# CT or P Problem and Symmetric Gapped Fermion Solution

Juven Wang<EMAIL_ADDRESS>Center of Mathematical Sciences and
Applications, Harvard University, MA 02138, USA

###### Abstract

An analogous “Strong CP problem” is identified in a toy model in two
dimensional spacetime: a general 1+1d abelian U(1) anomaly-free chiral fermion
and chiral gauge theory with a generic theta instanton term
$\frac{\theta}{2\pi}\int F$. The theta term alone violates the charge-
conjugation-time-reversal CT and the parity P discrete symmetries. The
analogous puzzle here is the CT or P problem in 1+1d: Why can the
$\bar{\theta}$ angle (including the effect of $\theta$ and the complex phase
of a mass matrix) be zero or small for a natural reason? We show that this CT
or P problem can be solved by a Symmetric Mass Generation mechanism (SMG,
namely generating a mass or energy gap while preserving an anomaly-free
symmetry). This 1+1d toy model mimics several features of the 3+1d Standard
Model: chiral matter contents, confinement, and Anderson-Higgs mass by Yukawa-
Higgs term. One solution enriches this toy model by introducing several new
physics beyond the Standard Model: a parity-reflection PR discrete symmetry
mapping between the chiral and mirror fermions as fermion doubling localized
on two domain walls at a high energy, and SMG dynamically generates mass to
the mirror fermion while still preserving the anomaly-free chiral symmetry at
an intermediate energy scale, much before the Higgs mechanism generates mass
to the chiral fermion in at a lower energy. Another solution no longer
requires the mirror fermion sector, but directly replaces the chiral fermion’s
Higgs mass to the SMG mass. Without loss of generality, an arguably simplest
1+1d U(1) symmetric $3_{L}$-$4_{L}$-$5_{R}$-$0_{R}$ anomaly-free chiral
fermion and chiral gauge theory is demonstrated in detail.

###### Contents

  1. I Introduction and the Problem
  2. II A 1+1d model with the CT or P problem
  3. III A Solution to the CT or P problem via Symmetric Mass Generation on the mirror fermions
  4. IV Another CT or P solution via replacing the Higgs phase to SMG for the chiral fermions
  5. V Conclusion
    1. V.1 Thought Experiment
    2. V.2 Massless Fermion vs Mean-Field Order Mass vs Interacting or Disorder Mass Fermion
  6. A Fermion to Boson Field Theory
    1. A.1 Fermions
    2. A.2 Bosonization
    3. A.3 Light Sectors vs Dark Sector: Anomaly-free Condition vs Topological Gapping Condition

##  I Introduction and the Problem

One open puzzle in the 3+1d Standard Model (SM) is the strong CP problem Smith
_et al._ (1957); Abel  _et al._ (2020a); Dine (2000, 2000); Hook (2019). The
corresponding strong gauge force SU(3) should naturally allow a theta
$\theta_{3}$ term inserted into the SM path integral with a weight factor
$\exp(\hskip 1.0pt\mathrm{i}\hskip 1.0pt\theta_{3}n^{(3)})$, with the
appropriately quantized instanton number
$n^{(3)}\equiv\frac{1}{8\pi^{2}}\int{\mathrm{Tr}}[F^{(3)}\wedge
F^{(3)}]\in\mathbb{Z}$ Belavin  _et al._ (1975); ’t Hooft (1976a) of a
nonabelian SU(3) field strength $F^{(3)}$. The Cabibbo-Kobayashi-Maskawa (CKM)
matrix CP violating angle $\delta_{\rm CP}$ is experimentally verified to be
order 1. Näively, due to Naturalness without fine-tuning ’t Hooft (1980), the
other CP violating theta angle

$\displaystyle{\bar{\theta}_{3}}\equiv{\theta_{3}}+\arg(\det(M_{u}M_{d}))$ (1)

shall also be generically order 1. (Here $M_{u}$ and $M_{d}$ are two rank-3
matrices specifying the Yukawa-Higgs coupling, for $u,c,t$-type quarks and
$d,s,b$-type quarks respectively.) A generic theoretical
${\bar{\theta}_{3}}\in[0,2\pi)$ can violate CP thus T (except at
${\bar{\theta}_{3}}=0$ or $\pi$), but the surprising experimental fact
measured by the real-world neutron electric dipole moment shows that
$|{\bar{\theta}_{3}}|<10^{-10}\simeq 0$ is nearly zero Baker  _et al._ (2006);
Tanabashi  _et al._ (2018); Abel  _et al._ (2020b), making the quantum
chromodynamics (QCD) almost a CP and T invariant — this puzzle is known as the
strong CP problem. Since there is no particular reason (not even an anthropic
reason) for the SM to have ${\bar{\theta}_{3}}\simeq 0$, the typical strong CP
solutions proposed in the past literature tend to modify or enlarge the SM to
include some assumptions: (1) some of quarks (e.g., up quark) is massless ’t
Hooft (1976b), (2) extra continuous U(1) symmetry, then based on dynamical
arguments on the spontaneous symmetry breaking to relax to
${\bar{\theta}_{3}}\simeq 0$, e.g., Peccei-Quinn symmetry with axions Peccei
and Quinn (1977a, b); Weinberg (1978); Wilczek (1978) (3) extra discrete P or
CP symmetry imposed at a high energy Nelson (1984); Barr (1984); Babu and
Mohapatra (1989, 1990); Barr  _et al._ (1991). The purpose of this present
work is to propose a new type of CP problem’s solution by involving the
Symmetric Mass Generation (SMG) mechanism Wang and You (2022) — fermions can
become massive (also known as gapped) by a symmetric deformation from a
massless (aka gapless) theory, without involving any symmetry breaking within
an anomaly-free symmetry group.

In this work, we investigate the analogous problem in a 1+1d toy model to
mimic the strong CP problem in the 3+1d SM. Our toy model is a chiral fermion
theory coupled to U(1) gauge field, as a chiral gauge theory version of the
modified Schwinger’s 1+1D quantum electrodynamics (QED) Schwinger (1962). Then
we provide a solution to this problem in 1+1d. (In a companion work Hamada and
Wang , we will propose the similar SMG solution to the Strong CP problem for
the 3+1d models, including the SM.) The plan of this article goes as follows.

