# Symmetric Mass Generation

Juven Wang  Center of Mathematical Sciences and Applications, Harvard
University, MA 02138, USA<EMAIL_ADDRESS>Yi-Zhuang You  Department
of Physics, University of California, San Diego, CA 92093, USA<EMAIL_ADDRESS>
###### Abstract

The most well-known mechanism for fermions to acquire a mass is the Nambu-
Goldstone-Anderson-Higgs mechanism, i.e. after a spontaneous symmetry
breaking, a bosonic field that couples to the fermion mass term condenses,
which grants a mass gap for the fermionic excitation. In the last few years,
it was gradually understood that there is a new mechanism of mass generation
for fermions without involving any symmetry breaking within an anomaly-free
symmetry group. This new mechanism is generally referred to as the “Symmetric
Mass Generation (SMG).” It is realized that the SMG has deep connections with
interacting topological insulator/superconductors, symmetry-protected
topological states, perturbative local and non-perturbative global anomaly
cancellations, and deconfined quantum criticality. It has strong implications
for the lattice regularization of chiral gauge theories. This article defines
the SMG, summarizes current numerical results, introduces a novel unifying
theoretical framework (including the parton-Higgs and the s-confinement
mechanisms, as well as the symmetry-extension construction), and overviews
various features and applications of SMG.

###### Contents

  1. I Introduction
  2. II Example Models
    1. II.1 (0+1)D SMG: Fidkowski-Kitaev Majorana Fermion Model
    2. II.2 (1+1)D SMG: 3-4-5-0 Chiral Fermion Model
      1. II.2.1 Proof on the equivalence between the anomaly-free and SMG gapping conditions
    3. II.3 (2+1)D SMG: Honeycomb Lattice Model
    4. II.4 (3+1)D SMG: Chiral Fermion Model
    5. II.5 SMG in General Dimensions
  3. III Numerical Investigations
    1. III.1 Existence of SMG Phases
    2. III.2 Nature of the SMG Transitions
  4. IV Theoretical Understandings
    1. IV.1 Fluctuating Bilinear Mass Picture
    2. IV.2 Fermion Fractionalization Field Theory
      1. IV.2.1 Parton-Higgs mechanism
      2. IV.2.2 s-confinement mechanism
    3. IV.3 Examples of Fermion Fractionalization
      1. IV.3.1 (2+1)D Honeycomb Lattice Model
      2. IV.3.2 (3+1)D Chiral Fermion Model
    4. IV.4 Symmetry Extension Construction
      1. IV.4.1 Symmetry Breaking vs Symmetry Extension vs SMG
      2. IV.4.2 More on Symmetry Extension Construction
  5. V Features and Applications
    1. V.1 Green’s Function Zeros
    2. V.2 Deconfined Quantum Criticality
    3. V.3 Symmetric Mass Generation of the Standard Model
      1. V.3.1 Symmetry Extension of the 15$N_{f}$\- or 16$N_{f}$-Weyl-Fermion Standard Model
      2. V.3.2 Symmetric Mass Generation in a 27$N_{f}$-Weyl-Fermion Left-Right Model
  6. VI Summary and Outlook

##  I Introduction

Global symmetry is a central concept in quantum field theories (QFTs). One of
the most immediate implications of symmetry is that it restricts the terms
that can appear in a field theory action in the path integral formulation, as
the partition function ${Z}$ must remain invariant under the symmetry
transformation — the partition function invariance however can be up to an
invertible complex phase factor ${Z}\to\mathrm{e}^{\mathrm{i}\alpha}{Z}$ known
as a quantum anomaly [1, 2]. The quantum anomaly associated with the global
symmetry is also known as the ’t Hooft anomaly [3], which has profound
consequences. The invertible phase $\mathrm{e}^{\mathrm{i}\alpha}$ that
detects the anomaly also specifies a cobordism class of the partition function
of one-higher-dimensional invertible topological quantum field theory
(invertible TQFT) [4] via the anomaly inflow [5, 6].

The anomaly provides a concise and powerful organization principle to classify
quantum field theories, and to dictate the influence of the ultraviolet (UV)
kinematics on the infrared (IR) dynamics of a field theory, because the
anomaly does not change under symmetric deformations of the theory, including
the renormalization group (RG) flow. The preservation of the anomaly index
from UV to IR is known as the anomaly matching. If a quantum field theory with
a global symmetry $G$ has a non-vanishing ’t Hooft anomaly in $G$, its IR
dynamics cannot be trivially gapped while preserving $G$. As a consequence,
either the symmetry $G$ is spontaneously broken in IR [7, 8, 9, 10, 11], or
the anomaly must be saturated by anomalous _gapless_ quantum criticality
(e.g., the IR limit of the system corresponds to massless free theories or
interacting conformal field theories [CFT]) or _gapped_ topological order
(e.g., the IR limit of the system is described by a topological quantum field
theory [TQFT]) [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24].

Many examples of anomalies involve massless fermions, such as the chiral
fermion Adler-Bell-Jackiw anomaly [25, 26]. The anomaly causes an obstruction
to open a gap in the fermion spectrum (i.e. an obstruction to make the fermion
correlation length finite) without breaking the symmetry. The symmetry-
breaking mechanism to generate a mass gap is known as Nambu-Goldstone-
Anderson-Higgs mechanism [27, 28, 29, 30, 31, 32, 33]. Here Nambu-Goldstone
refers to the spontaneous symmetry breaking, while Anderson-Higgs refers to
giving a mass by the elementary or composite boson condensation. However,
there are instances where the anomaly vanishes for a collection of massless
fermions, yet the symmetry is still restrictive enough to forbid any fermion
bilinear mass term. In this case, although there is no obstruction towards
gapping the fermions symmetrically, the mechanism to achieve the symmetric
gapped state must go beyond the free-fermion (perturbatively free or weak-
coupling) approach, which potentially leads to a non-perturbative strong-
coupling approach in order to generate a finite excitation gap in the fermion
many-body spectrum by non-trivial interaction effects. The _strong-coupling_
[34] here refers to the coupling in the continuum field theory being non-
perturbative, or the interaction energy being of the same order as the kinetic
energy on the lattice scale (which may also be called the _intermediate_
-strength interaction on a lattice). This phenomenon of gapping out massless
fermions by interactions in an anomaly-free system without breaking the
anomaly-free symmetry is now called the _symmetric mass generation_ (SMG) [35,
34].

The idea of gapping out massless fermions by interaction dates back to Eichten
and Preskill [36] in an attempt to regularize chiral fermions on the lattice.
The understanding of SMG is significantly deepened over the past few years,
following the development in condensed matter theory regarding symmetry-
protected topological (SPT) states [37, 38, 39]. SPT states are short-range-
entangled quantum many-body states respecting certain global symmetry $G$. The
bulk of a SPT state is featureless (i.e. gapped, symmetric and non-
degenerated). A non-trivial SPT state is most explicitly characterized by its
non-trivial boundary features, which are endowed by the non-vanishing ’t Hooft
anomaly of $G$ (or mixed $G$ symmetry gauge-gravity anomaly in broader cases)
in the boundary effective theory of the SPT state. The one-to-one
correspondence between the bulk of a SPT state and its boundary anomaly [40,
41, 42, 43, 44] provides the basis to classify distinct SPT states by their
distinct boundary anomalies.

The connection between SPT state and anomaly relates SMG to another topic: the
interaction-reduced classification of fermionic SPT states. Fermionic SPT
states [45, 46, 47, 48, 4, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58] are SPT
states of fermionic systems with (at least) fermion parity symmetry
$\mathbb{Z}_{2}^{F}$ (or more generally, invertible topological phases of
fermions [4, 57]). In the free-fermion limit [59, 60, 61, 62, 63], non-trivial
fermionic SPT states are characterized by the symmetry-protected gapless
fermion boundary modes, if the symmetry forbids any fermion bilinear mass on
the boundary. However, the free-fermion analysis does not rule out the
possibility of gapping out the boundary fermions by interaction though the SMG
mechanism. Such situation can indeed happen when the boundary fermions are
actually anomaly free, such that the bulk state should be classified as a
trivial gapped state under symmetric interactions, even though it looks like a
non-trivial SPT state in the non-interacting limit. This leads to the
interaction-reduced classification of fermionic SPT states in the bulk, which
is closely related to the SMG for gapless fermions on the boundary.

The first example of interaction-reduced classification was provided by
Fidkowski and Kitaev [64, 65] in (1+1)D fermion systems, where an explicit
interaction was proposed to drive the SMG among fermion zero modes on the
(0+1)D boundary. This motivated a sequence of works generalizing the
discussion to (2+1)D [66, 67, 68, 69], (3+1)D [15, 16, 17, 70, 71], and higher
dimensions [72, 73, 74]. Studies along this direction reveals families of
interactions that could potentially drive the SMG in different dimensions,
paving ways for numerical verifications in concrete lattice models. It is also
realized that the interaction must be carefully designed to drive the SMG:
some symmetric interactions are helpful towards this goal, while other
symmetric interactions are not [75].

Currently most known examples of interaction-reduced classifications all
correspond to cancellations of _non-perturbative global anomalies_. However,
there are also cases of SMG that involve cancellations of _perturbative local
anomalies_. Important examples of such are the chiral fermions [76, 77, 78,
79] in even dimensional spacetime. Regularizing chiral fermions on a lattice
is a long-standing problem in the lattice gauge theory. The Nielsen-Ninomiya
no-go theorem [80, 81, 82] states that it is not possible to gap the fermion
doubler in a non-interacting local lattice model without breaking the chiral
symmetry. One possibility to circumvent the no-go theorem is to consider
interaction effects [36], which introduces fermion interaction to gap out the
fermion doubler (the mirror fermion) in local lattice fermion models, leaving
the normal fermion (the light fermion) untouched. However, the early attempts
[83, 84, 85, 86, 87, 88, 89, 90, 91, 79, 92] were not successful, either
because certain anomalies were not carefully cancelled or because the
appropriate gapping interaction has not been found. With a deeper
understanding of the SMG mechanism, the problem was revisited by Wen [93] for
$\mathrm{Spin}(10)$ chiral fermions in (3+1)D. The idea is further developed
by subsequent works in the same dimension [94, 95, 96, 97, 98] as well as in
lower-dimensional analogs [75, 99, 100, 101]. More recent numerical works have
successfully shown that the SMG indeed provides a feasible solution to
regularize chiral fermions [102, 103, 104, 105].

The article is organized as follows. We will start by introducing some
selective representative models of SMG (one in each spacetime dimension, in
(0+1)D, (1+1)D, (2+1)D and (3+1)D respectively) in Sec. II. We conclude with
Sec. II.5 by providing a general definition of SMG in all dimensions. We then
review the numerical efforts in Sec. III, which is mainly focused on two
tasks: (i) to establish the existence of SMG phases in Sec. III.1 and (ii) to
investigate the nature of SMG transitions in Sec. III.2. Based on these
backgrounds, we then summarize the recent theoretical progress in Sec. IV,
which aims to (i) understand the SMG phase by the fluctuating bilinear mass
picture in Sec. IV.1 and (ii) describe the SMG transition by fermion
fractionalization field theory in Sec. IV.2. In particular, we unify two
currently existing SMG mechanisms (namely the parton-Higgs mechanism and the
s-confinement mechanism) under the same theoretical framework of fermion
fractionalization. We also make connection to further understanding of SMG
based on the symmetry extension construction [23, 24] in Sec. IV.4. In Sec. V,
we discuss other aspects of SMG including Green’s function zeros Sec. V.1, the
deconfined quantum criticality Sec. V.2, and the Standard Model regularization
Sec. V.3. Finally, we summarize our review in Sec. VI.

##  II Example Models

###  II.1 (0+1)D SMG: Fidkowski-Kitaev Majorana Fermion Model

The simplest example of SMG happens in (0+1)D spacetime among a collection of
Majorana fermion zero modes, as first shown by Fidkowski and Kitaev [64, 65].
The model concerns a system of eight Majorana fermion modes, described by the
Majorana fermion operators $\chi_{a}$ ($a=1,2,\cdots,8$) satisfying
$\\{\chi_{a},\chi_{b}\\}=2\delta_{ab}$. Consider an anti-unitary (time-
reversal) symmetry
$\mathbb{Z}_{2}^{T}:\chi_{a}\to\chi_{a},\mathrm{i}\to-\mathrm{i}$ and the
fermion parity symmetry $\mathbb{Z}_{2}^{F}:\chi_{a}\to-\chi_{a}$. Without
involving interactions, the Hamiltonian must take a fermion bilinear form to
preserve the fermion parity symmetry $\mathbb{Z}_{2}^{F}$. But any Majorana
fermion bilinear term $\mathrm{i}\chi_{a}\chi_{b}$ (one needs $\mathrm{i}$ to
keep the operator Hermitian) will break the time-reversal symmetry
$\mathbb{Z}_{2}^{T}$, so the free-fermion Hamiltonian has to vanish, e.g.
$H=0$, under both symmetry requirements, and these Majorana modes cannot be
gapped on the free-fermion level. Eight Majorana fermion modes form four
qubits, hence the system has a $2^{4}=16$ fold degeneracy, and this degeneracy
is protected by the $\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ symmetry.

However, it is possible to create a many-body excitation gap by fermion
interaction, leaving a unique ground state of this (0+1)D system. As
introduced by Fidkowski and Kitaev [64], the following four-fermion
interaction suffices to gap out all eight Majorana fermion modes without
breaking the time-reversal symmetry $\mathbb{Z}_{2}^{T}$

$H_{\text{FK}}=-\sum_{a<b<c<d}V_{abcd}\chi_{a}\chi_{b}\chi_{c}\chi_{d}.$ (1)

The coefficient is specified by
$V_{abcd}=\langle\mathsf{4e}|\chi_{a}\chi_{b}\chi_{c}\chi_{d}|\mathsf{4e}\rangle$,
where $|\mathsf{4e}\rangle=(|0000\rangle+|1111\rangle)/\sqrt{2}$ is a many-
body reference state written in the Fock state basis
$|n_{1}n_{2}n_{3}n_{4}\rangle$, labeled by the fermion occupation numbers
$n_{i}=(1+\mathrm{i}\chi_{2i-1}\chi_{2i})/2$ (for $i=1,2,3,4$). Here the
fermion number operator $n_{i}=c_{i}^{\dagger}c_{i}$ can also be expressed in
terms of the complex fermion annihilation operator
$c_{i}=(\chi_{2i-1}+\mathrm{i}\chi_{2i})/2$, which are constructed by pairing
up the Majorana operators. The reference state $|\mathsf{4e}\rangle$ describes
the quartet condensation of the complex fermions $c_{i}$ (where four fermions
are created or annihilated together in the quantum superposition), which is
also known as the charge-4e superconducting state [106, 107, 108, 109, 110,
111, 112] in condensed matter physics. It turns out that the state
$|\mathsf{4e}\rangle$ respects the $\mathbb{Z}_{2}^{T}$ symmetry, and is the
unique ground state of $H_{\text{FK}}$ with a finite excitation gap of $14$
(energy units).

The fact that the system has _eight_ Majorana zero modes is crucial for the
SMG to occur. In contrast, if there are only four Majorana zero modes, the
only Hamiltonian allowed by symmetry is

$H=-g\chi_{1}\chi_{2}\chi_{3}\chi_{4}=-gP_{1}^{F}P_{2}^{F},$ (2)

where each pair of Majorana fermion operators defines a fermion parity
operator $P_{i}^{F}=\mathrm{i}\chi_{2i-1}\chi_{i}=2n_{i}-1$ (associated with
the $i$th complex fermion mode). This Hamiltonian always has a two-fold ground
state degeneracy, regardless of the sign of $g$. When $g<0$, the ground states
$|P_{1}^{F}=P_{2}^{F}=\pm 1\rangle$ are bosonic (as the total fermion parity
$P_{1}^{F}P_{2}^{F}=+1$ is even). When $g>0$, the ground states
$|P_{1}^{F}=-P_{2}^{F}=\pm 1\rangle$ are fermionic (as
$P_{1}^{F}P_{2}^{F}=-1$). In either cases, the two-fold degenerated ground
states transform into each other under $\mathbb{Z}_{2}^{T}$, and form a
_Kramers doublet_ [113]. The ground state degeneracy implies that the
$\mathbb{Z}_{2}^{T}$ symmetry is spontaneously broken. Hence even with
interactions, _four_ Majorana zero modes still _cannot_ be symmetrically
gapped. However, by doubling to eight Majorana fermions, it is then possible
to couple the two Kramers doublets (originated from the first- and the last-
four Majorana fermions under interaction) together via a Heisenberg-type spin-
spin interaction as [72, 114]

$H=-P_{1}^{F}P_{2}^{F}-P_{3}^{F}P_{4}^{F}+{\bm{S}}_{\text{I}}\cdot{\bm{S}}_{\text{II}},$
(3)

where the first two terms stabilizes two Kramers doublets, and
${\bm{S}}_{\text{I/II}}$ stands for the effective spin-operator for each
Kramers doublet. This will end up with a unique spin-singlet bosonic ground
state with a finite gap to all excitations, which successfully gaps out all
fermions without breaking the symmetry. Eq. (3) can be rewritten as
$H_{\text{FK}}$ in Eq. (1) under some appropriate basis choice.

From the perspective of quantum anomaly, the (0+1)D Majorana fermions with a
$\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$
111$\mathbb{Z}_{2}^{F}:\chi_{i}\to-\chi_{i}$ denotes the fermion parity
symmetry, which is always assumed implicitly for local fermion systems.
internal symmetry (or the $\text{Pin}^{-}$ spacetime-internal symmetry [48,
114], or the BDI symmetry class [116, 117]) has a $\mathbb{Z}_{8}$ class of
non-perturbative global anomaly, whose anomaly index corresponds to the number
of Majorana modes. With eight Majorana modes, the anomaly vanishes, meaning
that the system can be trivially gapped without breaking symmetry. However,
the $\mathbb{Z}_{2}^{T}$ symmetry is restrictive enough to rule out any
fermion bilinear mass, making interaction a necessary ingredient in the
fermion mass generation, which corresponds to the SMG mechanism.

