{Z}_{k}=U(1)_{n}\times
SU(n)_{1}$ is equivalent as an UMTC to the discrete $\mathbb{Z}_{n}$ gauge
theory $D^{[k]}(\mathbb{Z}_{n})$ deformed by the Dijkgraaf-Witten invariant
$[k]$ in $H^{3}(\mathbb{Z}_{n},U(1))=\mathbb{Z}_{n}$ (see appendix C). To see
this, we first notice when $n$ is divisible by 8, $SU(n)_{1}$ has the same
topological order as $\overline{U(1)_{n}}$ – the time-reversal conjugate of
$U(1)_{n}$. This is because $\left[\Psi^{k-1}\right]$ shares the same fusion
and statistical properties of $\bar{e}$, the primitive anyon in
$\overline{U(1)_{n}}$, and all anyons in $SU(n)_{1}$ can be generated by
powers of $\left[\Psi^{k-1}\right]$ since $n=2k$ and $k-1$ are relatively
prime. Consequently, the $SO(2n)_{1}/\mathbb{Z}_{k}$ twist liquid has the
identical topological order of the non-chiral product
$D^{[k]}(\mathbb{Z}_{n})=U(1)_{n}\times\overline{U(1)_{n}}$, which can be
described by a two-component Chern-Simons field theory
$\mathcal{S}=\int_{2+1}K_{IJ}a^{I}\wedge da^{J}/(4\pi)$, where
$K=n\sigma_{z}$. This $K$-matrix is inequivalent to $n\sigma_{x}$, which
instead corresponds to the un-deformed discrete gauge theory
$D^{[0]}(\mathbb{Z}_{n})$.

For $k=n/2\equiv 2$ modulo 4, the chiral central charge $c=n$ is equivalent to
4 modulo 8, and therefore the orbifold phase
$SO(2n)_{1}/\mathbb{Z}_{n}=U(1)_{n}\times SU(n)_{1}$ cannot be identified to a
non-chiral gauge theory. Instead, it is equivalent as an UMTC to the relative
tensor product $SO(8)_{1}\boxtimes D^{[k]}(\mathbb{Z}_{n})$, where the product
$\boxtimes$ involves a fermion pair condensation. To see this, we observe the
UMTC equivalence $SU(n)_{1}\equiv SO(8)_{1}\boxtimes\overline{U(1)_{n}}$,
where the vector fermion $\psi$ in $SO(8)_{1}$ and the self-conjugate fermion
$\bar{\psi}=\bar{e}^{n/2}$ in $\overline{U(1)_{n}}$ are pair condensed in the
product, i.e. $\psi\bar{e}^{n/2}=1$. The $SU(n)_{1}$ anyon
$\left[\Psi^{k-1}\right]$ can be identified with the product $s_{+}\bar{e}$,
where $s_{+}$ is the even spinor of $SO(8)_{1}$. This is because they both
carry spin $1/2-1/(2n)$ modulo 1, have order $n$, and generate all anyons that
are not confined by the condensate. The non-chiral product
$U(1)_{n}\times\overline{U(1)_{n}}$ is identical to the deformed discrete
gauge theory $D^{[k]}(\mathbb{Z}_{n})$ for the same reasons in the $k\equiv 0$
mod 4 case presented above.

Overall, we summarize the $\mathbb{Z}_{k}$ orbifold phases for general integer
$k$ by the UMTC equivalence

$\displaystyle\frac{SO(2n)_{1}}{\mathbb{Z}_{k}}=U(1)_{l}\times SU(n)_{1}$ (70)
$\displaystyle=\left\\{\begin{array}[]{*{20}l}SO(2n)_{1}\times
D^{[0]}(\mathbb{Z}_{k}),&\mbox{for odd $k$}\\\
D^{[k]}(\mathbb{Z}_{2k}),&\mbox{for $k\equiv 0$ mod 4}\\\ SO(8)_{1}\boxtimes
D^{[k]}(\mathbb{Z}_{2k}),&\mbox{for $k\equiv 2$ mod 4}\end{array}\right.$ (74)

where $k=n$ and $l=4n$ when $n$ is odd, or $k=n/2$ and $l=n$ when $n$ is even.
The $\mathbb{Z}_{2k}$ group is the “non-symmorphic” $\mathbb{Z}_{2}$ central
extensions of $\mathbb{Z}_{k}$ and associates to non-trivial cohomology
elements in $H^{2}(\mathbb{Z}_{k},\mathcal{A})$. The super-script $[k]$ in
$D^{[k]}(\mathbb{Z}_{2k})$ labels the Dijkgraaf-Witten invariant in the
cohomology group $H^{3}(\mathbb{Z}_{2k},U(1))=\mathbb{Z}_{2k}$ that deforms
the gauge theory. The two even $k$ cases in (70) agree with (3) and admit the
decomposition $SO(2n)_{1}\boxtimes D^{[k]}(\mathbb{Z}_{2k})$. This is because
(i) when $k=n/2\equiv 2$ modulo 4, $SO(2n)_{1}$ is equivalent to $SO(8)_{1}$
as an UMTC. (ii) When $k=n/2\equiv 0$ modulo 4, $SO(2n)_{1}$ is equivalent to
$D^{[0]}(\mathbb{Z}_{2})$ as an UMTC, and
$D^{[k]}(\mathbb{Z}_{2k})=D^{[0]}(\mathbb{Z}_{2})\boxtimes
D^{[k]}(\mathbb{Z}_{2k})$ under the $\mathbb{Z}_{2}$ charge pair condensation
in the relative tensor product.

##  IV Dihedral twist liquids

We construct coupled-wire models of the following non-Abelian twist liquid
topological phases (see figure 1(b))

$\displaystyle\frac{SU(n)_{1}}{\mathbb{Z}_{2}}$
$\displaystyle=SO(n)_{2},\quad\frac{U(1)_{l}}{\mathbb{Z}_{2}}=\frac{SO(n)_{1}\times
SO(n)_{1}}{SO(n)_{2}}$ $\displaystyle\frac{SO(2n)_{1}}{D_{k}}$
$\displaystyle=\frac{SU(n)_{1}}{\mathbb{Z}_{2}}\boxtimes\frac{U(1)_{l}}{\mathbb{Z}_{2}},$
(75)

where $k=n/2=l/2$ when $n$ is even or $k=n=l/4$ when $n$ is odd, and the
$\mathbb{Z}_{2}$ charge pair is condensed in the relative tensor product
$\boxtimes$. The $SU(n)_{1}/\mathbb{Z}_{2}$ and $U(1)_{l}/\mathbb{Z}_{2}$
orbifolds are components of the coset decomposition Ginsparg (1988a); Di
Francesco  _et al._ (1999); Blumenhagen (2009)

$\displaystyle\left[SO(n)_{1}\right]^{2}=SO(n)_{2}\times\frac{\left[SO(n)_{1}\right]^{2}}{SO(n)_{2}}$
(76)

where the $SO(n)_{2}$ WZW KM theory is the diagonal sub-algebra of
$\left[SO(n)_{1}\right]^{2}$. From section III.3, we identified
$\left[SO(n)_{1}\right]^{2}$ with the $\mathbb{Z}_{2}$ twist liquid
$SO(2n)_{1}/\mathbb{Z}_{2}$. Therefore the coset decomposition (76) is the
$\mathbb{Z}_{2}$ orbifold version of the conformal embedding
$SO(2n)_{1}\supseteq U(1)_{l}\times SU(n)_{1}$. In section III.3, we discussed
the global $D_{k}=\mathbb{Z}_{2}\ltimes\mathbb{Z}_{k}$ symmetry of the
$SO(2n)_{1}$ WZW CFT , where the $\mathbb{Z}_{k}$ and $\mathbb{Z}_{2}$ actions
were presented in (50). We also showed that the decoupled pair $U(1)_{l}\times
SU(n)_{1}$ carried a local $\mathbb{Z}_{k}$ gauge symmetry and could be
identified as the $SO(2n)_{1}/\mathbb{Z}_{k}$ orbifold phase. In addition,
they carry a global $\mathbb{Z}_{2}$ conjugation symmetry

$\displaystyle\mathbb{Z}_{2}:\quad\phi_{y\perp}^{\sigma}\to-\phi_{y\perp}^{\sigma},\quad\phi^{\sigma}_{ya}-\phi^{\sigma}_{yb}\to\phi^{\sigma}_{yb}-\phi^{\sigma}_{ya}$
(77)

where $\phi_{a}$, for $a=1,\ldots,n$, are the bosonized variables of the Dirac
fermions $d_{a}\sim e^{i\phi_{a}}$ in $SO(2n)_{1}$,
$\phi_{\perp}=\phi_{1}+\ldots+\phi_{n}$ is the diagonal part that generates
$U(1)_{l}$, and $e^{i(\phi_{a}-\phi_{b})}$ are the roots of the $SU(n)_{1}$
WZW KM algebra. Here the $\mathbb{Z}_{2}$ symmetry applies globally to all
species $a,b=1,\ldots,n$, propagation directions $\sigma=L,R$ on all wires
$y=1,\ldots,\mathsf{L}$.