First, in Sec. II, we define the analogous problem called CT or P problem in
this 1+1d toy model with chiral fermions and with a seemly generic nonzero
$\theta$ term, written as $\frac{\theta}{2\pi}\int F$ for some abelian field
strength $F$. Under the standard unitary C, P, and anti-unitary T active
transformations on the fields (e.g., Wang (2021a)), the two components of 1+1d
abelian gauge field $A=(A_{0},A_{1})$ and its field strength $F_{01}$ at the
spacetime coordinates $x^{\mu}=(x^{0},x^{1})=(t,x)$ are transformed as

$\displaystyle\mathbb{Z}_{2}^{\rm C}$ $\displaystyle:$ $\displaystyle
A_{0}(t,x)\mapsto-A_{0}(t,x),\quad A_{1}(t,x)\mapsto-A_{1}(t,x),\quad
F_{01}(t,x)\mapsto-F_{01}(t,x).{}$ $\displaystyle\mathbb{Z}_{2}^{\rm P}$
$\displaystyle:$ $\displaystyle A_{0}(t,x)\mapsto+A_{0}(t,-x),\quad
A_{1}(t,x)\mapsto-A_{1}(t,-x),\quad F_{01}(t,x)\mapsto-F_{01}(t,-x).$ (2)
$\displaystyle\mathbb{Z}_{2}^{\rm T}$ $\displaystyle:$ $\displaystyle
A_{0}(t,x)\mapsto+A_{0}(-t,x),\quad A_{1}(t,x)\mapsto-A_{1}(-t,x),\quad
F_{01}(t,x)\mapsto+F_{01}(-t,x),\quad\hskip 1.0pt\mathrm{i}\hskip
1.0pt\mapsto-\hskip 1.0pt\mathrm{i}\hskip 1.0pt.{}$

The only nonzero field strength
$F_{01}=-F_{01}=\partial_{0}A_{1}-\partial_{1}A_{0}\equiv E_{1}$ is the
electric field along the spatial $x$ direction. The electric field $E_{1}$ is
C odd, P odd, and T even. So the $\frac{\theta}{2\pi}\int F$ violates both CT
and P for a generic $\theta\in[0,2\pi)$ within the $2\pi$ periodicty, unless
at the special $\theta=0$ or $\pi$ that preserves CT and P. Hence we shall
call this a CT problem or P problem in 1+1d: How can the analogous 1+1d
$\bar{\theta}$ angle be zero or small for a natural reason?

Then, in Sec. III, we provide a CT solution to the 1+1d toy model to argue why
$\bar{\theta}\simeq 0$ by introducing the mirror fermion doubling to the
chiral fermion theory, and gapping the mirror fermion by SMG Wang and You
(2022). In particular, here our 1+1d CT solution consists of the following
inputs:
1\. Fermion doubling at a high energy above an energy scale ${\rm
E}>\Lambda_{\rm SMG}$, due to Nielsen-Ninomiya (NN) argument Nielsen and
Ninomiya (1981). Our model contains a chiral fermion sector and a mirror
fermion sector on two 1+1d branes (or two domain walls) separated by a finite-
width 2+1d bulk, imposed at the high energy.
2\. Parity-reflection symmetry $\mathbb{Z}_{2}^{\rm PR}$ is imposed at a high
energy ${\rm E}>\Lambda_{\rm SMG}$, reminiscent to the parity solution of the
strong CP problem in 3+1d Babu and Mohapatra (1989, 1990); Barr  _et al._
(1991); Craig  _et al._ (2021). Both chiral and mirror fermion sectors are
mapped to each other under the $\mathbb{Z}_{2}^{\rm PR}$ symmetry. However,
our approach is still _different_ from the parity solution Babu and Mohapatra
(1989, 1990); Barr  _et al._ (1991); Craig  _et al._ (2021) for at least two
reasons: (1) Chiral and mirror fermions are fermion doublings with opposite
chiralities Nielsen and Ninomiya (1981) (applicable in even dimensional
spacetime in general). (2) We do _not_ double copy the gauge group of the
chiral sector to the mirror sector as Ref. [Babu and Mohapatra, 1989, 1990;
Barr  _et al._ , 1991; Craig  _et al._ , 2021] did. Our chiral and mirror
fermions couple to the exactly same set of gauge fields.
3\. The Symmetric Mass Generation (SMG) Wang and You (2022) fully gaps the
mirror fermion sector but still fully preserves the chiral symmetry on both
the chiral and mirror fermion side. The idea is an incarnation of the Eichten-
Preskill approach Eichten and Preskill (1986), but in the modern setup
enriched by the SMG deformation Wang and You (2022). We will follow the
explicit 1+1d multi-fermion interaction construction in Wang and Wen (2013,
2019); Zeng  _et al._ (2022). The SMG is sometimes called the Kitaev-Wen
mechanism Fidkowski and Kitaev (2011, 2010); Wen (2013); Wang and Wen (2020)
or a mass gap without mass term You  _et al._ (2014); You and Xu (2015);
BenTov (2015); BenTov and Zee (2016). In general, we can induce the SMG energy
gap (or mass gap) without any explicit or spontaneously $G$-symmetry-breaking
mass term if and only if $G$ is ’t Hooft anomaly-free, although such SMG
deformation often requires to introduce either dangerously irrelevant
operators with nonperturbative coupling Wang and Wen (2013, 2019); Zeng  _et
al._ (2022), or new matter or gauge sectors brought down from the high-energy
or short distance Seiberg (2019); Wang (2020a); Razamat and Tong (2021); Tong
(2021); Wang and You (2021). Thus SMG deformations are largely overlooked in
the previous literature. In general, in order to check the quantum field
theory (QFT) is anomaly-free, we require a classification of perturbative
local and nonperturbative global anomalies by one-higher dimensional
invertible topological field theories (iTFTs), which becomes systematic
especially due to the work of Freed-Hopkins Freed and Hopkins (2021).

In Sec. IV, we provide another solution to this 1+1d CT or P problem of the
chiral fermion theory via replacing its gapped Higgs phase to the SMG phase.
Namely, once we assume the gapped chiral fermion theory is not due to a mean-
field mass or Higgs transition, but due to instead to a certain interacting
SMG deformation, then the $\theta$ angle can be absorbed to the SMG
interaction terms, thus the $\bar{\theta}$ angle is zero.

In Sec. V, we conclude by explaining some physics intuitions behind our
solutions. We demonstrate the interplay between the SMG Wang and You (2022)
and the Laughlin’s style of the flux insertion thought experiment Laughlin
(1981), namely threading a magnetic flux through the hole of the annulus or
cylinder strip to induce the electric field and the 1+1d boundary anomalous
current transport through the 2+1d bulk Wang and Wen (2013); Santos and Wang
(2014).

##  II A 1+1d model with the CT or P problem

In this section, we will show that the CT or P problem is a general issue for
a 1+1d theory with U(1) gauge field.111Here U(1) field can be either treated
as a _background gauge field_ (i.e., the path integral $Z[A]$ depends on the
specific choices of gauge bundle and connection $A$) or as a _dynamically
gauge field_ (i.e., summed over the gauge bundle and connection in the path
integral $Z=\int[DA]Z[A]$), depending on the context that we will illuminate.
Our CT or P solution will be applicable to the most general 1+1d U(1) anomaly-
free theory involving chiral fermions or chiral bosons following the setup in
Wang and Wen (2013, 2019).

