# Fermionic quantum criticality through the lens of topological holography

Sheng-Jie Huang  Max Planck Institute for the Physics of Complex Systems,
Nöthnitzer Str. 38, 01187 Dresden, Germany

###### Abstract

We utilize the topological holographic framework to characterize and gain
insights into the nature of quantum critical points and gapless phases in
fermionic quantum systems. Topological holography is a general framework that
describes the generalized global symmetry and the symmetry charges of a local
quantum system in terms of a slab of a topological order, termed as the
symmetry topological field theory (SymTFT), in one higher dimension. In this
work, we consider a generalization of the topological holographic picture for
(1+1)$d$ fermionic quantum phases of matter. We discuss how spin structures
are encoded in the SymTFT and establish the connection between the formal
fermionization formula in quantum field theory and the choice of fermionic
gapped boundary conditions of the SymTFT. We demonstrate the identification
and the characterization of the fermionic gapped phases and phase transitions
through detailed analysis of various examples, including the fermionic systems
with $\mathbb{Z}_{2}^{F}$, $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$,
$\mathbb{Z}_{4}^{F}$, and the fermionic version of the non-invertible
$\text{Rep}(S_{3})$ symmetry. Our work uncovers many exotic fermionic quantum
critical points and gapless phases, including two kinds of fermionic symmetry
enriched quantum critical points, a fermionic gapless symmetry protected
topological (SPT) phase, and a fermionic gapless spontaneous symmetry breaking
(SSB) phase that breaks the fermionic non-invertible symmetry.

###### Contents

  1. I Introduction
    1. I.1 Summary of results
  2. II Review: topological holography for bosonic systems
  3. III Fermionization in topological holography
  4. IV Fermionic chains with $\mathbb{Z}_{2}^{F}$ symmetry
    1. IV.1 Gapped phases
    2. IV.2 Free Majorana fermion
  5. V Fermionic chains with $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ symmetry
    1. V.1 Gapped phases
    2. V.2 Quantum critical points
      1. V.2.1 $\text{Trivial}\leftrightarrow(\text{SSB}+\text{Kitaev})=\text{Ising}^{2}\bf{/Z_{2}^{D}}$
      2. V.2.2 $\text{Kitaev}\leftrightarrow(\text{Kitaev}+\text{GW})=\text{Ising}^{2}\bf{/Z_{2}^{D}}$
      3. V.2.3 $\text{Trivial}\leftrightarrow\text{GW}=\text{Majorana}^{2}$
      4. V.2.4 $\text{Kitaev}\leftrightarrow\text{SSB}=\text{Ising}\times\text{Majorana}$
      5. V.2.5 $\text{Trivial}\leftrightarrow\text{Kitaev}=\text{Majorana}$
  6. VI Fermionic symmetry enriched quantum critical points
    1. VI.1 $\text{GW}\leftrightarrow(\text{SSB}+\text{Kitaev})=\text{Ising}^{2*}\bf{/Z_{2}^{D}}$
    2. VI.2 $\text{GW}\leftrightarrow\text{SSB}=\text{Ising}\times\mathcal{F}_{\text{GW}}$
    3. VI.3 ($\text{Kitaev}+\text{GW})\leftrightarrow\text{SSB}=(\text{Ising}\times\text{Majorana})^{*}$
  7. VII Fermionic chains with $\mathbb{Z}_{4}^{F}$ symmetry
    1. VII.1 Fermionization
    2. VII.2 Gapped phases
    3. VII.3 Quantum critical points
      1. VII.3.1 $\mathbb{Z}_{4}^{F}\ \text{Trivial}\leftrightarrow(\mathbb{Z}_{2}\ \text{SSB}+\text{Kitaev})=\text{Majorana}^{2}$
      2. VII.3.2 $\mathbb{Z}_{4}^{F}\ \text{Trivial}\leftrightarrow\mathbb{Z}_{2}\ \text{SSB}=\text{Ising}\times\mathcal{F}_{0}$
      3. VII.3.3 $\mathbb{Z}_{2}\ \text{SSB}\leftrightarrow(\mathbb{Z}_{2}\ \text{SSB}+\text{Kitaev})=2\ \text{Majorana}$
    4. VII.4 The fermionic intrinsically gapless SPT phase with $\mathbb{Z}_{4}^{F}$ symmetry
  8. VIII Fermionic chains with super fusion category $\text{SRep}(\mathcal{H}_{S_{3}})$ symmetry
    1. VIII.1 Fermionization
    2. VIII.2 Gapped phases
    3. VIII.3 Quantum critical points
      1. VIII.3.1 Input critical theories
      2. VIII.3.2 $\text{SRep}(\mathcal{H}_{S_{3}})$ Trivial $\leftrightarrow$ ($\text{SRep}(\mathcal{H}_{S_{3}})/\mathbb{Z}_{2}$ SSB + Kitaev) = fermionic minimal model $(m=5)$
      3. VIII.3.3 $\text{SRep}(\mathcal{H}_{S_{3}})$ Kitaev $\leftrightarrow$ $\text{SRep}(\mathcal{H}_{S_{3}})/\mathbb{Z}_{2}$ SSB = fermionic minimal model $(m=5)$
      4. VIII.3.4 $\text{SRep}(\mathcal{H}_{S_{3}})$ Trivial $\leftrightarrow$ $\text{SRep}(\mathcal{H}_{S_{3}})/\mathbb{Z}_{2}$ SSB = 3-Potts $\times$ $\mathcal{F}_{0}$
      5. VIII.3.5 $\text{SRep}(\mathcal{H}_{S_{3}})$ Trivial $\leftrightarrow$ $\text{SRep}(\mathcal{H}_{S_{3}})$ Kitaev = Majorana
      6. VIII.3.6 $\text{SRep}(\mathcal{H}_{S_{3}})/\mathbb{Z}_{2}$ SSB $\leftrightarrow$ ($\text{SRep}(\mathcal{H}_{S_{3}})/\mathbb{Z}_{2}$ SSB + Kitaev) = Ising + Majorana
  9. IX Discussion
  10. A Fermionic gapped boundaries and super Lagrangian algebra
  11. B Fermionic minimal models

##  I Introduction

Understanding the phases and transitions in condensed matter physics stands as
a central theme in condensed matter physics. Landau paradigm provides a
unified frameworks that characterized phases and phase transitions by the
symmetry and the patterns of their breaking. However, recent developments in
the field have led to the discovery of novel phases of matter that cannot be
described by Landau theory such as topological orders with long-rang
entanglement [1] as well as phase transitions that are forbidden by the Landau
theory [2].

In developing theoretical frameworks that characterizes and classifies phase
and phase transitions, symmetry has been played a very important role as an
organizing principle. It provides valuable insights into classifications and
also yields significant physical implications, including the emergence of
conservation laws and constraints on low-energy dynamics. The generalization
of the notion of symmetry builds on a fact that there is an one-to-one
correspondence between the global symmetry described by a symmetry group $G$
and codimension-$1$ topological defects satisfying group multiplication fusion
rules. There are however various different kinds of topological defects that
could have higher-codimensions, satisfying non-invertible fusion rules, and
have a non-trivial interplay among themselves, which have been conjectured to
have a higher categorical description [3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. The
modern perspective of the symmetry is that every topological defect should be
viewed as a generalized global symmetry [13, 14, 15].

Recently, a novel topological holographic framework has emerged, offering a
holographic perspective on symmetry. At its core, this framework encodes
symmetry data within a topological order residing in one higher dimension,
thereby extending earlier developments within the context of $(1+1)d$ rational
CFTs [16, 17, 18, 19]. The central picture is given by the “sandwich”
construction, where we can view the $D$ spacetime dimensional quantum system
of interest as a slab of topological order in $D+1$ dimensions with
appropriately chosen boundary conditions. This topological order is commonly
called a _symmetry topological field theory_ (SymTFT) or a _symmetry
topological order_ (SymTO) [6, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31,
32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49]. This
approach essentially separates the generalized symmetry and the generalized
symmetry charges from the dynamics of a quantum system and encode such
information in the bulk topological order. It can describe both gapped and
gapless phases in a unified framework, and also allows one to make non-
perturbative statements about phases, phases transitions, and dualities.
Recent developments towards physical applications include the generalization
to continuous or non-finite symmetries [50, 51, 52, 53], the studying of the
gapped phases with non-invertible fusion category symmetries [46, 47],
understanding exotic quantum critical points and gapless SPT phases in bosonic
systems [37, 48, 54, 55, 49], and constructing lattice models with non-
invertible symmetries [56, 57, 58].

Our main focus in this work is to study quantum criticality in fermionic
systems in (1+1)$d$ from the point of view of topological holography.
Traditionally, one invaluable tool in tackling this challenge is the
bosonization and the Jordan–Wigner transformation. This technique provide a
bridge between the fermions and bosons, facilitating the exploration of
fermionic systems by mapping them onto more tractable bosonic counterparts.
However, identifying the correct conformal field theory (CFT) description of a
phase transition remains to be a non-trivial task, which often required
numerical simulations. It’s thus desirable to develop a theoretical framework
that can help to identify possible CFTs of a phase transition. Previous works
have applied the SymTFT to characterize phase transitions and gapless SPT
phases for bosonic systems [37, 48, 54, 55, 49]. However, much less is known
regarding the application of this framework to phase transitions and gapless
phases for fermionic systems and the resulting physical implications.

In this work, we apply the idea of topological holography to characterize
fermionic gapped phases and identify possible CFT descriptions for the phase
transitions. In order to proceed, we need a generalization of the sandwich
construction for fermionic systems in the topological holography framework.
Recall the central picture of the “sandwich” construction is that we can view
the (1+1)$d$ quantum system of interest as a slab of (2+1)$d$ topological
order with appropriately chosen boundary conditions. The left boundary for
bosonic systems is chosen to be a gapped topological boundary, on which some
bosonic anyons in the bulk SymTFT condense, and the confined anyon lines on
the left boundary implement the global symmetry of the original (1+1)$d$
system. To apply the idea of topological holography to fermoinic systems, one
needs to choose a fermionic gapped boundary condition involving the
condensation of the fermionic anyons [59, 60, 61] on the symmetry boundary.
This generalized SymTFT picture for fermions was proposed in Ref. [62] (see
also Ref. [33, 63, 64, 65] for recent developments). The confined anyons on
the fermion condensed boundary implement the global symmetry of the fermoinic
system. By choosing a physical boundary condition on the right, the sandwich
produces the (1+1)$d$ fermionic system that we would like to study. In this
work, we show how to apply this picture to study fermionic quantum critical
points and gapless SPT phases. In applying the SymTFT framework to study phase
transitions, we always need some input critical theories, which lives at the
physical boundary, that describe the transitions between the topological
gapped boundaries of the SymTFT. Such input critical theories usually can be
obtained from some familiar critical points such as the symmetry breaking
transitions or more systematically by performing the “holographic modular
bootstrap” [66, 31, 67]. Many fermionic quantum critical points can be
obtained with very little input.

###  I.1 Summary of results

We summarize the main results of this work below. In Sec. III, we discuss the
fermionization in the topological holography framework. In general, performing
the fermionization corresponds to change the symmetry boundary to a fermionic
gapped boundary conditions such that some fermionic anyons in the bull SymTFT
are condensed. We review the fermionic gapped boundaries and super Lagrangian
algebra in Appendix A. We establish the connection between the formal
fermionization formula that is commonly used in quantum field theory and the
choice of the fermionic gapped boundary conditions on the symmetry boundary in
Eq. (13)-(16). We explain how spin structures are encoded in the SymTFT, and
give explicit general formulas in Eq. (20) for fermionic partition functions
on the torus with different spin structures in terms of the partition
functions in the SymTFT anyon basis.

We discuss one of the simplest example in Sec. IV: Fermionic systems with
$\mathbb{Z}_{2}^{F}$ fermion parity symmetry. We show that the bulk SymTFT is
given by the $\mathbb{Z}_{2}$ toric code and the symmetry boundary is the
fermion $f$ condensed boundary. We explain how to identify the only two
possible gapped phases, trivial and the Kitaev phases, with the bosonic gapped
boundaries of the $\mathbb{Z}_{2}$ toric code by investigating the behavior of
the twisted parition functions on the torus.

We then apply the topological holographic picture to understand the transition
between the trivial and the Kitaev phases. The simplest example is given by
the free Majorana CFT, which is obtained by assuming that there is an Ising
CFT living at the physical boundary of the $\mathbb{Z}_{2}$ toric code SymTFT.
The partition functions with different spin structures are calculated
explicitly in this framework. We can also take the input to be any minimal
model CFTs sitting at the physical boundary and the sandwich construction will
produce the fermionic minimal model CFTs. The general discussion for the
fermoinic minimal models is presented in Appendix B.

We discuss fermionic systems with $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$
symmetry in Sec. V with the SymTFT to be the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ toric code. We choose the symmetry
boundary to be one of the fermion condensed boundary that corresponds to
fermionizing with respecting to the diagonal $\mathbb{Z}_{2}$ subgroup in the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry from the bosonic side. All the
gapped phases are identified with the bosonic gapped boundary by examining the
twisted partition functions with different spin structures.

We discuss fermionic quantum critical points with
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ symmetry in Sec. V and Sec. VI. We
consider the input boundary critical theory to be the Ising and the
$\text{Ising}^{2}$ CFTs. The Ising CFT describes the transition between the
gapped boundaries with the Lagrangian algebra $1\oplus m_{1}\oplus e_{2}\oplus
m_{1}e_{2}$ and the $m$-condensed boundary $1\oplus m_{1}\oplus m_{2}\oplus
m_{1}m_{2}$. The $\text{Ising}^{2}$ CFT describes the transition between the
$e$\- and the $m$-condensed boundaries. With these two input CFTs, we obtain
various fermionic quantum critical points between the gapped phases. The
fermionic quantum critical points discussed in Sec. V are described by common
fermionic CFTs. We give a detailed discussion on how to identify these CFTs
and calculate their partition functions explicitly.

Some transitions that could happen in the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ fermionic chains are described by the
symmetry enriched quantum critical points, where there are non-trivial SPT
signatures in the twisted sectors, which are discussed in Sec. VI. These
exotic quantum critical points generally happens at the transition between a
SSB phase, where the bosonic symmetry is broken, and a fermionic SPT phase. We
find that there are two types of the fermionic symmetry enriched CFTs. The
first type of the fermionic symmetry enriched CFTs can be understood as the
stacking of the usual SSB transitions with an fermionic SPT state (T
operation). The twisted sectors have addition phase factors corresponding to
the SPT invariants. The second type of the fermionic symmetry enriched CFTs
are obtained by performing a “twisted gauging” to the usual ones, where the
twisted gauging procedure can be understood as a combination of gauging and
stacking a SPT state (TS operation). We find a non-trivial example that
describes the transition between the $\mathbb{Z}_{2}$ SSB phase and the gapped
phase with coexisting Kitaev and the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$
fermionic SPT phases (we refer it as the “Kitaev + GW” phase later). It
belongs to the universality class of the $\text{Ising}\times\text{Majorana}$
CFT. However, there are non-trivial phenomena in the twisted sectors such as
the insertion of a $\mathbb{Z}_{2}$ defect inducing the change of spin
structure.

In Sec. VII we discuss fermionic systems with $\mathbb{Z}_{4}^{F}$ symmetry.
We show that such systems can be obtained by fermionizing with respecting to
the $\mathbb{Z}_{2}$ subgroup of a $\mathbb{Z}_{4}$ symmetry on the bosonic
side, or by fermionizing with respecting to one of the $\mathbb{Z}_{2}$
subgroup of an anomalous $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry with
the mixed anomaly. We identify all the gapped phases and the critical points
by using the SymTFT picture. We also discuss a fermionic example of the
intrinsically gapless SPT (igSPT) phases with the microscopic
$\mathbb{Z}_{4}^{F}$ symmetry such that the low-energy sector realizes the
bosonic $\mathbb{Z}_{2}$ anomaly in $H^{3}(\mathbb{Z}_{2},U(1))$.

In Sec. VIII we discuss fermionic systems with non-invertible symmetry. We
focus on a fermionic non-invertible symmetry which is obtained by fermoinizing
the $\mathbb{Z}_{2}$ subgroup in the $\text{Rep}(S_{3})$ symmetry. The
resulting symmetry is described by a super fusion category that is generated
by the $\mathbb{Z}_{2}^{F}$ and a single non-invertible symmetry. It shares
the same fusion rule as the $\text{Rep}(S_{3})$ symmetry (refer to as
$\text{SRep}(\mathcal{H}_{S_{3}})$ later). We first classify the gapped phases
with the $\text{SRep}(\mathcal{H}_{S_{3}})$ symmetry by using the topological
holography where the SymTFT is given by the $S_{3}$ gauge theory. There are
four possible gapped phases depending on whether the non-invertible symmetry
is broken, as well as whether the system is in the Kitaev phase or not. Some
of the critical points are governed by non-trivial critical theories. We find
the transition between the $\text{SRep}(\mathcal{H}_{S_{3}})$ symmetric
trivial and the “$\text{SRep}(\mathcal{H}_{S_{3}})/\mathbb{Z}_{2}^{F}\
\text{SSB}+\text{Kitaev}$” phases 111We use
“$\text{SRep}(\mathcal{H}_{S_{3}})/\mathbb{Z}_{2}^{F}$ SSB” to denote the
phase that breaks the non-invertible symmetry in
$\text{SRep}(\mathcal{H}_{S_{3}})$ spontaneously but preserves the
$\mathbb{Z}_{2}^{F}$ symmetry can be described by the fermionic $m=5$ minimal
model CFT. The non-trivial automorphism in the $S_{3}$ gauge theory that
exchanges the non-abelian charge $C$ and flux $F$ guarantees that the the
transition between the $\text{SRep}(\mathcal{H}_{S_{3}})$ symmetric Kitaev and
the $\text{SRep}(\mathcal{H}_{S_{3}})/\mathbb{Z}_{2}^{F}\ \text{SSB}$ phases
are described by the same fermionic CFT. Another non-trivial critical theory
happens at the transition between the
$\text{SRep}(\mathcal{H}_{S_{3}})/\mathbb{Z}_{2}^{F}\ \text{SSB}$ and the
$\text{SRep}(\mathcal{H}_{S_{3}})/\mathbb{Z}_{2}^{F}\
\text{SSB}+\text{Kitaev}$ phases. This critical theory is the “sum” of the
Ising and the Majorana CFTs. This is an example of the _gapless SSB_ phases
introduced in Ref. [68], where the spontaneous breaking of the non-invertible
symmetry leads to two superselection sectors that support two distinct CFTs.