In this section, we construct a new $D_{k}$ twist liquid phase by further
promoting the $\mathbb{Z}_{2}$ symmetry into a local gauge symmetry.

###  IV.1 The $SU(n)_{1}/\mathbb{Z}_{2}$ twist liquid

The $SU(n)_{1}/\mathbb{Z}_{2}$ orbifold phase is constructed based on the
identification $SU(n)_{1}/\mathbb{Z}_{2}=SO(n)_{2}$. Schellekens (1999) The
coupled-wire model is built on a 2D array of wires, each consists of the
$SO(n)_{1}\times SO(n)_{1}$ pair. The kinetic Hamiltonian density is the
decoupled direct sum

$\displaystyle\mathcal{H}_{0}$
$\displaystyle=\mathcal{H}_{0}^{A}\oplus\mathcal{H}_{0}^{B}$ (78)
$\displaystyle=\frac{iv}{2}\sum_{y=1}^{\mathsf{L}}\sum_{a=1}^{n}\sum_{\sigma=+,-}\sigma\left(\psi^{A\sigma}_{ya}\partial_{x}\psi^{A\sigma}_{ya}+\psi^{B\sigma}_{ya}\partial_{x}\psi^{B\sigma}_{ya}\right)$

where $\psi^{A\sigma}_{ya}=\psi^{\sigma}_{ya}$ and
$\psi^{B\sigma}_{ya}=\psi^{\sigma}_{y,n+a}$, for $a=1,\ldots,n$, are the two
sets of Majorana fermions of the $SO(n)_{1}^{A}\times SO(n)_{1}^{B}$ pair at
the $y^{\mathrm{th}}$ wire, and $\sigma=L,R=+,-$ labels the two propagation
directions. Here, the two $SO(n)_{1}$ CFTs originate independently from two
distinct sets of interacting 1D electrons. (See section II for the electronic
spin liquid and superconductor origins.) Consequently, only fermion pairs
$\psi_{ya}^{\sigma}\psi_{ya^{\prime}}^{\sigma^{\prime}}$ and
$\psi_{y,n+a}^{\sigma}\psi_{y,n+a^{\prime}}^{\sigma^{\prime}}$ from the same
$A/B$ sector on the same wire are integral combinations of local electrons.
Fermion pairs from distinct sectors
$\psi_{y,a}^{\sigma}\psi_{y,n+a^{\prime}}^{\sigma^{\prime}}$ are all non-
local. In general, an operator is an integral combination of local electrons
(up to intra-wire ground state expectation values) if and only if it is even
under the $\mathbb{Z}_{2}^{A}\times\mathbb{Z}_{2}^{B}$ gauge symmetry defined
in (52). Grouping Majorana fermions from opposite sectors into Dirac fermions
$d^{\sigma}_{ya}=(\psi^{A\sigma}_{ya}+i\psi^{B\sigma}_{ya})/\sqrt{2}\sim
e^{i\phi^{\sigma}_{ya}}$, the $\mathbb{Z}_{2}^{B}$ symmetry is the conjugation

$\displaystyle\mathbb{Z}_{2}^{B}(y):\quad\begin{split}&\phi^{\sigma}_{y^{\prime}a}\to(-1)^{\delta_{yy^{\prime}}}\phi^{\sigma}_{y^{\prime}a}\\\
&\psi^{A\sigma}_{y^{\prime}a}\to\psi^{A\sigma}_{y^{\prime}a},\quad\psi^{B\sigma}_{y^{\prime}a}\to(-1)^{\delta_{yy^{\prime}}}\psi^{B\sigma}_{y^{\prime}a}\end{split}$
(79)

that flips $d^{\sigma}_{ya}\to{d^{\sigma}_{ya}}^{\dagger}$. In the following,
we refer to the $\mathbb{Z}_{2}^{B}$ symmetry simply as the $\mathbb{Z}_{2}$
symmetry.

The coset decomposition (76) separates the $[SO(n)_{1}]^{2}$ CFT into the
decoupled $U(1)_{l}/\mathbb{Z}_{2}=[SO(n)_{1}]^{2}/SO(n)_{2}$ and
$SU(n)_{1}/\mathbb{Z}_{2}=SO(n)_{2}$ sectors. The $U(1)_{l}$ is generated by
the diagonal bosonized variable $\phi_{\perp}=\phi_{1}+\ldots+\phi_{n}$.
Unlike previously in section III.2 when the $U(1)_{l}$ coupled-wire model was
constructed based on $SO(2n)_{1}$ wires, now the new notion of locality from
the $SO(n)_{1}\times SO(n)_{1}$ pair and its local $\mathbb{Z}_{2}$ gauge
symmetry (79) enforce the $U(1)_{l}/\mathbb{Z}_{2}$ orbifold CFT on each wire.
This is because the physical Hilbert space is even under
$\mathbb{Z}_{2}:\phi_{\perp}\to-\phi_{\perp}$. The $U(1)_{l}$ current

$\displaystyle\partial\phi_{\perp}\sim\sum_{a=1}^{n}\psi_{a}^{A}\psi_{a}^{B}$
(80)

is no longer local as it is odd under $\mathbb{Z}_{2}$. It takes the role of
the $\mathbb{Z}_{2}$ gauge charge in $U(1)_{l}/\mathbb{Z}_{2}$ and must appear
in pairs. The $U(1)_{l}/\mathbb{Z}_{2}$ sector on a given wire is gapped under
the intra-wire backscattering interaction borrowed from (31)

$\displaystyle\mathcal{U}^{y}_{U(1)_{l}/\mathbb{Z}_{2}}=-u_{\perp}^{\mathrm{intra}}\partial_{x}\phi_{y\perp}^{L}\partial_{x}\phi_{y\perp}^{R}-\Delta_{\perp}^{\mathrm{intra}}\cos\Theta_{y}^{\perp},$
(81)

where $\Theta_{y}^{\perp}=\phi^{L}_{y\perp}-\phi^{R}_{y\perp}$. Here, both the
density interaction and the sine-Gordon potential are even under (79) and are
integral combinations of local electrons. When
$u_{\perp}^{\mathrm{intra}}>v(n^{2}-4)/[2\pi n(n^{2}+4)]$, the sine-Gordon
potential is relevant in the RG sense and introduces a finite excitation
energy gap for the $U(1)_{l}/\mathbb{Z}_{2}$ sector.

The remaining $SO(n)_{2}$ sector of the decomposition (76) is the diagonal WZW
KM sub-algebra of the $SO(n)_{1}^{A}\times SO(n)_{1}^{B}$ pair. It generates
the simultaneous orthogonal rotation of both the $A$ and $B$ Majorana fermions
$\psi^{A/B}_{a}\to O_{a}^{b}\psi^{A/B}_{b}$. The $SO(n)_{2}$ currents are the
sums
$J_{ab}=J_{ab}^{A}+J_{ab}^{B}=i\psi_{a}^{A}\psi_{b}^{A}+i\psi_{a}^{B}\psi_{b}^{B}$,
for $1\leq a<b\leq n$. Focusing on the $\sigma=L$ chiral sector, the currents
obey the operator product expansion Ginsparg (1988a); Di Francesco  _et al._
(1999); Blumenhagen (2009)

$\displaystyle
J_{ab}(z)J_{cd}(w)=\frac{2\delta_{ac}\delta_{bd}}{(z-w)^{2}}+\frac{if_{(ab)(cd)(ef)}}{z-w}J_{ef}(w)+\ldots.$
(82)