In particular, for simplicity but without loss of generality, we shall focus
on arguably the simplest example of 1+1d U(1) symmetric chiral fermion theory
known as the $3_{L}$-$4_{L}$-$5_{R}$-$0_{R}$ model or the 3-4-5-0 model. This
1+1d Lorentz invariant and U(1) symmetric chiral fermion 3-4-5-0 theory has
two left-moving and two right-moving complex Weyl fermions
$\psi_{L,3},\psi_{L,4},\psi_{R,5},\psi_{R,0}$ with U(1) charges 3,4 and 5,0.
Relevant data of this 1+1d theory is organized in Table 1. (The model can be
straightforward generalized to other 1+1d models, which we report in a
companion work.)

| Weyl fermion |  $\begin{array}[]{c}\text{disorder}\\\ \text{scalar}\\\ \end{array}$ | Higgs
---|---|---|---
| $\psi_{L,3}$ | $\psi_{L,4}$ | $\psi_{R,5}$ | $\psi_{R,0}$ | $\phi_{1}$ | $\phi_{2}$ | $\phi_{H}$
${\rm U}(1)_{\rm 1st}$ $q_{{1}}$ | 3 | 4 | 5 | 0 | $-4$ | $-2$ | 1
${\rm U}(1)_{\rm 2nd}$ $q_{{2}}$ | 0 | 5 | 4 | 3 | $-2$ | $-4$ | $-1$
${\rm U}(1)_{\rm 3rd}$ $q_{{3}}=Kq_{1}$ | 3 | 4 | $-5$ | 0 |  |  |
${\rm U}(1)_{\rm 4th}$ $q_{{4}}=Kq_{2}$ | 0 | 5 | $-4$ | $-3$ |  |  |
${\rm U}(1)_{\rm vector}$ $q_{\rm v}=\frac{1}{3}(q_{1}+q_{2})$ | 1 | 3 | 3 | 1 |  |  | 0
Yukawa-Higgs $\phi_{H}$ Vector mass term’s power coefficient
$m_{3,0}$ : $\phi_{H}^{3}\psi_{L,3}^{\dagger}\psi_{R,0}$ | $-1$ | 0 | 0 | $1$ | 0 | 0 | 3
$m_{4,5}$ : $\phi_{H}^{\dagger}\psi_{L,4}^{\dagger}\psi_{R,5}$ | 0 | $-1$ | 1 | 0 | 0 | 0 | $-1$
Multi-fermion (power exponent) or sine-Gordon (coefficient) interactions
$\ell_{1}=\frac{1}{3}(q_{3}-2q_{4})$ | 1 | $-2$ | $1$ | $2$ |  |  |
$\ell_{2}=\frac{1}{3}(2q_{3}-q_{4})$ | 2 | 1 | $-2$ | $1$ |  |  |
Disorder Yukawa-scalar $\phi_{1}$ and $\phi_{2}$ term’s power exponent
$\phi_{1}^{2}\psi_{L,3}\psi_{R,5}$ | 1 | $0$ | 1 | 0 | 2 | 0 |
$\phi_{1}^{\dagger}\psi_{L,4}^{\dagger}\psi_{R,0}$ | 0 | $-1$ | 0 | 1 | $-1$ | 0 |
$\phi_{2}^{\dagger}\psi_{L,3}\psi_{R,5}^{\dagger}$ | 1 | 0 | $-1$ | 0 | 0 | $-1$ |
$\phi_{2}^{2}\psi_{L,4}\psi_{R,0}$ | 0 | 1 | 0 | 1 | 0 | 2 |
Table 1:  Without loss of generality, we present an arguably simplest 1+1d
U(1) symmetric anomaly-free chiral fermion theory, the
$3_{L}$-$4_{L}$-$5_{R}$-$0_{R}$ or 3-4-5-0 model with two left-moving and two
right-moving Weyl fermions (3) and a choice of ${\rm U}(1)_{\rm 1st}\times{\rm
U}(1)_{\rm 2nd}\times{\rm U}(1)_{\rm 3rd}\times{\rm U}(1)_{\rm 4th}$ symmetry
charges. The field contents, their quantum numbers, and the coefficient or
power exponent of interactions are organized above.
$\bullet$ The Yukawa-Higgs vector mass term
$\phi_{H}^{3}\psi_{L,3}^{\dagger}\psi_{R,0}+\phi_{H}^{\dagger}\psi_{L,4}^{\dagger}\psi_{R,5}+\text{h.c.}$
of (8) breaks the ${\rm U}(1)^{4}$ symmetry down to ${\rm U}(1)_{\rm vector}$
in the $\phi_{H}$ Higgs condensed phase. We _cannot_ rotate the CT and P
violating $\theta$ angle away in the chiral sector alone without introducing a
compensating complex phase to the vector mass term.
$\bullet$ The Symmetric Mass Generation (SMG) to fully gap this chiral fermion
theory can be achieved by the point-splitting multi-fermion interactions
$\big{(}{g}_{1}({\psi}_{L,3})({\psi}_{L,4}^{\dagger})^{2}({\psi}_{R,5})({\psi}_{R,0})^{2}+{g}_{2}({\psi}_{L,3})^{2}({\psi}_{L,4})({\psi}_{R,5}^{\dagger})^{2}({\psi}_{R,0})+\text{h.c.}\big{)}$
in (49), whose bosonized counterpart ${\psi\equiv:\exp(\pm\hskip
1.0pt\mathrm{i}\hskip 1.0pt\varphi):}$ contains two cosine sine-Gordon
interactions $g_{1}\cos(\ell_{1}\varphi)+g_{2}\cos(\ell_{2}\varphi)$ with the
$\ell_{1}$ and $\ell_{2}$ vectors as in (81). Alternatively, we can introduce
the disorder scalars $\phi_{1}$ and $\phi_{2}$ as
$(\phi_{1}^{2}\psi_{L,3}\psi_{R,5}+\phi_{1}^{\dagger}\psi_{L,4}^{\dagger}\psi_{R,0}+\phi_{2}^{\dagger}\psi_{L,3}\psi_{R,5}^{\dagger}+\phi_{2}^{2}\psi_{L,4}\psi_{R,0}+\text{h.c.}+\frac{1}{\sqrt{|g_{1}|}}\phi_{1}^{\dagger}\phi_{1}+\frac{1}{\sqrt{|g_{2}|}}\phi_{2}^{\dagger}\phi_{2}$
in (51), where disorder scalars can be integrated out to derive the multi-
fermion interactions (49).
$\bullet$ See Appendix A for the mathematical structure behind the general
class of these models.