The structure of the paper is as follows. In Sec. II, we review the central
idea of the topological holography for bosonic systems. We discuss the
generalization of such framework to fermionic systems in Sec. III and
establish the connection between the formal fermionization formula in quantum
field theory and the choice of the fermionic gapped boundary conditions of the
SymTFT. Fermionic gapped boundaries are reviewed in Appendix A. We also
explain how spin structures are encoded in the SymTFT. In Sec. IV, we discuss
fermionic systems with $\mathbb{Z}_{2}^{F}$ fermion parity symmetry as a
simple example. We show how to identify the gapped phases and how to use the
SymTFT to characterize the phase transition. The simplest critical theory is
given by the free Majorana CFT, which is discussed in Sec. IV.2. Other
fermionic minimal models are discussed in Appendix B. In Sec. V, we consider
ferminic chains with $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ symmetry and
discuss fermionic quantum critical points that are described by commnn
fermionic CFTs in Sec. V.2. In Sec. VI, we focus on the fermionic symmetry
enriched quantum critical points in the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ fermionic systems. We find that there
are two types of the fermionic symmetry enriched quantum critical points with
distinct signatures in the twisted sectors. In Sec. VII, we discuss fermionic
systems with $\mathbb{Z}_{4}^{F}$ symmetry and to obtain such symmetry by
fermionizing bosonic systems. We then classify the gapped phases, and
characterize the phase transitions and a gapless SPT phase. In Sec. VIII we
discuss fermionic systems with non-invertible symmetry. We focus on a
fermionic non-invertible $\text{SRep}(\mathcal{H}_{S_{3}})$ symmetry, which is
obtained by fermoinizing the $\mathbb{Z}_{2}$ subgroup in the
$\text{Rep}(S_{3})$ symmetry. We classify all possible gapped phases and
discuss their phase transitions. We conclude with a discussion in Sec. IX.

##  II Review: topological holography for bosonic systems

Figure 1: The sandwich picture in the topological holography. A (1+1)$d$
theory can be viewed as a sandwich built from a topological order
$\mathcal{T}$ in the bulk with a topological gapped boundary conditions on the
left and a potentially non-topological boundary condition on the right.

Here we review the basic idea of the topological holography for bosonic
systems. The essential idea of the topological holography is to encode the
symmetry data and the corresponding symmetry charges of a quantum phase of
matter into a _sandwich_ , which is a topological order $\mathcal{T}$ defined
on a slab with appropriate boundary conditions. One of the main benefits of
this approach is to separate the topological data from the dynamical contents
in the theory of interest. Fig. 1 illustrates the main idea. The bulk of the
slab is a topological order that support topological gapped boundary
conditions, which will be called a _SymTFT_. The left boundary is chosen to be
a topological gapped boundary of the topological order SymTFT, on which some
bosonic anyons condensed. The confined topological defect lines on the left
topological boundary encodes all the topological data of the symmetry in the
bosonic systems. We will thus call the topological boundary on the left of the
sandwich the _symmetry boundary_. Fixing the symmetry boundary, different
choices of the right boundary result in different bosonic quantum phases. We
will refer the right boundary as the _physical boundary_. We note that the
right boundary could be non-topological, such as a CFT. Different phases can
be analysed on the same footing only when we fix a choice of the symmetry
boundary.

##### Sandwich

Focusing the discussion on the (1+1)$d$ bosonic systems, consider a (2+1)$d$
SymTFT on an open 3D manifold $M\times I$, where $I=[0,1]$ is a 1D
interval222In this work, we work in Euclidean signature., and $M$ is a closed
surface. We take $M\times\\{0\\}$ as the symmetry boundary, and
$M\times\\{1\\}$ as the physical boundary. The partition function on this
(2+1)$d$ manifold can be viewed as the inner product of two states:

$Z=\braket{\mathcal{A}}{\Psi},$ (1)

where $|\mathcal{A}\rangle$ is the boundary state on the symmetry boundary,
which is topological, and $|\Psi\rangle$ is the boundary state on the physical
boundary. When the surface $M$ is not a sphere (i.e. nonzero genus), the
Hilbert space of the bulk topological theory on $M$ has dimension greater than
1. There is a canonical choice for the basis states $|\alpha\rangle$. When $M$
is a torus, the label $\alpha$ corresponds to an anyon type in the bulk, and
the state $|\alpha\rangle$ can be obtained as the basis of the Hilbert space
on the solid torus $D^{2}\times S^{1}$, where $D^{2}$ is a disk, with an
insertion of anyon $\alpha$ wrapping around $S^{1}$.

The physical boundary state $|\Psi\rangle$, which may or may not be
topological, can always be expanded in the anyon basis $|\alpha\rangle$:

$|\Psi\rangle=\sum_{\alpha\in\mathcal{T}}Z_{\alpha}|\alpha\rangle.$ (2)

When $M$ is a torus $T^{2}$, the coefficient $Z_{\alpha}$ in this expansion is
equal to the partition function defined on the solid torus $D^{2}\times S^{1}$
with $\Psi$ boundary condition and an insertion of anyon $\alpha$ around
$S^{1}$.

Gapped boundaries of a (2+1)$d$ topological order are classified by the
Lagrangian algebra
$\mathcal{A}=\bigoplus_{\alpha\in\mathcal{T}}w_{\alpha}\alpha$, where
$w_{\alpha}$ are some non-negative integers (see Appendix A for a brief
review). Each topological gapped boundary corresponds to a state

$|\mathcal{A}\rangle=\sum_{\alpha\in\mathcal{T}}w_{\alpha}|\alpha\rangle.$ (3)

Physically, the Lagrangian algebra describes condensation of anyons on the
boundary: if $w_{\alpha}>0$, then an anyon $\alpha$ can condense on the
boundary.

In the anyon basis, the partition function of the (1+1)$d$ theory can be
computed by taking the inner product:

$\displaystyle Z$ $\displaystyle=\langle\mathcal{A}|\Psi\rangle$
$\displaystyle=\sum_{\alpha,\beta\in\mathcal{T}}w_{\beta}Z_{\alpha}\langle\beta|\alpha\rangle$
$\displaystyle=\sum_{\alpha\in\mathcal{T}}w_{\alpha}Z_{\alpha}.$ (4)

Therefore, the partition function of the (1+1)$d$ theory can be written as a
linear combination of the partition function of the SymTFT $\mathcal{T}$ on
the solid torus with an insertion of of Lagrangian algebra $\mathcal{A}$
around $S^{1}$. The construction can also be generalized to higher genus
surfaces.

##### Symmetries and operators

In the sandwich construction, the topological defect lines confined on the
left symmetry boundary implement the symmetry of our system. In general, the
topological defect lines are described by a fusion category $\mathcal{C}$. The
anyons in the bulk SymTFT are then described by a modular tensor category
called the Drinfeld center $\mathcal{Z}(\mathcal{C})$.

Local operators can be organized according to their transformations under the
fusion category symmetry. As a result, a local operator can be uniquely
attached a bulk anyon label $\alpha$ with $w_{\alpha}>0$. Intuitively, the
local operator $\mathcal{O}_{\alpha}$ corresponds to a bulk anyon line
$\alpha$ that condenses on the left topological boundary, and stretches across
the sandwich.

The symmetry $\mathcal{C}$ acts on the local operator as the _half braiding_
between the bulk anyon line $\alpha\in\mathcal{Z}(\mathcal{C})$ and the
boundary topological line $m\in\mathcal{C}$ [69, 70]. For abelian SymTFT, it’s
usually convenient to move the boundary symmetry line into the bulk and the
symmetry charges are encoded in the bulk anyon braiding phases.

##### Phases

Topological holography provides a unified framework to describe gapped and
gapless phases. Once the symmetry boundary is fixed with a fusion category
symmetry $\mathcal{C}$, different choices of the physical boundary realize
different phases of matter. If the physical boundary is also a topological
gapped boundary, the sandwich construction realizes a gapped phase in bosonic
systems with the symmetry $\mathcal{C}$. If we choose the physical boundary to
be gapless, such as a CFT, the sandwich construction realizes a critical point
or a gapless phase with the fusion category symmetry $\mathcal{C}$. Such
gapless boundaries are also characterized by condensable algebras of the form
$\mathcal{B}=\bigoplus_{\beta\in\mathcal{T}}w_{\beta}\beta$ [49], which
satisfy the conditions listed in Appendix A except the Lagrangian condition.

##### Dualities

Topological holography provides a powerful way to systematically understand
dualities between (1+1)$d$ theories. Given a bulk SymTFT $\mathcal{T}$, in
general there can be automorphisms that permutes the bulk anyons:

$\rho:\alpha\rightarrow\rho(\alpha).$ (5)

Each of this automorphism is an anyonic symmetry of the SymTFT and corresponds
to an invertible codimension-1 twist defect $\mathcal{D}$ in the bulk, such
that a bulk anyon is permuted when it passing through the defect. In the
sandwich construction, we can insert this twist defect parallel to the
sandwich, giving rise to a new (1+1)$d$ phase.

More specifically, consider two topological boundary states
$|\mathcal{L}_{1}\rangle$ and $|\mathcal{L}_{2}\rangle$ on the physical
boundary. The corresponding partition functions are given by
$Z_{1}=\langle\mathcal{A}|\mathcal{L}_{1}\rangle$ and
$\mathbb{Z}_{2}=\langle\mathcal{A}|\mathcal{L}_{2}\rangle$, where
$\langle\mathcal{A}|$ is the topological boundary state on the symmetry
boundary. We say that there is a duality between these two gapped phases, if
there exist a twist defect $\mathcal{D}$, such that
$\mathcal{D}|\mathcal{L}_{1}\rangle=|\mathcal{L}_{2}\rangle$. Written in terms
of the partition functions, this means that
$Z_{2}=\langle\mathcal{A}|\mathcal{L}_{2}\rangle=\langle\mathcal{A}|\mathcal{D}|\mathcal{L}_{1}\rangle$.

We can also consider dualities between two critical theories. For two critical
boundaries $\Psi_{1}$ and $\Psi_{2}$ related by the twist defect of this kind:
$\mathcal{D}|\Psi_{1}\rangle=|\Psi_{2}\rangle$, there is a duality between the
partition functions of these two critical theories:

$\displaystyle Z_{2}$ $\displaystyle=\langle\mathcal{A}|\Psi_{2}\rangle$
$\displaystyle=\langle\mathcal{A}|\mathcal{D}|\Psi_{1}\rangle$
$\displaystyle=\langle\mathcal{A}^{\prime}|\Psi_{1}\rangle,$ (6)

where $\langle\mathcal{A}^{\prime}|=\langle\mathcal{A}|\mathcal{D}$. This
shows that the partition function of theory $2$ is given by the sandwich of
theory $1$ but with a different choice of the symmetry boundary. We can thus
use this relation to write down the partition function for theory $2$ by using
the characters of theory $1$. This will be the main strategy of this work.

In general, even for any two topological gapped boundaries $\mathcal{A}$ and
$\mathcal{B}$ not related by an anyonic symmetry, there is a corresponding
duality between the two bosonic quantum systems, simply given by changing the
topological gapped boundary condition on the symmetry boundary. This class of
dualities is more general since the fusion category symmetries corresponding
to the gapped boundaries $\mathcal{A}$ and $\mathcal{B}$ might not be
equivalent in the sense of fusion categories. More formally, two quantum
systems with fusion category symmetries $\mathcal{S}$ and
$\mathcal{S}^{\prime}$ are dual to each other in this generalized sense if
$\mathcal{Z}(\mathcal{S})\cong\mathcal{Z}(\mathcal{S}^{\prime})$.

##  III Fermionization in topological holography

To specify a theory of fermions on a general Riemann surface, it requires a
choice of spin structure. The spin structure represents whether the fermions
are periodic (P) or anti-periodic (A) around each cycle333Fermion being anti-
periodic around a given cycle corresponds to the Neveu-Schwarz (NS) spin
structure and periodic corresponds to the Ramond (R) spin structure in string
theory.. We denote this spin structure by $\rho$.

Given a bosonic theory with a non-anomalous $\mathbb{Z}_{2}^{B}$ symmetry, a
fermionic theory can be obtained by the fermionization procedure:

$Z_{F}(\rho)=\frac{1}{2^{g}}\sum_{a}Z_{B}(a)(-1)^{\text{Arf}(\rho+a)},$ (7)

where $\rho$ is the spin structure of the Riemann surface with genus $g$, $a$
is the background gauge field for the $\mathbb{Z}_{2}^{B}$ symmetry, and
$\text{Arf}(\cdot)$ is the Arf invariant. This definition of the
fermionization corresponds to the Jordan-Wigner transformation on the lattice.

The inverse map is given by the bosonization:

$Z_{B}(a)=\frac{1}{2^{g}}\sum_{\rho}Z_{F}(\rho)(-1)^{\text{Arf}(\rho+a)}.$ (8)

The bosonization map however is not unique since there is a freedom of
stacking a $(1+1)d$ Kitaev chain with partition function
$(-1)^{\text{Arf}(\rho)}$, which produces anther fermion theory
$\mathcal{T}_{F^{\prime}}$. Performing the bosonization for
$\mathcal{T}_{F^{\prime}}$ givens another bosonic theory
$\mathcal{T}_{B^{\prime}}$:

$Z_{B^{\prime}}(a)=\frac{1}{2^{g}}\sum_{\rho}Z_{F}(\rho)(-1)^{\text{Arf}(\rho+a)+\text{Arf}(\rho)}.$
(9)

The two bosonic theories $\mathcal{T}_{B}$ and $\mathcal{T}_{B^{\prime}}$ are
related by gauging the $\mathbb{Z}_{2}^{B}$ symmetry, which corresponds to the
Kramers-Wannier duality. The relations between the bosonic and fermionic
theories are summarized in the diagram below:

${\mathcal{T}_{B}}$${\mathcal{T}_{B^{\prime}}}$${\mathcal{T}_{F}}$${\mathcal{T}_{F^{\prime}}}$B/F$\scriptstyle{\mathbb{Z}_{2}^{B}\text{gauging}}$B/FStacking
Kitaev (10)

where the vertical arrows represent the bosonization/fermionizaion procedure.

In general, performing the fermionization procedure in the sandwich picture
corresponds to change the symmetry boundary from a bosonic gapped boundary to
a fermionic one as discussed in Fig. 2 [62]. Fermionic gapped boundaries can
be described as the fermion condensation and are classified by super
Lagrangian algebra, which we review in Appendix. A. The confined defect lines
on the fermionic gapped boundary implement the symmetry in the fermionic
systems, which in general is described by a super fusion category [60, 61]
444We note that there is always a freedom to stack an invertible $(1+1)d$ Arf
TQFT, which describes the Kitaev chian, to the fermionic gapped boundary and
change the fermionic symmetry structure. See Ref. [71] for more details. In
this work, we don’t consider this stacking..

Figure 2: Performing the fermionization procedure in the sandwich picture
corresponds to change the symmetry boundary from a bosonic gapped boundary
$\mathcal{A}_{b}$ to a fermionic gapped boundary $\mathcal{A}_{f}$.

Since we will be working in the anyon basis, it is useful to express the
fermionization formula Eq. (7) in terms of the multi-components partition
functions in the anyon basis. On the torus, Eq. (7) can be written as

$Z_{F}[m,n]=\frac{1}{2}\sum_{r,s}(-1)^{(m+r)(n+s)}Z_{B}[r,s],$ (11)

where $(m,n)\in\\{0,1\\}$ labels the spin structure along the $x$\- and
$t$-direction on the torus, respectively, and $(r,s)\in\\{0,1\\}$ labels the
number of $\mathbb{Z}_{2}$ defects in the temporal and spatial directions,
respectively. Here we use the convention that $m,n=0$ is anti-periodic
denoting as the “A” structure and $m,n=1$ is periodic denoting the “P”
structure. The twisted partition function can be written in the anyon basis
[48]:

$\displaystyle Z[g,h]$ $\displaystyle=\langle
W^{t}_{\mu_{g}}\mathcal{A}|W^{x}_{\nu_{h}}|\psi\rangle$
$\displaystyle=\sum_{\alpha,\gamma\in\mathcal{Z}(\mathcal{C})}w_{\alpha}N_{\mu_{g},\alpha}^{\gamma}\frac{S_{\gamma\nu_{h}}}{S_{0\gamma}}Z_{\gamma},$
(12)

where $\mu_{g}$ and $\nu_{h}$ are anyons in the bulk SymTFT that implement the
$g$ and $h$ symmetry 555It’s possible that $\mu_{g}$ and $\nu_{h}$ have non-
trivial half-braiding phases as they are moved into the symmetry boundary
[45]. In this work, we always choose the bulk representatives such that the
half-braiding phases are trivial and the boundary lines can be freely pulled
into the bulk without any phase factors.. To perform the fermionization, we
first identify $\mu_{f}$ to be the bulk anyon line that implement the
$\mathbb{Z}_{2}$ symmetry participating in the fermionization and let
$\mathcal{A}_{0}$ denotes the Lagrangian algebra of the bosonic system.
Pluging Eq. (12) into Eq. (11), we obtain the fermionic partition function in
the $(\text{A},\text{A})$ spin structure:

$Z_{F}[\text{A},\text{A}]=\frac{1}{2}\left(\sum_{\alpha\in
A_{0}^{(+)}}w_{\alpha}Z_{\alpha}+\sum_{\alpha\in(\mu_{f}\times
A_{0})^{(-)}}w_{\alpha}Z_{\alpha}\right),$ (13)

where $A_{0}^{(+)}$ is the set of anyons in the Lagrangian algebra
$\mathcal{A}_{0}$ that braid trivially with $\mu_{f}$, and $(\mu_{f}\times
A_{0})^{(-)}$ is the set of anyons in $(\mu_{f}\times A_{0})$ that has a
$(-1)$ mutaul braiding phase with the anyon $\mu_{f}$. Partition functions in
the other sectors can be obtained similarly:

$Z_{F}[\text{P},\text{A}]=\frac{1}{2}\left(\sum_{\alpha\in
A_{0}^{(-)}}w_{\alpha}Z_{\alpha}+\sum_{\alpha\in(\mu_{f}\times
A_{0})^{(+)}}w_{\alpha}Z_{\alpha}\right),$ (14)
$Z_{F}[\text{A},\text{P}]=\frac{1}{2}\left(\sum_{\alpha\in
A_{0}^{(+)}}w_{\alpha}Z_{\alpha}-\sum_{\alpha\in(\mu_{f}\times
A_{0})^{(-)}}w_{\alpha}Z_{\alpha}\right),$ (15)
$Z_{F}[\text{P},\text{P}]=\frac{1}{2}\left(-\sum_{\alpha\in
A_{0}^{(-)}}w_{\alpha}Z_{\alpha}+\sum_{\alpha\in(\mu_{f}\times
A_{0})^{(+)}}w_{\alpha}Z_{\alpha}\right).$ (16)

We have defined $A_{0}^{(-)}$ to be the set of anyons in the Lagrangian
algebra $\mathcal{A}_{0}$ that has a $(-1)$ mutual braiding phase with
$\mu_{f}$, and $(\mu_{f}\times A_{0})^{(+)}$ is the set of anyons in
$(\mu_{f}\times A_{0})$ that braids trivially with $\mu_{f}$.

We note that, in writing Eq. (13)-(16), a specific fermionic gapped boundary
with the super Lagrangian algebra $A_{f}$ is chosen, where the anyons in the
set $A_{0}^{(+)}$ and $(\mu_{f}\times A_{0})^{(-)}$ are condensed on the
symmetry boundary. More formally, we find the set of anyons in the super
Lagrangian algebra $A_{f}=A_{0}^{(+)}\cup(\mu_{f}\times A_{0})^{(-)}$ and the
super Lagrangian algebra takes the form:

$\mathcal{A}_{f}=\oplus_{\alpha\in A_{f}}w_{\alpha}\alpha,$ (17)

where $w_{\alpha}$ is determined by the Lagrangian algebra on the symmetry
boundary before we perform the fermionization.

In general there are multiple choices of the fermionic gapped boundary to
perform the fermionization, which corresponds to fermionizing with respecting
to different $\mathbb{Z}_{2}$ subgroup and the freedom of the stacking with an
Kitaev chain. In this work, we will often choose the fermionic gapped boundary
that corresponds to the Jordan–Wigner transformation on the lattice.