Here, $z$ and $w$ are complex Euclidean space-time parameters
$e^{2\pi(v\tau+ix)/l}$, $\tau=it$ is the Wick rotated time, and $l$ is the
$x$-circumference of the closed circular wires. The structure constant of the
$SO(n)$ Lie algebra is

$\displaystyle f_{(ab)(cd)(ef)}$
$\displaystyle=\delta_{ea}\delta_{fd}\delta_{bc}-\delta_{ea}\delta_{fc}\delta_{bd}$
$\displaystyle\;\;\;+\delta_{eb}\delta_{ad}\delta_{fc}-\delta_{eb}\delta_{fd}\delta_{ac}.$
(83)

The factor of 2 in (82) sets the level of the $SO(n)_{2}$ WZW KM algebra. At
the same time, $SO(n)_{2}$ is also the WZW KM sub-algebra of $SU(n)_{1}$ that
is even under the $\mathbb{Z}_{2}$ gauge symmetry. Among the unitary rotations
$d_{a}\to U_{a}^{b}d_{b}$ of Dirac fermions, only the real orthogonal ones are
compatible and commutes with the $\mathbb{Z}_{2}$ conjugation $d\to
d^{\dagger}$. The $SO(n)_{2}$ currents are self-conjugate combinations of
$SU(n)_{1}$ root operators

$\displaystyle\begin{split}J_{ab}&=J_{ab}^{A}+J_{ab}^{B}=i\psi_{a}^{A}\psi_{b}^{A}+i\psi_{a}^{B}\psi_{b}^{B}\\\
&\sim 2\cos\left(\phi_{a}-\phi_{b}\right),\end{split}$ (84)

which are integral combinations of local electrons. The $SO(n)_{2}$ complement
of $SU(n)_{1}$ consists of the Cartan-Weyl generators and the anti-conjugate
root combinations

$\displaystyle\begin{split}&\partial\tilde{\phi}_{p}\sim\frac{i}{\sqrt{p(p+1)}}\left(-p\psi^{A}_{p+1}\psi^{B}_{p+1}+\sum_{a=1}^{p}\psi_{a}^{A}\psi_{a}^{B}\right)\\\
&S_{ab}=i\psi_{a}^{A}\psi_{b}^{B}-i\psi_{a}^{B}\psi_{b}^{A}\sim
2\sin\left(\phi_{a}-\phi_{b}\right)\end{split}$ (85)

both of which are odd under $\mathbb{Z}_{2}$ and are therefore fractional.
Similar to (80), operators in (85) take the role of the $\mathbb{Z}_{2}$ gauge
charge in $SU(n)_{1}/\mathbb{Z}_{2}$.

The $SO(n)_{2}$ sector is gapped using the inter-wire current backscattering
interaction

$\displaystyle\mathcal{U}^{y+1/2}_{SO(n)_{2}}=u^{\mathrm{inter}}{\bf
J}_{y+1,SO(n)_{2}}^{L}\cdot{\bf J}_{y,SO(n)_{2}}^{R}$ (86)
$\displaystyle=-\Delta^{\mathrm{inter}}\sum_{1\leq a<b\leq
n}\cos\left(\phi_{y+1,a}^{L}-\phi_{y+1,b}^{L}\right)\cos\left(\phi_{y,a}^{R}-\phi_{y,b}^{R}\right)$
$\displaystyle=-\frac{1}{2}\Delta^{\mathrm{inter}}\sum_{1\leq a<b\leq
n}\left(\cos\theta^{y+1/2}_{ab}+\cos\varphi^{y+1/2}_{ab}\right)$

where ${\bf J}_{SO(n)_{2}}=(J_{ab})_{1\leq a<b\leq n}$ are the $SO(n)_{2}$
currents defined in (84),
$\theta^{y+1/2}_{ab}=\phi^{L}_{y+1,a}-\phi^{L}_{y+1,b}-\phi^{R}_{y,a}+\phi^{R}_{y,b}$
and
$\varphi^{y+1/2}_{ab}=\phi^{L}_{y+1,a}-\phi^{L}_{y+1,b}+\phi^{R}_{y,a}-\phi^{R}_{y,b}$.
The interaction is marginally relevant Cardy (1996) when
$u^{\mathrm{inter}},\Delta^{\mathrm{inter}}>0$. Since the two sets of sine-
Gordon variables $\theta$ and $\varphi$ do not all mutually commute, they
cannot both take non-vanishing ground state expectation values. The current
backscattering interaction (86) condenses the bosons
$J_{y+1,ab}^{L}J_{y,ab}^{R}$, and when $\Delta^{\mathrm{inter}}>0$, it pins
the positive ground state expectation values

$\displaystyle\left\langle\cos\left(\phi_{y+1,a}^{L}-\phi_{y+1,b}^{L}\right)\cos\left(\phi_{y,a}^{R}-\phi_{y,b}^{R}\right)\right\rangle>0.$
(87)

At the same time, the $SO(n)_{2}$ currents are quadratic fusion products of
the $\mathbb{Z}_{2}$ gauge charges from (85), $S\times S=J$. Focusing on the
$\sigma=L$ chiral sector, the gauge charges obey the operator product
expansion

$\displaystyle S_{ab}(z)S_{cd}(w)$ (88)
$\displaystyle=\frac{2(\delta_{ac}\delta_{bd}+\delta_{bc}\delta_{ad})}{(z-w)^{2}}-\frac{ig_{(ab)(cd)(ef)}}{z-w}J_{ef}(w)+\ldots,$

where
$g_{(ab)(cd)(ef)}=\delta_{ac}\delta_{be}\delta_{df}+\delta_{bd}\delta_{ae}\delta_{cf}+\delta_{bc}\delta_{ae}\delta_{df}+\delta_{ad}\delta_{be}\delta_{cf}$.
Hence, the interaction (86) also condenses the $\mathbb{Z}_{2}$ charge pairs
$S_{y+1,ab}^{L}S_{y,ab}^{R}$, and the ground state expectation value

$\displaystyle
s_{y+1/2}=\left\langle\sin\left(\phi_{y+1,a}^{L}-\phi_{y+1,b}^{L}\right)\sin\left(\phi_{y,a}^{R}-\phi_{y,b}^{R}\right)\right\rangle$
(89)

may be positive or negative. Its sign determines which set of sine-Gordon
variables, $\theta$ or $\varphi$, in (86) is pinned.

$\displaystyle\begin{split}s_{y+1/2}>0:\quad\left\langle\cos\theta^{y+1/2}_{ab}\right\rangle=1,\quad\left\langle\cos\varphi^{y+1/2}_{ab}\right\rangle=0,\\\
s_{y+1/2}<0:\quad\left\langle\cos\theta^{y+1/2}_{ab}\right\rangle=0,\quad\left\langle\cos\varphi^{y+1/2}_{ab}\right\rangle=1.\end{split}$
(90)

The positive case corresponds to backscattering of $SU(n)_{1}$ roots
$\cos\theta_{ab}\sim
e^{i(\phi_{a}^{L}-\phi_{b}^{L})}e^{-i(\phi_{a}^{R}-\phi_{b}^{R})}+h.c.$ (c.f.
(35)), while the negative case corresponds to pairing of $SU(n)_{1}$ roots
$\cos\varphi_{ab}\sim
e^{i(\phi_{a}^{L}-\phi_{b}^{L})}e^{i(\phi_{a}^{R}-\phi_{b}^{R})}+h.c.$. Both
lead to a finite excitation energy gap in
$SU(n)_{1}/\mathbb{Z}_{2}=SO(n)_{2}$.

Overall, the coupled-wire model

$\displaystyle\mathcal{H}_{SU(n)_{1}/\mathbb{Z}_{2}}$
$\displaystyle=\mathcal{H}_{0}+\sum_{y=1}^{\mathsf{L}}\left(\mathcal{U}^{y}_{U(1)_{l}/\mathbb{Z}_{2}}+\mathcal{U}^{y+1/2}_{SO(n)_{2}}\right)$
(91)

– that combines the kinetic Hamiltonian (78), the $U(1)_{l}/\mathbb{Z}_{2}$
intra-wire interaction (81) and the $SU(n)_{1}/\mathbb{Z}_{2}$ inter-wire
interaction (86) – describes the $SU(n)_{1}/\mathbb{Z}_{2}=SO(n)_{2}$ twist
liquid topological phase. The model has a finite bulk excitation energy gap
and carries gapless chiral $SO(n)_{2}$ WZW CFT on the edges.