The free massless fermion part of action is

$\displaystyle S_{\Psi,\rm free}=\int\hskip 1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x\big{(}\hskip 1.0pt\mathrm{i}\hskip
1.0pt\psi^{\dagger}_{L,3}(\partial_{t}-\partial_{x})\psi_{L,3}+\hskip
1.0pt\mathrm{i}\hskip
1.0pt\psi^{\dagger}_{L,4}(\partial_{t}-\partial_{x})\psi_{L,4}+\hskip
1.0pt\mathrm{i}\hskip
1.0pt\psi^{\dagger}_{R,5}(\partial_{t}+\partial_{x})\psi_{R,5}+\hskip
1.0pt\mathrm{i}\hskip
1.0pt\psi^{\dagger}_{R,0}(\partial_{t}+\partial_{x})\psi_{R,0}\big{)}.$ (3)

Let us define the multiplet $\Psi_{\rm
I}\equiv\Big{(}\begin{smallmatrix}\psi_{L,3}\\\ \psi_{L,4}\\\ \psi_{R,5}\\\
\psi_{R,0}\end{smallmatrix}\Big{)}_{\rm I}$ and a diagonal unimodular rank-4
matrix $K={\rm diag}(1,1,-1,-1)$ and ${\rm I,J}\in\\{1,2,3,4\\}$, so Eq. (3)
is equivalent to $S_{\Psi,\rm free}=\int\hskip 1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x\hskip 1.0pt\mathrm{i}\hskip 1.0pt\Psi_{\rm
I}^{\dagger}(\delta_{\rm IJ}\partial_{t}-K_{\rm IJ}^{-1}\partial_{x})\Psi_{\rm
J}=\int\hskip 1.0pt\mathrm{d}t\hskip 1.0pt\mathrm{d}x\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Psi_{\rm I}^{\dagger}(\partial_{t}-K_{\rm
II}^{-1}\partial_{x})\Psi_{\rm I}$. Here the repeated indices are summed,
namely $\sum_{\rm I,J}$ and $\sum_{\rm I}$ respectively. The corresponding
charges of fermions are listed down in Table 1. Let us systematically
enumerate the properties and comment on this 1+1d model.

  1. 1.

Symmetry: The free massless fermion part of theory (3) has a spacetime Lorentz
symmetry ${\rm Spin}$ group and an enlarged internal symmetry ${\rm
U}(2)_{L}\times{\rm U}(2)_{R}$. The Spin group is a double cover of the
special orthogonal group SO, graded by the fermion parity
$\mathbb{Z}_{2}^{{\rm F}}:\Psi\mapsto-\Psi$. We can choose ${\rm
U}(2)_{L}\times{\rm U}(2)_{R}\supset{\rm U}(1)_{\rm 1st}\times{\rm U}(1)_{\rm
2nd}\times{\rm U}(1)_{\rm 3rd}\times{\rm U}(1)_{\rm 4th}$ to contain the
maximal torus ${\rm U}(1)^{4}$ abelian subgroup. The spacetime Spin and
internal ${\rm U}(1)^{4}$ symmetry combined together is ${{\rm Spin}\times{\rm
U}(1)^{4}}$ because the fermions as the spacetime spinors of Spin group are
allowed to have both odd or even charges of U(1). (However, the spacetime-
internal symmetry would be $\frac{{\rm Spin}\times{\rm
U}(1)^{4}}{\mathbb{Z}_{2}^{{\rm F}}}\equiv{{\rm
Spin}\times_{\mathbb{Z}_{2}^{{\rm F}}}{\rm U}(1)^{4}}$ if the fermions must
have odd charges of U(1) while the bosons must have even charges of U(1).) The
active Lorentz symmetry Spin(1,1) and U(1) symmetry transformations map on
fermions respectively as:

$\displaystyle{\rm Spin}$ $\displaystyle:$ $\displaystyle\Psi_{\rm
I}(x^{\mu})\mapsto\Lambda_{\frac{1}{2}}\Psi_{\rm
I}(\Lambda^{\mu}{}_{\nu}x^{\nu}).$ (4) $\displaystyle{\rm U}(1)$
$\displaystyle:$ $\displaystyle\Psi_{\rm I}(x^{\mu})\mapsto\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0ptq_{\rm I}\upalpha}\Psi_{\rm
I}(x^{\mu}),\quad\upalpha\in[0,2\pi).$ (5)

The $\Lambda_{\frac{1}{2}}$ and $\Lambda^{\mu}{}_{\nu}$ are respectively the
standard Lorentz transformations on the spin-$\frac{1}{2}$ left or right-
handed Weyl spinor and the spin-1 Lorentz vector. The U(1) can be any choice
from Table 1, the $q_{\rm I}$ has ${\rm I}\in\\{1,2,3,4\\}$ assigned to four
Weyl fermions $\psi_{L,3},\psi_{L,4},\psi_{R,5},\psi_{R,0}$ respectively.

Moreover, follow (2), the active discrete C-P-T symmetry transformations act
on Weyl fermion fields as:

$\displaystyle\mathbb{Z}_{2}^{\rm C}$ $\displaystyle:$
$\displaystyle\psi_{L}(t,x)\mapsto-\psi_{L}^{*}(t,x),\quad\psi_{R}(t,x)\mapsto+\psi_{R}^{*}(t,x).{}$
$\displaystyle\mathbb{Z}_{2}^{\rm P}$ $\displaystyle:$
$\displaystyle\psi_{L}(t,x)\mapsto+\psi_{R}(t,-x),\quad\psi_{R}(t,x)\mapsto+\psi_{L}(t,-x).$
(6) $\displaystyle\mathbb{Z}_{2}^{\rm T}$ $\displaystyle:$
$\displaystyle\psi_{L}(t,x)\mapsto+\psi_{R}(-t,x),\quad\psi_{R}(t,x)\mapsto+\psi_{L}(-t,x).{}$

The free fermion 3-4-5-0 theory preserves the charge conjugation C, but
violates the parity P (thus being chiral) and time-reversal T symmetry, hence
it is also named the $3_{L}$-$4_{L}$-$5_{R}$-$0_{R}$ chiral fermion theory.

  2. 2.