The Fidkowski-Kitaev (FK) interaction $H_{\text{FK}}$ in Eq. (1) has a
(unnecessarily high) flavor symmetry of $\mathrm{Spin}(7)$, which rotates
seven fermion bilinear operators
$\Phi^{\alpha}:=\chi_{a}\Gamma_{ab}^{\alpha}\chi_{b}$ (for
$\alpha=1,\cdots,7$) as a vector representation of $\mathrm{SO}(7)$, with
$\Gamma^{\alpha}=(\sigma^{123},\sigma^{203},\sigma^{323},\sigma^{211},\sigma^{021},\sigma^{231},\sigma^{002})$,
where $\sigma^{\mu\nu\cdots}=\sigma^{\mu}\otimes\sigma^{\nu}\otimes\cdots$
denotes the direct product of Pauli matrices. It can be shown that Eq. (1) can
be equivalently written as

$\begin{split}H_{\text{FK}}=-\frac{1}{4!}\sum_{\alpha=1}^{7}(\Phi^{\alpha}\Phi^{\alpha}-16),\end{split}$
(4)

which exhibits the $\mathrm{Spin}(7)$ symmetry explicitly. However, it is
possible to lower the symmetry to $\mathrm{Spin}(6)\cong\mathrm{SU}(4)$ for
example, without affecting the SMG physics [95],

$\begin{split}&H_{\mathrm{SU}(4)}=-\frac{1}{192}\sum_{\alpha=1}^{6}\Phi^{\alpha}\Phi^{\alpha}\\\
&=-(c_{1}c_{2}c_{3}c_{4}+\text{h.c.})-\frac{1}{3}\sum_{i<j}(n_{i}-\tfrac{1}{2})(n_{j}-\tfrac{1}{2}),\end{split}$
(5)

where $c_{i}$ (for $i=1,2,3,4$) are complex fermions in $\mathrm{SU}(4)$
fundamental representation and $n_{i}=c_{i}^{\dagger}c_{i}$ are their number
operators. $H_{\mathrm{SU}(4)}$ stabilizes the _same_ symmetric and non-
degenerated ground state $|\mathsf{4e}\rangle$ as $H_{\text{FK}}$ with a
finite gap to all excitations. This example illustrates that the interaction
that drives SMG is not unique, since the SMG is a generic phenomenon in
anomaly free fermion systems.

The decomposition of the four-fermion interaction as a product of two fermion-
bilinear operators $\Phi^{\alpha}\Phi^{\alpha}$ in both Eq. (4) and Eq.
(LABEL:eq:H_SU4) suggests a common physical picture to understand the SMG as
fluctuating bilinear masses. Under the Hubbard-Stratonovich transformation,
the SMG interactions in the above examples take the general form of Yukawa-
Higgs interaction

$H_{\text{YH}}=-\phi_{\alpha}\chi_{a}\Gamma^{\alpha}_{ab}\chi_{b}+\frac{1}{2g}\phi_{\alpha}\phi_{\alpha},$
(6)

where $\phi_{\alpha}$ is a bosonic Yukawa field that couples to the fermion
bilinear mass, and by integrating out $\phi^{\alpha}$, the desired four
fermion interaction is generated. If the Yukawa field condenses, i.e.
$\langle\phi_{\alpha}\rangle\neq 0$, the fermions will be gapped (with a gap
size proportional to the amplitude of $\langle\phi_{\alpha}\rangle$) and the
symmetry is also broken spontaneously. However, if it is possible to fluctuate
the orientation of the Yukawa field smoothly in the spacetime (which is only
the time here for the (0+1)D system) without bringing its local amplitude to
zero, the fermion could potentially retain the excitation gap while restoring
the required symmetry. This intuitive picture leads to fruitful understandings
of SMG in higher dimensions [16, 118, 100], which will be further discussed in
Sec. IV.1.

The SMG in (0+1)D is intimately related to the interaction-reduced
classification of fermionic SPT states in (1+1)D. In the non-interacting
limit, the (0+1)D Majorana zero modes can be viewed as the topological edge
modes of a (1+1)D fermionic SPT state protected by the same
$\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ internal symmetry. A specific
lattice model for such SPT root state is Kitaev’s Majorana fermionic chain
[119], which supports a dangling Majorana zero mode on each open end. So eight
Majorana zero modes can be viewed as the boundary state of eight copies of
Majorana chains. Without interaction, the
$\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ symmetric (BDI symmetry class)
_free_ fermionic SPT states are $\mathbb{Z}$ classified [59, 60, 61], where
the index corresponds to the number of Majorana edge modes [120]. The fact
that eight Majorana zero modes can be trivially gapped out by interaction
without breaking the symmetry implies that eight copies of Majorana chain
actually belong to the trivial SPT phase, as their interface with the vacuum
state can be made featureless (i.e. gapped and non-degenerated) by the SMG.
This indicates that the $\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ (or
$\text{Pin}^{-}$) symmetric _interacting_ fermionic SPT states are
$\mathbb{Z}_{8}$ classified [64, 65, 45], which is consistent with the formal
result $\Omega_{2}^{\text{Pin}^{-}}(\mathsf{pt})=\mathbb{Z}_{8}$ by the
cobordism classification [121, 48]. The phenomenon that the fermion SPT
classification is reduced from $\mathbb{Z}$ in the non-interacting limit to
$\mathbb{Z}_{8}$ under interaction is called the _interaction-reduced
classification_.

The $\mathbb{Z}\to\mathbb{Z}_{8}$ interaction-reduced classification implies
that eight copies of the Majorana chain can be smoothly tuned (without closing
the bulk gap) to the trivial phase under interaction. The bulk model is
defined on a one-dimensional lattice

$\begin{split}H&=H_{0}+H_{\text{int}},\\\
H_{0}&=\frac{\mathrm{i}}{2}\sum_{i}\sum_{a=1}^{8}(u\,\chi_{2i-1,a}\chi_{2i,a}+v\,\chi_{2i,a}\chi_{2i-1,a}),\\\
H_{\text{int}}&=-g\sum_{i}\sum_{a<b<c<d}V_{abcd}\chi_{i,a}\chi_{i,b}\chi_{i,c}\chi_{i,d},\end{split}$
(7)

where each site $i$ hosts eight Majorana modes $\chi_{i,a}$ ($a=1,\cdots,8$).
In the free-fermion limit ($g=0$), the $u<v$ and $u>v$ phases are separated by
a quantum phase transition at which the single-particle band gap closes in the
bulk. Ref. 64 shows that the quantum critical point can be circumvented by
applying the Fidkowski-Kitaev interaction on every site with strength $g$
(where $V_{abcd}$ in $H_{\text{int}}$ follows the same definition as that in
Eq. (1)). This can be argued by inspecting the limiting cases $(u=0,v=1)$ or
$(u=1,v=0)$ where Majorana fermions are fully dimerized along the lattice over
even or odd bonds, see Fig. 1. The two dimerized states can both be smoothly
tuned to the same strong coupling state
$\bigotimes_{i}|\mathsf{4e}\rangle_{i}$ as $g\to\infty$ without closing the
many-body gap, which can be explicitly verified by diagonalizing the local
Hamiltonian across a bond. This establishes a smooth deformation path between
the $u<v$ and $u>v$ phases by going through the strong coupling regime as
shown in Fig. 1, demonstrating that eight copies of the Majorana chain is
indeed in the trivial phase.

Figure 1: Phase diagram of the interacting Majorana chain model described by
Eq. (7)

Moreover, the argument also implies that the gapless Majorana fermions (in the
(1+1)D bulk) at the free-fermion critical point ($u=v$) can be gapped out by
turning on the interaction $g$ (see Fig. 1), leading to another example of SMG
222The anomalous chiral $\mathbb{Z}_{2}$ symmetry should be included to
protect the criticality.. The bulk SMG of the (1+1)D Majorana chain (at the
critical point) is dynamically equivalent to the SMG for the boundary mode of
a (2+1)D fermionic topological superconductor (TSC) made of eight copies of
the $p_{x}\pm\mathrm{i}p_{y}$ superconductor [123, 124, 125]. In general, each
interaction-reduced classification of the fermion SPT state implies the
existence of SMG mechanisms for the gapless fermions both on the _boundary_
and at the _bulk_ critical point between the SPT and the trivial state, where
the same local interaction that gaps out the boundary mode can be used to gap
out the bulk critical point as well. Using the connection of SMG phenomena
between the boundary and the bulk, Ref. 72 obtains a series of interaction-
reduced classification of fermionic SPT states in all spacetime dimensions
systematically, hence extending the phenomenon of SMG to higher dimensions.

###  II.2 (1+1)D SMG: 3-4-5-0 Chiral Fermion Model

The 3-4-5-0 $\mathrm{U}(1)$ chiral fermion model by Wang-Wen [75, 101]
provides an instructive example of SMG in (1+1)D, which involves the
cancellation of a perturbative anomaly and is not related to interaction-
reduced fermionic SPT classifications. The (1+1)D chiral fermion theory is
described by the Lagrangian

$\mathcal{L}=\sum_{a=1}^{4}\psi_{a}^{\dagger}\mathrm{i}(\partial_{t}-v_{a}\partial_{x})\psi_{a},$
(8)

where $\psi_{a}$ are complex fermion fields (for $a=1,2,3,4$) with different
velocities $v=(1,1,-1,-1)$. The model can be assigned a $\mathrm{U}(1)$
symmetry, under which the fermions transform as
$\mathrm{U}(1):\psi_{a}\to\mathrm{e}^{\mathrm{i}\theta q_{a}}\psi_{a}$ with
$q=(3,4,5,0)$. The two left-moving Weyl fermions $\psi_{1}$ and $\psi_{2}$ are
assigned with charge 3 and 4, and the two right-moving Weyl fermions
$\psi_{3}$ and $\psi_{4}$ are assigned with charge 5 and 0, hence the name of
the 3-4-5-0 model.

The seemingly peculiar charge assignment of the chiral fermions is designed to
cancel the $\mathrm{U}(1)$ ’t Hooft anomaly. The (1+1)D fermion with
$\mathrm{U}(1)$ internal symmetry (or
$\mathrm{Spin}^{c}\equiv(\mathrm{Spin}\times\mathrm{U}(1))/\mathbb{Z}_{2}$
spacetime-internal symmetry [48, 114], or the A symmetry class [116, 117]) has
a $\mathbb{Z}$ perturbative anomaly, whose anomaly index is given by
$\sum_{a}v_{a}q_{a}^{2}$, which vanishes for the charge assignment of the
3-4-5-0 model. The (1+1)D chiral fermions can be also viewed as the chiral
edge modes of a (2+1)D integer quantum Hall state, also known as the fermionic
SPT state in symmetry class A. The integer quantum Hall states are
$\mathbb{Z}$ classified by the quantized Hall conductance, and the
classification does not further reduce under interaction. The bulk Hall
conductance (in unit of $e^{2}/h$) matches the boundary anomaly index.

The fact that the $\mathrm{U}(1)$ ’t Hooft anomaly vanishes for the 3-4-5-0
model indicates that it should be possible to trivially gap out all fermions
without breaking the $\mathrm{U}(1)$ symmetry. However, the $\mathrm{U}(1)$
symmetry is restrictive enough to prevent the gapping to happen on the free-
fermion level, because any fermion bilinear term that produces a gap must take
the form of $\psi_{a}^{\dagger}\psi_{b}$ or $\psi_{a}\psi_{b}$ (with
$a\in\\{1,2\\}$ and $b\in\\{3,4\\}$) that mixes the left- and right-moving
fermions. Since the four flavors of fermions all carry distinct
$\mathrm{U}(1)$ charges that do not add or subtract to zero, any flavor mixing
fermion bilinear term will necessarily break the $\mathrm{U}(1)$ symmetry,
which makes it impossible to symmetrically gap out these chiral fermions on
the free-fermion level.

Nevertheless, the gapping can be achieved with fermion interaction, hence an
example of SMG. The interaction to achieve the SMG would involve at least six-
fermion terms, which can be derived using the null-vector condition [126, 127,
128, 129, 130] for quantum Hall edge states. By bosonization
$\psi_{a}\sim\mathrm{e}^{\mathrm{i}\varphi_{a}}$, the (1+1)D chiral fermion
system can be effectively described by a Luttinger liquid theory

$\mathcal{L}=\frac{1}{4\pi}(K_{ab}\partial_{t}\varphi_{a}\partial_{x}\varphi_{b}-V_{ab}\partial_{x}\varphi_{a}\partial_{x}\varphi_{b}),$
(9)

where $K=\text{diag}(1,1,-1,-1)$ and $V=\text{diag}(1,1,1,1)$ are diagonal
matrices. In the bosonization language, backscattering fermion interactions
can be introduced as

$\mathcal{L}_{\text{int}}=\sum_{\alpha}g_{\alpha}\cos(l_{\alpha,a}\varphi_{a}),$
(10)

each interaction labeled by a charge vector $l_{\alpha}$. It is possible to
find two charge vectors $l_{1}=(1,-2,1,2)$ and $l_{2}=(2,1,-2,1)$ that satisfy
the null-vector condition $l_{\alpha}^{\intercal}K^{-1}l_{\beta}=0$ (for
$\alpha,\beta=1,2$). The null-vector condition ensures that the vertex
operators $O_{\alpha}=\mathrm{e}^{\mathrm{i}\l_{\alpha}^{\intercal}\varphi}$
are both self-boson and mutual-boson, which can be simultaneously condensed at
large coupling $g_{\alpha}$. Once these vertex operators condense $\langle
O_{\alpha}\rangle\neq 0$, all fermion excitations will be gapped, since none
of the fermion operator braid trivially with any of the vertex operator (as
seen from $l_{\alpha}^{\intercal}K^{-1}\neq 0$). Furthermore, the interaction
term $\mathcal{L}_{\text{int}}$, as well as the condensate $\langle
O_{\alpha}\rangle$, preserves the $\mathrm{U}(1)$ symmetry since
$l_{\alpha}^{\intercal}q=0$ for $\alpha=1,2$. Therefore,
$\mathcal{L}_{\text{int}}$ provides a symmetric way to gap out all chiral
fermions in the 3-4-5-0 model, realizing the SMG. In terms of the fermion
field, the interaction can be translated from Eq. (10) to

$\begin{split}\mathcal{L}_{\text{int}}&=g_{1}(\psi_{1}\psi_{2}^{\dagger}\partial_{x}\psi_{2}^{\dagger}\psi_{3}\psi_{4}\partial_{x}\psi_{4}+\text{h.c.})\\\
&+g_{2}(\psi_{1}\partial_{x}\psi_{1}\psi_{2}\psi_{3}^{\dagger}\partial_{x}\psi_{3}^{\dagger}\psi_{3}\psi_{4}+\text{h.c.}),\end{split}$
(11)

where the $\partial_{x}$ operator (that generates infinitesimal translation)
is inserted as a point-splitting regularization to avoid identical fermion
operators appearing at the same spatial position.

The multi-fermion interaction in Eq. (11) can be mediated by two independent
Yukawa fields $\phi_{1},\phi_{2}$ via

$\begin{split}\mathcal{L}_{\text{YH}}=-&(\phi_{1}^{2}\psi_{1}\psi_{3}+\phi_{1}^{\dagger}\psi_{2}^{\dagger}\psi_{4}+\text{h.c.})+\frac{1}{\tilde{g}_{1}}\phi_{1}^{\dagger}\phi_{1}\\\
-&(\phi_{2}^{2}\psi_{2}\psi_{4}+\phi_{2}^{\dagger}\psi_{1}\psi_{3}^{\dagger}+\text{h.c.})+\frac{1}{\tilde{g}_{2}}\phi_{2}^{\dagger}\phi_{2}.\end{split}$
(12)

Integrating out the Yukawa fields 333Whenever two fermion operators overlap at
the same spacetime point, the replacement
$\psi_{a}\psi_{a}\to\psi_{a}\partial_{x}\psi_{a}$ is assumed to split the
fermion operators. will generate the interaction in Eq. (11) to the leading
order of $g_{\alpha}\sim\tilde{g}_{\alpha}^{2}$ (plus additional density-
density interactions like
$\psi_{1}^{\dagger}\psi_{1}\psi_{3}^{\dagger}\psi_{3}$ or
$\psi_{2}^{\dagger}\psi_{2}\psi_{4}^{\dagger}\psi_{4}$ whose effect is only to
renormalize the $V$ matrix in the Luttinger liquid theory and can be safely
ignored). The Yukawa fields $\phi_{1}$ and $\phi_{2}$ carry the
$\mathrm{U}(1)$ charges $-4$ and $-2$ respectively. Directly condensing the
Yukawa fields would provide Dirac/Majorana masses to all chiral fermions
$\psi_{a}$ at the price of breaking the $\mathrm{U}(1)$ symmetry.
Nevertheless, the SMG mechanism suggests an alternative scenario that these
Yukawa fields are fluctuating in the disordered (a.k.a. strong-coupling
symmetric) phase, such that the $\mathrm{U}(1)$ symmetry remains unbroken but
the chiral fermions could still acquire a spectral gap via the Yukawa
interaction in suitable parameter regimes. Similar idea was numerically
explored in Ref. 92 without success, due to the incorrect design of the
fermion interaction (which does not satisfy the null-vector condition). The
correct design of fermion interaction in Eq. (11) or Eq. (12) includes only a
restricted subset of symmetry-allowed interactions [75], which are helpful for
the SMG. More recent numerical study in Ref. 105 has confirmed that the
correct interaction indeed leads to the SMG phase.

The SMG transition happens when the interaction is beyond a finite critical
strength, because the interaction is perturbatively irrelevant at the free-
fermion fixed point, due to its high-order nature. However, strong enough
interaction could lead to non-perturbative effects. Increasing the interaction
strength generally tunes the Luttinger parameter (by renormalizing the $V$
matrix) and alters operator scaling dimensions in the Luttinger liquid theory
[105]. When the scaling dimension of the interaction term itself is tuned to
marginal, the SMG transition will be triggered, which drives the system from
the gapless phase to the featureless gapped phase [34, 105]. In this case, the
SMG transition will belong to the Berezinskii-Kosterlitz-Thouless (BKT)
universality class.

####  II.2.1 Proof on the equivalence between the anomaly-free and SMG gapping
conditions

Although the earlier discussions focus on the SMG of (1+1)D 3-4-5-0
$\mathrm{U}(1)$ chiral fermion model, there is no obstacle to generalize to
show any (1+1)D anomaly-free chiral fermion model with multiple
$\mathrm{U}(1)$ symmetries can allow SMG, following [75]. Ref. 75 proves that
the _anomaly-free condition_ of the (1+1)D multiple $\mathrm{U}(1)$ chiral
fermion theory is equivalent to the _SMG gapping condition_ of the same
theory. The proof can be achieved due to the exact bosonization-fermionization
techniques in (1+1)D. Here we recall the proof to complete the discussion of
(1+1)D SMG.

A generic (1+1)D anomaly-free multiple-$\mathrm{U}(1)$ chiral fermion theory
have equal numbers of left and right moving Weyl fermions,
$N_{L}=N_{R}=N\in\mathbb{Z}^{+}$, such that the total Weyl fermion number is
$2N=N_{L}+N_{R}=2\mathbb{Z}$, an even integer.