Figure 3: A non-Abelian $\mathbb{Z}_{2}$ gauge flux $\sigma$ or $\tau$ (green
star) sits at the domain wall where the order parameter $s_{y+1/2}$ in (a)
(89) for $SU(n)_{1}/\mathbb{Z}_{2}$ or (b) (109) for $U(1)_{l}/\mathbb{Z}_{2}$
changes sign. An orbiting anyon is conjugated, $\Psi_{m}\to\Psi_{n-m}$ or
$e^{m}\to e^{-m}$, when passing across the branch cut (red curvy line).

Domain wall between the two sets of ground states (90) where the order
parameter in (89) changes sign, $s_{y+1/2}(x)\sim\langle
S^{L}_{y+1,ab}(x)S^{R}_{y,ab}(x)\rangle\sim\pm\mathrm{sgn}(x-x_{0})$, traps a
$\mathbb{Z}_{2}$ gauge flux at $x_{0}$. (See figure 3(a).) In (85), we
identified $S^{\sigma}_{y,ab}$ as one of the non-local fields that represents
the $\mathbb{Z}_{2}$ gauge charge $S$. The sign flip of $s_{y+1/2}(x)$
corresponds to the $\pi$ monodromy braiding phase between the $\mathbb{Z}_{2}$
gauge flux and gauge charge. At the same time, the $\mathbb{Z}_{2}$ flux
conjugates any orbiting $SU(n)_{1}$ anyon, $\Psi^{m}\to\Psi^{n-m}$. To see
this, we first re-express the $SU(n)_{1}$ primary fields defined in (42) as
$\Psi^{m}\sim e^{i\sum_{j=1}^{m}\sum_{b=1}^{n}(\phi_{a_{j}}-\phi_{b})/n}$.
When $\Psi^{m}$ passes through the side of the domain wall with
$s_{y+1/2}(x)>0$, the backscattering term pins
$\langle\Psi^{m}_{L}(x)\Psi^{m}_{R}(x)^{\dagger}\rangle\sim
e^{i\sum_{jb}\langle\theta^{y+1/2}_{a_{j}b}(x)\rangle/n}\sim 1$ to a finite
value. Further moving $\Psi^{m}$ to the other side of the domain wall with
$s_{y+1/2}(x)<0$, there is no longer any backscattering term, but instead the
pairing term pins $\langle\Psi^{m}_{L}(x)\Psi^{m}_{R}(x)\rangle\sim
e^{i\sum_{jb}\langle\varphi^{y+1/2}_{a_{j}b}(x)\rangle/n}\sim 1$ to a finite
value. Therefore, $\Psi^{m}$ is conjugated to
$[(\Psi^{m})^{\dagger}]=[\Psi^{n-m}]$ as we complete this full circle.
Consequently, in the $SU(n)_{1}/\mathbb{Z}_{2}$ orbifold phase, any pair of
conjugated partners are grouped together into a single non-Abelian super-
selection sector

$\displaystyle\Phi^{i}=\Psi^{i}+\Psi^{n-i}$ (92)

for $1\leq i<n/2$. When $n$ is even, the Abelian anyon $\Psi^{n/2}$ in
$SU(n)_{1}$ is closed under the $\mathbb{Z}_{2}$ symmetry (79). It decomposes
into the $\mathbb{Z}_{2}$ even and odd sectors

$\displaystyle\begin{split}\left[\Psi^{n/2}_{+}\right]=\mathrm{span}\left\\{\cos\left(\frac{1}{2}\phi_{\perp}-\sum_{j=1}^{n/2}\phi_{a_{j}}\right)\right\\}_{1\leq
a_{1}<\ldots<a_{n/2}\leq n},\\\
\left[\Psi^{n/2}_{-}\right]=\mathrm{span}\left\\{\sin\left(\frac{1}{2}\phi_{\perp}-\sum_{j=1}^{n/2}\phi_{a_{j}}\right)\right\\}_{1\leq
a_{1}<\ldots<a_{n/2}\leq n},\end{split}$ (93)

each forms an irreducible representation of $SO(n)_{2}$.

When $n$ is odd, there are two $\mathbb{Z}_{2}$ fluxes that differ from each
other by a $\mathbb{Z}_{2}$ charge

$\displaystyle\sigma=\tau\times S,\quad\tau=\sigma\times S.$ (94)

They obey the fusion rule

$\displaystyle\sigma\times\sigma=\tau\times\tau=1+\sum_{i=1}^{(n-1)/2}\Phi^{i},$
$\displaystyle\Phi^{i}\times\sigma=\Phi^{i}\times\tau=\sigma+\tau.$ (95)

When $n$ is even, there are four $\mathbb{Z}_{2}$ fluxes related to each other
by

$\displaystyle\sigma^{\pm}=\tau^{\pm}\times
S,\quad\tau^{\pm}=\sigma^{\pm}\times S,$
$\displaystyle\sigma^{\pm}\times\Phi^{i}=\tau^{\pm}\times\Phi^{i}=\left\\{\begin{array}[]{*{20}l}\sigma^{\pm}+\tau^{\pm},&\mbox{if
$i$ even}\\\ \sigma^{\mp}+\tau^{\mp},&\mbox{if $i$ odd}\end{array}\right..$
(98)

They obey the pair fusion rules

$\displaystyle\sigma^{\pm}\times\sigma^{\pm}=\tau^{\pm}\times\tau^{\pm}=1+\Psi^{n/2}_{\pm}+\sum_{i\mbox{
}\mathrm{even}}\Phi^{i}$
$\displaystyle\sigma^{\pm}\times\sigma^{\mp}=\tau^{\pm}\times\tau^{\mp}=\sum_{i\mbox{
}\mathrm{odd}}\Phi^{i}$ (99)

when $n\equiv 0$ modulo 4, or

$\displaystyle\sigma^{\pm}\times\sigma^{\pm}=\tau^{\pm}\times\tau^{\pm}=\Psi^{n/2}_{\pm}+\sum_{i\mbox{
}\mathrm{odd}}\Phi_{i}$
$\displaystyle\sigma^{\pm}\times\sigma^{\mp}=\tau^{\pm}\times\tau^{\mp}=1+\sum_{i\mbox{
}\mathrm{even}}\Phi_{i}$ (100)

when $n\equiv 2$ modulo 4.

anyons | quantum dimension | spin (modulo 1)
---|---|---
1 | 1 | 0
$S$ | 1 | 1
$\Phi^{i}$ | 2 | $i(n-i)/(2n)$
$\sigma$ | $\sqrt{n}$ | $(n-1)/16$
$\tau$ | $\sqrt{n}$ | $(n+7)/16$
Table 1: Anyon content of $SU(n)_{1}/\mathbb{Z}_{2}=SO(n)_{2}$ when $n$ is odd. Here $i=1,\ldots,(n-1)/2$. anyons | quantum dimension | spin (modulo 1)
---|---|---
1 | 1 | 0
$S$ | 1 | 1
$\Psi_{\pm}^{n/2}$ | 1 | $n/8$
$\Phi^{i}$ | 2 | $i(n-i)/(2n)$
$\sigma^{\pm}$ | $\sqrt{n/2}$ | $(n-1)/16$
$\tau^{\pm}$ | $\sqrt{n/2}$ | $(n+7)/16$
Table 2: Anyon content of $SU(n)_{1}/\mathbb{Z}_{2}=SO(n)_{2}$ when $n$ is
even. Here $i=1,\ldots,n/2-1$.

The anyons of $SU(n)_{1}/\mathbb{Z}_{2}=SO(n)_{2}$ together with their quantum
dimensions and spins are summarized in table 1 and 2. The complete modular
anyon content of $SO(n)_{2}$ can be found in appendix B. When $n$ is odd, the
$\mathbb{Z}_{2}$ fluxes $\sigma$ and $\tau$ are referred to as metaplectic
anyons by Hastings, Nayak, and Wang in ref. Hastings  _et al._ , 2013, 2014,
and the $SO(n)_{2}$ topological order is known as a metaplectic topological
quantum field theory. The braiding operations performed by the metaplectic
anyons can be simulated by a classical computer. On the other hand, they
showed that the $d=2$ super-sectors $\Phi_{i}$ in (92) computes a $\\#P$-hard
link invariant, and therefore their braidings cannot be efficiently simulated
classically. This suggests the topological order we consider may have (non-
universal) quantum computing power beyond a classical computer.