Anomaly: Recent developments of the classification of invertible topological
field theories via cobordism groups based on Freed-Hopkins’ work Freed and
Hopkins (2021); Wan and Wang (2019) help to systematically classify the
anomalies of quantum field theories. The mathematically rigorous anomaly
classification of the spacetime-internal ${{\rm Spin}\times{\rm U}(1)^{4}}$
symmetry is obtained by the cobordism group $\mathrm{TP}_{3}^{{\rm
Spin}\times{\rm U}(1)^{4}}$ Wan and Wang (2019). The nonperturbative global
anomalies are classified by the finite subgroup part (namely the torsion
subgroup) of the bordism group $\Omega_{3}^{{\rm Spin}\times{\rm U}(1)^{4}}$.
The perturbative local anomalies are classified by the integer $\mathbb{Z}$
classes (namely the free subgroup) of the bordism group $\Omega_{4}^{{\rm
Spin}\times{\rm U}(1)^{4}}$. Follow the computation in Ref. [Wan and Wang,
2019], it can be shown that the $\mathrm{TP}_{3}^{{\rm Spin}\times{\rm
U}(1)^{4}}$ only contains the $\mathbb{Z}$ perturbative local anomalies —
either the gravitational local anomaly captured by the Feynman diagram  or the
U(1) local anomalies captured by the Feynman diagram . The gravitational local
anomaly coefficient is the chiral central charge $c_{-}\equiv c_{L}-c_{R}$
which is anomaly-free ($c_{-}=2-2=0$) for the 3-4-5-0 model. The U(1) local
anomalies can be expressed as a symmetric bilinear form using the $K$ matrix
shown below.

  3. 3.

The one-loop Feynman graph of two-point correlators of the background $A$
gauge fields of ${\rm U}(1)$ charges $q_{\alpha}$ and $q_{\beta}$ have the
anomaly coefficients

$\displaystyle q^{\intercal}_{\alpha}Kq_{\beta}\equiv\sum_{\rm
I,J}q_{\alpha,\rm I}K_{\rm IJ}q_{\beta,\rm J}$ (7)

with the transpose ${\intercal}$. The ${\rm U}(1)_{\rm 1st}\times{\rm
U}(1)_{\rm 2nd}$ are self and mutual ’t Hooft anomaly-free,
$q_{1}^{\intercal}Kq_{1}=q_{2}^{\intercal}Kq_{2}=q_{1}^{\intercal}Kq_{2}=0$.
Similarly, the ${\rm U}(1)_{\rm 3rd}\times{\rm U}(1)_{\rm 4th}$ are also self
and mutual ’t Hooft anomaly-free
$q_{3}^{\intercal}Kq_{3}=q_{4}^{\intercal}Kq_{4}=q_{3}^{\intercal}Kq_{4}=0$.
But mutually between ${\rm U}(1)_{\rm 1st}\times{\rm U}(1)_{\rm 2nd}$ and
${\rm U}(1)_{\rm 3rd}\times{\rm U}(1)_{\rm 4th}$, there are mixed anomalies,
because $q_{1}^{\intercal}Kq_{3}=q_{2}^{\intercal}Kq_{4}=50$ and
$q_{1}^{\intercal}Kq_{4}=q_{2}^{\intercal}Kq_{3}=40$.

  4. 4.

Analogy to the 3+1d SM: We now make an analogy between the 3+1d’s electroweak
Higgs symmetry breaking to the 1+1d model with Higgs symmetry
breaking.222Below we write the SM’s Lie algebra $su(3)\times su(2)\times
u(1)_{Y}$ because the SM’s Lie group has the four compatible versions, $({\rm
SU}(3)\times{\rm SU}(2)\times{\rm U}(1)_{Y})/\mathbb{Z}_{p}$ with $p=1,2,3,6$.
The 3+1d SM chiral fermion with the chiral $su(2)\times u(1)_{Y}$ symmetry for
the $su(2)$ chiral weak force and the chiral hypercharge $Y$ is analogous to
the 1+1d chiral fermion model with the chiral ${\rm U}(1)_{\rm 1st}\times{\rm
U}(1)_{\rm 2nd}$ symmetry.

In 3+1d, the SM Higgs condensation breaks the chiral electroweak $su(2)\times
u(1)_{Y}$ down to the vector $u(1)_{\rm EM}$ subgroup, since $su(2)\times
u(1)_{Y}\supset u(1)_{\rm EM}$. The electroweak Higgs carries nonzero $su(2)$
and $u(1)_{Y}$ charges, but carries a zero $u(1)_{\rm EM}$. Similarly, in
1+1d, we can introduce a scalar Higgs $\phi_{H}$ and the Yukawa-Higgs (YH)
term with a Higgs potential

$\displaystyle S_{\rm YH}=\int\hskip 1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x(\phi_{H}^{3}\psi_{L,3}^{\dagger}\psi_{R,0}+\phi_{H}^{\dagger}\psi_{L,4}^{\dagger}\psi_{R,5}+\text{h.c.}-\mu_{H}|\phi_{H}|^{2}-\lambda_{H}|\phi_{H}|^{4})$
(8)

to the action. We denote the first YH term as the $m_{3,0}$ term and the
second YH term as the $m_{4,5}$ term. Kinematically, the YH term preserves the
${\rm U}(1)_{\rm 1st}\times{\rm U}(1)_{\rm 2nd}\supset{\rm U}(1)_{\rm
vector}$. The $\phi_{H}$ carries nonzero ${\rm U}(1)_{\rm 1st}$ and ${\rm
U}(1)_{\rm 2nd}$ charges, but carries a zero ${\rm U}(1)_{\rm vector}$ charge.
Dynamically, when $\mu_{H}<0$, the Higgs condensation
$\langle\phi_{H}\rangle\neq 0$ breaks ${\rm U}(1)_{\rm 1st}\times{\rm
U}(1)_{\rm 2nd}$ down to the ${\rm U}(1)_{\rm vector}$ subgroup and induces a
vector mass term to all the fermions.

Notice that $\psi_{R,0}$ has a ${\rm U}(1)_{\rm 1st}$ charge 0 in the 1+1d toy
model, thus it is somehow analogous to the neutrino of the 3+1d SM. But in
this analogy, $\psi_{R,0}$ carries a ${\rm U}(1)_{\rm 2nd}$ charge 3 and a
${\rm U}(1)_{\rm vector}$ charge 1, so it is different from the SM’s _right-
handed neutrino_ that is neutral to all SM gauge force $su(3)\times
su(2)\times u(1)_{Y}$ and $u(1)_{\rm EM}$. The $\psi_{R,0}$ carries a ${\rm
U}(1)_{\rm vector}$ charge 1, thus it is also different from the SM’s _left-
handed neutrino_ that is neutral to the $u(1)_{\rm EM}$. The mass term of
$\psi_{R,0}$ (pairing with the charged $\psi_{L,3}$) is also different from
the unsettled neutrino mass mechanism in the SM (e.g., Dirac, Majorana, or
other causes from the interacting energy gap above the topological field
theory Wang (2020b, 2021b)).

  5. 5.