  1. 1.

The _SMG gapping condition_ requires to add $N$ independent compatible gapping
terms [126, 127, 128, 129] to preserve internal chiral $\mathrm{U}(1)$
symmetries. To prove the SMG gapping holds, we bosonize the fermionic theory

$\displaystyle\mathcal{L}=\sum_{a=1}^{N_{L}}\psi_{a}^{\dagger}\mathrm{i}(\partial_{t}-\partial_{x})\psi_{a}+\sum_{a=1}^{N_{R}}\psi_{a}^{\dagger}\mathrm{i}(\partial_{t}+\partial_{x})\psi_{a}$
(13)

to a multiplet chiral boson theory

$\displaystyle\mathcal{L}=\frac{1}{4\pi}(K_{ab}\partial_{t}\varphi_{a}\partial_{x}\varphi_{b}-V_{ab}\partial_{x}\varphi_{a}\partial_{x}\varphi_{b}),$
(14)

where $K=K_{f}\equiv\bigl{(}{\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}}\bigl{)}\oplus\bigl{(}{\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}}\bigl{)}\oplus\dots$ and the appropriate rescaled
$V=\mathbb{I}_{2N\times 2N}$ are diagonal rank-$2N$ matrices. The $K_{f}$ is
the unimodular symmetric bilinear canonical form for the fermionic system
(with $|\det(K)|=1$). The advantage of Eq. (14) is that choosing
$K=K_{b}\equiv\bigl{(}{\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}}\bigl{)}\oplus\bigl{(}{\begin{smallmatrix}0&1\\\
1&0\end{smallmatrix}}\bigl{)}\oplus\dots$ works also for the unimodular
symmetric bilinear canonical form for the bosonic system (with $|\det(K)|=1$).
The SMG gapping condition requires:
$\bullet$ To find a set of $N$ linear-independent of integer-valued
$2N$-component $l$ vectors such that

$\displaystyle
l_{\alpha}^{\intercal}K^{-1}l_{\beta}=\sum_{a,b}l_{\alpha,a}(K^{-1})_{ab}l_{\beta,b}=0,$
(15)

for $\alpha,\beta\in\\{1,2,\dots,N\\}$ and each $l_{\alpha}$ vector contains
$2N$ components ($l_{\alpha,a}$ with $a=1,\dots,2N$). Gapping Eq. (14)
requires to add the sine-Gordon deformation
$\mathcal{L}_{\text{int}}=\sum_{\alpha=1}^{N}g_{\alpha}\cos(l_{\alpha,a}\varphi_{a})$,
which can be fermionized to multi-fermion interactions.
$\bullet$ The massless Weyl fermion theory has at most a internal
$\mathrm{U}(N_{f})\times\mathrm{U}(N_{R})$ symmetry, which contains at most a
chiral $\mathrm{U}(1)^{2N}$ symmetry. But for SMG, one can preserve at most
$N$-linear independent chiral $\mathrm{U}(1)^{N}$ symmetries, labeled by a set
of charge vectors, $q_{\alpha}$ with $\alpha\in\\{1,2,\dots,N\\}$, such that
the fermions transform as
$\psi_{a}\to\psi_{a}\mathrm{e}^{\mathrm{i}{q_{\alpha,a}\theta}}$, and
bosonized fields $\varphi_{a}\to\varphi_{a}+{q_{\alpha,a}\theta}$ with
$a=1,\dots,2N$ and $\theta\in[0,2\pi)$. The symmetric sine-Gordon interactions
demand

$\displaystyle
l_{\alpha}^{\intercal}q_{\beta}=\sum_{a}l_{\alpha,a}q_{\beta,a}=0$ (16)

for any $\alpha,\beta\in\\{1,2,\dots,N\\}$.

  2. 2.

Its _anomaly-free condition_ , on the other hand, requires:
$\bullet$ Gravitational anomaly free (two-point one-loop Feynman diagram of
grav2 vertices vanish): The left and right chiral central charges ${\rm
c}_{L}={\rm c}_{R}$, which means $N_{L}=N_{R}$.
$\bullet$ Gauge anomaly free (two-point one-loop Feynman diagram of
$\mathrm{U}(1)^{2}$ vertices vanish): For each $\mathrm{U}(1)$ symmetry, with
left-handed and right-handed Weyl fermion charge vector $q_{L}$ and $q_{R}$
respectively, the anomaly-free requires the square sum of each component $\sum
q_{L}^{2}-q_{R}^{2}=0$. In terms of the symmetric bilinear form $K$ for both
bosonic ($K_{b}$) and fermionic ($K_{f}$) systems, the anomaly-free condition
demands that

$\displaystyle
q_{\alpha}^{\intercal}K^{-1}q_{\beta}=\sum_{a,b}q_{\alpha,a}(K^{-1})_{ab}q_{\beta,b}=0,$
(17)

again for $\alpha,\beta\in\\{1,2,\dots,N\\}$ and $a,b\in\\{1,2,\dots,2N\\}$.

$\bullet$ The above two anomaly-free conditions are perturbative local
anomalies. The (1+1)D nonperturbative global anomalies are classified by
cobordism groups (denoted $\mathrm{TP}_{3}$ or $\Omega_{3}$ with the special
orthogonal SO or Spin group and some internal $\mathrm{U}(1)$ symmetries, such
$\mathrm{SO}\times\mathrm{U}(1)$, $\mathrm{Spin}\times\mathrm{U}(1)$ and
$\mathrm{Spin}^{c}$) which turn out to always vanish [132].

  3. 3.

To explain that the SMG gapping condition holds implies the anomaly-free
condition also holds, Ref. 75 shows that given the set of $l_{\alpha}$
satisfying (15), one can find the set of $q_{\alpha}$ simultaneously
satisfying (16) and (17). This is true because that given $l_{\alpha}$, we can
choose $q_{\alpha}=Kl_{\alpha}=K^{-1}l_{\alpha}$ thanks to the property
$K=K^{-1}$.

  4. 4.

To explain that the anomaly-free condition holds implies the SMG gapping
condition also holds, Ref. 75 shows that given the set of $q_{\alpha}$
satisfying (17), one can find the set of $l_{\alpha}$ simultaneously
satisfying (16) and (15). This is true because that given $q_{\alpha}$, we can
choose $l_{\alpha}=Kq_{\alpha}=K^{-1}q_{\alpha}$ thanks to $K=K^{-1}$.

  5. 5.

The above two remarks prove that the _if and only if_ (sufficient and
necessary) conditions to the equivalence of the anomaly-free condition and the
SMG gapping condition. Once the set of $q_{\alpha}$ and $l_{\alpha}$ are
found, they form $2N$ linear-independent integer-valued vectors spanning
completely the $2N$-dimensional vector space (known as the Narain lattice
[133]).

  6. 6.

What remains to be explained is why the SMG gapping condition defines a gapped
boundary without any topological boundary degeneracy [128]. The idea is
viewing the (1+1)D theory (14) as the boundary theory of (2+1)D invertible
TQFT with a Chern-Simons action
$S_{\text{bulk}}=\frac{K_{ab}}{4\pi}\int_{\mathcal{M}^{3}}A_{a}\wedge\mathrm{d}A_{b}$
on a 3-manifold $\mathcal{M}^{3}$. The $A_{a}$ is a multiplet 1-form gauge
field. Hereafter all repeated indices are summed over. A stable boundary
condition requires the variation of $S_{\text{bulk}}$ on the boundary
2-manifold $(\partial\mathcal{M})^{2}$ vanished [127] under the boundary
1-form gauge field $A_{\partial}\to A_{\partial}+\delta A_{\partial}$
variation:
$\delta_{\text{bdry}}(S_{\text{bulk}})=\frac{K_{ab}}{4\pi}\int_{(\partial\mathcal{M})^{2}}A_{\partial,a}\wedge\delta
A_{\partial,b}.$ The differential $\delta$ of this variation is a symplectic
form
$\omega_{\text{Sp}}=\frac{K_{ab}}{4\pi}\int_{(\partial\mathcal{M})^{2}}\delta
A_{\partial,a}\wedge\delta A_{\partial,b}$ on the space of boundary gauge
fields. Consistent stable boundary conditions on $(\partial\mathcal{M})^{2}$
defines a Lagrangian submanifold with respect to the symplectic form
$\omega_{\text{Sp}}$ in symplectic geometry.

$\bullet$ One consistent boundary condition sets one component of
$A_{\partial}$ vanished, such as
$\big{(}K_{ab}A_{b,t}-V_{ab}A_{b,x}\big{)}\rvert_{\partial M}=0,$ which gives
a gapless (1+1)D CFT (14).

$\bullet$ Another boundary condition sets the gauge degrees of freedom
$l_{\alpha,a}A_{a}\rvert_{\partial\mathcal{M}}=0$ vanish [53]. The boson modes
$\varphi_{a}$, originally related by the gauge transformation $A_{a}\to
A_{a}+\mathrm{d}\lambda_{a}$ and $\varphi_{a}\to\varphi_{a}-\lambda_{a}$, now
may condense on the boundary with nonzero vacuum expectation values
$\langle\exp\big{(}\mathrm{i}(l_{\alpha,a}\varphi_{a})\big{)}\rangle\rvert_{\partial\mathcal{M}}\neq
0$, more precisely, indeed
$\langle\exp\big{(}\mathrm{i}(\frac{l_{\alpha,a}}{|\gcd({l_{\alpha}})|}\varphi_{a})\big{)}\rangle\rvert_{\partial\mathcal{M}}\neq
0,$ where
$\gcd({l_{\alpha}})\equiv\gcd(l_{\alpha,1},l_{\alpha,2},\dots,l_{\alpha,2N})$
is the greatest common divisor (gcd) of the all components of ${l_{\alpha}}$.
This condensation of $\varphi$ can be triggered by the earlier sine-Gordon
cosine term at a strong $g$ coupling. The boundary vertex operator and bulk
line operator are connected
$\exp\big{(}\mathrm{i}(l_{\alpha,a}\varphi_{a})\rvert_{\partial\mathcal{M}}+\mathrm{i}\int
l_{\alpha,a}A_{a}\rvert_{\mathcal{M}}\big{)}$. The gapped bulk and gapped
boundary demand that the partition function $Z$ evaluated on the 3-manifold
$M$ with the 2-boundary $\partial\mathcal{M}$ has a finite value (in fact
$Z=1$ when the $Z$ corresponds to counting the dimension of the Hilbert space
for the invertible TQFT). This means that arbitrary link configuration of the
bulk line operators should give a trivial braiding statistical phase to $Z$,
so there are no unwanted quantum fluctuations destabilizing the gapped vacuum
— namely, the mutual statistics $\exp(\mathrm{i}2\pi
l_{\alpha}^{\intercal}K^{-1}l_{\beta})=1$ and the self statistics
$\exp(\mathrm{i}\pi l_{\alpha}^{\intercal}K^{-1}l_{\alpha})=1$ are trivial for
all $\alpha,\beta\in\\{1,2,\dots,N\\}$. Hence we derive the correspondence
between the $N$ independent compatible SMG gapping terms and the $N$ null-
braiding statistics $l_{\alpha}$ vectors [126, 127, 128, 129]. This completes
the proof [75].

###  II.3 (2+1)D SMG: Honeycomb Lattice Model

The honeycomb lattice model is a simple lattice model for SMG in (2+1)D. The
model is defined on a honeycomb lattice, with each lattice site $i$ hosting
four complex fermion modes, denoted as $c_{ia}$ (for $a=1,2,3,4$). The model
is described by the Hamiltonian

$H=-\sum_{a=1}^{4}\sum_{\langle
ij\rangle}c_{ia}^{\dagger}c_{ja}-g\sum_{i}c_{i1}c_{i2}c_{i3}c_{i4}+\text{h.c.},$
(18)

where $\langle ij\rangle$ stands for the bond between nearest neighboring
sites $i$ and $j$ on the honeycomb lattice, see Fig. 2(a). The interaction
strength $g$ is the only tuning parameter of this model.

Figure 2: (a) Honeycomb lattice. (b) Graphene band structure (inset defines
high symmetry points in the Brillouin zone).

When $g=0$, the free-fermion hopping on the honeycomb lattice gives rise to
the graphene band structure [134], as shown in Fig. 2(b), that produces
$4\times 2=8$ gapless Dirac fermions $\psi_{Qa}$ at low energy, where $4$
stands for the four internal flavors ($a=1,2,3,4$) and $2$ comes of the
fermion doubling ($Q=K,K^{\prime}$) in the Brillouin zone. They can be
described by a low-energy effective field theory Lagrangian

$\mathcal{L}=\sum_{Q=K,K^{\prime}}\sum_{a=1}^{4}\bar{\psi}_{Qa}\gamma^{\mu}\partial_{\mu}\psi_{Qa},$
(19)

where $Q=K,K^{\prime}$ labels the valley (fermion doubling) freedom,
$\gamma^{\mu}=(\sigma^{2},\sigma^{1},\sigma^{3})$ and $\psi_{Qa}$ are complex
Grassmann spinors in (2+1)D (with
$\bar{\psi}_{Qa}=\psi_{Qa}^{\dagger}\gamma^{0}$). They can also be viewed as
16 gapless Majorana fermions equivalently, by decomposing each complex
Grassmann field into two real Grassmann fields.

The key physical symmetries that protect these gapless fermions in the non-
interacting limit are the lattice translation symmetry and the _anti-unitary_
sublattice symmetry $\mathbb{Z}_{2}^{S}:c_{i}\to(-)^{i}c_{i}^{\dagger}$,
$\mathrm{i}\to-\mathrm{i}$ (where $(-)^{i}$ stands for a sign factor that
takes $\pm$ on A/B sublattice of the honeycomb lattice) [59, 61, 63]. These
two symmetries can be combined to create an emergent anti-unitary symmetry at
low-energy
$\mathbb{Z}_{4}^{TF}:\psi_{Qa}\to\mathrm{i}\gamma^{0}\psi_{Qa}^{\dagger},\mathrm{i}\to-\mathrm{i}$
444Because $\mathcal{T}^{2}=-1$ on fermions, $\mathcal{T}^{2}$ should
correspond to the fermion parity operator which need a further square to
become identity, therefore $\mathcal{T}$ generates a four-fold cyclic group
$\mathbb{Z}_{4}^{TF}$., whose generator
$\mathcal{T}=T_{{\bm{R}}}^{3/4}\mathcal{S}$ consists of a sublattice symmetry
$\mathbb{Z}_{2}^{S}$ generator $\mathcal{S}$ followed by a $3/4$ fraction of
the unit-cell translation $T_{{\bm{R}}}$. Although the 2D lattice translation
symmetry $\mathbb{Z}^{2}$ (generated by $T_{\bm{R}}$ along two linearly
independent Bravais lattice vectors ${\bm{R}}$) cannot be fractionalized on
the lattice, yet for the low-energy effective theory Eq. (19), the translation
symmetry acting on the low-energy fermions $\psi_{Q}$ becomes an emergent
valley $\mathrm{U}(1)$ symmetry
$T_{{\bm{R}}}:\psi_{K/K^{\prime}}\to\mathrm{e}^{\pm\mathrm{i}2\pi/3}\psi_{K/K^{\prime}}$
($\pm$ signs are associated with $K$ and $K^{\prime}$ valleys respectively),
which can be fractionalized to
$T_{{\bm{R}}}^{3/4}:\psi_{K/K^{\prime}}\to\pm\mathrm{i}\psi_{K/K^{\prime}}$.
The anomalous nature of the combined symmetry $\mathbb{Z}_{4}^{TF}$ is
manifested by the fact that $\mathbb{Z}_{4}^{TF}$ is only an emergent symmetry
at low-energy and becomes ill-defined on the lattice level.

The (2+1)D Majorana fermion with a $\mathbb{Z}_{4}^{TF}$ internal symmetry (or
the $\text{Pin}^{+}$ spacetime-internal symmetry [48, 114], or the DIII
symmetry class [116, 117]) has a
$\Omega_{4}^{\text{Pin}^{+}}(\mathsf{pt})=\mathbb{Z}_{16}$ non-perturbative
global anomaly based on the cobordism, whose anomaly index corresponds to the
number of gapless Majorana fermions. The honeycomb lattice model precisely has
16 Majorana fermions at low-energy, which is free of the $\mathbb{Z}_{16}$
non-perturbative global anomaly. However, the $\mathbb{Z}_{4}^{TF}$ symmetry
is still restrictive enough to rule out all possible fermion bilinear gapping
terms from appearing in Eq. (19), which again calls for the SMG mechanism.

The four-fermion interaction $g$ in Eq. (18) is one choice of the interaction
that drives the desired SMG. On every site, this interaction is the same as
the $\mathrm{SU}(4)$ symmetric interaction in Eq. (LABEL:eq:H_SU4) (up to
unimportant density-density interactions, which do not affect the ground state
but only to renormalize the gap size). The interaction $g$ explicitly drives a
four-fermion condensation, also known as the charge-4e superconducting order
[106, 107, 108, 109, 110, 111, 112]. In the large $g$ limit, the many-body
ground state of the system is simply the product
$\bigotimes_{i}|\mathsf{4e}\rangle_{i}$ of on-site ground states
$|\mathsf{4e}\rangle_{i}=(|0000\rangle_{i}+|1111\rangle_{i})/\sqrt{2}$, which
is symmetric, non-degenerated and gapped, realizing the SMG phase. Therefore,
by tuning the strength $g$, one expects to drive an SMG transition at some
intermediate $g\sim 1$ (that is comparable with the bandwidth of the lattice
fermion). Various numerical simulations of this model (and its variants) [136,
137, 138, 118, 139] have suggested the existence of such a direct and
continuous SMG transition between the gapless and the gapped phases in (2+1)D.
A field theory description of the SMG quantum critical point was proposed in
Ref. 35, which will be further reviewed in Sec. IV.3.1.

The honeycomb lattice model also has an explicit $\mathrm{SU}(4)$ symmetry
$c_{ia}\to U_{ab}c_{ib}$ that rotates fermions among the four on-site flavors.
Although the $\mathrm{SU}(4)$ symmetry does not affect the anomaly analysis in
any essential way, it could help to remove all the $\mathrm{SU}(4)$-breaking
relevant perturbations (if there were any) at the critical point, which might
help to promote a continuous SMG transition. Thus the $\mathrm{SU}(4)$
symmetry can also be included in the discussion, and the total internal
symmetry becomes
$\mathrm{SU}(4)\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4}^{TF}$ (where
$\mathrm{SU}(4)$ and $\mathbb{Z}_{4}^{TF}$ share the fermion parity
$\mathbb{Z}_{2}^{F}$ subgroup).