###  IV.2 The $U(1)_{l}/\mathbb{Z}_{2}$ twist liquid

The $U(1)_{l}/\mathbb{Z}_{2}$ orbifold Ginsparg (1988b); Dijkgraaf  _et al._
(1988, 1989) phases were constructed by a coupled-wire model as fractional
quantum Hall states in ref. Kane and Stern, 2018; Tam  _et al._ , 2020 based
on a 2-fluid model Kane  _et al._ (2017). Here, we construct the orbifold
phases in the electron-based superconducting or spin liquid settings. The
coupled-wire model makes use of the coset identification 111L. Dixon, cited in
ref. Ginsparg, 1988a. $U(1)_{l}/\mathbb{Z}_{2}=[SO(n)_{1}]^{2}/SO(n)_{2}$. The
level is $l=n$ ($l=4n$) when $n$ is even (odd), and it sets the compactified
free boson circle radius $R=\sqrt{l}/2$. The model is the “particle-hole
conjugation” of the $SU(n)_{1}/\mathbb{Z}_{2}=SO(n)_{2}$ model in (91) in the
sense that the inter-wire and intra-wire backscattering directions of
$SO(n)_{2}$ and $U(1)_{l}/\mathbb{Z}_{2}$ are reversed.

$\displaystyle\mathcal{H}_{U(1)_{l}/\mathbb{Z}_{2}}$
$\displaystyle=\mathcal{H}_{0}+\sum_{y=1}^{\mathsf{L}}\left(\mathcal{U}^{y}_{SO(n)_{2}}+\mathcal{U}^{y+1/2}_{U(1)_{l}/\mathbb{Z}_{2}}\right).$
(101)

Like the previous $SU(n)_{1}/\mathbb{Z}_{2}$ case, the model here is based on
a 2D array of $SO(n)_{1}^{A}\times SO(n)_{1}^{B}$ wires, each carries two sets
of Majorana fermions $\psi^{A\sigma}_{ya}=\psi^{\sigma}_{ya}$ and
$\psi^{B\sigma}_{ya}=\psi^{\sigma}_{y,n+a}$, for $a=1,\ldots,n$,
$\sigma=L,R=+,-$ and $y=1,\ldots,\mathsf{L}$. All fermion pair from the same
$C=A,B$ sector on the same wire,
$\psi^{C\sigma}_{ya}\psi^{C\sigma^{\prime}}_{ya^{\prime}}$, is an integral
combination of local bosons. All fermion pairs from opposite sectors,
$\psi^{A\sigma}_{ya}\psi^{B\sigma^{\prime}}_{y^{\prime}a^{\prime}}$, are
fractional and must come in pairs. The kinetic Hamiltonian $\mathcal{H}_{0}$
is identical to the previous $SU(n)_{1}/\mathbb{Z}_{2}$ model and is given in
(78). The intra-wire $SO(n)_{2}$ current backscattering potential is

$\displaystyle\mathcal{U}^{y}_{SO(n)_{2}}=u^{\mathrm{intra}}\sum_{1\leq
a<b\leq n}J^{ab}_{y,SO(n)_{2}^{L}}J^{ab}_{y,SO(n)_{2}^{R}}$ (102)

where
$J^{ab}_{y,SO(n)_{2}^{\sigma}}=i(\psi^{A\sigma}_{ya}\psi^{A\sigma}_{yb}+\psi^{B\sigma}_{ya}\psi^{B\sigma}_{yb})$
are the $SO(n)_{2}$ currents. It gaps the $SO(n)_{2}$ degrees of freedom in
all wires for the same reasons (86) does and has been presented in the
previous subsection.

For the inter-wire interaction
$\mathcal{U}^{y+1/2}_{U(1)_{l}/\mathbb{Z}_{2}}$, it may be tempting to mimick
(31) and consider the sine-Gordon potentials
$\cos\left[q\left(\phi^{L}_{y+1,\perp}\pm\phi^{R}_{y,\perp}\right)\right]$,
where $e^{i\phi_{\perp}}=e^{i(\phi_{1}+\ldots+\phi_{n})}$ is a $SO(n)_{2}$
singlet that generates $U(1)_{l}/\mathbb{Z}_{2}$ and is the diagonal product
of Dirac fermions $e^{i\phi_{a}}=(\psi_{a}^{A}+i\psi_{a}^{B})/\sqrt{2}$.
However, the potentials switch

$\displaystyle\cos\left[q\left(\phi^{L}_{y+1,\perp}+\phi^{R}_{y,\perp}\right)\right]\leftrightarrow\cos\left[q\left(\phi^{L}_{y+1,\perp}-\phi^{R}_{y,\perp}\right)\right]$
(103)

under the local $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry (see (79))

$\displaystyle\begin{split}&\mathbb{Z}^{A}_{2}(y):\quad\phi^{\sigma}_{y^{\prime}\perp}\to(-1)^{\delta_{yy^{\prime}}}\phi^{\sigma}_{y^{\prime}\perp}+n\pi\delta_{yy^{\prime}}\\\
&\mathbb{Z}^{B}_{2}(y):\quad\phi^{\sigma}_{y^{\prime}\perp}\to(-1)^{\delta_{yy^{\prime}}}\phi^{\sigma}_{y^{\prime}\perp}\end{split}$
(104)

and are therefore not integral combinations of electrons. Instead, we
introduce the interactions

$\displaystyle\begin{split}\mathcal{U}^{y+1/2}_{U(1)_{l}/\mathbb{Z}_{2}}&=-u^{\mathrm{inter}}_{\perp}\left(\Sigma_{y+1/2}\right)^{2}\\\
&\quad-\Delta^{\mathrm{inter}}_{\perp}\cos\left(q\phi^{L}_{y+1,\perp}\right)\cos\left(q\phi^{R}_{y,\perp}\right)\\\
\Sigma_{y+1/2}&=g\partial_{x}\phi^{L}_{y+1,\perp}\partial_{x}\phi^{R}_{y,\perp}\\\
&\quad+\sin\left(q\phi^{L}_{y+1,\perp}\right)\sin\left(q\phi^{R}_{y,\perp}\right)\end{split}$
(105)

where $q=1$ when $n$ is even, or $q=2$ when $n$ is odd. The interactions are
local because $\cos\left(q\phi_{\perp}\right)$,
$\left(\partial_{x}\phi_{\perp}\right)^{2}$,
$\left[\sin\left(q\phi_{\perp}\right)\right]^{2}$, and
$\partial_{x}\phi_{\perp}\sin\left(q\phi_{\perp}\right)$ are symmetric under
(104) and are combinations of even products of fermions in both the $A$ and
$B$ sectors.

The $\Delta^{\mathrm{inter}}_{\perp}$ term can be expressed as

$\displaystyle\cos\left(q\phi^{L}_{y+1,\perp}\right)\cos\left(q\phi^{R}_{y,\perp}\right)=\frac{1}{2}\left(\cos\theta^{y+\frac{1}{2}}_{\perp}+\cos\varphi^{y+\frac{1}{2}}_{\perp}\right).$
(106)

The angle variables
$\theta^{y+1/2}_{\perp}=q(\phi^{L}_{y+1,\perp}-\phi^{R}_{y,\perp})$ and
$\varphi^{y+1/2}_{\perp}=q(\phi^{L}_{y,\perp}+\phi^{R}_{y+1,\perp})$ do not
commute, and therefore the two sine-Gordon potentials compete. The
$u^{\mathrm{inter}}_{\perp}$ term can be re-expressed – using a Hubbard-
Stratonovich transformation Stratonovich (1958); Hubbard (1959) by introducing
a real bosonic auxiliary field $s_{y+1/2}$ – into

$\displaystyle\frac{\tilde{s}_{y+1/2}^{2}}{u^{\mathrm{inter}}_{\perp}}-u^{\mathrm{inter}}_{\perp}\left(\Sigma_{y+1/2}\right)^{2}=\frac{s_{y+1/2}^{2}}{u^{\mathrm{inter}}_{\perp}}-2s_{y+1/2}\Sigma_{y+1/2}.$
(107)

$\tilde{s}_{y+1/2}=s_{y+1/2}-u^{\mathrm{inter}}_{\perp}\Sigma_{y+1/2}$
completes the square and can be integrated out. In low-energy, the auxiliary
takes a finite ground state expectation value $\langle s_{y+1/2}\rangle$. The
interaction (105) admits the mean-field approximation