Suppose we have fixed a fermionic gapped boundary condition $\mathcal{A}_{f}$
on the symmetry boundary of the sandwich and denote the corresponding boundary
state as $|\mathcal{A}_{f}\rangle$. Let $\mu_{\rho}$ be the anyon line that
generates the fermion parity symmetry in the bulk SymTFT 666Again, we choose a
bulk representative $\mu_{\rho}$ such that the half-braiding phase is
trivial.. The spin structure $(m,n)$ can be understood as the insertion of $m$
and $n$ numbers of the fermion parity line along the temporal and the spatial
cycles, respectively. We define a map $\iota$ that maps the spin structures
$(m,n)$ to the bulk anyons:

$\displaystyle\iota:m$ $\displaystyle\rightarrow 1\text{ , if }m=0$ (18)
$\displaystyle m$ $\displaystyle\rightarrow\mu_{\rho}\text{ , if }m=1,$ (19)

and similarly for $n$. The fermionic partition function with different spin
structures are represented as the diagram shown in Fig. 3, and can be obtained
directly by the following formula:

$\displaystyle Z_{F}[m,n]$ $\displaystyle=\langle
W_{\iota}\mathcal{A}_{f}|V_{\iota}|\psi\rangle$
$\displaystyle=\sum_{\alpha,\gamma\in\mathcal{Z}(\mathcal{C})}w_{\alpha}N_{\iota,\alpha}^{\gamma}\frac{S_{\gamma\iota}}{S_{0\gamma}}Z_{\gamma},$
(20)

where $W_{\iota}$ and $V_{\iota}$ denotes the anyon line $\iota$ along $t$ and
$x$-cycles, respectively.

Figure 3: The fermionic partition functions with different spin structures are
represented as the twisted partition functions $Z_{F}[m,n]$ in the anyon
basis. We denote the representative anyon in the bulk SymTFT that implements
the fermion parity symmetry as $\mu_{\rho}$ and define a map $\iota$ from the
spin structures to the bulk anyons (see main text). The twisted partition
function is equivalent to the partition function of the $(2+1)$D SymTFT on a
solid torus with an insertion of a composite anyon
$\iota(m)\times\mathcal{A}_{f}$ along the time-direction, wrapping around by
$\iota(n)$ in the spatial direction. The temporal cycle is the vertical
direction and the spatial cycle is horizontal.

##  IV Fermionic chains with $\mathbb{Z}_{2}^{F}$ symmetry

Here we discuss the topological holographic picture for $(1+1)d$ fermionic
chain with only $\mathbb{Z}_{2}^{F}$ symmetry and free Majorana fermion CFT.
In the sandwich construction, we can always choose the bulk SymTFT to be
described by a $\mathbb{Z}_{2}$ toric code. The symmetry boundary is chosen to
be a fermionic gapped boundary condition. In the $\mathbb{Z}_{2}$ toric code,
the fermionic gapped boundary is described by the super Lagrangian algebra:

$\mathcal{A}_{f}=1\oplus f,$ (21)

The confined topological defect line on the symmetry boundary is given by

$X_{f}=e\oplus m,$ (22)

satisfying a $\mathbb{Z}_{2}$ fusion rule $X_{f}\otimes X_{f}=1$. Physically,
$X_{f}$ generates the fermion parity symmetry on the symmetry boundary. When
$X_{f}$ line be pulled into the bulk SymTFT, it has a choice of being an $e$
or $m$ line. Here we choose $m$ line as the bulk representative of the
$\mathbb{Z}_{2}^{F}$ fermion parity symmetry. The two choices are related by
the EM exchange automorphism $e\leftrightarrow m$ and will lead to a duality
between different gapped phases.

###  IV.1 Gapped phases

If we choose both symmetry and physical boundaries to be gapped boundary
conditions, the sandwich construction realizes $(1+1)d$ gapped phases with
$\mathbb{Z}_{2}^{F}$ fermion parity symmetry. There are only two
possibilities: the trivial and the Kitaev phase. If the physical boundary is
the $m$-condensed boundary, it corresponds to the trivial phase. This can be
derived from the fact that the partition functions with different spin
structures are all equal to $1$ by using Eq. (13)-(16).

The Kitaev chain corresponds to choosing the physical boundary to be the
$e$-condensed boundary. One can check that the partition function in the
$(P,P)$ spin structure is given by

$Z[\text{P},\text{P}]=\frac{S_{em}}{S_{0e}}\langle e|e\rangle=-1$ (23)

where we have used Eq. (20). The other partition functions are all equal to
$1$. Physically, this is the statement that the ground state of a Kitaev chain
carries an odd fermion parity with a periodic boundary condition. Note that
this identification of the phases is tied to the choice of the symmetry line
of the $\mathbb{Z}_{2}^{F}$ fermion parity being the $m$ anyon line in the
bulk SymTFT. If we choose $e$ to be the bulk representative of the fermion
parity symmetry, the identification of the gapped phases will reverse.

We note that our choice is consistent with the Jordan–Wigner transformation.
The Jordan–Wigner transformation maps the trivial phase of an Ising spin chain
to the trivial phase of the fermionic chain, and maps the $\mathbb{Z}_{2}$ SSB
phase to the Kitave phase. In the sandwich picture of bosonic systems with a
$\mathbb{Z}_{2}$ symmetry, we choose the $e$ condensed boundary as our
symmetry boundary on the left. It follows that choosing the right boundary to
be the $m$ condensed boundary produces the $\mathbb{Z}_{2}$ trivial phase and
choosing the $e$ condensed boundary on the right produces the $\mathbb{Z}_{2}$
SSB phases. Upon performing the fermionization, we change the symmetry
boundary on the left to be the $f$ condensed boundary and fix $m$ line to be
the $\mathbb{Z}_{2}^{F}$ symmetry generator. From the above discussion, we see
that the fermionization in the SymTFT framework indeed maps $\mathbb{Z}_{2}$
trivial phase to the trivial phase of the fermionic chain and maps the
$\mathbb{Z}_{2}$ SSB phase to the Kitaev phase.

###  IV.2 Free Majorana fermion

The free Majorana fermion CFT can be obtained by fermionizing an Ising CFT.
Upon performing the fermionization, we simply choose the symmetry boundary to
be the $f$-condensed boundary condition, and the physical boundary is
described by the Ising CFT. We denote the critical boundary state on the
physical boundary as $|\Psi_{\text{Ising}}\rangle$. The untwisted partition
function is given by the overlap:

$\displaystyle Z[\text{A},\text{A}]$
$\displaystyle=\langle\mathcal{A}_{f}|\Psi_{\text{Ising}}\rangle$
$\displaystyle=Z_{1}+Z_{f}$
$\displaystyle=|\chi_{1}|^{2}+|\chi_{\psi}|^{2}+\chi_{1}\overline{\chi}_{\psi}+\chi_{\psi}\overline{\chi}_{1},$
(24)

where in the second equality we have used the multi-component partition
functions of the Ising CFT in the $\mathbb{Z}_{2}$ toric code anyon basis [48]

$\displaystyle Z_{1}$ $\displaystyle=|\chi_{1}|^{2}+|\chi_{\psi}|^{2},$
$\displaystyle Z_{e}$ $\displaystyle=|\chi_{\sigma}|^{2},$ $\displaystyle
Z_{m}$ $\displaystyle=|\chi_{\sigma}|^{2},$ $\displaystyle Z_{f}$
$\displaystyle=\chi_{1}\overline{\chi}_{\psi}+\chi_{\psi}\overline{\chi}_{1},$
(25)

By comparing with Eq. (7), one can check that Eq. (24) is indeed the fermion
partition function of the free Majorana fermion with $(\text{A},\text{A})$
spin structure.

Now we consider the fermionic partition functions twisted by the fermion
parity. In the sandwich picture, they are represented by inserting $X_{f}$
line along $x$ and/or $t$ direction. Using Eq. (12), we find

$\displaystyle Z[\text{A},\text{P}]$ $\displaystyle=Z_{1}-Z_{f}$
$\displaystyle=|\chi_{1}|^{2}+|\chi_{\psi}|^{2}-(\chi_{1}\overline{\chi}_{\psi}+\chi_{\psi}\overline{\chi}_{1}),$
(26) $\displaystyle Z[\text{P},\text{A}]$ $\displaystyle=Z_{e}+Z_{m}$
$\displaystyle=2|\chi_{\sigma}|^{2},$ (27) $\displaystyle
Z[\text{P},\text{P}]$ $\displaystyle=Z_{e}-Z_{m}$ $\displaystyle=0.$ (28)

These are indeed the partition function for the free Majorana theory.

Since all minimal models has a non-anomalous $\mathbb{Z}_{2}$ symmetry, all of
them can be fermionized into _fermionic minimal models_. The sandwich picture
for the fermionic minimal models is discussed in Appendix B.

##  V Fermionic chains with $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ symmetry

We now discuss the topological holographic picture fermionic chains with
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ symmetry. The bulk of the sandwich is
a $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ gauge theory. For later convenience, we
first review some properties of the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ gauge
theory and the corresponding sandwich picture for the bosonic cases. We
sometimes denote the first $\mathbb{Z}_{2}$ by $\mathbb{Z}_{2}^{a}$ and the
second one by $\mathbb{Z}_{2}^{b}$.

There are 6 gapped boundaries as summarized in Table 1. The automorphism in
the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ gauge theory are generated by the EM
exchange for the $\mathbb{Z}_{2}^{a}$ and $\mathbb{Z}_{2}^{b}$ gauge theory:

$\sigma_{i}:e_{i}\leftrightarrow m_{i},$ (29)

and the twist defect from the gauged $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ SPT:

$\displaystyle s:e_{i}\rightarrow e_{i},\ m_{i}\rightarrow m_{i}e_{j},i\neq
j.$ (30)

Choosing the symmetry boundary to be the $e$ condensed boundary
$\mathcal{L}_{1}$ and physical boundary to be $\mathcal{L}_{i}$ for $i=1$ to
$6$ realizes bosonic gapped phases with the unbroken subgroup and the SPT
index $H^{2}(G,\mathrm{U}(1))$ shown in the second and third columns in Table.
1. The automorphisms correspond to the dualities between these gapped phases.

Lagrangian algebra | Unbroken subgroup | $H^{2}(G,\mathrm{U}(1))$ | Order parameters
---|---|---|---
$\mathcal{L}_{1}=1\oplus e_{1}\oplus e_{2}\oplus e_{1}e_{2}$ | $1$ | $1$ |  $\mathcal{O}_{\mathbb{Z}_{2}^{a}}=W_{e_{1}}$, $\mathcal{O}_{\mathbb{Z}_{2}^{b}}=W_{e_{2}}$
$\mathcal{L}_{2}=1\oplus m_{1}\oplus e_{2}\oplus m_{1}e_{2}$ | $\mathbb{Z}_{2}^{a}$ | 1 | $\mathcal{O}_{\mathbb{Z}_{2}^{b}}=W_{e_{2}}$
$\mathcal{L}_{3}=1\oplus e_{1}\oplus m_{2}\oplus e_{1}m_{2}$ | $\mathbb{Z}_{2}^{b}$ | 1 | $\mathcal{O}_{\mathbb{Z}_{2}^{a}}=W_{e_{1}}$
$\mathcal{L}_{4}=1\oplus e_{1}e_{2}\oplus m_{1}m_{2}\oplus f_{1}f_{2}$ | $\mathbb{Z}_{2}^{{\rm diag}}$ | 1 | $\mathcal{O}_{\mathbb{Z}_{2}^{D}}=W_{e_{1}e_{2}}$
$\mathcal{L}_{5}=1\oplus e_{1}m_{2}\oplus m_{1}e_{2}\oplus f_{1}f_{2}$ | $\mathbb{Z}_{2}^{a}\times\mathbb{Z}_{2}^{b}$ | $(-1)^{g_{1}h_{2}}$ | -
$\mathcal{L}_{6}=1\oplus m_{1}\oplus m_{2}\oplus m_{1}m_{2}$ | $\mathbb{Z}_{2}^{a}\times\mathbb{Z}_{2}^{b}$ | $1$ | -
Table 1: Gapped boundaries of the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ gauge
theory. The first column shows the Lagrangian algebra. The second and the
third columns show the symmetry and the SPT index realized in the sandwich
construction by choosing the symmetry boundary to be $\mathcal{L}_{1}$. The
fourth column shows the order parameters as the Wilson line.

To obtain the topological holographic picture of the fermionic chains, we need
to know the fermionic gapped boundary conditions of the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ gauge theory, which are summarized in
Table 2 [61]. The fermionization is performed by choosing the symmetry
boundary to be one of the fermionic gapped boundaries in Table 2. Here we
choose the symmetry boundary to be $\mathcal{L}^{f}_{1}=1\oplus
e_{1}e_{2}\oplus m_{1}f_{2}\oplus f_{1}m_{2}$. This choice corresponds to
perform the fermionization with respecting to the diagonal $\mathbb{Z}_{2}$
symmetry:

$Z_{F}(\rho)=\frac{1}{2^{g}}\sum_{b}Z_{B}(b_{1}=b_{2}=b)(-1)^{\text{Arf}(\rho+b)},$
(31)

where $b_{1}$ and $b_{2}$ are the $\mathbb{Z}_{2}$ gauge fields for the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry. The fermion parity
$\mathbb{Z}_{2}^{F}$ symmetry is implemented by the $m_{1}m_{2}$ line on the
symmetry boundary and also the bulk and the $\mathbb{Z}_{2}$ symmetry is
implemented by the $m_{1}$ line:

$\displaystyle\mathbb{Z}_{2}^{F}$ $\displaystyle:m_{1}m_{2},$
$\displaystyle\mathbb{Z}_{2}$ $\displaystyle:m_{1}.$ (32)

Super Lagrangian algebra
---
$\mathcal{L}^{f}_{1}=1\oplus e_{1}e_{2}\oplus m_{1}f_{2}\oplus f_{1}m_{2}$
$\mathcal{L}^{f}_{2}=1\oplus e_{1}f_{2}\oplus m_{1}m_{2}\oplus f_{1}m_{2}$
$\mathcal{L}^{f}_{3}=1\oplus f_{1}\oplus f_{2}\oplus f_{1}f_{2}$
$\mathcal{L}^{f}_{4}=1\oplus f_{1}\oplus e_{2}\oplus f_{1}e_{2}$
$\mathcal{L}^{f}_{5}=1\oplus e_{1}\oplus f_{2}\oplus e_{1}f_{2}$
$\mathcal{L}^{f}_{6}=1\oplus m_{1}\oplus f_{2}\oplus m_{1}f_{2}$
$\mathcal{L}^{f}_{7}=1\oplus f_{1}\oplus m_{2}\oplus f_{1}m_{2}$
$\mathcal{L}^{f}_{8}=1\oplus e_{1}f_{2}\oplus m_{1}e_{2}\oplus f_{1}m_{2}$
$\mathcal{L}^{f}_{9}=1\oplus f_{1}e_{2}\oplus e_{1}m_{2}\oplus m_{1}f_{2}$
Table 2: Fermionic gapped boundaries of the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ gauge theory described by the super
Lagrangian algebra.

###  V.1 Gapped phases

By choosing the physical boundary to be one of the bosonic gapped boundaries
in Table. 1, the sandwich construction realizes fermionic gapped phases with
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ symmetry, which are summarized in
Table 3.

The identification of the fermionic gapped phases is achieved by calculating
the partition functions by using Eq. (12) and Eq. (20). As a demonstration,
let’s focus on the case where the physical boundary is given by
$\mathcal{L}_{2}$ and calculate the fermionic partition functions. The
partition function in the $(\text{P},\text{P})$ structure without any
$\mathbb{Z}_{2}$ twist is given by the diagram Fig. 3 with $\rho=m_{1}m_{2}$:

$Z[\text{P},\text{P}]=\langle
m_{1}m_{2}\times\mathcal{L}^{f}_{1}|V^{t}_{m_{1}m_{2}}|\mathcal{L}_{2}\rangle=-1.$
(33)

One can check that the partition functions in the $(\text{P},\text{P})$
structure equal to $-1$ in any of the $\mathbb{Z}_{2}$ twist sectors. All
other twisted partition functions equal to $1$. These are precisely the
behavior of the partition functions of the Kitaev phase. We thus conclude that
choosing the physical boundary to be $\mathcal{L}_{2}$ realizes the Kitaev
phase. The same conclusion can be achieved by using the fermionization formula
Eq. (7) directly. We thus use $Z[\text{P},\text{P}]=-1$ to identify the phases
that coexist with the Kitaev phase, which happens for $\mathcal{L}_{1}$ and
$\mathcal{L}_{3}$ boundary.

For phases that break the $\mathbb{Z}_{2}$ symmetry spontaneously, the
prefactor in front of the partition functions is $2$, indicating the two-fold
ground state degeneracy. For example, for the $\mathcal{L}_{4}$ boundary, we
find that
$Z[\text{A},\text{A}]=\langle\mathcal{L}_{1}|\mathcal{L}_{4}\rangle=2$ and all
other twisted partition functions equal to $2$. Therefore, $\mathcal{L}_{4}$
corresponds to the $\mathbb{Z}_{2}$ SSB phase. We can use the absolute value
of the partition functions to identify any phases that could coexist with the
$\mathbb{Z}_{2}$ SSB phases, which happens for $\mathcal{L}_{1}$ boundary. We
therefore find that the $\mathcal{L}_{1}$ boundary corresponds to a gapped
phase that breaks the $\mathbb{Z}_{2}$ spontaneously with the remaining
fermionic sector in the Kitaev phase. We denote this phase as “SSB+Kitaev”.

The Gu-Wen fermionic SPT phase with $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$
symmetry [62] is characterized by the partition function on the torus:

$(-1)^{\int_{X}q_{\rho}(a)},$ (34)

where $\rho$ is the spin structure, $a\in H^{1}(X,\mathbb{Z}_{2})$ is the
$\mathbb{Z}_{2}$ gauge field, and $\int_{X}q_{\rho}(a)$ is a
$\mathbb{Z}_{2}$-valued quadratic function of $a$ that satisfies

$\int_{X}q_{\rho}(a+a^{\prime})=\int_{X}q_{\rho}(a)+\int_{X}q_{\rho}(a^{\prime})+a\cup
a^{\prime}.$ (35)

We can write the GW partition function Eq. (34) more explicitly in terms of
the quadratic form $p_{\rho}:H_{1}(X,\mathbb{Z}_{2})\rightarrow\mathbb{Z}_{2}$
that satisfies

$\displaystyle
p_{\rho}(\gamma_{1}+\gamma_{2})=p_{\rho}(\gamma_{1})+p_{\rho}(\gamma_{2})+\gamma_{1}\cdot\gamma_{2},$
(36)

$\gamma_{1},\gamma_{2}\in H_{1}(X,\mathbb{Z}_{2})$ and
$\gamma_{1}\cdot\gamma_{2}$ is the intersection number. In particular,

$\displaystyle p_{\rho}(\gamma)=0$ (37)

if the spin structure is anti-periodic around a non-trivial cycle $\gamma$,

$\displaystyle p_{\rho}(\gamma)=1$ (38)

if the spin structure is periodic around a non-trivial $\gamma$, and
$p_{\rho}(\gamma)=0$ if $\gamma$ is homologically trivial. The GW partition
function can then be written as

$(-1)^{\int_{X}q_{\rho}(a)}=(-1)^{a(\gamma_{1})a(\gamma_{2})+p_{\rho}(\gamma_{1})a(\gamma_{2})+a(\gamma_{1})p_{\rho}(\gamma_{2})},$
(39)

where now $\gamma_{1}$ and $\gamma_{2}$ are the two non-trivial cycle on the
torus. We denote the twisted partition functions as
$Z[(\rho_{x},\rho_{t});(g_{x},g_{t})]$ where $(\rho_{x},\rho_{t})$ are the
spin structure in the spatial and temporal directions and $(g_{x},g_{t})$ are
the spatial and temporal $\mathbb{Z}_{2}$ twists. Using Eq. (39), one can see
that

$\displaystyle Z[(A,A);(1,1)]=Z[(A,P);(1,0)]$
$\displaystyle=Z[(P,A);(0,1)]=Z[(P,P);(0,1)]$
$\displaystyle=Z[(P,P);(1,0)]=Z[(P,P);(1,1)]=-1,$ (40)

and the other partition functions are equal to $1$. We thus use Eq. (40) to
identify the GW phase, which happens for $\mathcal{L}_{3}$ and
$\mathcal{L}_{5}$ boundaries. The results are summarized in the third column
in Table 3.