The decomposition of the on-site $\mathrm{SU}(4)$-symmetric interaction into a
product of fermion bilinear operators $\Phi_{i}^{\alpha}\Phi_{i}^{\alpha}$ as
in Eq. (LABEL:eq:H_SU4) indicates that one can again introduce a Yukawa field
$\phi_{\alpha}$ to mediate the fermion interaction, [35]

$\mathcal{L}_{\text{YH}}=-\phi_{\alpha}(\psi_{Ka}^{\intercal}\mathrm{i}\gamma^{0}\tilde{\Gamma}^{\alpha}_{ab}\psi_{K^{\prime}b}+\text{h.c.})+\frac{1}{2\tilde{g}}\phi^{\ast}_{\alpha}\phi_{\alpha},$
(20)

where
$\tilde{\Gamma}^{\alpha}=(\sigma^{12},\sigma^{20},\sigma^{32},\mathrm{i}\sigma^{21},\mathrm{i}\sigma^{02},\mathrm{i}\sigma^{23})$
are the flavor-space matrices for the flavor-sextet pairing that transform as
representation $\mathbf{6}$ in $\mathrm{SU}(4)$. Importantly, the Yukawa field
must transform as $\mathbb{Z}_{4}^{TF}:\phi_{\alpha}\to-\phi_{\alpha}$ to keep
$\mathcal{L}_{\text{YH}}$ invariant under $\mathbb{Z}_{4}^{TF}$, meaning that
directly condensing the Yukawa field $\langle\phi_{\alpha}\rangle\neq 0$ will
necessarily break the protecting symmetry $\mathbb{Z}_{4}^{TF}$ (as well as
breaking the standing-by $\mathrm{SU}(4)$ symmetry). The SMG interaction thus
provides a mechanism to allow the Yukawa field to condense locally without
establishing long-range order, so as to maintain the fermion gap without
breaking the symmetry.

###  II.4 (3+1)D SMG: Chiral Fermion Model

Lattice regularization of chiral fermions in (3+1)D has always been an
important motivation to study SMG in lattice gauge theories. A rich class of
SMG transitions has been proposed and analyzed in different 4d chiral fermion
models recently [98, 34]. Here we will review one simplest example from [34]
to illustrate the essential features of these models. The example considers a
collection of right-handed massless Weyl fermions $\psi$ transforming under an
internal $\mathrm{SU}(N)$ symmetry in the
${\tiny\yng(2)}\oplus(N+4)\bar{\tiny\yng(1)}$ representation, described by the
Lagrangian

$\mathcal{L}=\psi^{\dagger}\mathrm{i}\bar{\sigma}^{\mu}\partial_{\mu}\psi,$
(21)

where $\psi$ is the collection of complex Grassmann spinor fields. More
explicitly, the fermion field can be split into
$\psi=\psi_{\lambda}\oplus\psi_{\psi}$, such that $\psi_{\lambda}$ denotes the
single $\tiny\yng(2)$ Weyl fermion and $\psi_{\psi}$ denotes the $(N+4)$
multiples of $\bar{\tiny\yng(1)}$ Weyl fermions. The goal is to find a path of
gapping these fermions without breaking the $\mathrm{SU}(N)$ symmetry.

First of all, the symmetric gapping is possible because all fermions together
cancel the $\mathrm{SU}(N)$ ’t Hooft anomaly, as can be verified by the
following anomaly index calculation

$\begin{split}&A_{\mathrm{SU}(N)}({\tiny\yng(2)})+(N+4)A_{\mathrm{SU}(N)}(\bar{\tiny\yng(1)})\\\
=&(N+4)+(N+4)\times(-1)=0.\end{split}$ (22)

Secondly, the $\mathrm{SU}(N)$ symmetry is restrictive enough to forbid all
possible fermion bilinear masses. For chiral fermions, the only available
fermion bilinear mass is the Majorana mass that pairs up fermions like
$(\psi^{\intercal}\mathrm{i}\sigma^{2}\psi+\text{h.c.})$. Since the Majorana
mass is already antisymmetric in the spinor subspace, the flavor subspace must
be symmetric. However, the fermion representation of the $\mathrm{SU}(N)$
symmetry guarantees that its symmetric product does not contain a trivial
representation $\mathbf{1}$, as
$({\tiny\yng(2)}\oplus(N+4)\bar{\tiny\yng(1)})\times_{\mathcal{S}}({\tiny\yng(2)}\oplus(N+4)\bar{\tiny\yng(1)})\to\,{\tiny\yng(4)}\oplus(N+4){\tiny\yng(1)}\oplus(N+4){\bar{\tiny\yng(1)}\tiny\yng(2)}\oplus(N+4)^{2}{\overline{\tiny\yng(2)}}\,$.
So any bilinear mass term will transform non-trivially under $\mathrm{SU}(N)$
and cannot be condensed in the presence of the $\mathrm{SU}(N)$ symmetry.
Therefore, one must rely on fermion interactions to achieve symmetric gapping,
i.e. the SMG mechanism.

To design the appropriate interaction to realize the SMG transition, one might
search for candidate four-fermion interactions by looking for trivial
representations in $({\tiny\yng(2)}\oplus(N+4)\bar{\tiny\yng(1)})^{\times 4}$.
However, $({\tiny\yng(2)}\oplus(N+4)\bar{\tiny\yng(1)})^{\times
4}\not\to\mathbf{1}$ in general (unless for $N=2,4$), meaning that one might
need to look into higher-order fermion interactions, which are even more
irrelevant and less interesting to explore. Nevertheless, the three-fermion
combination $({\tiny\yng(2)}\oplus(N+4)\bar{\tiny\yng(1)})^{\times 3}$
contains trivial representations of the form
$(\psi_{\lambda}^{\intercal}\mathrm{i}\sigma^{2}\psi_{\psi})\psi_{\psi}$, but
this term is fermionic and cannot appear directly in the Lagrangian (otherwise
the fermion parity symmetry would be broken). An interesting idea from [98,
34] is to bring down $(N+4)(N+3)/2$ additional fermions $\psi_{\chi}$ from the
high-energy spectrum which transforms trivially under the $\mathrm{SU}(N)$
symmetry, such that a four-fermion interaction

$\mathcal{L}_{\text{int}}=g((\psi_{\lambda}^{\intercal}\mathrm{i}\sigma^{2}\psi_{\psi})(\psi_{\chi}^{\intercal}\mathrm{i}\sigma^{2}\psi_{\psi})+\text{h.c.})$
(23)

can be constructed. In this interaction, the fermion flavors are contracted in
the way such that $\psi_{\lambda}^{\intercal}\mathrm{i}\sigma^{2}\psi_{\psi}$
and $\psi_{\chi}^{\intercal}\mathrm{i}\sigma^{2}\psi_{\psi}$ transform as
$\tiny\yng(1)$ and $\bar{\tiny\yng(1)}$ under $\mathrm{SU}(N)$ respectively.

Table 1: Representations of the fermion $\psi$ and Yukawa $\phi$ fields in the chiral fermion model under $\mathrm{SU}(N)\times\mathrm{SU}(N+4)$. |  | $\mathrm{SU}(N)$ | $\mathrm{SU}(N+4)$ |
---|---|---|---|---
$\psi\Bigg{\\{}$ | $\psi_{\lambda}$ | $\tiny\yng(2)$ | $\mathbf{1}$ | physical fermion
$\psi_{\psi}$ | $\bar{\tiny\yng(1)}$ | $\bar{\tiny\yng(1)}$
$\psi_{\chi}$ | $\mathbf{1}$ | $\tiny\yng(1,1)$
$\phi$ |  | $\bar{\tiny\yng(1)}$ | $\tiny\yng(1)$ | Yukawa boson

It is instructive to note that the chiral fermion model has an additional
internal symmetry $\mathrm{SU}(N+4)$, under which $\psi_{\lambda}$,
$\psi_{\psi}$ and $\psi_{\chi}$ transform as $\mathbf{1}$,
$\bar{\tiny\yng(1)}$ and $\tiny\yng(1,1)$ respectively, as summarized in Tab.
1. The representations are so assigned that the fermions are also free of the
$\mathrm{SU}(N+4)$ anomaly, as

$\begin{split}&NA_{\mathrm{SU}(N+4)}(\bar{\tiny\yng(1)})+A_{\mathrm{SU}(N+4)}({\tiny\yng(1,1)})\\\
=&N\times(-1)+(N+4-4)=0.\end{split}$ (24)

Moreover, there is no mixed anomaly between $\mathrm{SU}(N)$ and
$\mathrm{SU}(N+4)$. The interaction $\mathcal{L}_{\text{int}}$ in Eq. (23)
also respects the larger $\mathrm{SU}(N)\times\mathrm{SU}(N+4)$ symmetry. Ref.
34 suggests that $\mathcal{L}_{\text{int}}$ is a plausible SMG interaction
that drives all fermions
$\psi=\psi_{\lambda}\oplus\psi_{\psi}\oplus\psi_{\chi}$ to the gapped phase
without breaking the $\mathrm{SU}(N)\times\mathrm{SU}(N+4)$ symmetry. The
mechanism will be further reviewed in Sec. IV.3.2.

The interaction $\mathcal{L}_{\text{int}}$ in Eq. (23) can be decomposed into
the following Yukawa couplings

$\begin{split}\mathcal{L}_{\text{YH}}=&-(\phi_{1}^{\dagger}(\psi_{\chi}^{\intercal}\mathrm{i}\sigma^{2}\psi_{\psi})+\phi_{2}(\psi_{\lambda}^{\intercal}\mathrm{i}\sigma^{2}\psi_{\psi})+\text{h.c.})\\\
&+\frac{1}{\tilde{g}}(\phi_{1}^{\dagger}\phi_{2}+\text{h.c.}),\end{split}$
(25)

where the Yukawa fields $\phi_{1}$ and $\phi_{2}$ are both in the
$(\bar{\tiny\yng(1)},\tiny\yng(1))$ representation of
$\mathrm{SU}(N)\times\mathrm{SU}(N+4)$. The above Yukawa decomposition again
provides an alternative way to understand the SMG transition by driving the
Yukawa field into a strong-coupling symmetric phase.

###  II.5 SMG in General Dimensions

For SMG to happen in a fermion system with a choice of internal symmetry $G$
in $d$-dimensional spacetime, the following two _necessary_ conditions must be
satisfied:

  * •

The anomaly index $\nu\in\text{TP}_{d+1}(G)$ of the system must vanish
($\nu=0$), where $\text{TP}_{d+1}(G)$ denotes the classification of invertible
topological phases (with low-energy invertible topological field theories) in
$(d+1)$-dimensional spacetime [4]. (We may also denote $d+1\equiv D$ as the
bulk dimension.)

  * •

The single fermion must be in a representation $\mathbf{r}_{\psi}^{G}$ of $G$,
such that the _antisymmetric_ product (denoted by $\times_{\mathcal{A}}$)
representation
$\mathbf{r}_{\psi}^{G}\times_{\mathcal{A}}\mathbf{r}_{\psi}^{G}$ does not
contain the trivial representation $\mathbf{1}^{G}$ in its direct sum
decomposition.

If the first condition is violated ($\nu\neq 0$), trivially and symmetrically
gapping out the fermions is impossible with or without interaction, due to the
anomaly obstruction. However, it may be possible to symmetrically gap out the
fermions with a symmetric anomalous topological order described by its
underlying low-energy topological quantum field theory (TQFT), see Sec. IV.4.
If the second condition is violated, a trivial and symmetric gap can already
be achieved at the free-fermion level by a fermion bilinear condensation
(through the trivial channel given by
$\mathbf{r}_{\psi}^{G}\times_{\mathcal{A}}\mathbf{r}_{\psi}^{G}\to\mathbf{1}^{G}$),
such that fermion interaction is not necessary. When both conditions are
satisfied, an interacting (strong-coupling) mechanism is needed to generate
the fermion mass without breaking the $G$ symmetry, i.e. the SMG mechanism.

These conditions are also sufficient for the existence of an interacting SMG
phase (i.e. the featureless gapped phase). However, they do not necessarily
imply a direct and continuous SMG transition of fermions from the gapless
phase to the gapped phase. A general design of the four fermion interaction
that leads to a single continuous transition between the gapless fermions and
the SMG remains a challenging problem. Much of the progress relies on
numerical simulations, as to be reviewed in the next section Sec. III.

##  III Numerical Investigations

###  III.1 Existence of SMG Phases

Because the SMG is generally a non-perturbative interaction effect (with
respect to the gapless free-fermion fixed point), numerical simulations play
important roles in the study of SMG. Given an anomaly-free gapless fermion
system with a proposed gapping interaction, the numerical study has two major
goals:

  * •

Establish the existence of the SMG phase in the strongly interacting limit.

  * •

Investigate the nature of the SMG transition at the critical interaction
strength.

The _SMG phase_ refers to the strongly interacting trivial gapped state of
fermions without symmetry breaking. It was also called the paramagnetic
strong-coupling (PMS) phase [137, 140], the strongly coupled symmetric gapped
(SCSG) phase [94, 95], or the featureless gapped phase [35, 141] in the
literature. Correspondingly, the gapless Dirac/Weyl/Majorana fermion phase in
the weak interaction regime is sometimes also called the paramagnetic weak-
coupling (PMW) phase [137, 140], or the semi-metal (SM) phase [35, 141]. The
phase transition separating the SMG phase and the free-fermion phase is called
the _SMG transition_.

Table 2: Summary of numerical studies of SMG phases. (Dim. - spacetime dimension, Sym. - internal symmetry in terms of Lie algebra). Dim. | Sym. | Model |  Method |  Reference
---|---|---|---|---
(1+1)D | $\mathfrak{u}(1)$ | $\psi^{6}$ |  DMRG |  [105]
| $\mathfrak{su}(2)$ | YH |  disorder average |  [100]
| $\mathfrak{so}(4)$ | $\psi^{4}$ |  QMC |  [136]
| $\mathfrak{su}(4)$ | $\psi^{4}$ |  QMC |  [142]
| $\mathfrak{so}(7)$ | YH |  RHMC |  [104]
(2+1)D | $\mathfrak{so}(4)$ | $\psi^{4}$ |  RHMC |  [118]
| $\mathfrak{so}(5)$ | $\psi^{4}$ |  QMC |  [136]
| $\mathfrak{su}(4)$ | $\psi^{4}$ |  FBMC |  [137, 138]
|  | $\psi^{4}$ |  QMC |  [139]
(3+1)D | $\mathfrak{so}(4)$ | YH |  RHMC |  [140, 143]
|  | QCD |  RHMC |  [144]
| $\mathfrak{su}(4)$ | $\psi^{4}$ |  RHMC |  [145, 146]
|  | $\psi^{4}$ |  FBMC |  [147, 148]

Numerical study of the SMG phases and phase transitions has been performed in
various spacetime dimensions with different symmetries, as summarized in Tab.
2. The model Hamiltonian $H=H_{0}+H_{\text{int}}$ takes the following general
form

$\begin{split}H_{0}=&-\sum_{i,j}\sum_{a}t_{ij}\psi_{ia}\psi_{ja}\\\
H_{\text{int}}=&g\sum_{i}\sum_{a,b,c,d,\cdots}V_{abcd\cdots}\psi_{ia}\psi_{ib}\psi_{ic}\psi_{id}\cdots,\\\
\text{or
}H_{\text{int}}=&\sum_{i}\sum_{a,b,\alpha}\phi_{i\alpha}\psi_{ia}\Gamma_{ab}^{\alpha}\psi_{ib}+H[\phi_{i\alpha}],\end{split}$
(26)

where $\psi_{ia}$ denotes the fermion of the flavor $a$ on the site $i$ (or in
the unit cell $i$). The free-fermion Hamiltonian $H_{0}$ generates gapless
fermions at low-energy, and the interaction Hamiltonian $H_{\text{int}}$
applies properly designed interactions to drive the SMG. The interaction
strength will be generally denoted as $g$ in the following discussion.
Different models are mainly distinct in the following aspects:

  * •

The low-energy fermions can be realized in the lattice model $H_{0}$ either as
the gapless boundary modes of a fermionic SPT state in one higher spacetime
dimension [105], or as the gapless bulk mode of a semi-metal state in the
designated spacetime dimension (such as the honeycomb lattice fermion in
(2+1)D [136, 139] and the stagger fermion in general dimensions [140, 143]).

  * •

The interaction $H_{\text{int}}$ can either be explicit given by multi-fermion
local interaction terms (denoted as $\psi^{4}$ for four-fermion interactions
or $\psi^{6}$ for six-fermion interactions in Tab. 2), or mediated by
intermediate Yukawa-Higgs fields (denoted as YH) or non-Abelian gauge fields
(denoted as QCD).

  * •

The $\mathfrak{so}(4)$ (or more precisely
$\mathrm{Spin}(3)\times_{\mathbb{Z}_{2}^{F}}\mathrm{Spin}(4)$),
$\mathfrak{so}(5)$ (or
$\mathrm{U}(1)\times_{\mathbb{Z}_{2}^{F}}\mathrm{Spin}(5)$),
$\mathfrak{su}(4)$ (or $\mathrm{SU}(4)\cong\mathrm{Spin}(6)$) interactions can
all be viewed as lower-symmetry descendants of the $\mathfrak{so}(7)$ (or
$\mathrm{Spin}(7)$) Fidkowski-Kitaev interaction, whose relations are
discussed in Appendix B of Ref. 95.

  * •

In even spacetime dimensions, the interaction can be restricted to part of the
fermions (forming the mirror sector) to study the SMG in chiral fermion
systems [105, 104].

Many different numerical methods have been used to study these models, which
include: the density matrix renormalization group (DMRG) [105], auxiliary-
field quantum Monte Carlo (QMC) [136], rational hybrid Monte Carlo (RHMC)
[149], and fermion bag Monte Carlo (FBMC) [150].

The SMG phase has been successfully achieved in studies listed in Tab. 2,
which demonstrate the generality of the SMG mechanism in different spacetime
dimensions with various symmetry assignments and under different forms of
interaction. The SMG phase has the following defining features that can be
checked in numerics:

  * •

Unique ground state with a gap to all excitations (including both fermionic
and bosonic excitations),

  * •

Absence of spontaneous symmetry breaking (no fermion bilinear condensation),

  * •

Formation of the four-fermion (or higher multi-fermion) condensate that
preserves the symmetry.