$\displaystyle\mathcal{U}^{y+1/2}_{U(1)_{l}/\mathbb{Z}_{2}}$
$\displaystyle=\frac{\langle
s_{y+1/2}\rangle^{2}}{u^{\mathrm{inter}}_{\perp}}-2g\langle
s_{y+1/2}\rangle\partial_{x}\phi^{L}_{y+1,\perp}\partial_{x}\phi^{R}_{y,\perp}$
$\displaystyle-\left(\frac{\Delta^{\mathrm{inter}}_{\perp}}{2}+\langle
s_{y+1/2}\rangle\right)\cos\theta^{y+1/2}_{\perp}$
$\displaystyle\quad\quad-\left(\frac{\Delta^{\mathrm{inter}}_{\perp}}{2}-\langle
s_{y+1/2}\rangle\right)\cos\varphi^{y+1/2}_{\perp}.$ (108)

When $\langle s_{y+1/2}\rangle$ is positive (negative) and for large $g$,
$g|\langle s_{y+1/2}\rangle|>v(q^{4}n^{2}-4)/[4\pi n(q^{4}n^{2}+4)]$,
$\cos\theta^{y+1/2}_{\perp}$ is relevant (irrelevant) while
$\cos\varphi^{y+1/2}_{\perp}$ is irrelevant (relevant) in the RG sense.
Therefore, depending on the sign of the order parameter

$\displaystyle\mathrm{sgn}\langle
s_{y+1/2}\rangle=\mathrm{sgn}\left\langle\sin\left(m\phi^{L}_{y+1,\perp}\right)\sin\left(m\phi^{R}_{y,\perp}\right)\right\rangle,$
(109)

either $\cos\theta^{y+1/2}_{\perp}$ or $\cos\varphi^{y+1/2}_{\perp}$ dominates
and takes non-vanishing ground state expectation value.

$\displaystyle\begin{split}s_{y+1/2}>0:\quad\left\langle\cos\theta^{y+1/2}_{\perp}\right\rangle=1,\quad\left\langle\cos\varphi^{y+1/2}_{\perp}\right\rangle=0,\\\
s_{y+1/2}<0:\quad\left\langle\cos\theta^{y+1/2}_{\perp}\right\rangle=0,\quad\left\langle\cos\varphi^{y+1/2}_{\perp}\right\rangle=1.\end{split}$
(110)

Both lead to a finite bulk excitation energy gap.

The ground state expectation values pattern (110) is similar to (90) in the
previous $SU(n)_{1}/\mathbb{Z}_{2}$ case, and consequently, the anyon
structure also resembles. The non-local bosonic fields
$\partial_{x}\phi_{\perp}$ and $\sin(q\phi_{\perp})$ belong in the
$\mathbb{Z}_{2}$ charge sector $S$ of the $U(1)_{l}/\mathbb{Z}_{2}$ orbifold
theory. The domain wall separating the two sets of ground states in (110),
where $\langle
s_{y+1/2}(x)\rangle\sim\langle\sin(q\phi^{L}_{y+1,\perp}(x))\sin(q\phi^{R}_{y,\perp}(x))\rangle\sim\pm\mathrm{sgn}(x-x_{0})$
changes sign, traps a $\mathbb{Z}_{2}$ gauge flux (twist field) at $x_{0}$. It
associates a $\pi$ monodromy braiding phase for an orbiting $\mathbb{Z}_{2}$
charge. It also conjugates an orbiting vertex field (39) in $U(1)_{l}$ between
$e^{m}=e^{im\phi_{\perp}/(qn)}\leftrightarrow e^{-m}=e^{-im\phi_{\perp}/(qn)}$
(see figure 3(b)) according to the $\mathbb{Z}_{2}^{A/B}$ action in (104).

When $e^{m}$ passes through the side of the domain wall with $s_{y+1/2}(x)>0$,
the backscattering term pins $\langle
e^{m}_{L}(x)e^{m}_{R}(x)^{\dagger}\rangle\sim
e^{im\langle\theta^{y+1/2}_{\perp}(x)\rangle/(q^{2}n)}\sim 1$ to a finite
value. Further moving $e^{m}$ to the other side of the domain wall with
$s_{y+1/2}(x)<0$, the pairing term pins $\langle
e^{m}_{L}(x)e^{m}_{R}(x)\rangle\sim
e^{im\langle\varphi^{y+1/2}_{\perp}(x)\rangle/(q^{2}n)}\sim 1$. Therefore,
$e^{m}$ is conjugated to $(e^{m})^{\dagger}=e^{-m}$ in a complete cycle. For
$m=1,\ldots,l/2-1$ modulo $l$,

$\displaystyle\mathcal{E}^{m}=e^{m}+e^{-m}$ (111)

is now a single super-selection sector. The $\mathbb{Z}_{2}$ invariant
$e^{l/2}=e^{iq\phi_{\perp}/2}$ decomposes into the even and odd sectors

$\displaystyle e^{l/2}_{+}=\cos(q\phi_{\perp}/2),\quad
e^{l/2}_{-}=\sin(q\phi_{\perp}/2),$ (112)

where $q=1$ for $n$ even, or $q=2$ for $n$ odd. There are four
$\mathbb{Z}_{2}$ twist fields $\sigma^{\pm}$ and $\tau^{\pm}$. They obey a set
of fusion rules identical to those in $SU(n)_{1}/\mathbb{Z}_{2}$ and can be
obtained by replacing $\Phi^{i},\Psi^{n/2}_{\pm}$ by
$\mathcal{E}^{i},e^{l/2}_{\pm}$ in (94), (98), (99) and (100). The spin
statistics and quantum dimensions of the orbifold anyons are listed in table 3
and the complete modular data is summarized in appendix B.

anyons | quantum dimension | spin (modulo 1)
---|---|---
1 | 1 | 0
$S$ | 1 | 1
$e_{\pm}^{l/2}$ | 1 | $l/8$
$\mathcal{E}^{i}$ | 2 | $i^{2}/(2l)$
$\sigma^{\pm}$ | $\sqrt{l/2}$ | $1/16$
$\tau^{\pm}$ | $\sqrt{l/2}$ | $9/16$
Table 3: Anyons in $U(1)_{l}/\mathbb{Z}_{2}$. Here $i=1,\ldots,l/2-1$.

The $U(1)_{l}/\mathbb{Z}_{2}=[SO(n)_{1}]^{2}/SO(n)_{2}$ topological order can
be related to $SU(n)_{1}/\mathbb{Z}_{2}=SO(n)_{2}$ described in the previous
subsection IV.1 by the UMTC equivalence

$\displaystyle\frac{[SO(n)_{1}]^{2}}{SO(n)_{2}}=[SO(n)_{1}]^{2}\boxtimes\overline{SO(n)_{2}}.$
(113)

Here $\overline{SO(n)_{2}}$ is the time-reversal conjugate (or spatial
reflection) of $SO(n)_{2}$ that hosts edge modes in the opposite direction and
anyons with reversed spins $h\to-h$. The relative tensor product $\boxtimes$
treats the $\mathbb{Z}_{2}$ charge pair $\zeta\bar{S}$ as an integral
combination that “condenses” in the vacuum sector in the anyon condensation
picture Bais and Slingerland (2009); Kong (2014b); Neupert  _et al._ (2016);
Burnell (2018), where $\zeta=\psi^{A}\psi^{B}$ is the fermion pair from
$[SO(n)_{1}]^{2}$ (see section III.3.1) and $\bar{S}$ is the $\mathbb{Z}_{2}$
charge of $\overline{SO(n)_{2}}$. In addition, when $n$ is even, the bosonic
spinor product $s_{\pm}^{A}s_{\pm}^{B}\overline{\Psi^{n/2}_{\pm}}$ are also
“condensed” in $\boxtimes$, where $s^{A}s^{B}$ are spinor pairs in
$[SO(2n)_{1}]^{2}$ and $\overline{\Psi^{n/2}_{\pm}}$ are defined in (93). (A
more detailed description of the UMTC equivalence (113) in the anyon
condensation picture can be found in appendix B.) Eq.(113) demonstrates a
generalized notion of “particle-hole” conjugation. The
$U(1)_{l}/\mathbb{Z}_{2}$ state can be regarded as the particle-hole conjugate
of $SU(n)_{1}/\mathbb{Z}_{2}$, where “holes” that occupy a $SO(n)_{2}$ state
are subtracted from a “filled” $[SO(n)_{1}]^{2}$ state. Similar particle-hole
relation also applies to the Abelian unitary family described in section III.2
where $SU(n)_{1}$ and $U(1)_{l}$ are conjugate dual so that one can be
obtained from subtracting the other from $SO(2n)_{1}$. This is analogous to
the anti-Pfaffian fractional quantum Hall state Levin  _et al._ (2007); Lee
_et al._ (2007), which is the particle-hole conjugate of the Moore-Read
Pfaffian state Moore and Read (1991); Greiter  _et al._ (1991) and is obtained
by subtracting from the filled Landau level, holes that occupy the Moore-Read
Pfaffian state. Here, $[SO(n)_{1}]^{2}$ for the orbifold theories (or
$SO(2n)_{1}$ for the Abelian unitary family) takes the role of the filled
Landau level and is considered to be the base of the “particle-hole”
conjugation.