We can dynamically gauge ${\rm U}(1)_{\rm 1st}\times{\rm U}(1)_{\rm 2nd}$ by
replacing the background field $A$ to dynamical gauge fields $a_{1}$ and
$a_{2}$. Namely, we promote the spacetime derivative
$\partial_{t}+v_{x}\partial_{x}$ (with $v_{x}=-1$ for the left-moving
$\psi_{L}$ while $v_{x}=+1$ for the right-moving $\psi_{R}$) to the covariant
derivative $D_{t}+v_{x}D_{x}\equiv(\partial_{t}-\hskip 1.0pt\mathrm{i}\hskip
1.0ptqa_{t})+v_{x}(\partial_{x}-\hskip 1.0pt\mathrm{i}\hskip 1.0ptqa_{x})$
where the charge $q$ of ${\rm U}(1)_{\rm 1st}\times{\rm U}(1)_{\rm 2nd}$ can
be read from Table 1.333In our case, the velocity $v_{x,\rm I}$ for the I-th
Weyl fermion is determined by its unimodular $K$ matrix component: $v_{x,\rm
I}=-K_{\rm II}^{-1}$. We also map $S_{\Psi,\rm free}\mapsto S_{\Psi,\rm
gauge}$ by replacing $\partial_{t}+v_{x}\partial_{x}\mapsto D_{t}+v_{x}D_{x}$
in the action. Under the ${\rm U}(1)_{\rm 1st}\times{\rm U}(1)_{\rm 2nd}$
symmetry transformations, $\Psi_{\rm I}\mapsto\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0ptq_{{\rm 1},\rm I}\upalpha_{\rm 1st}}\Psi_{\rm I}$
and $\Psi_{\rm I}\mapsto\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0ptq_{{\rm 2},\rm I}\upalpha_{\rm 2nd}}\Psi_{\rm I}$ with the quantized
$q_{{\rm 1},\rm I},q_{{\rm 2},\rm I}$ charges in Table 1, the corresponding
currents are conserved even at the quantum level when ${\rm U}(1)_{\rm
1st}\times{\rm U}(1)_{\rm 2nd}$ are dynamically gauged:

$J^{\mu}_{\rm 1st}\equiv\Psi_{\rm I}^{\dagger}q_{{\rm 1},\rm I}\tau^{\mu}_{\rm
L/R}\Psi_{\rm I}\equiv\sum_{\rm I,J}\Psi_{\rm
I}^{\dagger}\Big{(}\begin{smallmatrix}3\tau^{\mu}_{\rm L}&&&\\\
&4\tau^{\mu}_{\rm L}&&\\\ &&5\tau^{\mu}_{\rm R}&\\\ &&&0\tau^{\mu}_{\rm R}\\\
\end{smallmatrix}\Big{)}_{\rm IJ}\Psi_{\rm J},\quad J^{\mu}_{\rm
2nd}\equiv\Psi_{\rm I}^{\dagger}q_{{\rm 2},{\rm I}}\tau^{\mu}_{\rm
L/R}\Psi_{\rm I}\equiv\sum_{\rm I,J}\Psi_{\rm
I}^{\dagger}\Big{(}\begin{smallmatrix}0\tau^{\mu}_{\rm L}&&&\\\
&5\tau^{\mu}_{\rm L}&&\\\ &&4\tau^{\mu}_{\rm R}&\\\ &&&3\tau^{\mu}_{\rm R}\\\
\end{smallmatrix}\Big{)}_{\rm IJ}\Psi_{\rm J}.$ (9)

Here in 1+1d, we denote

$\displaystyle\tau^{\mu}_{\rm
L/R}\equiv\left\\{\begin{array}[]{l}\tau^{\mu}_{\rm
L}\equiv\bar{\tau}^{\mu}=(1,-1),\quad\mu=0,1.\\\ \tau^{\mu}_{\rm
R}\equiv\tau^{\mu}=(1,1),\quad\mu=0,1.\end{array}\right.$ (12)

On the other hand, under the classical ${\rm U}(1)_{\rm 3rd}\times{\rm
U}(1)_{\rm 4th}$ symmetry transformations, $\Psi_{\rm I}\mapsto\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0ptq_{{\rm 3},\rm
I}\upalpha_{\rm 3rd}}\Psi_{\rm I}$ and $\Psi_{\rm I}\mapsto\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0ptq_{{\rm 4},\rm
I}\upalpha_{\rm 4th}}\Psi_{\rm I}$ with the quantized $q_{{\rm 3},\rm
I},q_{{\rm 4},\rm I}$ charges in Table 1. Spoiled by the mixed anomaly between
${\rm U}(1)_{\rm 1st}\times{\rm U}(1)_{\rm 2nd}$ and ${\rm U}(1)_{\rm
3rd}\times{\rm U}(1)_{\rm 4th}$, we can verify their corresponding currents

$J^{\mu}_{\rm 3rd}\equiv\Psi_{\rm I}^{\dagger}q_{{\rm 3},\rm I}\tau^{\mu}_{\rm
L/R}\Psi_{\rm I}\equiv\sum_{\rm I,J}\Psi_{\rm
I}^{\dagger}\Big{(}\begin{smallmatrix}3\tau^{\mu}_{\rm L}&&&\\\
&4\tau^{\mu}_{\rm L}&&\\\ &&-5\tau^{\mu}_{\rm R}&\\\ &&&0\tau^{\mu}_{\rm R}\\\
\end{smallmatrix}\Big{)}_{\rm IJ}\Psi_{\rm J},\quad J^{\mu}_{\rm
4th}\equiv\Psi_{\rm I}^{\dagger}q_{{\rm 4},{\rm I}}\tau^{\mu}_{\rm
L/R}\Psi_{\rm I}\equiv\sum_{\rm I,J}\Psi_{\rm
I}^{\dagger}\Big{(}\begin{smallmatrix}0\tau^{\mu}_{\rm L}&&&\\\
&5\tau^{\mu}_{\rm L}&&\\\ &&-4\tau^{\mu}_{\rm R}&\\\ &&&-3\tau^{\mu}_{\rm
R}\\\ \end{smallmatrix}\Big{)}_{\rm IJ}\Psi_{\rm J},$ (13)

are not conserved, when ${\rm U}(1)_{\rm 1st}\times{\rm U}(1)_{\rm 2nd}$ are
dynamically gauged by summing over their gauge connections and bundles in the
path integral measure $\int[{\cal D}a_{1}][{\cal D}a_{2}]$.