Directly computing the many-body excitation gap is challenging for most
numerical approaches. One way to probe the excitation gap is to measure the
correlation function in different channels

$\begin{split}C_{ij}^{(1)}&=\langle\psi_{ia}\psi_{ja}\rangle\sim\mathrm{e}^{-|{\bm{x}}_{i}-{\bm{x}}_{j}|/\xi_{1}},\\\
C_{ij}^{(2)}&=\langle\psi_{ia}\psi_{ib}\psi_{ja}\psi_{jb}\rangle\sim\mathrm{e}^{-|{\bm{x}}_{i}-{\bm{x}}_{j}|/\xi_{2}}.\\\
\end{split}$ (27)

An exponential decaying correlation function implies a finite excitation gap
$\Delta_{n}\sim 1/\xi_{n}$ inversely proportional to the correlation length
$\xi_{n}$. In particular, the single-fermion correlation $C^{(1)}$ probes the
single-particle gap $\Delta_{1}$ for fermionic excitations, and the fermion-
bilinear correlation $C^{(2)}$ probes the gap $\Delta_{2}$ for collective
bosonic excitations. Various studies [136, 118, 145, 105] have used the
correlation function approach to demonstrate the gap opening (mass generation)
in the SMG phase, as illustrated in Fig. 3(a,b). Ref. 136 observed that the
fermion single-particle gap $\Delta_{1}$ is a bit larger than the bosonic
fluctuation gap $\Delta_{2}$ in the SMG phase, which seems to be consistent
with the fluctuating bilinear mass picture (where the bosonic excitations are
typically softer than the fermionic excitations), as to be discussed in Sec.
IV.1.

Figure 3: Typical features of SMG observed in numerics. Continuous gap opening
in (a) (1+1)D SMG driven by four-fermion interactions and (b) higher
dimensional SMG in general. (c) Vanishing fermion bilinear expectation
$\langle\bar{\psi}\Gamma\psi\rangle$ with its source field $m$ in the SMG
phase. (d) Diverging Yukawa field susceptibility $\chi$ at the SMG transition.

More explicitly, Ref. 100 investigated the fermion single-particle spectrum of
(1+1)D chiral gauge theory coupled to a fluctuating Higgs field with finite
spacetime correlation. By choosing the Higgs coupling strength and the Higgs
field correlation length appropriately, Ref. 100 was able to show that the
fermion excitation gap remains open in the SMG phase, despite of the absence
of Higgs condensation.

Many numerical works have also confirmed that there is no fermion bilinear
condensation in the SMG phase, such that the mass generation is not due to
spontaneous symmetry breaking. To check this statement, one can introduce a
small source field $m$ that couples to the fermion bilinear term
$\bar{\psi}\Gamma\psi$ of interest. By showing that in the large system size
limit ($L\to\infty$) the induced fermion bilinear expectation value vanishes
as the source field $m$ is turned off

$\mathop{\mathrm{lim}}_{m\to
0}\mathop{\mathrm{lim}}_{L\to\infty}\langle\bar{\psi}\Gamma\psi\rangle=0,$
(28)

one can rule out the fermion bilinear condensation, as illustrated in Fig.
3(c). Such check should be performed for all fermion bilinear terms that
transform as different representations under the symmetry. This approach was
adopted in various studies [145, 139, 146, 140, 143, 104]. Another approach is
to measure the correlation function of fermion bilinear operators. If the
correlation function decays exponentially, then the bilinear operator is not
long-ranged ordered (not condensed). This approach is used in Ref. 136, 105.

###  III.2 Nature of the SMG Transitions

Much effort of the numerical study has been focused on understanding the
nature of the SMG transition. Two questions can be asked:

  * •

Is the SMG transition a _direct_ transition (i.e. without any intermediate
phases setting in)?

  * •

Is the SMG transition _continuous_ (i.e. not first-order)?

If the answers to both questions were yes, the SMG transition would be a
quantum critical point which admits a field-theory description in the
continuum limit. Such case will be highly interesting from the theoretical
perspective, because the phases on both sides cannot be distinguished by a
symmetry-breaking order parameter, hence a direct and continuous SMG
transition is necessarily an exotic quantum phase transition beyond the
Landau-Ginzburg-Wilson paradigm.

Table 3: Summary of numerical studies of SMG transitions. (Dir. - direct transition or not, Con. - continuous transition or not). Dim. | Sym. | Dir. | Con. |  Remarks |  Reference
---|---|---|---|---|---
(1+1)D | $\mathfrak{u}(1)$ | yes | yes |  BKT, $g_{c}\sim 1$ |  [105]
| $\mathfrak{so}(4)$ | yes | yes |  $g_{c}=0$ |  [136]
| $\mathfrak{su}(4)$ | yes | yes |  $g_{c}=0$ |  [142]
(2+1)D | $\mathfrak{so}(4)$ | yes | yes |  |  [118]
| $\mathfrak{so}(5)$ | yes | yes |  |  [136]
| $\mathfrak{su}(4)$ | yes | yes |  $\eta=1.05$, $\nu=1.30$ |  [137, 138]
|  |  |  |  $\eta=0.7\pm 0.1$ |  [139]
(3+1)D | $\mathfrak{so}(4)$ | yes | yes |  by frustrating the Yukawa field |  [140, 143, 144]
| $\mathfrak{su}(4)$ | no | - |  small intermediate SSB phase |  [146, 147, 148]

The nature of the SMG transition depends on the spacetime dimension and the
symmetry. Tab. 3 summarizes numerical results in the literature. SMG
transitions in (1+1)D are most well understood. As four-fermion interactions
are marginal perturbations for (1+1)D massless fermions, the RG equation for
the interaction strength $g$ with respect to the RG scale $\ell=\ln\Lambda$
takes the general form of $\mathrm{d}g/\mathrm{d}\ell=\alpha g^{2}$ to the
leading order of $g$. If the coefficient $\alpha>0$, $g>0$ is marginally
relevant, which flows towards infinity for any finite $g$. This happens to any
SMG transition driven by four-fermion interactions in (1+1)D: the system
immediately enters the symmetric gapped phase as long as the interaction is
turned on (with the correct sign), such that the SMG critical point is at
$g_{c}=0$. The RG equation predicts that the excitation gap should open as
(see Fig. 3(a))

$\Delta\sim\exp(-\tfrac{1}{\alpha g}),$ (29)

which has been verified in Ref. 136, 142 with different models.

However, SMG can also be driven by higher-order fermion interactions. For
example, the SMG in (1+1)D $\mathrm{U}(1)$-symmetric chiral fermion 3-4-5-0
model is driven by six-fermion interactions, which are irrelevant at the
massless free-fermion fixed point. As shown in Ref. 105, the SMG transition in
this case happens at a finite interaction strength, and the transition is
shown to be in the BKT universality class.

Numerical studies have also found evidence for direct and continuous SMG
transitions in higher dimensions, as listed in Tab. 3. A direct transition can
be established by ruling out any intermediate phase between the gapless
fermion phase and the SMG phase. The most probable intermediate phase is the
spontaneous symmetry breaking (SSB) phase in which the fermion bilinear mass
(the Yukawa field) condenses. Such intermediate SSB phase is often observed in
(3+1)D systems.[146, 147, 148] However, there are also examples showing that
it is possible to shrink the intermediate SSB phase by frustrating the Yukawa
field (i.e. introducing local couplings of the Yukawa field in conflict with
its natural ordering tendency), which could foster a direct SMG
transition.[140, 143, 144]

Given a direct SMG transition, one can further check whether the transition is
continuous or first-order (discontinuous). The numerical evidence for a
continuous transition include:

  * •

Continuous gap opening across the SMG transition.

  * •

Universal scaling behavior of physical quantities near the transition.

The excitation gap (or the inverse correlation length) can be extracted from
correlation functions (as previously explained around Eq. (27)). Ref. 136, 118
have demonstrated that the gap opens smoothly across the SMG transition, as
depicted in Fig. 3(b), in support of a continuous transition (i.e. a quantum
critical point). As the correlation length diverges, physical quantities
should exhibit universal scaling behaviors near the quantum critical point.
The quantity that is often studied in numerics is the uniform static
susceptibility of the Yukawa field, which is defined as

$\chi=\frac{1}{L^{d}}\int\mathrm{d}^{d}x\langle\phi_{\alpha}(x)\phi_{\alpha}(0)\rangle,$
(30)

where the integration is over the spacetime and $L^{d}$ stands for the
spacetime volume. The following universal behavior is observed in Ref. 137,
138

$\chi=L^{2-\eta}f((g-g_{c})L^{1/\nu}),$ (31)

where $f$ is a universal function, and $\eta$ and $\nu$ are critical
exponents. Such that the susceptibility generally diverges as
$\chi\sim|g-g_{c}|^{-\nu(2-\eta)}$ near the SMG critical point, as illustrated
in Fig. 3(d). The exponent $\eta$ can also be determined from the power-law
fitting of the correlation function at the critical point,

$\langle\phi_{i\alpha}\phi_{j\alpha}\rangle\sim|{\bm{x}}_{i}-{\bm{x}}_{j}|^{-(d-2+\eta)}.$
(32)

This approach is used in Ref. 139. Currently, the scaling analysis has only
been performed for (2+1)D systems, where the simulation can achieve a linear
system size up to $L=60$ [138], which enables a rather reliable estimate of
critical exponents. As a comparison, for (3+1)D systems, the simulation can
only reach $L=16$ [146]. Numerical studies [137, 138, 139] of the (2+1)D
$\mathfrak{su}(4)$ symmetric SMG found $\eta\sim 1.05$ and $\nu\sim 1.30$,
which is close to the (large-$N_{f}$ limit) exponents ($\eta=\nu=1$) of the
(2+1)D Gross-Neveu-Yukawa universality class [151, 152]. However, the SMG
mechanism is physically distinct from the symmetry-breaking mass generation
described by the Gross-Neveu-Yukawa model. Their similar critical exponents
motivate the idea to view the SMG transition as a hidden Gross-Neveu-Yukawa
transition of fermionic partons, as to be discussed in Sec. IV.2.

##  IV Theoretical Understandings

###  IV.1 Fluctuating Bilinear Mass Picture

The _fluctuating bilinear mass_ (Yukawa field) provides an intuitive physical
picture for SMG. It suggests that the SMG can be generally understood in two
steps: starting with gapless Weyl/Majorana fermions, first condense a Yukawa
field that couples to fermion bilinear mass terms to gap out the fermions at
the price of breaking the protecting symmetry, then fluctuate the phase (or
direction) of the Yukawa field in the spacetime to restore the symmetry while
maintaining the local amplitude of the Yukawa field finite to keep the fermion
gap open. The picture can be described by the Yukawa-Higgs model with a
symmetry $G$ [145, 100]

$\begin{split}Z&=\int\mathcal{D}[\psi,\phi]\mathrm{e}^{-S_{\text{Y}}[\psi,\phi]-S_{\text{H}}[\phi]},\\\
S_{\text{Y}}[\psi,\phi]&=\int\mathrm{d}^{d}x\;(\bar{\psi}_{a}{\mathrm{i}}\gamma^{\mu}\partial_{\mu}\psi_{a}+\phi_{\alpha}\bar{\psi}_{a}\Gamma^{\alpha}_{ab}\psi_{b}),\\\
S_{\text{H}}[\phi]&=\int\mathrm{d}^{d}x\;((\partial_{\mu}\phi)^{2}+V_{\text{H}}(\phi)+\cdots),\end{split}$
(33)

where $\psi_{a}$ are the Weyl/Majorana fermions (written as real spinor
Grassmann fields with $\bar{\psi}_{a}=\psi_{a}^{\intercal}\gamma^{0}$) and
$\phi_{\alpha}$ are the Yukawa bosons (written as real scalar fields) 555For
generality, we assume that all (no matter real or complex) fields are
automatically translated into their minimal real embeddings to avoid
unnecessary complication of complex conjugations in the discussion.. Both the
fermion $\psi$ and the boson $\phi$ fields are in non-trivial representations
of the protecting symmetry $G$, denoted as $\mathbf{r}_{\psi}^{G}$ and
$\mathbf{r}_{\phi}^{G}$ respectively. The Yukawa coupling will be loosely
denoted as $\phi\,\bar{\psi}\Gamma\psi$ for generality, where the vertex
tensor $\Gamma^{\alpha}_{ab}$ is set by the Clebsch-Gordan coefficients of the
fusion channel
$\mathbf{r}_{\psi}^{G}\times_{\mathcal{A}}\mathbf{r}_{\psi}^{G}\to\mathbf{r}_{\phi}^{G}$.
Precise forms of the Yukawa coupling can be found in Eqs. (6, 12, 20, 25),
which vary from model to model. $S_{\text{Y}}[\psi,\phi]$ describes the
gapless fermion $\psi$ coupled to the Yukawa field $\phi$, and
$S_{\text{H}}[\phi]$ describes the dynamics of the Yukawa field with
$V_{\text{H}}(\phi)$ being some Higgs potential.

Integrating out the Yukawa field $\phi$ leads to a pure fermion model of
$\psi$. Assuming $S_{\text{H}}[\phi]=\frac{1}{2g}\phi^{2}$ takes the Gaussian
form at the ultraviolet (UV) level, the fermion model will be

$\begin{split}Z&=\int\mathcal{D}[\psi]\mathrm{e}^{-S[\psi]},\\\
S[\psi]&=\int\mathrm{d}^{d}x(\bar{\psi}{\mathrm{i}}\gamma^{\mu}\partial_{\mu}\psi+g(\bar{\psi}\Gamma^{\alpha}\psi)(\bar{\psi}\Gamma^{\alpha}\psi)).\end{split}$
(34)

It is assumed that the model is free of $G$-anomaly, but the symmetry $G$ is
still restrictive enough to forbid all possible bilinear mass terms, paving
ways for SMG. The fluctuating Yukawa field effectively mediates the fermion
interaction $g$ that is responsible to drive the SMG.

The appearance of the SMG in the Yukawa-Higgs model Eq. (33) requires some
delicate design of $S_{\text{H}}[\phi]$ to achieve the appropriate infrared
(IR) dynamics. Assuming the Higgs potential $V_{\text{H}}(\phi)$ pins the
Yukawa field $\phi$ to a finite amplitude (e.g. $|\phi|=1$) throughout the
spacetime (as in a non-linear $\sigma$-model), while allows its orientation to
fluctuate in its internal flavor space, numerical evidences [100] show that
the fluctuating Yukawa field is possible to gap out the fermions $\psi$
without spontaneously breaking the $G$ symmetry, if the Yukawa field
fluctuation has a finite but large correlation length $\xi\gg 1$ (s.t.
$\langle\phi(x)\phi(0)\rangle\sim\mathrm{e}^{-|x|/\xi}$). The fluctuation must
be smooth enough so as not to create sharp domain walls that trap gapless
domain-wall fermions and close the fermion gap. But the fluctuation must also
not be too smooth to establish long-range order of the Yukawa field and break
the symmetry. The SMG should be achieved by balancing these two factors.

An alternative way to argue for the SMG is the _topological defect
condensation_ approach [17, 16, 94, 95, 154]. This approach also starts by
condensing the Yukawa field with a finite amplitude, but then disordering the
Yukawa field orientation by condensing topological point defects (e.g.
vortices in 2D space or monopoles in 3D space) of the Yukawa field. Although
the point defects could trap fermion zero modes and leads to gapless fermion
excitations in the spectrum, these fermion zero modes can be gapped out by
local interactions (such as the FK interaction) uniformly applied throughout
the system. A non-trivial check in this approach is to show that for anomaly-
free fermion systems (that admits SMG), the point defect always trap $8n$
Majorana zero modes that can be trivialized by interaction. The advantage of
the topological defect condensation approach is that it does not rely on a
delicate tuning of the smoothness of the Yukawa field fluctuation.

Further justification of the fluctuating bilinear mass picture comes from a
more explicit trial wave function construction for the SMG state in (0+1)D
[35]. Let $|\Psi[\phi]\rangle$ be the quantum many-body ground state of
fermions $\psi$ on a background configuration of the Yukawa field $\phi$. The
fluctuating bilinear mass picture Eq. (33) suggest that the SMG state (the
featureless gapped state in the strong interacting limit) should be described
by

$|\Psi_{\text{SMG}}\rangle\propto\int\mathcal{D}[\phi]\mathrm{e}^{-S_{\text{H}}[\phi]}|\Psi[\phi]\rangle.$
(35)

In the (0+1)D example, the ground state of the Yukawa Hamiltonian
$H_{\text{Y}}=-\phi_{\alpha}\chi_{a}\Gamma_{ab}^{\alpha}\chi_{b}$ is
explicitly given by
$|\Psi[\phi]\rangle=(1+\frac{1}{|\phi|}\phi_{\alpha}\chi_{a}\Gamma_{ab}^{\alpha}\chi_{b}+c_{1}^{\dagger}c_{2}^{\dagger}c_{3}^{\dagger}c_{4}^{\dagger})|0000\rangle$
using notations introduced in Sec. II.1. Assuming $S_{\text{H}}[\phi]$
restricts the $\phi$ vector uniformly on a sphere, the spherical average
results in
$|\Psi_{\text{SMG}}\rangle\propto(1+c_{1}^{\dagger}c_{2}^{\dagger}c_{3}^{\dagger}c_{4}^{\dagger})|0000\rangle$
which indeed matches the ground state $|\mathsf{4e}\rangle$ of the FK
interaction. The construction also applies to $\mathfrak{su}(4)$ symmetric
SMGs in all higher dimensions, where the on-site interaction is the same as
the (0+1)D case. In the limit that the Yukawa coupling is strong and the
Yukawa field $\phi$ fluctuates independently on each site, the trial wave
function $|\Psi_{\text{SMG}}\rangle$ in Eq. (35) approaches to product state
$\bigotimes_{i}|\mathsf{4e}\rangle_{i}$ — the exact ground state in the strong
coupling ($g\to\infty$) limit. Away from the strong coupling limit,
$|\Psi_{\text{SMG}}\rangle$ might still serve as a variational approximation
of the SMG state, but this variational ansatz is less well-controlled beyond
$(0+1)$D.

Although the fluctuating bilinear mass picture provides a simple understanding
for the SMG phase/state, it breaks down near the SMG transition, where the
Yukawa field amplitude is no longer large and stable. A major theoretical
challenge is to develop field theory descriptions for the continuous SMG
transition, which will be the topic of the following subsection Sec. IV.2.