Lastly, we conclude this subsection by commenting on alternative coupled-wire
models where $SO(n)_{2}=SU(n)_{1}/\mathbb{Z}_{2}$ and
$[SO(n)_{1}]^{2}/SO(n)_{2}=U(1)_{l}/\mathbb{Z}_{2}$ are both back-scattered
within a wire or in-between wires. Similar to (44), a trivial topological
phase of decoupled wires is obtained when all degrees of freedom are gapped by
complete intra-wire interactions,

$\displaystyle\mathcal{H}_{\mathrm{trivial}}=\mathcal{H}_{0}+\sum_{y=1}^{\mathsf{L}}\left(\mathcal{U}^{y}_{U(1)_{l}/\mathbb{Z}_{2}}+\mathcal{U}^{y}_{SO(n)_{2}}\right).$
(114)

Here, $\mathcal{H}_{0}$ is the kinectic Hamiltonian (78) of the
$[SO(2n)_{1}]^{2}$ wires, and $\mathcal{U}^{y}_{U(1)_{l}/\mathbb{Z}_{2}}$ and
$\mathcal{U}^{y}_{SO(n)_{2}}$ are given in (81) and (102) respectively. On the
other hand, a non-trivial topological phase can be constructed using solely
inter-wire interactions (c.f. (45))

$\displaystyle\mathcal{H}_{[SO(n)_{1}]^{2}}=\mathcal{H}_{0}+\sum_{y=1}^{\mathsf{L}}\left(\mathcal{U}^{y+1/2}_{SO(n)_{2}}+\mathcal{U}^{y+1/2}_{U(1)_{l}/\mathbb{Z}_{2}}\right),$
(115)

where $\mathcal{U}^{y+1/2}_{SO(n)_{2}}$ and
$\mathcal{U}^{y+1/2}_{U(1)_{l}/\mathbb{Z}_{2}}$ are presented in (86) and
(105). It does not carry the topological order of a decoupled product
$SO(n)_{2}\times U(1)_{l}/\mathbb{Z}_{2}$. This is because there are product
fields, $\Phi^{2}\times\mathcal{E}^{2q}$ and its higher powers, that are
integral electronic combinations and belong to the vacuum sector. (See (92)
and (111) for the anyon definitions and compare with the anyon pair in (48).)
Therefore, the anyon pair splits into
$\Phi^{2}\times\mathcal{E}^{2q}=1+$(confined components) and is “condensed” in
the anyon condensation picture. The resulting UMTC of deconfined anyons is
identical to the $[SO(n)_{1}]^{2}$ topological order.

###  IV.3 The $SO(2n)_{1}/D_{k}$ twist liquid

In section III.3, we defined the global dihedral
$D_{k}=\mathbb{Z}_{2}\ltimes\mathbb{Z}_{k}$ symmetry of $SO(2n)_{1}$ in (50),
where $k=n$ ($k=n/2$) for $n$ odd (even). We demonstrated the gauging of the
cyclic $\mathbb{Z}_{k}$ symmetry leads to the twist liquid orbifold phase
$SO(2n)_{1}/\mathbb{Z}_{k}=U(1)_{l}\times SU(n)_{1}$, where $l=4n$ ($l=n$) for
$n$ odd (even). The construction was based on altering the notion of locality
so that only the current operators in the $\mathbb{Z}_{k}$-invariant
$U(1)_{l}\times SU(n)_{1}$ WZW sub-algebra in $SO(2n)_{1}$ were integral
electronic combinations. In this subsection, we construct the $D_{k}$ twist
liquid orbifold phase

$\displaystyle\frac{SO(2n)_{1}}{D_{k}}=\frac{U(1)_{l}\times
SU(n)_{1}}{\mathbb{Z}_{2}}$ (116)

by further gauging the conjugation symmetry, which corresponds to the quotient
group $\mathbb{Z}_{2}=D_{k}/\mathbb{Z}_{k}$.

${SO(2n)_{1}}$${SO(2n)_{1}/\mathbb{Z}_{k}}$${SO(2n)_{1}/D_{k}}$$\scriptstyle{\mathbb{Z}_{k}}$$\scriptstyle{\mathrm{gauging}}$$\scriptstyle{\mathbb{Z}_{2}}$$\scriptstyle{\mathrm{gauging}}$
(117)

The local fields in $SO(2n)_{1}/D_{k}$ are the $D_{k}$-invariant local fields
in $SO(2n)_{1}$, or equivalently, the $\mathbb{Z}_{2}$-invariant local fields
in $SO(2n)_{1}/\mathbb{Z}_{k}=U(1)_{l}\times SU(n)_{1}$ (recall the
conjugation action (77)). They are generated by (1) the $SO(n)_{2}$ WZW
currents, which are the self-conjugate $SU(n)_{1}$ currents,
$\cos\left(\phi_{a}-\phi_{b}\right)$, for $1\leq a<b\leq n$ (see (84)), (2)
the $\mathbb{Z}_{2}$ even local fields in $U(1)_{l}/\mathbb{Z}_{2}$,
$\cos\left(q\phi_{\perp}\right)$, $\left(\partial_{x}\phi_{\perp}\right)^{2}$,
$\left[\sin\left(q\phi_{\perp}\right)\right]^{2}$, and
$\partial_{x}\phi_{\perp}\sin\left(q\phi_{\perp}\right)$ (see below (105)), as
well as (3) the pairs of $\mathbb{Z}_{2}$ odd local fields in $U(1)_{l}$ and
$SU(n)_{1}$, $\partial\phi_{\perp}\partial\tilde{\phi}_{p}$,
$\partial\phi_{\perp}\sin\left(\phi_{a}-\phi_{b}\right)$,
$\sin\left(q\phi_{\perp}\right)\partial\tilde{\phi}_{p}$, and
$\sin\left(q\phi_{\perp}\right)\sin\left(\phi_{a}-\phi_{b}\right)$. If the
local fields were only generated by sets (1) and (2), the $SO(2n)_{1}/D_{k}$
theory would be identical to the decoupled product $U(1)/\mathbb{Z}_{2}\times
SU(n)_{1}/\mathbb{Z}_{2}$. Local fields in set (3) are pairs of
$\mathbb{Z}_{2}$ charges from both $U(1)_{l}/\mathbb{Z}_{2}$ and
$SU(n)_{1}/\mathbb{Z}_{2}$. Therefore, the $D_{k}$ twist liquid (116), as an
UMTC, is identical to the relative tensor product
$U(1)/\mathbb{Z}_{2}\boxtimes SU(n)_{1}/\mathbb{Z}_{2}$, where the
$\mathbb{Z}_{2}$ charge pair
$S_{U(1)/\mathbb{Z}_{2}}S_{SU(n)_{1}/\mathbb{Z}_{2}}$ is “condensed” in the
anyon condensation Bais and Slingerland (2009); Kong (2014b); Neupert  _et
al._ (2016); Burnell (2018) context. This notion of locality requires the
$\mathbb{Z}_{2}$ fluxes from $U(1)_{l}/\mathbb{Z}_{2}$ and
$SU(n)_{1}/\mathbb{Z}_{2}$ to appear simultaneously. In other words, the
$\mathbb{Z}_{2}$ local symmetry applies simultaneously on both $U(1)_{l}$ and
$SU(n)_{1}$ in (116).