Lagrangian algebra | Bosonic partition function | Fermionic gapped phases
---|---|---
$\mathcal{L}_{1}=1\oplus e_{1}\oplus e_{2}\oplus e_{1}e_{2}$ | $4\delta(A_{1})\delta(A_{2})$ | SSB + Kitave
$\mathcal{L}_{2}=1\oplus m_{1}\oplus e_{2}\oplus m_{1}e_{2}$ | $2\delta(A_{2})$ | Kitaev
$\mathcal{L}_{3}=1\oplus e_{1}\oplus m_{2}\oplus e_{1}m_{2}$ | $2\delta(A_{1})$ | Kitaev+GW
$\mathcal{L}_{4}=1\oplus e_{1}e_{2}\oplus m_{1}m_{2}\oplus f_{1}f_{2}$ | $2\delta(A_{1}+A_{2})$ | SSB
$\mathcal{L}_{5}=1\oplus e_{1}m_{2}\oplus m_{1}e_{2}\oplus f_{1}f_{2}$ | $(-1)^{A_{1}\cup A_{2}}$ | GW
$\mathcal{L}_{6}=1\oplus m_{1}\oplus m_{2}\oplus m_{1}m_{2}$ | 1 | Trivial
Table 3: Fermionic gapped phases with $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$
symmetry in topological holography. The left symmetry boundary is chosen to be
the fermionic gapped boundary $\mathcal{L}^{f}_{1}$. The first column shows
the physical boundary on the right of the sandwich. The second column shows
that bosonic TQFT partition function of the bosonic gapped phases before the
fermionization. The third column shows the fermoinic gapped phases realized by
the sandwich construction. “SSB” means that the $\mathbb{Z}_{2}$ symmetry is
broken spontaneously. “Kitaev” denotes the fermionic chain is in the Kitaev
phase. “GW” denotes the “Gu-Wen” phase, which is the non-trivial fermionic SPT
phase with $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ symmetry. “Trivial” means
the fermoinic chain is in the completely trivial symmetric phase. The
fermoinic gapped phases obtained by the sandwich picture match with the
fermionization results obtained directly by applying Eq. (7).

###  V.2 Quantum critical points

Figure 4: Summary of the phase transitions in the fermionic systems with
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ symmetry discussed in Sec. V.2 and
Sec. VI. The red arrow indicate the input boundary critical theory on the
physical boundary of the sandwich, which is the $\text{Ising}^{2}$ CFT.

We now discuss various quantum critical points in fermionic systems with
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ symmetry by using the sandwich
picture. As we will see below, the inputs that we require are the CFT
description of boundary phase transitions on the physical boundary. The input
CFT that we focus on heavily in this and the next section is the
$\text{Ising}^{2}$ CFT that describes the boundary critical point between the
topological gapped boundary $\mathcal{L}_{1}$ and $\mathcal{L}_{6}$.

Some of the critical points in the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$
fermionic systems have non-trivial topological responses in the twisted
sectors. Such critical points will be referred to as the symmetry enriched
quantum critical points and will be discussed in Sec. VI. The results are
summarized in Fig. 4.

####  V.2.1
$\text{Trivial}\leftrightarrow(\text{SSB}+\text{Kitaev})=\text{Ising}^{2}\bf{/Z_{2}^{D}}$

We consider the transition between the trivial and the “SSB + Kitaev” phase,
where the $\mathbb{Z}_{2}$ symmetry is spontaneously broken during the
transition. The trivial phase is realized by choosing the physical boundary to
be $\mathcal{L}_{1}$ and the “SSB + Kitaev” phase corresponds to
$\mathcal{L}_{6}$.

We now discuss the input boundary critical theory. It is helpful to consider
the bosonic side first. From Table. 1, we see that $\mathcal{L}_{1}$ and
$\mathcal{L}_{6}$ corresponds to complete SSB phase and the trivial symmetric
phase respectively if we choose the symmetry boundary to be $\mathcal{L}_{1}$.
If we tune the right boundary of the sandwich to be at the critical point
between the gapped boundaries $\mathcal{L}_{1}$ and $\mathcal{L}_{6}$, we know
that this critical point is described by the $\text{Ising}^{2}$ CFT. The CFT
boundary state of the sandwich is then given by

$|\psi_{16}\rangle=\sum_{\alpha,\beta\in\text{TC}}Z_{\alpha}Z_{\beta}|\alpha\beta\rangle,$
(41)

where $Z_{a}$ and $Z_{b}$ are the partition functions of the Ising CFT in the
anyon sector $a$ and $b$. This is our input boundary critical theory on the
physical boundary of the sandwich. Taking the overlap
$\langle\mathcal{L}_{1}|\psi_{16}\rangle$ produces the partition function for
the $\text{Ising}^{2}$ CFT as expected.

We now fermionize the theory by changing the symmetry boundary to be
$\mathcal{L}_{1}^{f}$ and calculate the partition function as the overlap:

$\displaystyle Z^{\text{Ising}^{2}\bm{/Z_{2}^{D}}}[\text{A},\text{A}]$
$\displaystyle=\langle\mathcal{L}_{1}^{f}|\psi_{16}\rangle$
$\displaystyle=Z_{1}^{2}+Z_{e}^{2}+2Z_{m}Z_{f}$
$\displaystyle=(|\chi_{1}|^{2}+|\chi_{\psi}|^{2})^{2}+2|\chi_{\sigma}|^{2}(\chi_{1}\overline{\chi}_{\psi}+\chi_{\psi}\overline{\chi}_{1})+4|\chi_{\sigma}|^{2}.$
(42)

We now show that Eq. (42) is the fermionization with respecting to the
diagonal $\mathbb{Z}_{2}$ symmetry. We rewrite Eq. (42) as

$\displaystyle Z^{\text{Ising}^{2}\bm{/Z_{2}^{D}}}[\text{A},\text{A}]$
$\displaystyle=(|\chi_{1}|^{2}+|\chi_{\psi}|^{2})^{2}+2|\chi_{\sigma}|^{2}(\chi_{1}\overline{\chi}_{\psi}+\chi_{\psi}\overline{\chi}_{1})+4|\chi_{\sigma}|^{2}$
$\displaystyle=\frac{1}{2}\left(Z_{0,0}^{2}+Z_{0,1}^{2}+Z_{1,0}^{2}-Z_{1,1}^{2}\right)$
$\displaystyle=\frac{1}{2}\sum_{a}Z(a_{1}=a,a_{2}=a)(-1)^{\text{Arf}(\rho+a)},$
(43)

where, in the second equality, we have used the twisted partition functions
for the Ising CFT:

$\displaystyle Z_{0,1}$
$\displaystyle=|\chi_{1}|^{2}+|\chi_{\psi}|^{2}-|\chi_{\sigma}|^{2},$ (44)
$\displaystyle Z_{1,0}$
$\displaystyle=|\chi_{\sigma}|^{2}+\chi_{1}\overline{\chi}_{\psi}+\chi_{\psi}\overline{\chi}_{1},$
(45) $\displaystyle Z_{1,1}$
$\displaystyle=|\chi_{\sigma}|^{2}-\chi_{1}\overline{\chi}_{\psi}-\chi_{\psi}\overline{\chi}_{1}.$
(46)

Therefore, we see that the theory given by Eq. (42) can be obtained by
performing the fermionization with respect to the diagonal $\mathbb{Z}_{2}$
symmetry. We denote this theory as $\text{Ising}^{2}\bm{/Z_{2}^{D}}$777We use
the symbol “$\bm{/Z_{2}}$” to denote fermionizing with respecting to a
$\mathbb{Z}_{2}$ symmetry to distinguish with gauging a $\mathbb{Z}_{2}$
symmetry..

####  V.2.2
$\text{Kitaev}\leftrightarrow(\text{Kitaev}+\text{GW})=\text{Ising}^{2}\bf{/Z_{2}^{D}}$

We proceed to discuss the transition between the Kitaev and the “Kitaev + GW”
phase. The Kitaev phase is realized by choosing the physical boundary to be
$\mathcal{L}_{2}$ and the “‘Kitaev + GW” phase corresponds to
$\mathcal{L}_{3}$. We denote the corresponding state as $|\psi_{2,3}\rangle$
and the partition function of the SPT transition is
$Z[{\text{A},\text{A}}]=\langle\mathcal{L}_{1}^{f}|\psi_{2,3}\rangle$. We
would like to relate the partition function of this transition to the input
boundary transition $|\psi_{1,6}\rangle$. Such duality transformation is
implemented by the insertion of the $D_{\sigma_{2}}$ twist defect. The
partition function can thus be written as

$\displaystyle Z[\text{A},\text{A}]$
$\displaystyle=\langle\mathcal{L}_{1}^{f}|\psi_{2,3}\rangle$
$\displaystyle=\langle\mathcal{L}_{1}^{f}|\mathcal{D}_{2}|\psi_{1,6}\rangle$
$\displaystyle=\langle\mathcal{L}_{2}^{f}|\psi_{1,6}\rangle$
$\displaystyle=Z_{1}^{2}+Z_{e}Z_{m}+Z_{m}Z_{f}+Z_{e}Z_{f}$
$\displaystyle=Z^{\text{Ising}^{2}\bm{/Z_{2}^{D}}}[\text{A},\text{A}],$ (47)

where $Z^{\text{Ising}^{2}\bm{/Z_{2}^{D}}}[\text{A},\text{A}]$ is given in Eq.
(42). We find that the transition is also described by the
$\text{Ising}^{2}\bm{/Z_{2}^{D}}$ theory.

####  V.2.3 $\text{Trivial}\leftrightarrow\text{GW}=\text{Majorana}^{2}$

Here we consider the transition between the trivial and the Gu-Wen SPT phases.
This critical point is realized by tuning the physical boundary of the
sandwich to be at the critical point between the gapped boundaries
$\mathcal{L}_{5}$ and $\mathcal{L}_{6}$, while choosing the symmetry boundary
to be the fermionic gapped boundary $\mathcal{L}_{1}^{f}$. We denote the
corresponding state as $|\psi_{5,6}\rangle$ and the partition function of the
SPT transition is
$Z[{\text{A},\text{A}}]=\langle\mathcal{L}_{1}^{f}|\psi_{5,6}\rangle$. We
would like to obtain the partition function of this transition by applying
some duality transformation. This is achieve by considering the twist defect
$\mathcal{D}=\mathcal{D}_{\sigma_{1}\sigma_{2}}\circ\mathcal{D}_{s}$. The
twist defect $\mathcal{D}$ acts on the gapped boundaries as

$\mathcal{D}|\mathcal{L}_{1}\rangle=|\mathcal{L}_{6}\rangle,\
\mathcal{D}|\mathcal{L}_{6}\rangle=|\mathcal{L}_{5}\rangle.$ (48)

Therefore, we have

$|\psi_{5,6}\rangle=\mathcal{D}|\psi_{1,6}\rangle.$ (49)

The partition function for the transition is given by inserting the twist
defect $\mathcal{D}$:

$\displaystyle Z[\text{A},\text{A}]$
$\displaystyle=\langle\mathcal{L}_{1}^{f}|\psi_{5,6}\rangle$
$\displaystyle=\langle\mathcal{L}_{1}^{f}|\mathcal{D}|\psi_{1,6}\rangle$
$\displaystyle=\langle\mathcal{L}_{3}^{f}|\psi_{1,6}\rangle$
$\displaystyle=Z_{1}^{2}+2Z_{1}Z_{f}+Z_{f}^{2}$
$\displaystyle=(|\chi_{1}|^{2}+|\chi_{\psi}|^{2}+\chi_{1}\overline{\chi}_{\psi}+\chi_{\psi}\overline{\chi}_{1})^{2},$
(50)

where, in the third equality, we have used the fact that
$\langle\mathcal{L}_{1}^{f}|\mathcal{D}=\langle\mathcal{L}_{3}^{f}|$. Eq. (50)
is the partition function of two copies of free Majorana theories in the
$(\text{A},\text{A})$ spin structure, which is equivalent to a Dirac theory.

####  V.2.4
$\text{Kitaev}\leftrightarrow\text{SSB}=\text{Ising}\times\text{Majorana}$

We consider the transition between the Kitaev and the $\mathbb{Z}_{2}$ SSB
phases in this section, which corresponds to the boundary $\mathcal{L}_{2}$
and $\mathcal{L}_{4}$, respectively. We denote the corresponding critical
boundary state as $|\psi_{2,4}\rangle$ and the partition function of the SPT
transition is
$Z[{\text{A},\text{A}}]=\langle\mathcal{L}_{1}^{f}|\psi_{2,4}\rangle$. We
would like to relate the critical boundary state $|\psi_{2,4}\rangle$ to
$|\psi_{1,6}\rangle$, which we know is described by the $\text{Ising}^{2}$
CFT. To perform the duality, we consider the defect
$\mathcal{D}_{\sigma_{1}}\circ\mathcal{D}_{s}$ such that the action is given
by
$\mathcal{D}_{\sigma_{1}}\circ\mathcal{D}_{s}|\psi_{1,6}\rangle=|\psi_{2,4}\rangle$.
The partition function for the transition is given by

$\displaystyle Z[\text{A},\text{A}]$
$\displaystyle=\langle\mathcal{L}_{1}^{f}|\psi_{2,4}\rangle$
$\displaystyle=\langle\mathcal{L}_{1}^{f}|\mathcal{D}_{\sigma_{1}}\circ\mathcal{D}_{s}|\psi_{1,6}\rangle$
$\displaystyle=\langle\mathcal{L}_{6}^{f}|\psi_{1,6}\rangle$
$\displaystyle=(Z_{1}+Z_{m})(Z_{1}+Z_{f})$
$\displaystyle=(|\chi_{1}|^{2}+|\chi_{\sigma}|^{2}+|\chi_{\psi}|^{2})(|\chi_{1}|^{2}+|\chi_{\psi}|^{2}+\chi_{1}\overline{\chi}_{\psi}+\chi_{\psi}\overline{\chi}_{1}),$
(51)

which is the stacking of the Ising CFT and a single Majorana theory.

####  V.2.5 $\text{Trivial}\leftrightarrow\text{Kitaev}=\text{Majorana}$

Thus far, the transitions discussed in this section are based on the input
boundary transition between $\mathcal{L}_{1}$ and $\mathcal{L}_{6}$, which is
the $\text{Ising}^{2}$ CFT. Here we demonstrate that it’s possible to obtain
the other phase transitions by using different input boundary transitions.

We consider the transition between the Trivial and the Kitaev phases. From
Table. 3, we see that this transition sits between the gapped boundary
$\mathcal{L}_{2}$ and $\mathcal{L}_{6}$. We now discuss a possible boundary
phase critical theory as our input. We note that $\mathcal{L}_{2}$ and
$\mathcal{L}_{6}$, on the bosonic side, corresponds to partial SSB phase
(break the second $\mathbb{Z}_{2}$) and the trivial
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetric phase respectively if we choose
the symmetry boundary to be $\mathcal{L}_{1}$. Since only the second
$\mathbb{Z}_{2}$ symmetry is broken, we can view the system as two layers of
toric code sandwich. The transition described by the Ising CFT only happens at
the physical boundary for second layer, and the first layer stays in the
m-condensed gapped boundary. The input CFT boundary state on the physical
boundary of the sandwich then takes the following form:

$|\psi_{26}\rangle=\sum_{\alpha,\beta\in\text{TC}}v_{\alpha}Z_{\beta}|\alpha\beta\rangle,$
(52)

where TC is the set of anyons in the toric code (we always order the anyons as
$(1,e,m,f)$), $v_{\alpha}=(1,0,1,0)$ describes the condensed anyons ($1$ and
$m$) on the physical boundary for the first copy, $Z_{\beta}$ is the Ising
partition function in the anyon sector $\beta\in$ TC for the second copy. If
we calculate the partition function for the bosonic system as the overlap
$Z=\langle\mathcal{L}_{1}|\psi_{26}\rangle$, it gives the partition function
of the Ising CFT.

Now we perform the fermionization by choosing the symmetry boundary to be
$\mathcal{L}_{1}^{f}$ and calculate the overlap:

$\displaystyle Z[\text{A},\text{A}]$
$\displaystyle=\langle\mathcal{L}_{1}^{f}|\psi_{26}\rangle$
$\displaystyle=v_{1}Z_{1}+v_{e}Z_{e}+v_{m}Z_{f}+v_{f}Z_{m}$
$\displaystyle=Z_{1}+Z_{f}$
$\displaystyle=|\chi_{1}|^{2}+|\chi_{\psi}|^{2}+\chi_{1}\overline{\chi}_{\psi}+\chi_{\psi}\overline{\chi}_{1},$
(53)

where we have used $v_{\alpha}=(1,0,1,0)$ in the third equality. This is
precisely the partition function of the free Majorana theory.

##  VI Fermionic symmetry enriched quantum critical points

Figure 5: The sandwich pictures for two different types of the fermionic
SEQCPs. (a) The first type of fermionic SEQCPs is obtain from the stacking of
the usual SSB transitions with a SPT state. It corresponds to an insertion of
a SPT twist defect $\mathcal{D}_{\textbf{T}}$ in the sandwich picture with the
symmetry boundary that can “absorb” the SPT defect. (b) The second type of
fermionic SEQCPs is obtain from the a twisted gauging TS. It corresponds to an
insertion of a twist defect
$\mathcal{D}_{\textbf{T}}\circ\mathcal{D}_{\textbf{S}}$, where
$\mathcal{D}_{\textbf{S}}$ is a defect that implements the S gauging.

Fermionic symmetry enriched quantum critical points (SEQCPs) are quantum
critical points generally happens at the transition between a SSB phase, where
the bosonic symmetry is completely broken, and a fermionic SPT phase. An
important property of these quantum critical points is that the untwisted
partition functions are the same as the partition function of the transition
between the SSB and the trivial symmetric phases. However, the twisted
partition functions differ from the usual SSB transitions and encode the SPT
invariants.