The $S_{\rm YH}$ also transforms to $S_{\rm YH}^{(\upalpha)}$ with the
modified YH term $\hskip 1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip
1.0pt3(\upalpha_{\rm 3rd}+\upalpha_{\rm
4th})}\phi_{H}^{3}\psi_{L,3}^{\dagger}\psi_{R,0}+\hskip
1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip 1.0pt9(\upalpha_{\rm
3rd}+\upalpha_{\rm
4th})}\phi_{H}^{\dagger}\psi_{L,4}^{\dagger}\psi_{R,5}+\text{h.c.}$ Under the
variation $\upalpha_{\rm 3rd}$ and $\upalpha_{\rm 4th}$, overall the path
integral $Z$ is transformed to

$\displaystyle Z$ $\displaystyle\mapsto$ $\displaystyle\int[{\cal
D}a_{1}][{\cal D}a_{2}][{\cal D}\psi][{\cal D}\bar{\psi}]\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt(S_{\Psi,\rm gauge}+S_{\rm
YH}^{(\upalpha)}+\int(\frac{1}{2e^{2}}F^{\rm 1st}\wedge\star F^{\rm
1st})+\int(\frac{1}{2e^{2}}F^{\rm 2nd}\wedge\star F^{\rm
2nd})+\int(\upalpha_{\rm 3rd}\hskip 1.0pt\mathrm{d}\star J_{\rm
3rd}+\upalpha_{\rm 4th}\hskip 1.0pt\mathrm{d}\star J_{\rm 4th}))}$ (15)
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\int\frac{1}{2\pi}\big{(}(\theta_{\rm 1st}-\upalpha_{\rm 3rd}(q_{{\rm
1}}^{\intercal}\cdot K\cdot q_{{\rm 3}})-\upalpha_{\rm 4th}(q_{{\rm
1}}^{\intercal}\cdot K\cdot q_{{\rm 4}}))F^{\rm 1st}+(\theta_{\rm
2nd}-\upalpha_{\rm 3rd}(q_{{\rm 2}}^{\intercal}\cdot K\cdot q_{{\rm
3}})-\upalpha_{\rm 4th}(q_{{\rm 2}}^{\intercal}\cdot K\cdot q_{{\rm
4}}))F^{\rm 2nd}\big{)}}.$

Note that $\hskip 1.0pt\mathrm{d}\star J=(\partial_{\mu}J^{\mu})\hskip
1.0pt\mathrm{d}t\hskip 1.0pt\mathrm{d}x$, and
$\frac{\theta}{2\pi}F\equiv\frac{\theta}{2\pi}\hskip
1.0pt\mathrm{d}a=\frac{\theta}{4\pi}\epsilon^{\mu\nu}F_{\mu\nu}\hskip
1.0pt\mathrm{d}^{2}x=\frac{\theta}{4\pi}\epsilon^{\mu\nu}(\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu})\hskip
1.0pt\mathrm{d}^{2}x=\frac{\theta}{2\pi}\epsilon^{\mu\nu}\partial_{\mu}a_{\nu}\hskip
1.0pt\mathrm{d}^{2}x$. We take $F=F^{\rm 1st}$ or $F^{\rm 2nd}$ for the field
strength of ${\rm U}(1)_{\rm 1st}$ or ${\rm U}(1)_{\rm 2nd}$ gauge field. The
periodicity constrains $\theta_{\rm 1st},\theta_{\rm 2nd},\upalpha_{\rm
3rd},\upalpha_{\rm 4th}\in[0,2\pi)$.

Here the Maxwell term $\int\frac{1}{2e^{2}}F\wedge\star F=\int\hskip
1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x(-\frac{1}{4e^{2}}F_{\mu\nu}F^{\mu\nu})=\int\hskip
1.0pt\mathrm{d}t\hskip 1.0pt\mathrm{d}x(-\frac{1}{2e^{2}}F_{01}F^{01})$ with a
minus sign $(-1)$ on the right-hand side due to the Lorentzian volume form. It
is equal to $\int\hskip 1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x\frac{1}{2e^{2}}(F_{01})^{2}=\int\hskip 1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x\frac{1}{2e^{2}}(E_{1})^{2}$, the spacetime integral of the
electric field square. The $e$ is the unit charge coupling for the ${\rm
U}(1)$ gauge theory.

For the moment, let us turn off the YH term $S_{\rm YH}=0$ (thus $S_{\rm
YH}^{(\upalpha)}=0$), so we can analyze the largest chiral symmetry.
$\bullet$ When ${\rm U}(1)_{\rm 1st}\times{\rm U}(1)_{\rm 2nd}$ and ${\rm
U}(1)_{\rm 3rd}\times{\rm U}(1)_{\rm 4th}$, are all treated as global
symmetries, they have mixed ’t Hooft anomaly because the aforementioned
$q_{1}^{\intercal}Kq_{3}=q_{2}^{\intercal}Kq_{4}=50$ and
$q_{1}^{\intercal}Kq_{4}=q_{2}^{\intercal}Kq_{3}=40$. This can be signaled by
coupling to non-dynamical background fields. ’t Hooft anomaly implies that
there is an obstruction to gauge ${\rm U}(1)^{4}$ altogether. But we can gauge
either of the two anomaly-free subgroups, which we choose ${\rm U}(1)_{\rm
1st}\times{\rm U}(1)_{\rm 2nd}$.
$\bullet$ When ${\rm U}(1)_{\rm 1st}\times{\rm U}(1)_{\rm 2nd}$ are
dynamically gauged by coupling to the dynamical $a_{1}$ and $a_{2}$, the
$J^{\mu}_{\rm 3rd}$ and $J^{\mu}_{\rm 4th}$ are indeed non-conserved due to
the Adler-Bell-Jackiw (ABJ) anomaly Adler (1969); Bell and Jackiw (1969). Eq.
(15) shows that:
If we gauge ${\rm U}(1)_{\rm 1st}$ only, ABJ anomaly breaks ${\rm U}(1)_{\rm
3rd}\times{\rm U}(1)_{\rm 4th}$ to the discrete $\mathbb{Z}_{50,{\rm
3rd}}\times\mathbb{Z}_{40,{\rm 4th}}$.
If we gauge ${\rm U}(1)_{\rm 2nd}$ only, ABJ anomaly breaks ${\rm U}(1)_{\rm
3rd}\times{\rm U}(1)_{\rm 4th}$ to $\mathbb{Z}_{40,{\rm
3rd}}\times\mathbb{Z}_{50,{\rm 4th}}$.
If we gauge both ${\rm U}(1)_{\rm 1st}\times{\rm U}(1)_{\rm 2nd}$, ABJ anomaly
breaks ${\rm U}(1)_{\rm 3rd}\times{\rm U}(1)_{\rm 4th}$ to
$\mathbb{Z}_{10,{\rm 3rd}}\times\mathbb{Z}_{10,{\rm 4th}}$.

  6. 6.

There is no CT or P problem if there is no YH or quadratic mass term. Namely,
we can always use the anomalous symmetry transformation with appropriate two
variables $\upalpha_{\rm 3rd}$ and $\upalpha_{\rm 4th}$ to solve two
constraints $\theta_{\rm 1st}-\upalpha_{\rm 3rd}(q_{{\rm 1}}^{\intercal}\cdot
K\cdot q_{{\rm 3}})-\upalpha_{\rm 4th}(q_{{\rm 1}}^{\intercal}\cdot K\cdot
q_{{\rm 4}})=0$ and $\theta_{\rm 2nd}-\upalpha_{\rm 3rd}(q_{{\rm
2}}^{\intercal}\cdot K\cdot q_{{\rm 3}})-\upalpha_{\rm 4th}(q_{{\rm
2}}^{\intercal}\cdot K\cdot q_{{\rm 4}})=0$. The $\theta_{\rm 1st}$ and
$\theta_{\rm 2nd}$ can be redefined by the anomalous chiral transformations to
0, so the massless theory has no CT nor P violation.