The coupled-wire model, that respects this notion of locality, is constructed
by an array of $SO(2n)_{1}\times SO(2n)_{1}$ wires. (See section II for the
electronic spin liquid and superconductor origins.) In low-energy, any given
wire at $y$ carries the chiral Majorana fermions $\psi^{A\sigma}_{ya}$ and
$\psi^{B\sigma}_{ya}$, where $C=A,B$ distinguishes the two $SO(2n)_{1}$
sectors, $a=1,\ldots,2n$, and $\sigma=L,R=+,-$. Like the previous orbifold
phases, only fermion pairs
$\psi^{C\sigma}_{ya}\psi^{C\sigma^{\prime}}_{ya^{\prime}}$ from the same
$C=A,B$ sector on the same wire $y$ are integral combinations of electrons.
The $SO(2n)_{1}/D_{k}$ twist liquid model is constructed by populating the
$SU(n)_{1}/\mathbb{Z}_{2}$ Hamiltonian (91) with
$\psi^{C\sigma}_{y,1},\ldots\psi^{C\sigma}_{y,n}$ and the
$U(1)_{l}/\mathbb{Z}_{2}$ Hamiltonian (101) with
$\psi^{C\sigma}_{y,n+1},\ldots\psi^{C\sigma}_{y,2n}$.

$\displaystyle\mathcal{H}_{SO(2n)_{1}/D_{k}}$ (118)
$\displaystyle=\mathcal{H}_{SU(n)_{1}/\mathbb{Z}_{2}}\left[\psi_{1},\ldots\psi_{n}\right]+\mathcal{H}_{U(1)_{l}/\mathbb{Z}_{2}}\left[\psi_{n+1},\ldots\psi_{2n}\right]$

The local $\mathbb{Z}_{2}$ symmetry

$\displaystyle\mathbb{Z}_{2}^{B}(y):\quad\psi^{A}\to\psi^{A},\quad\psi^{B}_{y^{\prime}}\to(-1)^{\delta_{yy^{\prime}}}\psi^{B}_{y^{\prime}}$
(119)

acts simultaneously on both $SU(n)_{1}/\mathbb{Z}_{2}$ and
$U(1)_{l}/\mathbb{Z}_{2}$. The $\mathbb{Z}_{2}$ charge
$S_{U(1)_{l}/\mathbb{Z}_{2}}\sim\psi^{A}_{a}\psi^{B}_{b}$ and
$S_{SU(n)_{1}/\mathbb{Z}_{2}}\sim\psi^{A}_{n+a}\psi^{B}_{n+b}$, for $1\leq
a<b\leq n$, are individually non-local and must come in pairs. On the other
hand, the $\mathbb{Z}_{2}$ charge pair
$S_{U(1)/\mathbb{Z}_{2}}S_{SU(n)_{1}/\mathbb{Z}_{2}}$ is integral because
$\psi^{A}_{a}\psi^{A}_{n+a}$ and $\psi^{B}_{b}\psi^{B}_{n+b}$ are local bosons
by construction. $S_{U(1)/\mathbb{Z}_{2}}$ and $S_{SU(n)_{1}/\mathbb{Z}_{2}}$
belong in the same anyon class $S$ that represents the $\mathbb{Z}_{2}$ gauge
charge in $SO(2n)_{1}/D_{k}$. Although
$\mathcal{H}_{SU(n)_{1}/\mathbb{Z}_{2}}$ and
$\mathcal{H}_{U(1)_{l}/\mathbb{Z}_{2}}$ in (118) involve distinct sets of
fermions, their combination is not a direct sum, unlike (49). This is because
the Hilbert space does not decompose thanks to the intertwined fermion pair
locality.

The topological order of $SO(2n)_{1}/D_{k}$ is closely related to that of the
discrete gauge theory of the dihedral group and its $\mathbb{Z}_{2}$
extensions. A review of discrete gauge theory can be found in appendix C. The
relation depends on $n$ modulo 8 because of the eight-fold periodicity of the
$SO(2n)_{1}$ UMTC and is summarized in (139). The $SO(2n)_{1}$ anyon classes
form an Abelian group $\mathcal{A}=\\{1,\psi,s_{+},s_{-}\\}$ under fusion
product, where $\mathcal{A}=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ when $n$ is
even, or $\mathbb{Z}_{4}$ when $n$ is odd. The fusion group $\mathcal{A}$
extends the global symmetry group $G=D_{k}$ of $SO(2n)_{1}$ in to a quantum
symmetry group Wen (2002); Etingof  _et al._ (2010); Teo  _et al._ (2015);
Barkeshli  _et al._ (2019) $\widehat{G}$ (see (54) and (190)). Inequivalent
group extensions are classified by the group cohomology Brown (1982)
$H^{2}(G,\mathcal{A})$ (see appendix C and D). When $n=k$ is odd,
$\widehat{G}=G\ltimes\mathcal{A}=D_{k}\ltimes\mathbb{Z}_{4}$ is “symmorphic”
and corresponds to the trivial element in $H^{2}(D_{k},\mathcal{A})$. The
twofold conjugation symmetry in $D_{k}$ is an outer automorphism of $SO(2n)$.
It acts non-trivially on $\mathcal{A}$ in the semi-direct product and flips
between the even and odd spinor, $s_{+}\leftrightarrow s_{-}$. When $n$ is
even, the $D_{k}$ symmetries are all inner automorphisms and act trivially on
$\mathcal{A}$. However, the global quantum symmetry group $\widehat{G}$ is
“non-symmorphic” and does not decompose in a direct product
$G\times\mathcal{A}$. When $k=n/2$ is even, we saw in section III.3.2 that the
$\mathbb{Z}_{k}$ flux had order $2k$ and its $k^{\mathrm{th}}$ power was
identified as the even spinor $m^{k}=s_{+}$ in $SO(2n)_{1}$. This extends
$D_{k}$ into $D_{2k}$, and the quantum symmetry group is
$\widehat{G}=D_{2k}\times\mathbb{Z}_{2}$. The non-symmorphic central extension
corresponds to the non-trivial cohomology element $[s_{\pm},1,1]$ (or
$[\psi,1,1]$) in $H^{2}(D_{k},\mathcal{A})=\mathcal{A}^{3}$ when $n\equiv 0$
(resp. $n\equiv 4$) modulo 8. When $k=n/2$ is odd, we showed in section
III.3.1 that the $\mathbb{Z}_{2}$ symmetry was extended to $\mathbb{Z}_{4}$
because the twofold flux squared to the $SO(2n)_{1}$ fermion
$s^{2}_{\pm}=\psi$. This extends the global dihedral symmetry group $D_{k}$
into the dicyclic group $Q_{4k}$ (defined in (125) below), and the quantum
symmetry group is $\widehat{G}=Q_{4k}\times\mathbb{Z}_{2}$. The non-symmorphic
central extension corresponds to a non-trivial cohomology element $[\psi]$ in
$H^{2}(D_{k},\mathcal{A})=\mathcal{A}$.

We first present the topological order of $SO(2n)_{1}/D_{k}$ when $n\equiv 0$
modulo 8. From (70) in section III.3, we saw that
$SO(2n)_{1}/\mathbb{Z}_{k}=U(1)_{l}\times SU(n)_{1}$ is equivalent, as an
UMTC, to the $D^{[k]}(\mathbb{Z}_{2k})$ discrete gauge theory, where $k=n/2$.
Gauging the $\mathbb{Z}_{2}$ conjugation symmetry leads to the
$D^{[k,0,0]}(D_{2k})$ discrete gauge theory, where

$\displaystyle D_{2k}$
$\displaystyle=\mathbb{Z}_{2}\ltimes\mathbb{Z}_{2k}=\left\\{\mu^{A}m^{a}:A=0,1,a=-k+1,\ldots,k\right\\}$
$\displaystyle=\left\langle m,\mu|m^{2k}=\mu^{2}=(\mu m)^{2}=1\right\rangle$
(120)

is the dihedral group that contains $\mathbb{Z}_{2k}$ as a normal subgroup and
$\mathbb{Z}_{2}=D_{2k}/\mathbb{Z}_{2k}$. Since the twist liquid descends from
$SO(2n)_{1}$, which has the identical topological order as the
$D^{[0]}(\mathbb{Z}_{2})$ discrete gauge theory, the sequence of gauging (117)
is the quantum double Kassel (1995); Bakalov and Kirillov (2001); Levin and
Wen (2005) (or Drinfeld center) of the group filtration

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