We found that there are two different types of the fermionic SEQCPs. The first
type of fermionic SEQCPs can be understood as the stacking of the usual SSB
transitions with a SPT state (an operation we label as T). These kind of the
SEQCPs are also called the gapless SPT phases [72, 73, 74, 75, 76]. In
general, we expect the following form of the partition function:

$\displaystyle Z^{\text{SEQCP}}[g,h]=Z^{\text{SSB}}[g,h]Z^{\text{SPT}}[g,h],$
(54)

where $Z^{\text{SPT}}[g,h]$ is the SPT invariant and $Z^{\text{SSB}}[g,h]$ is
the partition function of the symmetry breaking transition. In the sandwich
picture, it corresponds to a SymTFT with an insertion of a SPT twist defect
$\mathcal{D}_{\textbf{T}}$ and, in particular, the symmetry boundary can
“absorb” the defect $\mathcal{D}_{\textbf{T}}$, as illustrated in Fig. 5(a).

The other type of the fermionic SEQCPs is obtained from the usual SSB
transition by a twisted gauging TS, where S is understood as gauging a finite
subgroup in the fermionic symmetry group $G_{f}$. In the sandwich picture, it
corresponds to an insertion of a twist defect
$\mathcal{D}_{\textbf{TS}}=\mathcal{D}_{\textbf{T}}\circ\mathcal{D}_{\textbf{S}}$,
where $\mathcal{D}_{\textbf{S}}$ is a defect that implements the S gauging and
$\mathcal{D}_{\textbf{T}}$ is a SPT twist defect, as shown in Fig. 5(b). We
will present examples of both types of the fermionic SEQCPs in the
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ fermoinic systems in the following
sections.

###  VI.1
$\text{GW}\leftrightarrow(\text{SSB}+\text{Kitaev})=\text{Ising}^{2*}\bf{/Z_{2}^{D}}$

Our first example of the symmetry enriched quantum critical point is the
transition between the Gu-Wen and the “SSB+Kitaev” phases. We tune the
physical boundary of the sandwich to be at the critical point between the
gapped boundaries $\mathcal{L}_{1}$ and $\mathcal{L}_{5}$. We denote the
corresponding boundary state as $|\psi_{1,5}\rangle$ and the partition
function of the transition is given by the overlap:

$Z^{\text{Ising}^{2*}\bm{/Z_{2}^{D}}}[\text{A},\text{A}]=\langle\mathcal{L}_{1}^{f}|\psi_{1,5}\rangle.$
(55)

Now consider a twist defect $\mathcal{D}_{s}$ obtained from gauging the non-
trivial $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ SPT state, which implements the
anyon permutations shown in Eq. (30). The untwisted partition function (in the
(A,A) spin structure) of the symmetry enriched CFT can be obtained as follows:

$\displaystyle Z^{\text{Ising}^{2*}\bm{/Z_{2}^{D}}}[\text{A},\text{A}]$
$\displaystyle=\langle\mathcal{L}_{1}^{f}|\psi_{1,5}\rangle$
$\displaystyle=\langle\mathcal{L}_{1}^{f}|\mathcal{D}_{s}|\psi_{1,6}\rangle$
$\displaystyle=\langle\mathcal{L}_{1}^{f}|\psi_{1,6}\rangle$
$\displaystyle=Z^{\text{Ising}^{2}\bm{/Z_{2}^{D}}}[\text{A},\text{A}],$ (56)

where we have used the fact that
$\langle\mathcal{L}_{1}^{f}|\mathcal{D}_{s}=\langle\mathcal{L}_{1}^{f}|$ in
the third equality. We see that the untwisted partition function are the same
as the $\text{Ising}^{2}\bm{/Z_{2}^{D}}$ CFT, which describes the transition
between the trivial and the “SSB+Kitaev” phases.

Now we would like to show that the twisted sectors of the
$\text{Ising}^{2*}\bm{/Z_{2}^{D}}$ theory is different from the
$\text{Ising}^{2}\bm{/Z_{2}^{D}}$ CFT. We begin with the twisted partition
function of the $\text{Ising}^{2}\bm{/Z_{2}^{D}}$ CFT with both spatial and
temporal twists:

$\displaystyle Z^{\text{Ising}^{2}\bm{/Z_{2}^{D}}}[(\text{A},\text{P});(1,0)]$
$\displaystyle=\langle
m_{1}\times\mathcal{L}_{1}^{f}|W_{m_{1}m_{2}}|\psi_{16}\rangle$
$\displaystyle=Z_{m_{1}}+Z_{f_{1}e_{2}}-Z_{f_{2}}-Z_{e_{1}m_{2}}$
$\displaystyle=(|\chi_{1}|^{2}+|\chi_{\psi}|^{2})|\chi_{\sigma}|^{2}+(\chi_{1}\overline{\chi}_{\psi}+\chi_{\psi}\overline{\chi}_{1})|\chi_{\sigma}|^{2}$
$\displaystyle\qquad-(|\chi_{1}|^{2}+|\chi_{\psi}|^{2})(\chi_{1}\overline{\chi}_{\psi}+\chi_{\psi}\overline{\chi}_{1})-|\chi_{\sigma}|^{4}.$
(57)

While, in the symmetry enriched CFT $\text{Ising}^{2*}\bm{/Z_{2}^{D}}$, the
partition function with the same twists is given by

$\displaystyle
Z^{\text{Ising}^{2*}\bm{/Z_{2}^{D}}}[(\text{A},\text{P});(1,0)]$
$\displaystyle=\langle
m_{1}\times\mathcal{L}_{1}^{f}|W_{m_{1}m_{2}}|\psi_{15}\rangle$
$\displaystyle=\langle
m_{1}\times\mathcal{L}_{1}^{f}|W_{m_{1}m_{2}}\mathcal{D}_{s}|\psi_{16}\rangle$
$\displaystyle=\langle
m_{1}e_{2}\times\mathcal{L}_{1}^{f}|W_{f_{1}f_{2}}|\psi_{16}\rangle$
$\displaystyle=-Z_{m_{2}}-Z_{e_{1}f_{2}}+Z_{f_{1}}+Z_{m_{1}e_{2}}$
$\displaystyle\begin{split}&=-((|\chi_{1}|^{2}+|\chi_{\psi}|^{2})|\chi_{\sigma}|^{2}+(\chi_{1}\overline{\chi}_{\psi}+\chi_{\psi}\overline{\chi}_{1})|\chi_{\sigma}|^{2}\\\
&\qquad-(|\chi_{1}|^{2}+|\chi_{\psi}|^{2})(\chi_{1}\overline{\chi}_{\psi}+\chi_{\psi}\overline{\chi}_{1})-|\chi_{\sigma}|^{4})\end{split}$
$\displaystyle=-Z^{\text{Ising}^{2}\bm{/Z_{2}^{D}}}[(\text{A},\text{P});(1,0)],$
(58)

where, in the third equality, we push the twist defect $\mathcal{D}_{s}$ into
the left boundary, which implements the corresponding anyon permutation for
the spatial and temporal twists. We see that the twist partition functions Eq.
(58) and Eq. (57) differ by a minus sign, which comes from the SPT invariant
of the Gu-Wen state. Physically, it means that the non-trivial
$\mathbb{Z}_{2}$ flux carries an additional fermion. Similarly, one can show
that the fermion parity flux is decorated by a non-trivial $\mathbb{Z}_{2}$
charge. The twisted partition function takes the following form:

$Z^{\text{Ising}^{2*}\bm{/Z_{2}^{D}}}[\rho;(g,h)]=Z^{\text{Ising}\bm{/Z_{2}^{D}}}[\rho;(g,h)]Z^{\text{GW}}[\rho;(g,h)],$
(59)

where $Z^{GW}[\rho;(g,h)]$ is the partition function of the Gu-Wen fermionic
SPT phase. This gives an example of the first type of the fermionic SEQCPs.

###  VI.2
$\text{GW}\leftrightarrow\text{SSB}=\text{Ising}\times\mathcal{F}_{\text{GW}}$

Here we discuss another example of the symmetry enriched quantum critical
points. We consider the transition between the Gu-Wen and the SSB phases. In
the sandwich picture, the physical boundary is tuned to be sitting at the
transition between the gapped boundaries $\mathcal{L}_{4}$ and
$\mathcal{L}_{5}$. We denote the corresponding boundary state as
$|\psi_{45}\rangle$. The untwisted partition function is given by the overlap

$Z^{\text{Ising}\times\mathcal{F}_{\text{GW}}}[\text{A},\text{A}]=\langle\mathcal{L}_{1}^{f}|\psi_{45}\rangle,$
(60)

where $\mathcal{F}_{\text{GW}}$ denotes the theory of the Gu-Wen SPT phase. As
we will show below, this symmetry enriched critical theory carries the
topological responses coming from the Gu-Wen phase. We refer to this symmetry
enriched critical theory as $\text{Ising}\times\mathcal{F}_{\text{GW}}$ with
the $\mathbb{Z}_{2}$ symmetry in the Ising and $\mathcal{F}_{\text{GW}}$ being
identified.

To proceed, we consider the duality implemented by the defect
$\mathcal{D}_{s}\circ\tilde{\mathcal{D}}$ where
$\tilde{\mathcal{D}}=\mathcal{D}_{\sigma_{1}}\circ\mathcal{D}_{s}\circ\mathcal{D}_{\sigma_{1}\sigma_{2}}$.
The defect $\tilde{\mathcal{D}}$ acts on the gapped boundaries as

$\tilde{\mathcal{D}}|\mathcal{L}_{1}\rangle=|\mathcal{L}_{4}\rangle,\
\tilde{\mathcal{D}}|\mathcal{L}_{2}\rangle=|\mathcal{L}_{6}\rangle.$ (61)

We can now pull the defect $\mathcal{D}_{s}\circ\tilde{\mathcal{D}}$ from the
right physical boundary and write the partition function Eq. (60) as

$\displaystyle
Z^{\text{Ising}\times\mathcal{F}_{\text{GW}}}[\text{A},\text{A}]$
$\displaystyle=\langle\mathcal{L}_{1}^{f}|\psi_{45}\rangle$
$\displaystyle=\langle\mathcal{L}_{1}^{f}|\mathcal{D}_{s}\circ\tilde{\mathcal{D}}|\psi_{12}\rangle.$
(62)

Using the fact that the symmetry boundary $\mathcal{L}_{1}^{f}$ can absorb the
gauged SPT defect $\mathcal{D}_{s}$ and action of the defect
$\tilde{\mathcal{D}}$:

$\langle\mathcal{L}_{1}^{f}|\tilde{\mathcal{D}}=\langle\mathcal{L}_{5}^{f}|,$
(63)

the partition function becomes

$Z^{\text{Ising}\times\mathcal{F}_{\text{GW}}}[\text{A},\text{A}]=\langle\mathcal{L}_{5}^{f}|\psi_{12}\rangle.$
(64)

Now we need to know the boundary state $|\psi_{12}\rangle$. We see from Table.
1 that, on the bosonic side, this is the critical point between the partially
$\mathbb{Z}_{2}$ ordered phase and a completely
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry breaking phase, where the
remaining $\mathbb{Z}_{2}$ symmetry is broken spontaneously during the
transition. Therefore, if we view the system as two layers of which each layer
has $\mathbb{Z}_{2}$ symmetry, we expect that the transition only happens in
the first layer and it is described by the Ising CFT. The second layer remains
in the $\mathbb{Z}_{2}$ SSB phase during the transition.

Upon making this assumption, we can write the boundary state
$|\psi_{12}\rangle$ as

$|\psi_{12}\rangle=\sum_{\alpha,\beta\in\text{TC}}Z_{\alpha}u_{\beta}|\alpha\beta\rangle,$
(65)

where TC is the set of anyons in the toric code, $Z_{\alpha}$ is the Ising
partition function in the anyon sector $\alpha\in$ TC for the first layer, and
$u_{\beta}=(1,1,0,0)$ describes the condensed anyons ($1$ and $e$) on the
right physics boundary for the second layer. By taking the overlap
$\langle\mathcal{L}_{1}|\psi_{12}\rangle$, we recover the partition function
of the Ising CFT stacking with a $\mathbb{Z}_{2}$ SSB gapped sector:

$\displaystyle Z_{\mathcal{B}}=\langle\mathcal{L}_{1}|\psi_{12}\rangle$
$\displaystyle=Z_{1}u_{1}+Z_{e}u_{1}+Z_{1}u_{e}+Z_{e}u_{e}$
$\displaystyle=2(|\chi_{1}|^{2}+|\chi_{\psi}|^{2}+|\chi_{\sigma}|^{2}),$ (66)

where we have used $u_{1}=u_{e}=1$, and Eq. (25) in the second equality, which
is also called a gapless SSB phase [68].

Using the boundary state Eq. (65), we obtain the untwisted partition function
for the Gu-Wen to SSB transition by calulating the overlap Eq. (64):

$\displaystyle
Z^{\text{Ising}\times\mathcal{F}_{\text{GW}}}[\text{A},\text{A}]$
$\displaystyle=\langle\mathcal{L}_{5}^{f}|\psi_{12}\rangle$
$\displaystyle=Z_{1}u_{1}+Z_{e}u_{1}+Z_{1}u_{f}+Z_{e}u_{f}$
$\displaystyle=|\chi_{1}|^{2}+|\chi_{\psi}|^{2}+|\chi_{\sigma}|^{2},$ (67)

where we have used $u_{1}=1$, $u_{f}=0$, and Eq. (25) in the second equality.
Since the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$ symmetry is broken
spontaneously down to $\mathbb{Z}_{2}^{F}$, we expect that the transition is
described by an Ising CFT, which is consistent with Eq. (67). The transition
from the Gu-Wen phase to the SSB phase is in the same universality class as
the critical point between the trivial and the SSB phases, of which the
partition function is given by the overlap:

$\displaystyle
Z^{\text{Ising}\times\mathcal{F}_{\text{0}}}[\text{A},\text{A}]$
$\displaystyle=\langle\mathcal{L}_{1}^{f}|\psi_{46}\rangle$
$\displaystyle=\langle\mathcal{L}_{1}^{f}|\tilde{\mathcal{D}}|\psi_{12}\rangle$
$\displaystyle=\langle\mathcal{L}_{5}^{f}|\psi_{12}\rangle$
$\displaystyle=|\chi_{1}|^{2}+|\chi_{\psi}|^{2}+|\chi_{\sigma}|^{2}.$ (68)

We note that Eq. (68) is a Ising CFT stacking with a trivial fermionic theory
$\mathcal{F}_{0}$. From Eq. (67) and Eq. (68), we confirm that the untwisted
partition functions are indeed the same for both transitions. However, we will
show that there are non-trivial topological responses in the twisted sectors.

We begin with the twisted partition function of the Trivial-to-SSB transition.
We consider the twist by inserting a $\mathbb{Z}_{2}$ defect along the time
cycle and measure the fermion parity by inserting the fermion parity defect
along the spatial cycle. This twisted partition function is given by the
overlap:

$\displaystyle
Z^{\text{Ising}\times\mathcal{F}_{\text{0}}}[(\text{A},\text{P});(1,0)]$
$\displaystyle=\langle
m_{1}\times\mathcal{L}_{1}^{f}|W_{m_{1}m_{2}}|\psi_{46}\rangle$
$\displaystyle=\langle
m_{1}\times\mathcal{L}_{1}^{f}|W_{m_{1}m_{2}}\circ\tilde{\mathcal{D}}|\psi_{12}\rangle$
$\displaystyle=\langle
m_{1}\times\mathcal{L}_{5}^{f}|W_{e_{2}}|\psi_{12}\rangle$
$\displaystyle=Z_{m}u_{1}+Z_{f}u_{1}-Z_{m}u_{f}-Z_{f}u_{f}$
$\displaystyle=Z_{m}+Z_{f}$
$\displaystyle=|\chi_{\sigma}|^{2}+\chi_{1}\overline{\chi}_{\psi}+\chi_{\psi}\overline{\chi}_{1},$
(69)

where, in the fifth equality, we have used $u_{1}=u_{e}=1$ and
$u_{f}=u_{m}=0$. Eq. (69) is the same as the twisted partition function
$Z_{1,0}$ in Eq. (45) with only an insertion of the $\mathbb{Z}_{2}$ defect
along the time cycle in the Ising CFT. This is expected since the Ising CFT is
essentially decoupled from the trivially gapped fermionic sector
$\mathcal{F}_{0}$.

Now we consider the same twist in the
$\text{Ising}\times\mathcal{F}_{\text{GW}}$ theory, which is given by

$\displaystyle
Z^{\text{Ising}\times\mathcal{F}_{\text{GW}}}[(\text{A},\text{P});(1,0)]$
$\displaystyle=\langle
m_{1}\times\mathcal{L}_{1}^{f}|W_{m_{1}m_{2}}|\psi_{45}\rangle$
$\displaystyle=\langle
m_{1}\times\mathcal{L}_{1}^{f}|W_{m_{1}m_{2}}\circ\mathcal{D}_{s}\circ\tilde{\mathcal{D}}|\psi_{12}\rangle$
$\displaystyle=\langle
m_{1}e_{2}\times\mathcal{L}_{5}^{f}|W_{e_{1}e_{2}}|\psi_{12}\rangle$
$\displaystyle=Z_{m}u_{m}+Z_{f}u_{m}-Z_{m}u_{e}-Z_{f}u_{e}$
$\displaystyle=-(Z_{m}+Z_{f})$
$\displaystyle=-(|\chi_{\sigma}|^{2}+\chi_{1}\overline{\chi}_{\psi}+\chi_{\psi}\overline{\chi}_{1})$
$\displaystyle=-Z^{\text{Ising}\times\mathcal{F}_{\text{0}}}[(\text{A},\text{P});(1,0)].$
(70)

The minus sign in Eq. (70) is the topological invariant coming from the Gu-Wen
phase. It means that the $\mathbb{Z}_{2}$ flux in the Ising CFT is decorated
by an addition fermion from the gapped sector. Similarly, one can show that
the fermion parity flux in the gapped sector is decorated by a non-trivial
$\mathbb{Z}_{2}$ charge in the Ising CFT. This is also an example of the first
type of the fermionic SEQCPs and can be understood as the stacking of the GW-
Wen ferminic SPT state.

###  VI.3
($\text{Kitaev}+\text{GW})\leftrightarrow\text{SSB}=(\text{Ising}\times\text{Majorana})^{*}$

Here we discuss a new type of the symmetry enriched quantum critical point. We
consider the transition between the “Kitaev \+ GW” phase and the
$\mathbb{Z}_{2}$ SSB phases. This corresponds to the transition between the
$\mathcal{L}_{3}$ and $\mathcal{L}_{4}$ boundaries on the physical boundary
and the symmetry boundary is still $\mathcal{L}_{1}^{f}$. We denote the
corresponding critical boundary state as $|\psi_{3,4}\rangle$ and the
partition function of the SPT transition is
$Z[{\text{A},\text{A}}]=\langle\mathcal{L}_{1}^{f}|\psi_{3,4}\rangle$.

We would like to relate the critical boundary state $|\psi_{3,4}\rangle$ to
$|\psi_{1,6}\rangle$, which we know is described by the $\text{Ising}^{2}$
CFT. To perform the duality, we consider the defect
$\mathcal{D}_{\sigma_{2}}\circ\mathcal{D}_{s}$ such that the action is given
by
$\mathcal{D}_{\sigma_{2}}\circ\mathcal{D}_{s}|\psi_{1,6}\rangle=|\psi_{3,4}\rangle$.
The untwisted partition function for the transition is given by

$\displaystyle Z[\text{A},\text{A}]$
$\displaystyle=\langle\mathcal{L}_{1}^{f}|\psi_{3,4}\rangle$
$\displaystyle=\langle\mathcal{L}_{1}^{f}|\mathcal{D}_{\sigma_{2}}\circ\mathcal{D}_{s}|\psi_{1,6}\rangle$
$\displaystyle=\langle\mathcal{L}_{7}^{f}|\psi_{1,6}\rangle$
$\displaystyle=(Z_{1}+Z_{m})(Z_{1}+Z_{f})$
$\displaystyle=(|\chi_{1}|^{2}+|\chi_{\sigma}|^{2}+|\chi_{\psi}|^{2})(|\chi_{1}|^{2}+|\chi_{\psi}|^{2}+\chi_{1}\overline{\chi}_{\psi}+\chi_{\psi}\overline{\chi}_{1}),$
(71)

which is the same as the Ising $\times$ Majorana CFT.