###  IV.2 Fermion Fractionalization Field Theory

The _fermion fractionalization_ [35, 141] provides an overarching theoretical
framework to understand various continuous SMG transitions in different
dimensions. It unifies different theoretical pictures of SMG in literature.
Its key proposal is that the physical fermion $\psi\sim\upeta^{n}\uppsi$
fractionalizes into bosonic $\upeta$ and fermionic $\uppsi$ partons that
deconfine _at and only at_ the transition, where the bosonic parton $\upeta$
may appear $n$ multiple times in the fractionalization scheme. Note that
upright greek letters will be used to denote parton fields in the parton
theory.

The fermionic parton $\uppsi$ is generally put in the same gapless phase as
the physical fermion $\psi$. The physical free fermion phase corresponds to
the condensed phase of the bosonic parton $\upeta$, and the SMG phase
corresponds to the disordered (gapped) phase of the bosonic parton $\upeta$.
Accompanied with the fractionalization, an additional gauge field $a$ must be
introduced to bind the partons together and to remove the redundant degrees of
freedom introduced by fractionalization. Let $K$ be the emergent gauge group
(typically non-Abelian), then the partons are generally charged under an
enlarged symmetry-gauge group $G\times K$ (where $G$ is the symmetry group of
the physical fermion), as summarized by Tab. 4.

Table 4: Representations of different fields under $G\times K$, also under spacetime symmetry $\mathrm{Spin}(d)$. Representation assignments must be consistent with $\psi\sim\upeta^{n}\uppsi$, $\phi\sim\bar{\psi}\Gamma\psi$, and $\upphi\sim\bar{\uppsi}\Gamma\uppsi$. | $G$ | $K$ | $\mathrm{Spin}$ | meaning
---|---|---|---|---
$\psi$ | $\mathbf{r}_{\psi}^{G}$ | $\mathbf{1}^{K}$ | spinor | physical fermion
$\phi$ | $\mathbf{r}_{\phi}^{G}$ | $\mathbf{1}^{K}$ | scalar | Yukawa boson
$\upeta$ | $\mathbf{r}_{\upeta}^{G}$ | $\bar{\tiny\yng(1)}^{K}$ | scalar | bosonic parton
$\uppsi$ | $\mathbf{r}_{\uppsi}^{G}$ | $\mathbf{r}_{\uppsi}^{K}$ | spinor | fermionic parton (like $\psi$)
$a$ | $\mathbf{1}^{G}$ | $\bar{\tiny\yng(1)}\tiny\yng(1)^{K}$ | vector | gauge boson
$\upphi$ | $\mathbf{r}_{\upphi}^{G}$ | $\mathbf{r}_{\upphi}^{K}$ | scalar | parton-Higgs boson (like $\phi$)

The choice of representations should satisfy the following general rules:

  * •

The bosonic parton $\upeta$ is always in the (anti)-fundamental representation
$\bar{\tiny\yng(1)}^{K}$ of $K$, such that its condensation can fully Higgs
the gauge group $K$.

  * •

The representation must be assigned in consistent with the fermion
fractionalization $\psi\sim\upeta^{n}\uppsi$, such that the following fusion
channel must exist

$(\mathbf{r}_{\upeta}^{G},\bar{\tiny\yng(1)}^{K})^{\times
n}\times(\mathbf{r}_{\uppsi}^{G},\mathbf{r}_{\uppsi}^{K})\to(\mathbf{r}_{\psi}^{G},\mathbf{1}^{K}).$
(36)

  * •

The $K$-gauge field $a$ is always in the trivial representation
$\mathbf{1}^{G}$ of $G$ and the adjoint representation
$\bar{\tiny\yng(1)}\tiny\yng(1)^{K}$ of $K$.

Given the representation $(\mathbf{r}_{\psi}^{G},\mathbf{1}^{K})$ of the
physical fermion $\psi$, there is still the freedom to choose the gauge group
$K$ and the representation $(\mathbf{r}_{\uppsi}^{G},\mathbf{r}_{\uppsi}^{K})$
of the fermionic parton $\uppsi$. Once the choice is made, the multiplicity
$n$ and the representation $(\mathbf{r}_{\upeta}^{G},\bar{\tiny\yng(1)}^{K})$
of the bosonic parton $\upeta$ are fixed by Eq. (36). There could be multiple
parton theories that describe the same SMG transition. Within the present
framework, different parton theories are specified by different choices of $K$
and $(\mathbf{r}_{\uppsi}^{G},\mathbf{r}_{\uppsi}^{K})$.

The parton theory for the SMG transition generally takes the form of [35, 141]

$\begin{split}Z&=\int\mathcal{D}[\upeta,\uppsi,a]\mathrm{e}^{-S_{\text{B}}[\upeta,a]-S_{\text{F}}[\uppsi,a]},\\\
S_{\text{B}}[\upeta,a]&=\int\mathrm{d}^{d}x\;(|D_{\mu}\upeta|^{2}+r|\upeta|^{2}+u|\upeta|^{4}),\\\
S_{\text{F}}[\uppsi,a]&=\int\mathrm{d}^{d}x\;\bar{\uppsi}{\mathrm{i}}\gamma^{\mu}D_{\mu}\uppsi,\end{split}$
(37)

where $\uppsi$ and $\upeta$ are treated as real fields (with
$\bar{\uppsi}=\uppsi^{\intercal}\gamma^{0}$) to avoid unnecessary
complications of complex conjugation, and
$D_{\mu}=\partial_{\mu}-\mathrm{i}a_{\mu}$ denotes the covariant derivative
(assuming the gauge connection $a$ is automatically represented in the
representation of the matter field that it couples to). The parton theory
describes the bosonic $\upeta$ and the fermionic $\uppsi$ partons coupled
together by the gauge field $a$, and is expected to provide an effective
description of the low-energy physics around the SMG critical point. The
bosonic parton mass $r$ is treated as the only driving parameter in the parton
theory, and is responsible for tuning the SMG transition. $r<0$ corresponds to
the gapless phase (the free-fermion fixed point), $r>0$ corresponds to the
trivially gapped phase (the strong-coupling fixed point), and $r=0$
corresponds to the critical point where the SMG transition happens. The states
of different fields in different cases are summarized in Tab. 5.

Table 5: States of different fields across the SMG transition. | gapless phase | SMG | gapped phase
---|---|---|---
| $(r<0)$ | $(r=0)$ | $(r>0)$
$\psi$ | gapless | fractionalized | gapped
$\phi$ | gapped ($\xi\sim 1$) | gapped | gapped
$\upeta$ | condensed | critical | gapped
$\uppsi$ | gapless | gapless | gapped
$a$ | Higgs | deconfined | Higgs/confined
$\upphi$ | gapped | critical | condensed

When $r<0$, the bosonic parton $\upeta$ condenses (i.e.
$\langle\upeta\rangle\neq 0$). As $\upeta$ carries the anti-fundamental
representation of the gauge group $K$, the gauge structure is fully Higgs down
by the bosonic parton condensation. At the same time, the fermionic parton
$\uppsi$ becomes the physical fermion $\psi\sim\langle\upeta\rangle^{n}\uppsi$
effectively (as the fields $\uppsi$ and $\psi$ only differ by a constant
$\langle\upeta\rangle^{n}$ which simply serves as a quasi-particle weight).
The parton theory then reduces to the physical fermion theory
$S[\psi]=\int\mathrm{d}^{d}x\;\bar{\psi}{\mathrm{i}}\gamma^{\mu}\partial_{\mu}\psi$
in the gapless free-fermion phase.

When $r>0$, the bosonic parton $\upeta$ is gapped and decouples from the gauge
field at low-energy. Below the gap of $\upeta$ field, the parton theory is
effectively a quantum chromodynamics (QCD) theory $S_{\text{F}}[\uppsi,a]$
involving the fermionic parton $\uppsi$ coupled to the gauge field $a$. Then
it relies on an assumption that the QCD dynamics leads to a featureless gapped
ground state (a.k.a. a trivial ground state), such that both the fermionic
partons and the gauge bosons are gapped without breaking the $G$ symmetry or
generating topological orders. Of course, one necessary condition for the QCD
theory to trivialize is that the theory must have vanishing $G\times K$
anomaly. However, even if the QCD theory is anomaly free, it still depends on
the dynamical details to achieve a trivially gapped ground state. The major
theoretical effort to understand SMG lies in designing appropriate mechanism
to trivialize the QCD theory. Within the fermion fractionalization framework,
two mechanisms has been proposed in the literature: (i) parton-Higgs (weak
coupling) [35, 141, 34] and (ii) s-confinement (strong coupling) [98, 34]. In
some cases, the QCD trivialization may be consistently achieved by both
mechanisms.

####  IV.2.1 Parton-Higgs mechanism

The parton-Higgs mechanism introduces a collection of scalar fields $\upphi$
(parton-Higgs fields) that couple to the fermionic parton $\uppsi$ via Yukawa
interactions, such that the QCD theory $S_{\text{F}}[\uppsi,a]$ is extended to
a QCD-Yukawa-Higgs theory

$\begin{split}S_{\text{F}}[\uppsi,\upphi,a]=\int\mathrm{d}^{d}x\;(&\bar{\uppsi}{\mathrm{i}}\gamma^{\mu}D_{\mu}\uppsi+\upphi_{\alpha}\bar{\uppsi}\Gamma^{\alpha}\uppsi\\\
+&(D_{\mu}\upphi)^{2}+V_{\text{H}}(\upphi)),\end{split}$ (38)

where the parton-Higgs field $\upphi$ (as fermionic parton bilinear mass) is
in the representation $(\mathbf{r}_{\upphi}^{G},\mathbf{r}_{\upphi}^{K})$ of
$G\times K$. The representation
$(\mathbf{r}_{\upphi}^{G},\mathbf{r}_{\upphi}^{K})$ must be obtained from an
antisymmetric product of fermionic parton representations for consistency

$(\mathbf{r}_{\uppsi}^{G},\mathbf{r}_{\uppsi}^{K})\times_{\mathcal{A}}(\mathbf{r}_{\uppsi}^{G},\mathbf{r}_{\uppsi}^{K})\to(\mathbf{r}_{\upphi}^{G},\mathbf{r}_{\upphi}^{K}).$
(39)

The Yukawa vertex tensor $\Gamma^{\alpha}$ is fixed by this fusion channel.
The QCD-Yukawa-Higgs theory in Eq. (38) is reminiscent of the Yukawa-Higgs
theory in Eq. (33) by promoting the physical fermion $\psi$ and physical
Yukawa field $\phi$ to their parton counterparts $\uppsi$ and $\upphi$.
However, the key difference lies in the different $G\times K$ representations
of $\psi,\phi$ comparing to $\uppsi,\upphi$, as listed in Tab. 4.

Unlike the physical Yukawa field $\phi$, which is solely charged under $G$ in
a non-trivial representation $\mathbf{r}_{\phi}^{G}$, such that the
condensation of $\phi$ inevitably breaks the $G$ symmetry; the parton-Higgs
field $\upphi$ is in a joint representation
$(\mathbf{r}_{\upphi}^{G},\mathbf{r}_{\upphi}^{K})$ of $G\times K$, which
could admit a $G$-symmetric condensation of $\upphi$. The sufficient and
necessary condition for the existence of a condensed configuration
$\langle\upphi\rangle$ that preserves the $G$ symmetry is that there exists a
subgroup $G\times K^{\prime}$ of $G\times K$, such that when $G\times K$ is
broken to $G\times K^{\prime}$, the representation of $\upphi$ admits a
trivial branching channel:

$(\mathbf{r}_{\upphi}^{G},\mathbf{r}_{\upphi}^{K})\to(\mathbf{1}^{G},\mathbf{1}^{K^{\prime}}).$
(40)

If the condition is met, a Higgs potential $V_{\text{H}}(\upphi)$ can be
constructed to drive the condensation of $\upphi$ in the above trivial
branching channel. Such a condensation of the parton-Higgs field $\upphi$ will
(i) gap out all fermionic partons $\uppsi$ (that couple to it), (ii) Higgs
down the gauge group from $H$ to its subgroup $K^{\prime}\subseteq K$, and
(iii) preserving the symmetry group $G$. Then the only freedom remaining in
the theory is the pure gauge fluctuation of $K^{\prime}$. If $K^{\prime}$
remains a non-Abelian group, its gauge coupling can flow strong into the
confine phase and gap out the remaining gauge bosons. In this way, all
freedoms in the QCD-Yukawa-Higgs theory are symmetrically gapped and the
system ends up in the SMG phase. The physical fermion $\psi\sim\upeta\uppsi$
will also be gapped, as its partons are gapped (both $\upeta$ and $\uppsi$ are
gapped).

####  IV.2.2 s-confinement mechanism

The s-confinement mechanism (“s” for “smooth”) refers to the strong coupling
dynamics of the QCD theory that confines fermionic partons and gauge bosons
all together without breaking the $G$ symmetry. This is possible if the ’t
Hooft anomaly of $G$ is matched up in the theory [3]. One motivation for the
s-confinement theory comes from the observation that the four-fermion SMG
interaction $\psi^{4}$ (for physical fermions $\psi$) can be broken up into
“3+1” as $(\psi^{3})\psi$. If there is some strong-coupling mechanism that
binds the first three fermions into a composite fermion
$\psi_{\text{comp}}\sim\psi^{3}$ at low-energy, the interaction can then be
viewed as a fermion bilinear mass like
$\bar{\psi}_{\text{free}}\psi_{\text{comp}}$ (where
$\bar{\psi}_{\text{free}}\sim\psi$ stands for the last dangling fermion),
which provides a free-fermion understanding for the SMG. This picture is
alternative to the “2+2” fluctuation bilinear mass picture, where the
interaction is broken up into $(\psi^{2})(\psi^{2})$ with each $\psi^{2}$
being a composite boson (i.e. the Yukawa boson $\phi$). However, the Yukawa
boson is still interacting, which complicates the analysis of its low-energy
dynamics. In contrast, the composite fermion flows to a free fermion critical
point, which enables simple argument for gapping. Nevertheless, this still
relies on a strong-coupling mechanism for the composite fermion to form in the
first place.

The fermion fractionalization plays an important role in understanding the
formation of composite fermions in the s-confinement picture. After
fractionalization, physical fermions become fermionic partons that couple to
an emergent gauge field, then the emergent gauge force can be employed to
confine the fermions into the desired composite. Not all fermions are
fractionalized in the s-confinement theory. The physical fermions
$\psi=\psi_{\text{frac}}\oplus\psi_{\text{free}}$ is first divided into two
sectors, where $\psi_{\text{frac}}\sim\upeta^{n}\uppsi$ are part of the
physical fermions that will be fractionalized to bosonic $\upeta$ and
fermionic $\uppsi$ partons, and $\psi_{\text{free}}$ are the remaining
physical fermions that do not fractionalize. To drive the SMG transition, the
first step is still to gap the bosonic parton $\upeta$, leaving the low-energy
freedom in a QCD theory with the fermionic parton $\uppsi$ coupled to an
emergent (non-Abelian) gauge field $a$. Under gauge confinement, the fermionic
partons $\uppsi$ are bound together in the IR to form fermion composites,
denoted as $\psi_{\text{comp}}\sim(\uppsi)^{3}$, just like fundamental quarks
forming baryons. The necessary condition for the confinement to happen is the
existence of a trivializing fusion channel in the gauge sector

$(\mathbf{r}_{\uppsi}^{K})^{\times 3}\to\mathbf{1}^{K},$ (41)

such that the gauge charges of $\uppsi^{3}$ can be neutralized to produce the
physical composite fermion $\psi_{\text{comp}}$. Then the fermion bilinear
mass $\bar{\psi}_{\text{free}}\psi_{\text{comp}}$ can be introduced to gap out
all fermions together, realizing the SMG. Moreover, the s-confinement
mechanism admits the supersymmetric extension [34], where addition of
supersymmetry provides extra control for the strong-coupling confinement
dynamics at low-energy [155].

###  IV.3 Examples of Fermion Fractionalization

In the following, we will briefly exemplify the fermion fractionalization
theory in two SMG models introduced in Sec. II.

####  IV.3.1 (2+1)D Honeycomb Lattice Model

The SMG in the (2+1)D honeycomb lattice model can be understood using the
parton-Higgs mechanism within the framework of fermion fractionalization [35].
The internal symmetry in consideration is

$G=\mathrm{SU}(4)\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4}^{TF},$ (42)

where $\mathbb{Z}_{4}^{TF}$ is the essential protecting symmetry. The physical
fermion $\psi$ is in the $\mathbf{4}_{1}$ representation of $G$. The proposed
fermion fractionalization scheme $\psi\sim\upeta\uppsi$ introduces an emergent
gauge group

$K=\mathrm{SU}(4),$ (43)

and put the fermionic parton $\uppsi$ in the $(\mathbf{1}_{1},\mathbf{4})$
representation of $G\times K$. This choice fully specifies the parton theory.
The physical and fractionalized fields in the parton construction are
summarized in Tab. 6, such that all consistency conditions are met among their
symmetry representations.

Table 6: Representations of different fields under $(\mathrm{SU}(4)\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4}^{TF})_{G}\times\mathrm{SU}(4)_{K}$, as well as the Euclidean spacetime symmetry $\mathrm{Spin}(3)$ , which are relevant to the SMG in the (2+1)D honeycomb lattice model. | $(\mathrm{SU}(4)\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4}^{TF})_{G}$ | $\mathrm{SU}(4)_{K}$ | $\mathrm{Spin}$
---|---|---|---
$\psi$ | $\tiny\yng(1)_{1}=\mathbf{4}_{1}$ | $\mathbf{1}$ | spinor (fermion)
$\phi$ | $\tiny\yng(1,1)_{2}=\mathbf{6}_{2}$ | $\mathbf{1}$ | scalar (boson)
$\upeta$ | $\tiny\yng(1)_{0}=\mathbf{4}_{0}$ | $\bar{\tiny\yng(1)}=\bar{\mathbf{4}}$ | scalar (boson)
$\uppsi$ | $\mathbf{1}_{1}$ | $\tiny\yng(1)=\mathbf{4}$ | spinor (fermion)
$a$ | $\mathbf{1}_{0}$ | $\bar{\tiny\yng(1)}\tiny\yng(1)=\mathbf{15}$ | vector (boson)
$\upphi$ | $\mathbf{1}_{2}$ | $\tiny\yng(1,1)=\mathbf{6}$ | scalar (boson)

The SMG can be achieved through the parton-Higgs mechanism by gapping out the
bosonic parton $\upeta$ and condensing the parton Yukawa field
$\upphi\sim\bar{\uppsi}\Gamma\uppsi$. This breaks $G\times K$ to $G\times
K^{\prime}$, where the new gauge group is

$K^{\prime}=\mathrm{Sp}(2)\subseteq\mathrm{SU}(4)=K.$ (44)

In particular, because the parton Yukawa field $\upphi$ charges 2 under both
the $\mathbb{Z}_{4}^{TF}$ symmetry group and the $\mathbb{Z}_{4}^{K}$ center
of the gauge group $K=\mathrm{SU}(4)$, condensing $\upphi$ necessarily locks
the $\mathbb{Z}_{4}^{TF}$ and $\mathbb{Z}_{4}^{K}$ generators together,
breaking $\mathbb{Z}_{4}^{TF}\times\mathbb{Z}_{4}^{K}$ down to
$\mathbb{Z}_{4}^{TF}\times\mathbb{Z}_{2}^{K^{\prime}}$, which is consistent
with the $\mathbb{Z}_{2}^{K^{\prime}}$ center of residual gauge group
$K^{\prime}=\mathrm{Sp}(2)$. As $G\times K$ is broken to $G\times K^{\prime}$,
the representation of $\upphi$ splits into

$(\mathbf{1}_{2},\mathbf{6})\to(\mathbf{1}_{0},\mathbf{1})\oplus(\mathbf{1}_{0},\mathbf{5}),$
(45)

which admits a trivial representation $(\mathbf{1}_{0},\mathbf{1})$ in the
decomposition, satisfying the trivializing condition in Eq. (40) for the
parton-Higgs mechanism to work.