The twisted partition functions however differ from the Ising $\times$
Majorana CFT as we now show. We consider the twisted partition function with a
$\mathbb{Z}_{2}$ twist along the spatial direction and the fermion parity
twist along the time direction, which is given by

$\displaystyle Z[(\text{A},\text{P});(1,0)]$ $\displaystyle=\langle
m_{1}\times\mathcal{L}_{1}^{f}|W_{m_{1}m_{2}}|\psi_{3,4}\rangle$
$\displaystyle=\langle
m_{1}\times\mathcal{L}_{1}^{f}|W_{m_{1}m_{2}}\circ\mathcal{D}_{2}\circ\mathcal{D}_{s}|\psi_{1,6}\rangle$
$\displaystyle=\langle
m_{1}e_{2}\times\mathcal{L}_{7}^{f}|W_{m_{1}}|\psi_{1,6}\rangle$
$\displaystyle=(Z_{e}+Z_{f})(Z_{m}-Z_{e})$
$\displaystyle=Z^{\text{Majorana}}[(\text{P},\text{P})]Z^{\text{Ising}}[(1,0)]$
$\displaystyle=Z[(\text{P},\text{P});(1,0)].$ (72)

We find that the $\mathbb{Z}_{2}$ twist along the spatial direction induces a
change of the spin structure along the same direction. This happens since the
$\mathbb{Z}_{2}$ defect line is attached with an additional
$\mathbb{Z}_{2}^{F}$ fermion parity line along the same direction, which
causes the change of the spin structure. This is an example of the second type
of the fermionic SEQCPs, which can be understood as performing the twist
gauging ST for the $\text{Ising}^{2*}\bm{/Z_{2}^{D}}$ theory where S here is
the gauging of the $\mathbb{Z}_{2}$ symmetry in
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}^{F}$.

##  VII Fermionic chains with $\mathbb{Z}_{4}^{F}$ symmetry

We discuss the quantum criticality in the fermionic chains with
$\mathbb{Z}_{4}^{F}$ symmetry in this section. In the sandwich construction,
we choose the bulk to be the $\mathbb{Z}_{4}$ gauge theory (also called the
$\mathbb{Z}_{4}$ toric code topological order).

We briefly review the basic properties of the $\mathbb{Z}_{4}$ toric code
topological order below. The anyons satisfy a
$\mathbb{Z}_{4}\times\mathbb{Z}_{4}$ fusion group, generated by the charge $e$
and the flux $m$. There is a non-trivial mutual braiding statistics between
$e$ and $m$

$\frac{S_{e,m}}{S_{e,0}}=\frac{S_{e,m}}{S_{0,m}}=i.$ (73)

The self-statistics of a generic anyon $e^{p}m^{q}$ with
$p,q\in\mathbb{Z}_{4}$ is given by

$\theta(e^{p}m^{q})=i^{pq}.$ (74)

The topological symmetry of the anyon theory is
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. The first $\mathbb{Z}_{2}$ is generated
by

$\sigma:m\leftrightarrow e,$ (75)

which corresponds to the EM exchange automorphism of the $\mathbb{Z}_{4}$
toric code. The other automorphism is given by

$s:e\rightarrow e^{3},m\rightarrow m^{3}$ (76)

which comes from the automorphism of the $\mathbb{Z}_{4}$ group. The
automorphism $s$ can also be viewed as the charge conjugation symmetry of the
$\mathbb{Z}_{4}$ toric code. There are three gapped boundaries with the
following Lagrangian algebra of condensed anyons

$\displaystyle\mathcal{L}_{1}$ $\displaystyle=1\oplus e\oplus e^{2}\oplus
e^{3},$ $\displaystyle\mathcal{L}_{2}$ $\displaystyle=1\oplus e^{2}\oplus
m^{2}\oplus e^{2}m^{2},$ $\displaystyle\mathcal{L}_{3}$ $\displaystyle=1\oplus
m\oplus m^{2}\oplus m^{3}.$ (77)

Choosing the symmetry boundary to be $\mathcal{L}_{1}$ corresponds to the
bosonic systems with $\mathbb{Z}_{4}$ symmetry, while choosing
$\mathcal{L}_{2}$ corresponds to the bosonic systems with an anomalous
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry with the mixed anomaly in
$\mathcal{H}^{2}(\mathbb{Z}_{2},\mathcal{H}^{1}(\mathbb{Z}_{2},\mathrm{U}(1)))$.
The second choice describes the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
deconfined quantum critical points [67, 48].

In order to describe fermionic systems, we need to know the fermionic gapped
boundaries of the $\mathbb{Z}_{4}$ toric code. There are two super Lagrangian
algebra:

$\displaystyle\mathcal{L}_{1}^{f}$ $\displaystyle=1\oplus em^{2}\oplus
e^{2}\oplus e^{3}m^{2},$ $\displaystyle\mathcal{L}_{2}^{f}$
$\displaystyle=1\oplus e^{2}m\oplus m^{2}\oplus e^{2}m^{3}.$ (78)

We choose $\mathcal{L}_{1}^{f}$ as the symmetry boundary, which corresponds to
the fermionic systems with $\mathbb{Z}_{4}^{F}$ symmetry. The
$\mathbb{Z}_{4}^{F}$ symmetry is generated by the $m$ line on the symmetry
boundary. The fermion parity symmetry is the $\mathbb{Z}_{2}$ normal subgroup
in $\mathbb{Z}_{4}^{F}$, which is generated by the $m^{2}$ line:

$\displaystyle\mathbb{Z}_{4}:m,\quad\mathbb{Z}_{2}^{F}:m^{2}.$ (79)

The mapping from the bulk lines to the boundary lines is given by:

$\displaystyle\\{m,em^{3},e^{2}m,e^{3}m^{3}\\}$ $\displaystyle\rightarrow m,$
$\displaystyle\\{m^{2},e,e^{2}m^{2},e^{3}\\}$ $\displaystyle\rightarrow
m^{2}.$ (80)

We note that the other choice of the symmetry boundary $\mathcal{L}_{2}^{f}$
is related by the EM exchange automorphism.

Lagrangian algebra | Fermionic gapped phases
---|---
$\mathcal{L}_{1}=1\oplus e\oplus e^{2}\oplus e^{3}$ |  $\mathbb{Z}_{2}$ SSB + Kitaev
$\mathcal{L}_{2}=1\oplus e^{2}\oplus m^{2}\oplus e^{2}m^{2}$ |  $\mathbb{Z}_{2}$ SSB
$\mathcal{L}_{3}=1\oplus m\oplus m^{2}\oplus m^{3}$ |  $\mathbb{Z}_{4}^{F}$ Trivial
Table 4: Fermionic gapped phases with $\mathbb{Z}_{4}^{F}$ symmetry. The left
symmetry boundary is chosen to be the fermionic gapped boundary
$\mathcal{L}^{f}_{1}$. The first column shows the physical boundary on the
right of the sandwich. The second column shows the fermoinic gapped phases
realized by the sandwich construction. “$\mathbb{Z}_{2}$ SSB” means that the
$\mathbb{Z}_{4}^{F}$ symmetry is broken spontaneously down to
$\mathbb{Z}_{2}^{F}$. “Kitaev” denotes the fermionic chain is in the Kitaev
phase. “$\mathbb{Z}_{4}^{F}$ Trivial” means the fermoinic chain is in the
$\mathbb{Z}_{4}^{F}$ symmetric trivial phase.

###  VII.1 Fermionization

If we start from a bosonic system with $\mathbb{Z}_{4}$ symmetry by choosing
the symmetry boundary to be the e-condensed boundary and change the symmetry
boundary to be the fermionic boundary $\mathcal{L}_{1}^{f}$, this process
implements the fermionization with respecting to the $\mathbb{Z}_{2}$ subgroup
in the $\mathbb{Z}_{4}$ symmetry. Here we derive the fermionization formula in
the topological holography. To proceed, we need the following twisted
partition functions on the bosonic side:

$\displaystyle Z_{0,0}$ $\displaystyle=Z_{1}+Z_{e}+Z_{e^{2}}+Z_{e^{3}},$
$\displaystyle Z_{0,2}$ $\displaystyle=Z_{1}-Z_{e}+Z_{e^{2}}-Z_{e^{3}},$
$\displaystyle Z_{2,0}$
$\displaystyle=Z_{m^{2}}+Z_{em^{2}}+Z_{e^{2}m^{2}}+Z_{e^{3}m^{2}},$
$\displaystyle Z_{2,2}$
$\displaystyle=Z_{m^{2}}-Z_{em^{2}}+Z_{e^{2}m^{2}}-Z_{e^{3}m^{2}},$ (81)

where we have used Eq. (12). In the sandwich construction, the fermionic
partition function in the $(\text{A},\text{A})$ spin structure is defined as
the partition function without inserting the fermion parity line, which can be
written in the anyon basis:

$Z_{\mathcal{F}}[\text{A},\text{A}]=Z_{1}+Z_{em^{2}}+Z_{e^{2}}+Z_{e^{3}m^{2}}.$
(82)

We can rewrite Eq. (82) by using Eq. (81):

$Z_{\mathcal{F}}[\text{A},\text{A}]=\frac{1}{2}(Z_{0,0}+Z_{0,2}+Z_{2,0}-Z_{2,2}).$
(83)

Similarly, we obtain the following fermionic partition functions in different
spin structures:

$\displaystyle Z_{\mathcal{F}}[\text{A},\text{P}]$
$\displaystyle=Z_{1}-Z_{em^{2}}+Z_{e^{2}}-Z_{e^{3}m^{2}}$
$\displaystyle=\frac{1}{2}(Z_{0,0}+Z_{0,2}-Z_{2,0}+Z_{2,2}),$ $\displaystyle
Z_{\mathcal{F}}[\text{P},\text{A}]$
$\displaystyle=Z_{e}+Z_{e^{2}m^{2}}+Z_{e^{3}}+Z_{m^{2}}$
$\displaystyle=\frac{1}{2}(Z_{0,0}-Z_{0,2}+Z_{2,0}+Z_{2,2}),$ $\displaystyle
Z_{\mathcal{F}}[\text{P},\text{P}]$
$\displaystyle=-Z_{e}+Z_{e^{2}m^{2}}-Z_{e^{3}}+Z_{m^{2}}$
$\displaystyle=\frac{1}{2}(-Z_{0,0}+Z_{0,2}+Z_{2,0}+Z_{2,2}),$ (84)

which are precisely given by the usual fermionization formula Eq. (11) with
respecting to the $\mathbb{Z}_{2}$ subgroup 888We note that, in calculating
the twisted partition functions, it’s possible to choose a bulk representative
of the fermion parity line from the set of bulk anyon lines
$\\{m^{2},e,e^{2}m^{2},e^{3}\\}$. All these choices give consistent results
once the half-braiding phase factors are taking into account..

If we start from a bosonic system with the anomalous
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry with the mixed anomaly by
choosing the symmetry boundary to be $\mathcal{L}_{2}$ and change the symmetry
boundary to be the fermionic boundary $\mathcal{L}_{1}^{f}$, the result is
that it implements the other fermionization procedure with an additional
stacking of the Kitaev chain. This result can be derived by following the
similar procedure with the help of the following twisted partition functions:

$\displaystyle Z_{(0,0),(0,0)}$
$\displaystyle=Z_{1}+Z_{e^{2}}+Z_{m^{2}}+Z_{e^{2}m^{2}},$ $\displaystyle
Z_{(0,0),(1,0)}$ $\displaystyle=Z_{1}+Z_{e^{2}}-Z_{m^{2}}-Z_{e^{2}m^{2}},$
$\displaystyle Z_{(1,0),(0,0)}$
$\displaystyle=Z_{e}+Z_{e^{3}}+Z_{em^{2}}+Z_{e^{3}m^{2}},$ $\displaystyle
Z_{(1,0),(1,0)}$ $\displaystyle=Z_{e}+Z_{e^{3}}-Z_{em^{2}}-Z_{e^{3}m^{2}}.$
(85)

For example, by using Eq. (82) and Eq. (85), the fermionic partition function
in the $(\text{P},\text{P})$ spin structure is given by

$\displaystyle Z_{\mathcal{F}}[\text{P},\text{P}]$
$\displaystyle=-Z_{e}+Z_{e^{2}m^{2}}-Z_{e^{3}}+Z_{m^{2}}$
$\displaystyle=-\frac{1}{2}(-Z_{(0,0),(0,0)}+Z_{(0,0),(1,0)}+Z_{(1,0),(0,0)}+Z_{(1,0),(1,0)}),$
(86)

which differs from the fermionization formula Eq. (11) by a minus sign, coming
from the additional Arf invariant. Following similar procedure, one can check
the fermionization formula for the other spin structures explicitly. We note
that the anomalous $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry becomes an
anomaly free $\mathbb{Z}_{4}^{F}$ symmetry generated by the lines in Eq. (79)
after the fermionization.

###  VII.2 Gapped phases

By choosing the physical boundary to be $\mathcal{L}_{i}$ for $i=1,2,3$, the
sandwich construction realizes fermionic gapped phases with
$\mathbb{Z}_{4}^{F}$ symmetry. There are only three gapped phases: the
$\mathbb{Z}_{4}^{F}$ symmetric trivial phase, the “$\mathbb{Z}_{2}$ SSB” phase
such that the $\mathbb{Z}_{4}^{F}$ symmetry is broken spontaneously down to
$\mathbb{Z}_{2}^{F}$ symmetry, and the “$\mathbb{Z}_{2}$ SSB + Kitaev” phase
where the $\mathbb{Z}_{4}^{F}$ symmetry is broken spontaneously down to
$\mathbb{Z}_{2}^{F}$ and the system is in the Kitaev phase.

The identification of the fermionic gapped phases can be obtained by
calculating the twisted partition functions by using Eq. (12) and Eq. (20).
The fermionic boundary $\mathcal{L}_{3}$ corresponds to the
$\mathbb{Z}_{4}^{F}$ symmetric trivial phase since the identity line
$\mathcal{W}_{1}$ is the only local operator. We check that all the twisted
partition functions equal to $1$.

The remaining boundaries $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ break the
symmetry down to $\mathbb{Z}_{2}^{F}$ since there is a non-trivial
$\mathbb{Z}_{2}$ order parameter given by $\mathcal{W}_{e^{2}}$.

To see that $\mathcal{L}_{1}$ is in the Kitaev phase, we calculate the
partition function in the $(\text{P},\text{P})$ spin structure by using Eq.
(84) and we obtain $Z_{\mathcal{F}}[\text{P},\text{P}]=-Z_{e}-Z_{e^{3}}=-2$.
The absolute value “2” indicates that there are two degenerate ground states,
which is consistent with the $\mathbb{Z}_{2}$ SSB. The $(-1)$ sign for the
$(\text{P},\text{P})$ spin structure is the signature of the Kitaev phase. For
$\mathcal{L}_{2}$, there is no addition minus sign in the
$(\text{P},\text{P})$ spin structure:
$Z_{\mathcal{F}}[\text{P},\text{P}]=Z_{e}+Z_{e^{3}}=2$, which means that it’s
the $\mathbb{Z}_{2}$ SSB phase. The identification of the fermionic gapped
phases are summarized in Table. 4.

###  VII.3 Quantum critical points

####  VII.3.1 $\mathbb{Z}_{4}^{F}\
\text{Trivial}\leftrightarrow(\mathbb{Z}_{2}\
\text{SSB}+\text{Kitaev})=\text{Majorana}^{2}$

Figure 6: Summary of the phase transitions in the fermionic systems with
$\mathbb{Z}_{4}^{F}$ symmetry.

Topological holography can be used to find the CFT description between any of
the gapped phases identified in the previous sections. To illustrate the idea,
we focus on the transition between the $\mathbb{Z}_{4}^{F}$ symmetric trivial
phase and the “$\mathbb{Z}_{2}$ SSB + Kitaev” phase. The input we are using is
the $U(1)_{4}$ CFT, which can be the DQCP transition for the anomalous
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ symmetry with the mixed anomaly.

Recall that, in the sandwich picture of the DQCP, we choose the symmetry
boundary to be $\mathcal{L}_{2}$ and tune the physical boundary to be sitting
at the transition between $\mathcal{L}_{1}$ and $\mathcal{L}_{3}$, which we
assume to be described by the $U(1)_{4}$ CFT. Now we perform the
fermionization by changing the symmetry boundary to be $\mathcal{L}_{1}^{f}$.
From the discussion of Sec. VII.1, we see that it corresponds to fermionizing
with respect to the first $\mathbb{Z}_{2}$ symmetry. By applying Eq. (11), we
obtain the fermionic partition function in the $(\text{A},\text{A})$ spin
structure:

$\displaystyle Z_{\mathcal{F}}[\text{A},\text{A}]$
$\displaystyle=(|K_{0}|^{2}+|K_{2}|^{2}+K_{0}\bar{K}_{2}+K_{2}\bar{K}_{0}),$
$\displaystyle=(|\chi_{1}|^{2}+|\chi_{\psi}|^{2}+\chi_{1}\bar{\chi}_{\psi}+\chi_{\psi}\bar{\chi}_{1})^{2},$
(87)

where $K_{n}$ with $n=-1,0,1,2$ are the characters of the $U(1)_{4}$ CFT, and
$\chi_{a}$ with $a=\\{1,\sigma,\psi\\}$ are the characters in the Ising CFT.
We recognize that Eq. (87) is the partition function for two copies of the
Majorana CFT, which is equivalent to a Dirac theory. We note that Ref. [28]
also discuss fermionizing the compact boson theory to obtain a Dirac theory
and the appearance of the $\mathbb{Z}_{4}^{F}$ symmetry by using pure CFT
analysis.