####  IV.3.2 (3+1)D Chiral Fermion Model

The (3+1)D chiral fermion model contains three sectors of physical fermions
$\psi=\psi_{\lambda}\oplus\psi_{\psi}\oplus\psi_{\chi}$ transformed under the
symmetry group

$G=\mathrm{SU}(N)\times\mathrm{SU}(N+4).$ (46)

The SMG in this model can be understood by fermion fractionalization, where
only part of the fermions $\psi_{\text{frac}}=\psi_{\psi}\oplus\psi_{\chi}$
are fractionalized: $\psi_{\psi}\sim\upeta^{\dagger}\uppsi_{\psi}$ and
$\psi_{\chi}\sim\upeta^{2}\uppsi_{\chi}$, while
$\psi_{\text{free}}=\psi_{\lambda}$ remains untouched. The fractionalization
introduces an emergent gauge group

$K=\mathrm{SU}(N+4).$ (47)

Tab. 7 summarizes the representation of fields satisfying all consistency
conditions.

Table 7: Representations of different fields under $(\mathrm{SU}(N)\times\mathrm{SU}(N+4))_{G}\times\mathrm{SU}(N+4)_{K}$, as well as the Euclidean spacetime symmetry $\mathrm{Spin}(4)$, which are relevant to the SMG in the (3+1)D chiral fermion model. | $(\mathrm{SU}(N)\times\mathrm{SU}(N+4))_{G}$ | $\mathrm{SU}(N+4)_{K}$ | $\mathrm{Spin}$
---|---|---|---
$\psi_{\lambda}$ | $(\tiny\yng(2),\mathbf{1})$ | $\mathbf{1}$ | spinor (fermion)
$\psi_{\psi}$ | $(\bar{\tiny\yng(1)},\bar{\tiny\yng(1)})$ | $\mathbf{1}$ | spinor (fermion)
$\psi_{\chi}$ | $(\mathbf{1},\tiny\yng(1,1))$ | $\mathbf{1}$ | spinor (fermion)
$\phi$ | $(\bar{\tiny\yng(1)},\tiny\yng(1))$ | $\mathbf{1}$ | scalar (boson)
$\upeta$ | $(\mathbf{1},\tiny\yng(1))$ | $\bar{\tiny\yng(1)}$ | scalar (boson)
$\uppsi_{\psi}$ | $(\bar{\tiny\yng(1)},\mathbf{1})$ | $\bar{\tiny\yng(1)}$ | spinor (fermion)
$\uppsi_{\chi}$ | $(\mathbf{1},\mathbf{1})$ | $\tiny\yng(1,1)$ | spinor (fermion)
$a$ | $(\mathbf{1},\mathbf{1})$ | $\bar{\tiny\yng(1)}\tiny\yng(1)$ | vector (boson)
$\upphi$ | $(\bar{\tiny\yng(1)},\mathbf{1})$ | $\tiny\yng(1)$ | scalar (boson)

To drive the chiral fermions $\psi$ to the SMG phase, the first step is to gap
out the bosonic parton $\upeta$. Then the theory contains a free fermion
theory of $\psi_{\lambda}$ and a QCD theory of the fermionic partons
$\uppsi_{\psi}\oplus\uppsi_{\chi}$ coupled to the $K$ gauge field $a$. The
trivialization of the QCD theory can be achieved by either the parton-Higgs
mechanism or the s-confinement mechanism (they are equivalent).

In the parton-Higgs mechanism, two parton-Higgs fields
$\upphi=\upphi_{1}\oplus\upphi_{2}$ are introduced and coupled to the fermions
as

$-(\upphi_{1}^{\dagger}(\uppsi_{\chi}^{\intercal}\mathrm{i}\sigma^{2}\uppsi_{\psi})+\upphi_{2}(\psi_{\lambda}^{\intercal}\mathrm{i}\sigma^{2}\uppsi_{\psi})+\text{h.c.}),$
(48)

which is closely reminiscent of the Yukawa-Higgs decomposition Eq. (25) of the
SMG interaction in the chiral fermion model. Both parton-Higgs fields
$\upphi_{1,2}$ are in the representation
$((\bar{\tiny\yng(1)},\mathbf{1}),{\tiny\yng(1)})$ of $G\times K$. Condensing
the parton-Higgs field $\upphi$ in a “color-flavor” locked form, the
$\mathrm{SU}(N)$ (flavor) subgroup in $G$ will be identified with the
$\mathrm{SU}(N)$ (color) subgroup in $K=\mathrm{SU}(N+4)$, such that the
color-flavor combined $\mathrm{SU}(N)$ rotation will form the diagonal
$\mathrm{SU}(N)$ group that leaves $\upphi$ invariant. The color-flavor
locking breaks the $G\times K$ group to its subgroup $G\times K^{\prime}$ with
666Some additional $\mathrm{U}(1)$ subgroups will be ignored to simplify the
discussion, without hurting the main idea. See [34] for a more rigorous
treatment.

$K^{\prime}=\mathrm{SU}(4)\subseteq\mathrm{SU}(N+4)=K.$ (49)

Under $G\times K\to G\times K^{\prime}$, the representation of $\upphi$
decomposes as

$\upphi:((\bar{\tiny\yng(1)},\mathbf{1}),{\tiny\yng(1)})\to((\mathbf{1},\mathbf{1}),\mathbf{1})\oplus((\bar{\tiny\yng(1)},\mathbf{1}),\mathbf{4})\oplus((\bar{\tiny\yng(1)}{\tiny\yng(1)},\mathbf{1}),\mathbf{1})$
(50)

which contains the trivial representation
$((\mathbf{1},\mathbf{1}),\mathbf{1})$ that corresponds to the condensed
configuration $\langle\upphi\rangle$, fulfilling the requirement of Eq. (40).
Moreover, representations of fermions branch as

$\begin{split}\psi_{\lambda}&:(({\tiny\yng(2)},\mathbf{1}),\mathbf{1})\to(({\tiny\yng(2)},\mathbf{1}),\mathbf{1}),\\\
\uppsi_{\psi}&:((\bar{\tiny\yng(1)},\mathbf{1}),\bar{\tiny\yng(1)})\to((\overline{\tiny\yng(2)},\mathbf{1}),\mathbf{1})\oplus((\bar{\tiny\yng(1,1)},\mathbf{1}),\mathbf{1})\oplus((\bar{\tiny\yng(1)},\mathbf{1}),\bar{\mathbf{4}}),\\\
\uppsi_{\chi}&:((\mathbf{1},\mathbf{1}),\tiny\yng(1,1))\to(({\tiny\yng(1,1)},\mathbf{1}),\mathbf{1})\oplus(({\tiny\yng(1)},\mathbf{1}),{\mathbf{4}})\oplus((\mathbf{1},\mathbf{1}),\mathbf{6}).\\\
\end{split}$ (51)

The fermion representation has transmuted, morphing from a chiral
representation of $G\times K$ into a vector-like representation of the
surviving $G\times K^{\prime}$. The parton-Higgs field condensation
$\langle\upphi\rangle\neq 0$ will provide mass for almost all fermions via the
Yukawa coupling in Eq. (48) (by pairing up conjugate fermion representations),
with the only exception of the $((\mathbf{1},\mathbf{1}),\mathbf{6})$ fermions
in $\uppsi_{\chi}$. But these fermions are fully neural under the symmetry
group $G$ and only charged under the remaining gauge group $K^{\prime}$. The
remaining $K^{\prime}$ gauge bosons and the
$((\mathbf{1},\mathbf{1}),\mathbf{6})$ fermions can be simultaneously removed
from the low-energy spectrum once the $K^{\prime}$ gauge fluctuation flows
strong and confines. Hence no low-energy freedom is left over, and the system
ends up in the SMG phase.

In the s-confinement mechanism, a gauge-neutral combination of fermionic
partons must first exist, as required by Eq. (41). As the fermionic partons
$\uppsi_{\psi}\oplus\uppsi_{\chi}$ carry the gauge charges
$\bar{\tiny\yng(1)}\oplus{\tiny\yng(1,1)}$ respectively (see Tab. 7), the
gauge neural three-fermion combination can be found in the fusion channel

${\tiny\yng(1,1)}\times\bar{\tiny\yng(1)}\times\bar{\tiny\yng(1)}\to\mathbf{1}$
(52)

So the fermionic partons can be confined into a composite fermion through this
confinement channel as

$\psi_{\text{comp}}\sim(\uppsi_{\chi}^{\intercal}\mathrm{i}\sigma^{2}\uppsi_{\psi})\uppsi_{\psi},$
(53)

which transforms as $((\overline{\tiny\yng(2)},\mathbf{1}),\mathbf{1})$ under
$G\times K$, precisely conjugate to the representation
$(({\tiny\yng(2)},\mathbf{1}),\mathbf{1})$ of the remaining physical fermion
$\psi_{\lambda}$. Therefore, the composite fermion $\psi_{\text{comp}}$ can
pair up with the remaining fermion $\psi_{\lambda}$ to produce a $G$-symmetric
mass term $\psi_{\lambda}^{\intercal}\mathrm{i}\sigma^{2}\psi_{\text{comp}}$,
that gaps out all fermions from the low-energy spectrum, leading to the SMG
state.

###  IV.4 Symmetry Extension Construction

####  IV.4.1 Symmetry Breaking vs Symmetry Extension vs SMG

From the symmetry perspective, the fermion mass generation generally falls
into two categories: (i) the _symmetry breaking_ mechanism and (ii) the
_symmetry extension_ mechanism [23, 24], see Fig. 4. At a high level, the SMG
is essentially an _symmetry extension_ mechanism in the _absence_ of the ’t
Hooft anomaly. (Strictly speaking, the SMG mechanism does not enlarge the
physical symmetry, but rather extends it to a symmetry-gauge structure.
Nevertheless, we will still call it by symmetry extension in a general sense.)

Figure 4: Classification of mass generation mechanisms.

  1. 1.

Symmetry/Gauge breaking: Anderson-Higgs mechanism, chiral symmetry breaking,
Dirac mass and Majorana mass are induced by the symmetry breaking — breaking
either global symmetries or gauge structures, by condensing a Yukawa-Higgs
field that couples to a fermion bilinear mass term. More precisely, starting
from a symmetry group $G$ (specifically here an internal symmetry, global or
gauged), $G$ is broken down to an appropriate subgroup $G^{\prime}\subseteq G$
to induce quadratic mass term for fermions. Mathematically, it can be
described by an _injective_ homomorphism ${\iota}$:

$G^{\prime}\xrightarrow{\iota}G.$ (54)

Here are some explicit examples:
$\bullet$ Bardeen-Cooper-Schrieffer (BCS) type $\mathbb{Z}_{2}$-gauged
superconductor with a low energy $\mathbb{Z}_{2}$ TQFT, we have
$G^{\prime}=\mathbb{Z}_{2}$ and $G=\mathrm{U}(1)$ electromagnetic gauge group.
$\bullet$ The Standard Model electroweak Higgs mechanism breaks
$G=G_{\text{SM}_{q}}\equiv(\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)_{\text{Y}})/\mathbb{Z}_{q}$
with $q=1,2,3,6$ and the appropriate greatest common divisor (gcd), down to
$G^{\prime}=(\mathrm{SU}(3)\times\mathrm{U}(1)_{\text{EM}})/\mathbb{Z}_{\gcd(q,3)}$.
$\bullet$ (3+1)D Dirac mass pairs two Weyl fermions ($\psi_{L}$ and
$\psi_{R}$) via the Dirac mass term $m_{\rm
D}(\psi_{L}^{\dagger}\psi_{R}+\psi_{R}^{\dagger}\psi_{L})$ which breaks the
unitary internal $G=\mathrm{U}(2)$ symmetry of two Weyl fermions down to a
vector $G^{\prime}=\mathrm{U}(1)$ symmetry.
$\bullet$ (1+1)D Dirac mass pairs two Weyl fermions ($\psi_{L}$ and
$\psi_{R}$) via the Dirac mass term $m_{\rm
D}(\psi_{L}^{\dagger}\psi_{R}+\psi_{R}^{\dagger}\psi_{L})$ which breaks the
unitary internal $G=\mathrm{U}(1)_{L}\times\mathrm{U}(1)_{R}$ symmetry of two
Weyl fermions down to a vector $\mathrm{U}(1)$ symmetry, so
$G^{\prime}=\mathrm{U}(1)$.
$\bullet$ (3+1)D Majorana mass pairs a single Weyl fermion $\psi_{L}$ to
itself, $\psi_{R}=\mathrm{i}\sigma^{2}\psi_{L}^{*}$, so the Majorana mass term
$m_{\rm M}\bar{\psi}\psi=m_{\rm
M}(\psi_{L}^{\dagger}(\mathrm{i}\sigma^{2})\psi^{*}_{L}+{\psi^{\rm
T}_{L}}(-\mathrm{i}\sigma^{2})\psi_{L})$, which breaks the unitary internal
$G=\mathrm{U}(1)$ symmetry of a Weyl fermion down to a fermion parity
$G^{\prime}=\mathbb{Z}_{2}^{F}$ symmetry.

  2. 2.

Symmetry/Gauge extension: In contrast, the symmetry extension [24] or gauge
enhancement [157] provides a fermion mass generation mechanism that preserves
the symmetry. Symmetry extension construction of gapped phases first appears
in [23] based on the gauge bundle descriptions, then Ref. 24 refines the idea
to the lattice models, group-cohomology cocycle or continuum field theory
descriptions. It extends the symmetry group $G$ (that can include the
spacetime-internal symmetry, global or gauged) to a larger group $\tilde{G}$
by enlarging the Hilbert space with additional/redundant degrees of freedom,
in order to trivialize the ’t Hooft anomaly or to lift any other symmetry
obstruction towards a gapped phase with a mass generation. Mathematically, it
can be described by a _surjective_ homomorphism ${r}$:

$\displaystyle\tilde{G}\xrightarrow{r}G,$ (55)

which can be understood as part of the group extension in an exact sequence
(see Eq. (56)) [24], or can be further generalized as the fibrations of their
classifying spaces and higher classifying spaces [158, 159, 160]. The symmetry
extension can induce the _topological mass_ in the presence of anomaly, and
the _symmetric mass_ in the absence of anomaly [114].

$\bullet$ Topological Mass Generation (TMG) — If the fermion system has a non-
trivial ’t Hooft anomaly in $G$, the anomaly will post an obstruction toward
_trivial_ gapping, which already rules out SMG, leaving the possibility to
TMG. A non-vanishing _perturbative local anomaly_ disallows any symmetric
gapped phase (even with topological order), also it can never be trivialized
by a symmetry extension. So, in order to implement the symmetry extension
construction, the non-vanishing $G$-anomaly must be a _nonperturbative global
anomaly_ in $G$.

For simplicity, the discussion below focuses on a limited special case of
(55). If the global anomaly in $G$ can become anomaly-free in $\tilde{G}$, by
pulling the $G$ group back to the extended $\tilde{G}$ group via a short exact
sequence

$1\to K\to\tilde{G}\xrightarrow{r}G\to 1,$ (56)

and if the normal subgroup $K$ is a discrete finite group, then the fermion
can acquire a topological mass upon gauging $K$, which gives rise to a
discrete $K$ gauge TQFT in appropriate spacetime dimensions and under
appropriate criteria detailed in Sec. IV.4.2. This is also called the group
extension of the original quotient group $G$ extended by a normal subgroup $K$
to the total group $\tilde{G}$. The topological mass refers to the energy gap
of a TQFT matching the ’t Hooft anomaly of $G$. Finding the group extension
$\tilde{G}$ often requires the essential use of algebraic topology criteria,
such as the Lydon-Hochschild-Serre spectral sequence method [24]. Ref. 24,
159, 161, 57, 114, 162 provide several explicit lattice Hamiltonian or lattice
path integral constructions.

$\bullet$ Symmetric Mass Generation (SMG) — In the case of SMG, the fermion
system is already anomaly free, but the physical symmetry $G$ is too
restrictive to allow any symmetric fermion-bilinear mass. However, with the
symmetry extension described by the following short exact sequence

$1\to K\to\tilde{G}=G\times K\xrightarrow{r}G\to 1,$ (57)

the fermion-bilinear mass (i.e. the parton-Higgs field $\upphi$) can be
charged under both $G$ and $K$, which can possibly be condensed, breaking only
$K$ to its subgroup $K^{\prime}$ without breaking $G$, as long as the
condensed fermion-bilinear mass transform trivially under $G\times
K^{\prime}\subseteq G\times K$. Moreover, the extended normal subgroup $K$
(and $K^{\prime}$) can be (and will always be) gauged, such that the actual
physical symmetry will not be enlarged by the symmetry extension. This is
simply a rephrasing of the parton-Higgs mechanism in the fermion
fractionalization framework discussed in Sec. IV.2.

Figure 5: Symmetry-gauge groups in different phases of the Yukawa-Higgs field
and the bosonic parton field.

Fig. 5 concludes how the symmetry-gauge group is extended/broken in different
phases. A few general requirements for the SMG to happen are summarized as
follows:

     * •

The full gauge group $K$ must be large enough to counteract any non-trivial
action of the symmetry group $G$ on the parton-Higgs field $\upphi$, i.e. $G$
must acts _projectively_ on $\upphi$, such that the $G$ symmetry can remain
unbroken under the condensation of $\upphi$.

     * •

To achieve SMG in this framework, the deformation path must pass through the
strong coupling regime where neither the bosonic parton $\upeta$ nor the
parton-Higgs field $\upphi$ has an expectation value, and the full gauge group
$K$ is unbroken.

     * •

After the parton-Higgs field $\upphi$ condenses, the remaining unbroken gauge
group $K^{\prime}$ would better be either trivial or non-Abelian, such that
either there is no residual $K^{\prime}$ gauge fluctuation or the residual
$K^{\prime}$ gauge fluctuation can be confined automatically. Otherwise, if
$K^{\prime}$ is Abelian, it becomes possible that the SMG critical point will
expand into a critical phase, described by an Abelian $K^{\prime}$-gauge
theory.

####  IV.4.2 More on Symmetry Extension Construction

The symmetry extension construction $\tilde{G}\xrightarrow{r}G$ in (55) based
on the pullback $G$-symmetry to the extended $\tilde{G}$-symmetry can be
interpreted in different languages for different communities [24]:

$\bullet$ For condensed matter, a nontrivial SPT state in the $G$ symmetry
cannot be deformed to a trivial tensor product state via a finite-depth of
local unitary transformations without breaking the $G$-symmetry. But the
successful symmetry extension means that we can find an appropriate
$\tilde{G}$ such that the SPT state in the extended $\tilde{G}$ symmetric
Hilbert space can be deformed to a trivial tensor product state via a finite-
depth of local unitary transformations still preserving the
$\tilde{G}$-symmetry.

$\bullet$ For quantum field theory or high-energy physics, the successful
symmetry extension means that the ’t Hooft anomaly in $G$-symmetry becomes
anomaly-free in $\tilde{G}$-symmetry.

$\bullet$ For mathematics, the successful symmetry extension means that a
nontrivial class of cocycle, cohomology, or cobordism of the $G$-symmetry
becomes a trivial class in the $\tilde{G}$-symmetry. Suppose the nontrivial
class of $G$-symmetry cocycle, cohomology, or cobordism in the $D$ dimensions
is labeled by $\omega_{D}^{G}$, then the trivialization means that its
pullback (namely $r^{*}\omega_{D}^{G}$) becomes a $\tilde{G}$-symmetry
coboundary in the $D$ dimensions (namely
$\omega_{D}^{\tilde{G}}=\delta\beta_{D-1}^{\tilde{G}}$) which splits to the
$\tilde{G}$ cochain (namely $\beta_{D-1}^{\tilde{G}}$) in the $D-1$ dimensions
[24]. In summary, given the $\omega_{D}^{G}$ of the $G$-symmetry, the
successful symmetry extension requires to find a solution of both the extended
$\tilde{G}$ and $\beta_{D-1}^{\tilde{G}}$ to satisfy

$\displaystyle
r^{*}\omega_{D}^{G}=\omega_{D}^{\tilde{G}}=\delta\beta_{D-1}^{\tilde{G}}.$
(58)

Here the bulk dimension is $D=d+1$, while the theory (on the boundary) with ’t
Hooft anomaly has its dimension $D-1=d$. Below some examples of the symmetry
extension construction of gapped phases in various dimensions based on the
simplest short exact sequence in (56)

$1\to K\to\tilde{G}\xrightarrow{r}G\to 1$

with a finite group $K$, are considered. In particular, two issues should be
addressed:
(1) Given a theory with ’t Hooft anomaly in $G$ symmetry (which we shall also
call it a $d$D anomalous boundary theory of a $(d+1)$D bulk SPT), can
$\tilde{G}$ be found to trivialize the anomaly? If so, what is the minimal
$\tilde{G}$?
(2) If the $\tilde{G}$ is found, there are two implications of the
construction based on (56) [24]:
$\bullet$ $\tilde{G}$-symmetric extended gapped phase (as a
$\tilde{G}$-symmetric gapped boundary of the bulk $G$-SPT). In this case, all
$K,\tilde{G}$ and $G$ are not dynamically gauged.
$\bullet$ ${G}$-symmetric gapped dynamical $K$-gauge theory with a ’t Hooft
anomaly in $G$. When $K$ is dynamically gauged, in some cases, the $G$ is
preserved at IR; in other cases, the $G$ becomes spontaneous symmetry breaking
(SSB) at IR. One should also be careful to distinguish the two different kinds
of dynamics.

Ref. 24 proves that “ _given any unitary or anti-unitary finite group $G$, for
$d\geq 1$, there always exists a finite $K$-extension to a
$\tilde{G}$-symmetric extended gapped phase to trivialize any nonperturbative
global anomaly in $G$_ (see also the proof given later in [158]).” It is
physically meaningful to consider gauging $K$ for $d\geq 2$. Once $K$ is
dynamically gauged, Ref. 24 finds that _${G}$ -symmetric gapped dynamical
$K$-gauge theory with a ’t Hooft anomaly in $G$_ can only be obtained for
$d\geq 3$, but also finds that the $G$-symmetry is spontaneously broken for
$d=2$.

The above result is established when $G$ is an ordinary 0-symmetry that acts
on the 0D point operator. The symmetry extension construction can also be
generalized to higher global symmetries [163] such as the 1-symmetry that acts
on the 1D line operator (see a recent review [164]). Here are some examples
(Table 8 for a summary) including ordinary and higher symmetries:

$\begin{array}[]{c}\text{Dim}\\\ \text{$d$}\end{array}$ | Original System. $G$ | $K\to\tilde{G}\to G$ |  $\begin{array}[]{c}\text{Reduced class in $\tilde{G}$}\\\ \text{of $d+1$D $G$-iTQFT}\\\ \text{or $d$D $G$-anomaly}\end{array}$ | $\begin{array}[]{c}\text{Gauge $K$}\\\ \text{Dynamics}\end{array}$
---|---|---|---|---
0+1 |  $\begin{array}[]{c}\text{bdry of Haldane chain:}\\\ \text{a doublet or qubit}\end{array}$ | $\mathrm{SO}(3)$ | $\mathbb{Z}_{2}\to\mathrm{SU}(2)\to G$ | $\mathbb{Z}_{2}\Rightarrow 0$ | No
$\mathbb{Z}_{2}^{T}$ | $\mathbb{Z}_{2}\to\mathbb{Z}_{4}^{T}\to G$
$\begin{array}[]{c}\text{bdry of 4 Kitaev chains:}\\\ \text{4 Majorana modes}\end{array}$ | $\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$ | $\mathbb{Z}_{2}\to\mathbb{Z}_{4}^{T}\times\mathbb{Z}_{2}^{F}\to G$ | $\mathbb{Z}_{8}\Rightarrow\mathbb{Z}_{4}$
$\begin{array}[]{c}\text{bdry of 2 Kitaev chains:}\\\ \text{2 Majorana modes}\end{array}$ | $\mathbb{Z}_{2}\to\mathbb{D}_{8}^{F,T}\to G$ | $\mathbb{Z}_{8}\Rightarrow\mathbb{Z}_{2}$
$\begin{array}[]{c}\text{bdry of 2 Kitaev chains:}\\\ \text{2 Majorana modes}\end{array}$ | $\mathbb{Z}_{4}^{TF}$ | $\mathbb{Z}_{4}\to\mathbb{M}_{16}^{F,T}\to G$ | $\mathbb{Z}_{2}\Rightarrow 0$
1+1 | $\begin{array}[]{c}\text{bdry of (2+1)D CZX model:}\\\ \text{(1+1)D edge modes}\end{array}$ | $\mathbb{Z}_{2}$ | $\begin{array}[]{c}\text{$\mathbb{Z}_{2}\to\mathbb{Z}_{4}\to G$}\\\ \text{Symm-extension, but SSB.}\end{array}$ | $\mathbb{Z}_{2}\Rightarrow 0$ |  $G$ SSB
2+1 | $\begin{array}[]{c}\text{bdry of (3+1)D $w_{1}(TM)^{4}$ SPT:}\\\ \text{(2+1)D surface state}\end{array}$ | $\mathbb{Z}_{2}^{T}$ | $\mathbb{Z}_{2}\to\mathbb{Z}_{4}^{T}\to G$ | $\mathbb{Z}_{2}\Rightarrow 0$ |  $\begin{array}[]{c}\text{$G$-symmetric}\\\ \text{$K$-gauge TQFT}\end{array}$
$\begin{array}[]{c}\text{bdry of (3+1)D $k\mathcal{P}(B_{2})$ higher-}\\\ \text{SPT: (2+1)D surface state}\end{array}$ | $\mathrm{SO}\times\mathbb{Z}_{2,[1]}$ | $\begin{array}[]{c}\text{even $k$:}\\\ \mathbb{Z}_{2}\to\mathrm{Spin}\times\mathbb{Z}_{2,[1]}\to G\\\ \text{but SSB}\end{array}$ | $k\in\mathbb{Z}_{4}\Rightarrow k\in\mathbb{Z}_{2}$ |  $G$ SSB
$\begin{array}[]{c}\text{odd $k$:}\\\ \text{No symm-extension.}\end{array}$ | No
3+1 | $\begin{array}[]{c}\text{bdry of (4+1)D $A^{5}$ SPT:}\\\ \text{(3+1)D bdry state}\end{array}$ | $\mathbb{Z}_{2}^{T}$ | $\mathbb{Z}_{2}\to\mathbb{Z}_{4}\to G$ | $\mathbb{Z}_{2}\Rightarrow 0$ |  $\begin{array}[]{c}\text{$G$-symmetric}\\\ \text{$K$-gauge TQFT}\end{array}$
$\begin{array}[]{c}\text{bdry of (4+1)D $w_{3}(TM)B_{2}^{e}$}\\\ \text{higher-SPT}\end{array}$ | $\mathrm{SO}\times\mathbb{Z}_{2,[1]}^{e}$ | $\mathbb{Z}_{2}\to\mathrm{Spin}\times\mathbb{Z}_{2,[1]}^{e}\to G$ | $\mathbb{Z}_{2}\Rightarrow 0$
$\begin{array}[]{c}\text{bdry of (4+1)D $kA\mathcal{P}(B_{2}^{e})$}\\\ \text{higher-SPT}\end{array}$ |  $\begin{array}[]{c}\text{$\mathrm{Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{8}$}\\\ \text{$\times\mathbb{Z}_{2,[1]}^{e}$}\end{array}$ | $\begin{array}[]{c}\text{even $k$:}\\\ \mathbb{Z}_{2}\to\mathrm{Spin}\times\mathbb{Z}_{8}\times\mathbb{Z}_{2,[1]}^{e}\to G\end{array}$ | $k\in\mathbb{Z}_{4}\Rightarrow k\in\mathbb{Z}_{2}$
$\begin{array}[]{c}\text{odd $k$:}\\\ \text{No symm-extension.}\\\ \text{No-go obstruction.}\end{array}$ | No
$\begin{array}[]{c}\text{bdry of (4+1)D $\frac{1}{2}\tilde{w}_{1}(TM)\mathcal{P}(B_{2}^{e})$}\\\ \text{higher-SPT:}\\\ \text{(3+1)D $\mathrm{SU}(2)_{\theta=\pi}$ YM}\end{array}$ | $\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2,[1]}^{e}$ | $\begin{array}[]{c}\text{$\mathbb{Z}_{2}\to\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{4,[1]}^{e}\to G$}\\\ \text{Symm-extension, but SSB.}\end{array}$ | $\mathbb{Z}_{2}\Rightarrow 0$ |  $G$ SSB
$\begin{array}[]{c}\text{bdry of (4+1)D}\\\ \text{$(-N_{f})\eta_{\rm 4d}(\mathrm{PD}(A))$-SPT:}\\\ \text{(3+1)D $15N_{f}$-fermion SM}\end{array}$ |  $\mathrm{Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4,X}$ | $\begin{array}[]{c}\text{even $N_{f}$:}\\\ \mathbb{Z}_{2}\to\mathrm{Spin}\times\mathbb{Z}_{4}\to G,\\\ \mathbb{Z}_{2}\to\mathrm{Spin}\times\mathbb{Z}_{8}\to\mathrm{Spin}\times\mathbb{Z}_{4}\end{array}$ |  $\mathbb{Z}_{16}\Rightarrow\mathbb{Z}_{2}$ |  $\begin{array}[]{c}\text{$G$-symmetric}\\\ \text{$K$-gauge TQFT}\end{array}$
$\begin{array}[]{c}\text{odd $N_{f}$:}\\\ \text{No symm-extension.}\\\ \text{No-go obstruction.}\end{array}$ | No
Table 8: Summary of the symmetry-extension construction based on $1\to
K\to\tilde{G}\xrightarrow{r}G\to 1$ in (56). The first and second column show
the spacetime dimension $d$ and the $G$-anomalous theory (as a boundary [bdry]
of a bulk $d+1$D SPT). The third column shows whether the
$\tilde{G}$-symmetry-extended gapped phase exists. The fourth column shows how
the boundary $G$-anomaly or bulk $G$-SPT classification is reduced in
$\tilde{G}$. The last column shows the dynamics after gauging $K$: Even if
$\tilde{G}$-symmetry-extended gapped phase exists, the $K$-gauged dynamics can
induce either a $G$ spontaneous symmetry-breaking (SSB) or a $G$-symmetric $K$
gauge theory.

  1. 1.

$d=1$, a (0+1)D anomalous theory and a (1+1)D bulk:

$\bullet$ Two Kitaev’s chains: In Sec. II.1, the discussion was limited to
gapping (0+1)D eight Majorana modes or (1+1)D eight Kitaev’s chains by the SMG
preserving the $\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$, which is free
from the $\mathbb{Z}_{8}$ global anomaly. Beyond SMG, even for (0+1)D two
Majorana modes or (1+1)D two Kitaev’s chains with $2\mod 8$ class of
$\mathbb{Z}_{8}$ global anomaly, it is still possible to gap the whole system,
leaving a single ground state _without breaking_
$G=\mathbb{Z}_{2}^{T}\times\mathbb{Z}_{2}^{F}$, but instead by _extending_ the
symmetry to a dihedral group of order 8 as
$\tilde{G}=\mathbb{D}_{8}^{F,T}\equiv\mathbb{Z}_{4}^{T}\rtimes\mathbb{Z}_{2}^{F}$
[114]. Namely, when $T^{2}=+1$ is extended to a time-reversal symmetry
fictionalization $T^{2}=-1$ and $T^{4}=+1$, the
$\tilde{G}=\mathbb{D}_{8}^{F,T}$ symmetric interactions can lift up the
degenerate Majorana zero modes. Crucially the time-reversal
$\mathbb{Z}_{2}^{T}$ generator $T$ does not commute with the fermion parity
$\mathbb{Z}_{2}^{F}$ generator $(-1)^{F}$, but $T(-1)^{F}T^{-1}=-(-1)^{F}$.
This means that $T$ switches a bosonic sector $|B\rangle$ and a fermionic
sector $|F\rangle$ in the Hilbert space. This is called _supersymmetry
extension_ that trivializes this (0+1)D fermionic anomaly and also trivializes
the (1+1)D fermionic SPT state [114]. Another related property is that the
preserved symmetry demands the anomalous boundary theory of $\pm 2\mod 8$
Majorana zero modes must be $\mathcal{N}=2$ supersymmetric quantum mechanics
with two supercharges [165].

Two Kitaev’s chains can also allow $G=\mathbb{Z}_{4}^{TF}$ symmetry with
$T^{2}=(-1)^{F}$ [166]. When $T^{2}=(-1)^{F}$ is extended to a time-reversal
symmetry fictionalization $T^{4}=-1$ and $T^{8}=+1$, a non-abelian order-16
finite $\tilde{G}=\mathbb{M}_{16}^{F,T}$ symmetric interactions can lift up
the degenerate Majorana zero modes [165].

$\bullet$ Four Kitaev’s chains and a Haldane chain: For (0+1)D four Majorana
modes or (1+1)D four Kitaev’s chains, its $4\mod 8$ class of $\mathbb{Z}_{8}$
global anomaly is actually equivalent to a single (1+1)D Haldane’s chain [72]
(tensor product with a trivial gapped fermionic product state) with $1\mod 2$
class of $\mathbb{Z}_{2}$ global anomaly. The $G=\mathbb{Z}_{2}^{T}$-symmetric
Haldane chain can be trivialized in a bosonic $\tilde{G}=\mathbb{Z}_{4}^{T}$
[161]. The $G=\mathrm{SO}(3)$-symmetric Haldane chain can be trivialized in a
$\tilde{G}=\mathrm{SU}(2)$ [161].

$\bullet$ Related studies on the fractionalized symmetries on the boundary of
the layers of Kitaev chains can also be found in [166, 167, 168, 169, 170]. In
particular, Ref. 167, 169 studies the pure $\mathbb{Z}_{2}$ class global
gravitational anomaly on the boundary of a single Kitaev chain (which is an
invertible fermionic topological order beyond the SPT, known as the
mathematical Arf invariant). The pure gravitational anomaly cannot be
trivialized by any symmetry extension, but may be “trivialized” by coupling to
a gravitational theory [167].

Here a (0+1)D theory has no parity $P$ but only at most time-reversal $T$, and
Majorana fermion has no charge conjugation $C$ symmetry; so only the $T$
fractionalization is found. In higher dimensions, the common theme along the
direction of this phenomenon is the $C$-$P$-$T$ fractionalization [171].

  2. 2.

$d=2$, a (1+1)D anomalous theory and a (2+1)D bulk:

$\bullet$ The (1+1)D edge of a (2+1)D CZX model as a $\mathbb{Z}_{2}$-SPT
state is known to allow a symmetric gapless or a symmetry-breaking gapped
boundary [172]. However, the $G=\mathbb{Z}_{2}$ can be extended to give a
$\tilde{G}=\mathbb{Z}_{4}$-symmetry-extended gapped boundary. Unfortunately,
gauging the normal subgroup $K=\mathbb{Z}_{2}$ results in a (1+1)D discrete
$K$-gauge theory with $G=\mathbb{Z}_{2}$ spontaneous symmetry breaking, which
is consistent with the standard lore that there is no (1+1)D non-invertible
intrinsic topological order, at least in the bosonic systems.

Other applications of the symmetry-extension construction on the (1+1)D gauge
theories and orbifolds can be found in a recent survey [173].

  3. 3.

$d=3$, a (2+1)D anomalous theory and a (3+1)D bulk:

$\bullet$ The (2+1)D surface of a (3+1)D $\mathbb{Z}_{2}^{T}$-SPT state
(topological superconductor) allows a symmetric gapless, symmetry-breaking, or
symmetric gapped surface topological order boundary [12, 174, 175]. The
$G=\mathbb{Z}_{2}^{T}$ can be extended to give a