####  VII.3.2 $\mathbb{Z}_{4}^{F}\
\text{Trivial}\leftrightarrow\mathbb{Z}_{2}\
\text{SSB}=\text{Ising}\times\mathcal{F}_{0}$

The CFT description for other critical points can be obtain by using the “club
sandwich” construction [55]. To study the transition between the
$\mathbb{Z}_{4}^{F}$ trivial and the $\mathbb{Z}_{2}$ SSB phases, we consider
the following club sandwich. The bulk of the club sandwich consists of a
$\mathbb{Z}_{4}$ toric code topological order on the left and a
$\mathbb{Z}_{2}$ toric code topological order on the right, separating by a
gapped domain wall on which the $m^{2}$ particle in the $\mathbb{Z}_{4}$ toric
code condenses. The particle $e^{2}$ and $m$ (and their fusion products with
the condensate $m^{2}$) becomes the $e^{\prime}$ and $m^{\prime}$ particles in
the $\mathbb{Z}_{2}$ toric code. We will denote the domain wall by
$\mathcal{I}_{m^{2}}$. If the physical boundary is the $e^{\prime}$-condensed
boundary $\mathcal{A}_{e^{\prime}}$ of the $\mathbb{Z}_{2}$ toric code, the
composition between the domain wall $\mathcal{I}_{m^{2}}$ and the boundary
$\mathcal{A}_{e^{\prime}}$ gives the $\mathcal{L}_{2}$ boundary in the
$\mathbb{Z}_{4}$ toric code. Similarly, the composition between the domain
wall $\mathcal{I}_{m^{2}}$ and the $m$-condensed boundary
$\mathcal{A}_{m^{\prime}}$ gives the $\mathcal{L}_{3}$ boundary. We then tune
the physical boundary to be sitting at the transition between the $e^{\prime}$
and $m^{\prime}$ condensed boundaries, which is described by the Ising CFT.
The fermionization is again implemented by changing the symmetry boundary to
be $\mathcal{L}_{1}^{f}$. The partition function is given by the overlap

$\displaystyle
Z^{\text{Ising}\times\mathcal{F}_{\text{0}}}[\text{A},\text{A}]$
$\displaystyle=\matrixelement{\mathcal{L}_{1}^{f}}{\mathcal{I}_{m^{2}}}{\Psi_{\text{Ising}}}$
$\displaystyle=Z_{1}+Z_{e^{\prime}},$
$\displaystyle=|\chi_{1}|^{2}+|\chi_{\psi}|^{2}+|\chi_{\sigma}|^{2},$ (88)

which is the Ising CFT stacking with a trivial fermionic theory
$\mathcal{F}_{0}$.

####  VII.3.3 $\mathbb{Z}_{2}\ \text{SSB}\leftrightarrow(\mathbb{Z}_{2}\
\text{SSB}+\text{Kitaev})=2\ \text{Majorana}$

For the transition between the $\mathbb{Z}_{2}$ SSB and the “$\mathbb{Z}_{2}$
SSB + Kitaev” phases, we consider the similar club sandwich consists of the
$\mathbb{Z}_{4}$ toric code and the $\mathbb{Z}_{2}$ toric code, separated by
a gapped domain wall on which the $e^{2}$ particle in the $\mathbb{Z}_{4}$
toric code condenses. We will denote the domain wall by $\mathcal{I}_{e^{2}}$.
The particle $e$ and $e^{3}$ becomes the $e^{\prime}$ particle and $m^{2}$
becomes the $m^{\prime}$ particles in the $\mathbb{Z}_{2}$ toric code. We also
tune the physical boundary to be sitting at the transition between the
$e^{\prime}$ and $m^{\prime}$ condensed boundaries, which is described by the
Ising CFT and change the symmetry boundary to be $\mathcal{L}_{1}^{f}$ to
perform the fermionization. The partition function is given by

$\displaystyle Z^{2\text{Majorana}}[\text{A},\text{A}]$
$\displaystyle=\matrixelement{\mathcal{L}_{1}^{f}}{\mathcal{I}_{e^{2}}}{\Psi_{\text{Ising}}}$
$\displaystyle=2(Z_{1}+Z_{f^{\prime}}),$
$\displaystyle=2(|\chi_{1}|^{2}+|\chi_{\psi}|^{2}+\chi_{1}\overline{\chi}_{\psi}+\chi_{\psi}\overline{\chi}_{1})$
$\displaystyle=2Z^{\text{Majorana}}[\text{A},\text{A}],$ (89)

where the prefactor “$2$” in the second equality comes from the summation over
the condensed particles $\\{1,m^{4},em^{2},e^{3}m^{2}\\}$ on the symmetry
boundary $\mathcal{L}_{1}^{f}$ which can pass through the gapped domain wall
$\mathcal{I}_{e^{2}}$. Physically, it signatures the 2-fold ground state
degeneracy since the $\mathbb{Z}_{4}^{F}$ symmetry has already broken down to
$\mathbb{Z}_{2}^{F}$. The transition is given by the free Majorana CFT in the
two superselection sectors, which is also called a gapless SSB phase [68].
This theory has the same primaries as the free Majorana CFT but every level in
the spectrum is at least two-fold degenerate. This concludes the discussion of
the phase transitions in the fermionic systems with $\mathbb{Z}_{4}^{F}$
symmetry and the results are summarized in Fig. 6.

###  VII.4 The fermionic intrinsically gapless SPT phase with
$\mathbb{Z}_{4}^{F}$ symmetry

Here we discuss an example of the intrinsically gapless fermionic SPT with
$\mathbb{Z}_{4}^{F}$ symmetry. We first briefly review the basic idea of the
bosonic igSPT phases.

An bosonic igSPT phase is a gapless phase that exhibiting an emergent IR
anomaly in a bosonic lattice model in $D-1$ spatial dimensions with an
anomaly-free UV symmetry [77, 78]. In general we have the following symmetry
structure. Let $\Gamma$ be the microscopic on-site symmetry, which is anomaly
free. Below an energy scale $\Delta$, there is an emergent IR symmetry $G$
that sits in the short exact sequence:

$1\rightarrow H\rightarrow\Gamma\rightarrow G\rightarrow 1,$ (90)

where $H$ is a normal subgroup of $\Gamma$ which acts trivially on the IR
degrees of freedom, and $G=\Gamma/H$. The emergent IR symmetry $G$ has an
anomaly in $\mathcal{H}^{D+1}(G,\mathrm{U}(1))$, where $D$ is the spacetime
dimensions. This anomaly becomes trivial when it is pulled back into
$\mathcal{H}^{D+1}(\Gamma,\mathrm{U}(1))$, so that the entire system can be
realized in the lattice models $D-1$ spatial dimensions with non-anomalous
$\Gamma$ symmetry.

An simple example is given by the $(1+1)d$ igSPT phases with
$\Gamma=\mathbb{Z}_{4}$, $H=\mathbb{Z}_{2}$, and $G=\mathbb{Z}_{2}$. The non-
trivial igSPT phase realized in the $\Gamma=\mathbb{Z}_{4}$ bosonic lattice
model shares the same (emergent) anomaly as the gapless edge modes of the
$\mathbb{Z}_{2}$ Levin-Gu SPT state.

The sandwich picture of the $\mathbb{Z}_{4}$ igSPT phase is given by the club
sandwich construction as discussed in Ref. [48, 55]. We have a
$\mathbb{Z}_{4}$ toric code topological order on the left and a double semion
(DS) order on the right. The two topological orders are separated by a gapped
domain wall on which the $e^{2}m^{2}$ particle in the $\mathbb{Z}_{4}$ toric
code condenses. We will denote the domain wall by $\mathcal{I}_{e^{2}m^{2}}$.
After the $e^{2}m^{2}$ condensation, the deconfined anyons in the
$\mathbb{Z}_{4}$ toric code are $\\{[1],[em],[em^{3}],[m^{2}]\\}$, which can
be identified with the anyons in the double semion order
$\\{1,s,\bar{s},b\\}$, respectively. For the bosonic $\mathbb{Z}_{4}$ igSPT
phase, the symmetry boundary is chosen to be the charge condensed boundary
$\mathcal{L}_{1}$.

Since $H=\mathbb{Z}_{2}$ is non-anomalous, we can perform the fermionization
with respecting the $H=\mathbb{Z}_{2}$ subgroup. This corresponds to change
the symmetry boundary to be the fermionic gapped boundary
$\mathcal{L}_{1}^{f}$ in the club sandwich construction and the global
symmetry becomes $\mathbb{Z}_{4}^{F}$.

On the physical boundary of the club sandwich, we have an edge theory of the
$\mathbb{Z}_{2}$ Levin-Gu SPT state, which can live at the edge of the DS
order. Upon dimensional reduction, the sandwich construction realizes the
igSPT state where the low-energy theory realizes the anomalous edge theory of
the $\mathbb{Z}_{2}$ Levin-Gu SPT state in a system with a
$\mathbb{Z}_{4}^{F}$ global symmetry.

As a concrete example, we can choose the physical boundary to be the
$\mathrm{U}(1)_{2}$ CFT [66]. The partition functions in the DS sectors are
given by:

$\displaystyle Z_{1}$ $\displaystyle=|\chi_{0}|^{2},$ $\displaystyle Z_{s}$
$\displaystyle=\chi_{1}\overline{\chi}_{0},$ $\displaystyle Z_{\bar{s}}$
$\displaystyle=\chi_{0}\overline{\chi}_{1},$ $\displaystyle Z_{b}$
$\displaystyle=|\chi_{1}|^{2}.$ (91)

The gapped domain wall $\mathcal{I}_{e^{2}m^{2}}$ in the club sandwich
construction can be written as

$\mathcal{I}_{e^{2}m^{2}}=\sum_{\alpha\in\mathbb{Z}_{4}\text{TC},\mu\in\text{DS}}W_{\alpha,\mu}|\alpha\rangle\langle\mu|,$
(92)

where $W_{\alpha,\mu}=1$ only for transparent anyons. The untwisted torus
partition function can then be obtained as the overlap with the insertion of
the gapped domain wall $\mathcal{I}_{e^{2}m^{2}}$:

$\displaystyle Z_{00}$
$\displaystyle=\matrixelement{\mathcal{L}_{1}^{f}}{\mathcal{I}_{e^{2}m^{2}}}{\Psi}$
$\displaystyle=\sum_{\alpha\in\mathbb{Z}_{4}\text{TC},\mu\in\text{DS}}w_{\alpha}W_{\alpha,\mu}Z_{\mu}$
$\displaystyle=Z_{1}+Z_{b},$ (93)
$\displaystyle=|\chi_{0}|^{2}+|\chi_{1}|^{2}.$ (94)

where in the second equality $w_{\alpha}=(1,0,1,0)$ indexed as
$(w_{1},w_{em^{2}},w_{e^{2}},w_{e^{3}m^{2}})$. We find that the untwisted
partition function is the same as the usual $\mathrm{U}(1)_{2}$ CFT.

The non-trivial characterization is in the twisted partition functions since
we can twist by the element in $\mathbb{Z}_{2}^{F}$. The analysis is parallel
to the bosonic $\mathbb{Z}_{4}$ igSPT phase by replacing the $\mathbb{Z}_{2}$
normal subgroup to $\mathbb{Z}_{2}^{F}$ [55]. Here we only give an example to
illustrate the main idea and other twisted partition functions can be derived
similarly. Consider the partition functions with a $g\in G=\mathbb{Z}_{2}$
twist in the space direction and a $h\in H=\mathbb{Z}_{2}^{F}$ twist in the
time direction:

$Z_{g,h}=\Tr_{\mathcal{H}_{g}}he^{-\beta\mathcal{H}}.$ (95)

Recall that the $g$ twist corresponds to an $m$ line on the left boundary that
generates the $\Gamma=\mathbb{Z}_{4}^{F}$ symmetry. The $h$ twist corresponds
to the $m^{2}$ line that generates the $\mathbb{Z}_{2}^{F}$ normal subgroup in
$\Gamma=\mathbb{Z}_{4}^{F}$. The twisted partition function can be calculated
in the club sandwich as

$\displaystyle Z_{g,h}$ $\displaystyle=\sum_{\alpha\in
m\times\mathcal{L}_{e},\mu\in\text{DS}}\frac{S_{\alpha,m^{2}}}{S_{0,\alpha}}W_{\alpha,\mu}Z_{\mu}$
$\displaystyle=-(Z_{s}+Z_{\bar{s}})$ $\displaystyle=-Z_{g,0},$ (96)

where $m\times\mathcal{L}_{e}=(m,em^{3},e^{2}m,e^{3}m^{3})$. The minus sign in
Eq. (96) means that there is an additional fermion decorated on the $g$ defect
line, and the twisted partition function $Z_{g,h}$ precisely measures the
fermion parity in the presence of $g$ defect.

##  VIII Fermionic chains with super fusion category
$\text{SRep}(\mathcal{H}_{S_{3}})$ symmetry

In this section we focus on fermionic systems with non-invertible symmetries.
In particular, we consider 1D fermionic systems with the super fusion category
$\text{SRep}(\mathcal{H}_{S_{3}})$ symmetry, where $\mathcal{H}_{S_{3}}$ is a
Hopf superalgebra [79]. This symmetry emerges out of the fermionization with
respecting to the $\mathbb{Z}_{2}$ symmetry in the $\text{Rep}(S_{3})$
symmetry. We give summary of the sandwich picture of the bosonic systems with
$S_{3}$ and $\text{Rep}(S_{3})$ symmetry below.

We begin with a review for the bosonic systems with
$S_{3}=\\{r,s|r^{3}=s^{2}=1,srs=r^{-1}\\}$ symmetry. The bulk of the sandwich
is the $S_{3}$ gauge theory with 8 anyon types. Following the convention in
Ref. [80], we label them as $A,B,\dots,H$. $A,B$ and $C$ are the three gauge
charges: $A$ corresponds to the identity representation, $B$ is the one-
dimensional representation and $C$ is the two-dimensional representation. $D$
and $E$ are the two fluxes with conjugacy class $[s]=\\{s,sr,sr^{2}\\}$.
$F,G,H$ are the three fluxes with conjugacy class $[r]=\\{r,r^{2}\\}$. The
$S$-matrix of the $S_{3}$ gauge theory is

$S=\frac{1}{6}\begin{pmatrix}1&1&2&3&3&2&2&2\\\ 1&1&2&-3&-3&2&2&2\\\
2&2&4&0&0&-2&-2&-2\\\ 3&-3&0&3&-3&0&0&0\\\ 3&-3&0&-3&3&0&0&0\\\
2&2&-2&0&0&4&-2&-2\\\ 2&2&-2&0&0&-2&-2&4\\\ 2&2&-2&0&0&-2&4&-2\\\
\end{pmatrix}.$ (97)

The topological spins are

$\\{0,0,0,0,\frac{1}{2},0,\frac{1}{3},\frac{2}{3}\\},$ (98)

for anyons $A,B,\dots,H$. There is a non-trivial automorphism given by
exchanging $C$ and $F$ anyons.

First we consider that the symmetry boundary is chosen by the pure charge
condensation $A\oplus B\oplus 2C$, on which the topological lines generate the
$S_{3}$ ($\text{Vec}_{S_{3}}$) symmetry. In this case, there are four physical
boundaries corresponding to four possible gapped phases:

  1. 1.

$\mathcal{L}_{1}=A\oplus B\oplus 2C$ and $H=\\{1\\}$: $S_{3}$ SSB phase. In
this case, the $S_{3}$ symmetry is completely broken, and there are 6 ground
states.

  2. 2.

$\mathcal{L}_{2}=A\oplus D\oplus F$ and $H=S_{3}$: $S_{3}$ symmetric phase.
There are no Wilson lines connecting the two boundaries, so the $S_{3}$
symmetry is unbroken. This is the $S_{3}$ symmetric state.

  3. 3.

$\mathcal{L}_{3}=A\oplus B\oplus 2F$ and $H=\mathbb{Z}_{3}$: $Z_{2}$ SSB
phase. The only nontrivial local operator is the local $B$ Wilson line, which
braids trivially with the $r$ and $r^{2}$ lines on the symmetry boundary. Thus
the remaining symmetry is $\mathbb{Z}_{3}$. There are 2 ground states
corresponding to the trivial local operator ${\cal O}_{A}$ and the nontrivial
local operator ${\cal O}_{B}$, as expected from the
$S_{3}\rightarrow\mathbb{Z}_{3}$ SSB.

  4. 4.

$\mathcal{L}_{4}=A\oplus C\oplus D$ and $H=\mathbb{Z}_{2}$: $Z_{3}$ SSB phase.
The nontrivial local operators are given by ${\cal O}_{C}^{1},{\cal
O}_{C}^{2}$, where $1$ and $2$ label the condensation channel on the reference
boundary (the channel is unique for $C$ on the right). They form an
irreducible representation under the $S_{3}$ symmetry, so the remaining global
symmetry is trivial. There are three ground states corresponding to the
trivial local operators ${\cal O}_{A}$ and the other two nontrivial local
operators ${\cal O}_{C}^{1},{\cal O}_{C}^{2}$. The three ground states form a
reducible representation, decomposing into the identity rep and the
2-dimensional rep.

If we choose the symmetry boundary to be the flux condensed boundary
$\mathcal{L}_{2}=A\oplus D\oplus F$, the topological lines generate the
$\text{Rep}(S_{3})$ symmetry, which come from the charges $A,B,C$ in the bulk.
The fusion rule of the $\text{Rep}(S_{3})$ symmetry is given by

$\displaystyle X_{B}\otimes X_{B}$ $\displaystyle=X_{A},$ $\displaystyle
X_{B}\otimes X_{C}$ $\displaystyle=X_{C}\otimes X_{B}=X_{C},$ $\displaystyle
X_{C}\otimes X_{C}$ $\displaystyle=X_{A}\oplus X_{B}\oplus X_{C}.$ (99)

The mapping from the boundary lines to the bulk anyon lines is given by

$\displaystyle X_{A}$ $\displaystyle\rightarrow A\oplus D\oplus F,$
$\displaystyle X_{B}$ $\displaystyle\rightarrow B\oplus E\oplus F,$
$\displaystyle X_{C}$ $\displaystyle\rightarrow C\oplus D\oplus E\oplus
G\oplus H.$ (100)

There are also four possible gapped phases by choosing the physical boundary
to be one of the four gapped boundary of the $S_{3}$ gauge theory, which is
studied in Ref. [81, 82]. For the sake of completeness, we summarize the
result below:

  1. 1.

$\mathcal{L}_{1}=A\oplus B\oplus 2C$: $\text{Rep}(S_{3})$ symmetric phase. In
this case, the $\text{Rep}(S_{3})$ symmetry is preserved and there is a unique
ground state.

  2. 2.

$\mathcal{L}_{2}=A\oplus D\oplus F$: $\text{Rep}(S_{3})$ SSB phase. There are
three ground states, corresponding to the local operators: ${\cal O}_{A},{\cal
O}_{D},{\cal O}_{F}$. Note that the local operators are formed by the pure
fluxes, which are labeled by conjugacy classes $[g]$. The symmetry actions of
the $\text{Rep}(S_{3})$ symmetry on these local operators is given by the
characters:

$U_{\rho}\circ{\cal O}_{[g]}=\chi_{\rho}(g){\cal O}_{[g]}.$ (101)

  3. 3.

$\mathcal{L}_{3}=A\oplus B\oplus 2F$: $\text{Rep}(S_{3})/\mathbb{Z}_{2}$ SSB
phase. There are three ground states, corresponding to the local operators:
${\cal O}_{A},{\cal O}_{F}^{1},{\cal O}_{F}^{2}$, where ${\cal O}_{F}^{1}$ and
${\cal O}_{F}^{2}$ come from the two possible condensation channel on the
physical boundary. The symmetry actions on the local operators are given by

$\displaystyle U_{B}$ $\displaystyle\circ{\cal O}_{A}={\cal O}_{A},\quad
U_{B}\circ{\cal O}_{F}^{i}={\cal O}_{F}^{i},$ $\displaystyle U_{C}$
$\displaystyle\circ{\cal O}_{A}=2{\cal O}_{A},\quad U_{C}\circ{\cal
O}_{F}^{i}=-{\cal O}_{F}^{i},$ (102)

where $i=1,2$. This implies that non-invertible symmetry $X_{C}$ is broken
spontaneously but the $\mathbb{Z}_{2}$ symmetry is unbroken.

  4. 4.

$\mathcal{L}_{4}=A\oplus C\oplus D$: $\mathbb{Z}_{2}$ SSB phase. The local
operators are given by ${\cal O}_{A},{\cal O}_{D}$ so that there are two
ground states. The local operator ${\cal O}_{D}$ transformed non-trivially
under both $X_{B}$ and $X_{C}$:

$U_{B}\circ{\cal O}_{D}=-{\cal O}_{D},\quad U_{C}\circ{\cal O}_{D}=0,$ (103)

which implies that the $\mathbb{Z}_{2}$ symmetry is broken spontaneously.

###  VIII.1 Fermionization

To perform the fermionization, we need to understand the fermionic gapped
boundaries for the $S_{3}$ gauge theory. There are two fermionic gapped
boundaries [61]:

$\displaystyle\mathcal{L}_{1}^{f}$ $\displaystyle=A\oplus E\oplus F,$ (104)
$\displaystyle\mathcal{L}_{2}^{f}$ $\displaystyle=A\oplus C\oplus E,$ (105)

where the two fermionic boundaries are related by the automorphism
$C\leftrightarrow F$.

We would like to fermionize the $\text{Rep}(S_{3})$ symmetry with respecting
to the $\mathbb{Z}_{2}$ symmetry. To do so, we need the following
$\mathbb{Z}_{2}$ twisted partition functions for systems with the
$\text{Rep}(S_{3})$ symmetry:

$\displaystyle Z_{0,0}=Z_{A}+Z_{D}+Z_{F},$ $\displaystyle
Z_{0,1}=Z_{A}-Z_{D}+Z_{F},$ $\displaystyle Z_{1,0}=Z_{B}+Z_{E}+Z_{F},$
$\displaystyle Z_{1,1}=Z_{B}-Z_{E}+Z_{F}.$ (106)

Applying the fermoinization formula Eq. (11), we find

$Z_{\mathcal{F}}[\text{A},\text{A}]=Z_{A}+Z_{E}+Z_{F},$ (107)

which means that the fermionization procedure is implemented by changing the
symmetry boundary from $\mathcal{L}_{2}=A\oplus D\oplus F$ to
$\mathcal{L}_{1}^{f}=A\oplus E\oplus F$.

The symmetry is generated by the line $X_{A},X_{B_{f}},X_{C}$, which satisfy
the same fusion rule as the $\text{Rep}(S_{3})$ symmetry [61]:

$\displaystyle X_{B_{f}}\otimes X_{B_{f}}$ $\displaystyle=X_{A},$
$\displaystyle X_{B_{f}}\otimes X_{C}$ $\displaystyle=X_{C}\otimes
X_{B_{f}}=X_{C},$ $\displaystyle X_{C}\otimes X_{C}$
$\displaystyle=X_{A}\oplus X_{B_{f}}\oplus X_{C}.$ (108)

Following Ref. [79], we refer to this super fusion category symmetry as
$\text{SRep}(\mathcal{H}_{S_{3}})$999This super fusion category is equivalent
to $\text{Rep}(S_{3})\boxtimes\text{SVec}$. Ref. [79] shows that in general
fermionizing a $\text{Rep}(G)$ symmetry results in a super fusion category
that is equivalent to $\text{Rep}(G)\boxtimes\text{SVec}$..

The fermion parity $\mathbb{Z}_{2}^{F}$ symmetry is generated by $X_{B_{f}}$.
The mapping from the boundary lines to the bulk anyon lines is given by

$\displaystyle X_{A}$ $\displaystyle\rightarrow A\oplus E\oplus F,$
$\displaystyle X_{B_{f}}$ $\displaystyle\rightarrow B\oplus D\oplus F,$
$\displaystyle X_{C}$ $\displaystyle\rightarrow C\oplus D\oplus E\oplus
G\oplus H.$ (109)

We will also choose their bulk representatives of $X_{A}$, $X_{B_{f}}$, and
$X_{C}$ to be $A$, $B$, and $C$ respectively. Lattice models with
$\text{SRep}(\mathcal{H}_{S_{3}})$ symmetry has been studied recently in Ref.
[58].

###  VIII.2 Gapped phases

Here we discuss the possible gapped phases in fermionic chians with the super
fusion category $\text{SRep}(\mathcal{H}_{S_{3}})$ symmetry, which are
realized by choosing the physical boundary to be $\mathcal{L}_{i}$ for
$i=1,2,3,4$.

  1. 1.

$\mathcal{L}_{1}=A\oplus B\oplus 2C$: $\text{SRep}(\mathcal{H}_{S_{3}})$
symmetric phase. In this case, the $\text{SRep}(\mathcal{H}_{S_{3}})$ symmetry
is preserved and there is a unique ground state.

  2. 2.

$\mathcal{L}_{2}=A\oplus D\oplus F$:
$\text{SRep}(\mathcal{H}_{S_{3}})/\mathbb{Z}_{2}$ SSB + Kitaev phase. There
are two ground states, corresponding to the local operators: ${\cal O}_{A}$
and ${\cal O}_{F}$. The symmetry actions on the local operators are given by

$\displaystyle U_{B}$ $\displaystyle\circ{\cal O}_{A}={\cal O}_{A},\quad
U_{B}\circ{\cal O}_{F}={\cal O}_{F},$ $\displaystyle U_{C}$
$\displaystyle\circ{\cal O}_{A}=2{\cal O}_{A},\quad U_{C}\circ{\cal
O}_{F}=-{\cal O}_{F}.$ (110)

This implies that non-invertible symmetry $X_{C}$ is broken spontaneously but
the $\mathbb{Z}_{2}^{F}$ symmetry is unbroken. In particular, the system is in
the Kitaev phase. This can be seen by calculating the partition function in
the $(\text{P},\text{P})$ spin structure:

$Z[\text{P},\text{P}]=-Z_{D}-Z_{F}=-2,$ (111)

where the factor “2” represents the 2-fold ground state degeneracy from the
non-invertible symmetry breaking and the minus sign signals the Kitaev phase.

  3. 3.

$\mathcal{L}_{3}=A\oplus B\oplus 2F$:
$\text{SRep}(\mathcal{H}_{S_{3}})/\mathbb{Z}_{2}$ SSB phase. There are three
ground states, corresponding to the local operators: ${\cal O}_{A},{\cal
O}_{F}^{1},{\cal O}_{F}^{2}$, where ${\cal O}_{F}^{1}$ and ${\cal O}_{F}^{2}$
come from the two possible condensation channel on the physical boundary. The
symmetry actions on the local operators are given by

$\displaystyle U_{B}$ $\displaystyle\circ{\cal O}_{A}={\cal O}_{A},\quad
U_{B}\circ{\cal O}_{F}^{i}={\cal O}_{F}^{i},$ $\displaystyle U_{C}$
$\displaystyle\circ{\cal O}_{A}=2{\cal O}_{A},\quad U_{C}\circ{\cal
O}_{F}^{i}=-{\cal O}_{F}^{i},$ (112)

where $i=1,2$. This implies that non-invertible symmetry $X_{C}$ is broken
spontaneously but the $\mathbb{Z}_{2}^{F}$ symmetry is unbroken.

  4. 4.

$\mathcal{L}_{4}=A\oplus C\oplus D$: $\text{SRep}(\mathcal{H}_{S_{3}})$
symmetric Kitaev phase. The $\text{SRep}(S_{3})$ symmetry is preserved and
there is a unique ground state. However, the system is in the Kitaev phase
since the partition function in the $(\text{P},\text{P})$ spin structure is
non-trivial:

$Z[\text{P},\text{P}]=-Z_{D}=-1,$ (113)

and the partition functions in the other spin structures are all equal to one.

###  VIII.3 Quantum critical points

Various quantum critical points between these fermionic gapped phases in the
fermionic systems with $\text{SRep}(\mathcal{H}_{S_{3}})$ symmetry can be
obtained by topological holography. We will begin with a review of the three
boundary critical theories in Sec VIII.3.1, which serve as our inputs to study
the fermionic quantum critical points. By choosing the symmetry boundary to be
appropriate bosonic gapped boundaries, we review how the sandwich construction
produces the CFT descriptions of the phase transitions for the bosonic systems
with $S_{3}$ and $\text{Rep}(S_{3})$ symmetry. We will proceed to discuss the
fermionic critical theories in Sec. VIII.3.2 \- VIII.3.6. The final result is
summarized in Fig. 7.

Figure 7: Summary of the phase transitions in the fermionic systems with
$\text{SRep}(\mathcal{H}_{S_{3}})$ symmetry. The red arrows indicate the
required input boundary critical theories on the physical boundary of the
sandwich discussed in Sec. VIII.3.1.

####  VIII.3.1 Input critical theories

#####  $S_{3}$ symmetric $\leftrightarrow$ $S_{3}$ SSB

The first input we need is the sandwich picture for the critical point between
the $S_{3}$ symmetric and $S_{3}$ SSB phases. In the sandwich picture, we tune
the physical boundary to be sitting at the critical point between the gapped
boundary $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ and the symmetry boundary is
the charge condensed boundary $\mathcal{L}_{1}$. We denote the boundary
critical theory as $\mathcal{P}_{12}$. It has been shown in Ref. [67] that one
of the possible CFT description for the boundary critical theory
$\mathcal{P}_{12}$ is a CFT with central charge $c=4/5$. It has following
multi-component partition functions in the anyon basis of the $S_{3}$ gauge
theory written in terms of the characters of the $(6,5)$ minimal model:

$\displaystyle Z_{A}^{(12)}$
$\displaystyle=|\chi_{0}|^{2}+|\chi_{3}|^{2}+|\chi_{\frac{2}{5}}|^{2}+|\chi_{\frac{7}{5}}|^{2},$
$\displaystyle Z_{B}^{(12)}$
$\displaystyle=\chi_{0}\bar{\chi}_{3}+\chi_{3}\bar{\chi}_{0}+\chi_{\frac{2}{5}}\bar{\chi}_{\frac{7}{5}}+\chi_{\frac{7}{5}}\bar{\chi}_{\frac{2}{5}},$
$\displaystyle Z_{C}^{(12)}$
$\displaystyle=|\chi_{\frac{2}{3}}|^{2}+|\chi_{\frac{1}{15}}|^{2},$
$\displaystyle Z_{D}^{(12)}$
$\displaystyle=|\chi_{\frac{1}{8}}|^{2}+|\chi_{\frac{13}{8}}|^{2}+|\chi_{\frac{1}{40}}|^{2}+|\chi_{\frac{21}{40}}|^{2},$
$\displaystyle Z_{E}^{(12)}$
$\displaystyle=\chi_{\frac{1}{8}}\bar{\chi}_{\frac{13}{8}}+\chi_{\frac{13}{8}}\bar{\chi}_{\frac{1}{8}}+\chi_{\frac{1}{40}}\bar{\chi}_{\frac{21}{40}}+\chi_{\frac{21}{40}}\bar{\chi}_{\frac{1}{40}},$
$\displaystyle Z_{F}^{(12)}$
$\displaystyle=|\chi_{\frac{2}{3}}|^{2}+|\chi_{\frac{1}{15}}|^{2},$
$\displaystyle Z_{G}^{(12)}$
$\displaystyle=\chi_{0}\bar{\chi}_{\frac{2}{3}}+\chi_{3}\bar{\chi}_{\frac{2}{3}}+\chi_{\frac{2}{5}}\bar{\chi}_{\frac{1}{15}}+\chi_{\frac{7}{5}}\bar{\chi}_{\frac{1}{15}},$
$\displaystyle Z_{H}^{(12)}$
$\displaystyle=\chi_{\frac{2}{3}}\bar{\chi}_{0}+\chi_{\frac{2}{3}}\bar{\chi}_{3}+\chi_{\frac{1}{15}}\bar{\chi}_{\frac{2}{5}}+\chi_{\frac{1}{15}}\bar{\chi}_{\frac{7}{5}}.$
(114)

Since the symmetry boundary is given by $\mathcal{L}_{1}=A\oplus B\oplus 2C$,
we have the partition function

$\displaystyle Z$ $\displaystyle=Z_{A}^{(12)}+Z_{B}^{(12)}+2Z_{C}^{(12)}$
$\displaystyle=|\chi_{0}+\chi_{3}|^{2}+|\chi_{\frac{2}{5}}+\chi_{\frac{7}{5}}|^{2}+2|\chi_{\frac{2}{3}}|^{2}+2|\chi_{\frac{1}{15}}|^{2},$
(115)

which is the of the partition function of the 3-state Potts model.

We note that if we change the symmetry boundary to be $\mathcal{L}_{2}=A\oplus
D\oplus F$, the physical boundary $\mathcal{L}_{1}$ then corresponds to the
$\text{Rep}(S_{3})$ symmetric phase and $\mathcal{L}_{2}$ corresponds to the
$\text{Rep}(S_{3})$ SSB phase. Now we tune the physical boundary to be
described by the theory $\mathcal{P}_{12}$ Eq. (114), the transition between
the $\text{Rep}(S_{3})$ symmetric and the $\text{Rep}(S_{3})$ SSB phases is
given by

$\displaystyle Z$ $\displaystyle=Z_{A}^{(12)}+Z_{D}^{(12)}+Z_{F}^{(12)}$
$\displaystyle=|\chi_{0}|^{2}+|\chi_{3}|^{2}+|\chi_{\frac{2}{5}}|^{2}+|\chi_{\frac{7}{5}}|^{2}+|\chi_{\frac{1}{8}}|^{2}+|\chi_{\frac{13}{8}}|^{2}+|\chi_{\frac{1}{40}}|^{2}+|\chi_{\frac{21}{40}}|^{2}+|\chi_{\frac{2}{3}}|^{2}+|\chi_{\frac{1}{15}}|^{2},$
(116)

which is the partition function of the tetracritical Ising model.

#####  $Z_{2}$ SSB $\leftrightarrow$ $S_{3}$ SSB

The second input is the sandwich picture for the transition between the
$Z_{2}$ SSB and $S_{3}$ SSB phases. The physical boundary is at the transition
between $\mathcal{L}_{1}$ and $\mathcal{L}_{3}$. We denote the boundary
critical theory as $\mathcal{P}_{13}$. Ref. [67] found that it can be
described by the following multi-component partition functions:

$\displaystyle Z_{A}^{(13)}$
$\displaystyle=|\chi_{0}+\chi_{3}|^{2}+|\chi_{\frac{2}{5}}+\chi_{\frac{7}{5}}|^{2},$
$\displaystyle Z_{B}^{(13)}$
$\displaystyle=|\chi_{0}+\chi_{3}|^{2}+|\chi_{\frac{2}{5}}+\chi_{\frac{7}{5}}|^{2},$
$\displaystyle Z_{C}^{(13)}$
$\displaystyle=2(|\chi_{\frac{2}{3}}|^{2}+|\chi_{\frac{1}{15}}|^{2}),$
$\displaystyle Z_{D}^{(13)}$ $\displaystyle=0,$ $\displaystyle Z_{E}^{(13)}$
$\displaystyle=0,$ $\displaystyle Z_{F}^{(13)}$
$\displaystyle=2(|\chi_{\frac{2}{3}}|^{2}+|\chi_{\frac{1}{15}}|^{2}),$
$\displaystyle Z_{G}^{(13)}$
$\displaystyle=\chi_{0}\bar{\chi}_{\frac{2}{3}}+\chi_{3}\bar{\chi}_{\frac{2}{3}}+\chi_{\frac{2}{5}}\bar{\chi}_{\frac{1}{15}}+\chi_{\frac{7}{5}}\bar{\chi}_{\frac{1}{15}},$
$\displaystyle Z_{H}^{(13)}$
$\displaystyle=\chi_{\frac{2}{3}}\bar{\chi}_{0}+\chi_{\frac{2}{3}}\bar{\chi}_{3}+\chi_{\frac{1}{15}}\bar{\chi}_{\frac{2}{5}}+\chi_{\frac{1}{15}}\bar{\chi}_{\frac{7}{5}}.$
(117)

The partition function for the $Z_{2}$ SSB $\leftrightarrow$ $S_{3}$ SSB
transition is then given by

$\displaystyle Z$ $\displaystyle=Z_{A}^{(13)}+Z_{B}^{(13)}+2Z_{C}^{(13)}$
$\displaystyle=2(|\chi_{0}+\chi_{3}|^{2}+|\chi_{\frac{2}{5}}+\chi_{\frac{7}{5}}|^{2}+2|\chi_{\frac{2}{3}}|^{2}+2|\chi_{\frac{1}{15}}|^{2}),$
(118)

where the prefactor “2” in front of the parenthesis corresponds the 2-fold
ground state degeneracy since the $\mathbb{Z}_{2}$ symmetry has been broken
spontaneously. The transition is given by 2 copies of the 3-state Potts model.

We note that we can also obtained the transition between the
$\text{Rep}(S_{3})$ symmetric and $\text{Rep}(S_{3})/\mathbb{Z}_{2}$ SSB
phases, at which the partition function is given by summing over the pure
fluxes $Z=Z_{A}^{(13)}+Z_{D}^{(13)}+Z_{F}^{(13)}$. One can check that it
equals to the partition function of the 3-state Potts model.

#####  $Z_{3}$ SSB $\leftrightarrow$ $S_{3}$ SSB

The third input is the transition between the $Z_{3}$ SSB and $S_{3}$ SSB
phases. The physical boundary is tuned to be sitting at the transition between
$\mathcal{L}_{1}$ and $\mathcal{L}_{4}$. We denote the boundary critical
theory as $\mathcal{P}_{14}$. Ref. [67] found that it can be described by the
following multi-component partition functions in terms of the characters of
the Ising CFT:

$\displaystyle Z_{A}^{(14)}$ $\displaystyle=|\chi_{1}|^{2}+|\chi_{\psi}|^{2},$
$\displaystyle Z_{B}^{(14)}$ $\displaystyle=|\chi_{\sigma}|^{2},$
$\displaystyle Z_{C}^{(14)}$
$\displaystyle=|\chi_{1}|^{2}+|\chi_{\sigma}|^{2}+|\chi_{\psi}|^{2},$
$\displaystyle Z_{D}^{(14)}$ $\displaystyle=|\chi_{\sigma}|^{2},$
$\displaystyle Z_{E}^{(14)}$
$\displaystyle=\chi_{1}\bar{\chi}_{\psi}+\chi_{\psi}\bar{\chi}_{1},$
$\displaystyle Z_{F}^{(14)}$ $\displaystyle=0,$ $\displaystyle Z_{G}^{(14)}$
$\displaystyle=0,$ $\displaystyle Z_{H}^{(14)}$ $\displaystyle=0.$ (119)

The partition function for the $Z_{3}$ SSB $\leftrightarrow$ $S_{3}$ SSB
transition is given by

$\displaystyle Z$ $\displaystyle=Z_{A}^{(14)}+Z_{B}^{(14)}+2Z_{C}^{(14)}$
$\displaystyle=3(|\chi_{1}|^{2}+|\chi_{\sigma}|^{2}+|\chi_{\psi}|^{2}),$ (120)

which is three copies of the Ising CFT as expected since the $\mathbb{Z}_{3}$
symmetry has been broken spontaneously.

We can also obtained the transition between the $\text{Rep}(S_{3})$ symmetric
and $\mathbb{Z}_{2}$ SSB phases. The partition function is given by summing
over the pure fluxes $Z=Z_{A}^{(14)}+Z_{D}^{(14)}+Z_{F}^{(14)}$, which is just
a single copy of the Ising CFT.

####  VIII.3.2 $\text{SRep}(\mathcal{H}_{S_{3}})$ Trivial $\leftrightarrow$
($\text{SRep}(\mathcal{H}_{S_{3}})/\mathbb{Z}_{2}$ SSB + Kitaev) = fermionic
minimal model $(m=5)$