  7. 7.

CT or P problem: Generic models with YH or mass terms suffer from the CT
problem. For example, even in the Higgs condensed phase, there is a
corresponding $\theta_{\rm v}$ term of a vector gauge field $a_{\rm v}$ for
the unbroken ${\rm U}(1)_{\rm vector}$ gauge group. We attempt to do the
generic chiral rotations on every chiral fermion to rotate the $\theta_{\rm
v}$ away:

$\displaystyle(\psi_{L,3},\psi_{L,4},\psi_{R,5},\psi_{R,0})\mapsto(\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\upalpha_{L,3}}\psi_{L,3},\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\upalpha_{L,4}}\psi_{L,4},\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\upalpha_{R,5}}\psi_{R,5},\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\upalpha_{R,0}}\psi_{R,0}),$ (16)

which we define this transformation as $\Psi_{\rm I}\mapsto\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\upchi_{\rm I}\delta_{\rm
IJ}}\Psi_{\rm J}$ with
$\upchi^{\intercal}\equiv({\upalpha_{L,3}},{\upalpha_{L,4}},{\upalpha_{R,5}},{\upalpha_{R,0}})$.
Under Eq. (16), the path integral $Z$ is transformed accordingly,

$\displaystyle Z$ $\displaystyle\mapsto$ $\displaystyle\int[{\cal D}a_{\rm
v}][{\cal D}\psi][{\cal D}\bar{\psi}]\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt(S_{\Psi,\rm gauge}+S_{\rm
YH}^{(\upchi)}+\int(\frac{1}{2e^{2}}F^{\rm v}\wedge\star F^{\rm
v})+\int(\upchi^{\intercal}\cdot\hskip 1.0pt\mathrm{d}\star
J_{\upchi}))}\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\int\frac{1}{2\pi}(\theta_{\rm v}-(q_{{\rm v}}^{\intercal}\cdot
K\cdot\upchi))F^{\rm v})}.$ (17) $\displaystyle S_{\rm YH}^{(\upchi)}$
$\displaystyle\equiv$ $\displaystyle\int\hskip 1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x(\phi_{H}^{3}\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip
1.0pt(-\upalpha_{L,3}+\upalpha_{R,0})}\psi_{L,3}^{\dagger}\psi_{R,0}+\phi_{H}^{\dagger}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt(-\upalpha_{L,4}+\upalpha_{R,5})}\psi_{L,4}^{\dagger}\psi_{R,5}+\text{h.c.}-\mu_{H}|\phi_{H}|^{2}-\lambda_{H}|\phi_{H}|^{4}).$
(18)

Here $\upchi^{\intercal}\cdot\hskip 1.0pt\mathrm{d}\star
J_{\upchi}\equiv\big{(}{\upalpha_{L,3}}\partial_{\mu}(\psi_{L,3}^{\dagger}\tau^{\mu}_{\rm
L}\psi_{L,3})+{\upalpha_{L,4}}\partial_{\mu}(\psi_{L,4}^{\dagger}\tau^{\mu}_{\rm
L}\psi_{L,4})+{\upalpha_{R,5}}\partial_{\mu}(\psi_{R,5}^{\dagger}\tau^{\mu}_{\rm
R}\psi_{R,5})+{\upalpha_{R,0}}\partial_{\mu}(\psi_{R,0}^{\dagger}\tau^{\mu}_{\rm
R}\psi_{R,0})\big{)}\hskip 1.0pt\mathrm{d}t\hskip 1.0pt\mathrm{d}x$.

To identify the CT problem of this model, we need to identify the associated
invariant $\bar{\theta}$ angle. The rank-2 mass matrix $M$ is defined by
rewriting the YH term in Eq. (18) as
$(\psi_{L,3}^{\dagger},\psi_{L,4}^{\dagger})M(\begin{smallmatrix}\psi_{R,5}\\\
\psi_{R,0}\end{smallmatrix})+\text{h.c.}$ So, we have $\arg(\det
M)\mapsto\arg(\langle\phi_{H}\rangle^{2})-\upalpha_{L,3}-\upalpha_{L,4}+\upalpha_{R,5}+\upalpha_{R,0}=\arg(\langle\phi_{H}\rangle^{2})-(1,1,1,1)\cdot
K\cdot\upchi$. For a single Higgs field, we can absorb the complex phase of
$\langle\phi_{H}\rangle$ to the chiral rotation, thus without losing
generality, below we choose the real condensate
$\langle\phi_{H}\rangle\in{\mathbb{R}}$, so
$\arg(\langle\phi_{H}\rangle^{2})=0$.

We can define $\arg({\rm diag}M)\mapsto\arg({\rm
diag}M)+(-\upalpha_{L,3}+\upalpha_{R,0},-\upalpha_{L,4}+\upalpha_{R,5})$ as a
2-component vector where angular value arg is evaluated along the diagonal of
$M$. Then analogous to the italic-form charge $q_{\rm v}\equiv(1,3,3,1)$, we
define the 2-component bold text-form charge ${\bf q}_{\rm v}\equiv(1,3)$
associated with the diagonal two component of $M$.

$\displaystyle\theta_{\rm v}$ $\displaystyle\mapsto$ $\displaystyle\theta_{\rm
v}-(q_{{\rm v}}^{\intercal}\cdot K\cdot\upchi)=\theta_{\rm
v}-\upalpha_{L,3}-3\upalpha_{L,4}+3\upalpha_{R,5}+\upalpha_{R,0}.{}$
$\displaystyle\arg(\det M)$ $\displaystyle\mapsto$ $\displaystyle\arg(\det
M)-(1,1,1,1)\cdot K\cdot\upchi.{}$ $\displaystyle{\bf q}_{{\rm
v}}^{\intercal}\cdot\arg({\rm diag}M)$ $\displaystyle\mapsto$
$\displaystyle{\bf q}_{{\rm v}}^{\intercal}\cdot\arg({\rm diag}M)-(q_{{\rm
v}}^{\intercal}\cdot K\cdot\upchi).{}$ $\displaystyle{\bar{\theta}_{\rm v}}$
$\displaystyle\mapsto$ $\displaystyle{\bar{\theta}_{\rm v}}\equiv{\theta_{\rm
v}}-{\bf q}_{{\rm v}}^{\intercal}\cdot\arg({\rm diag}M).$ (19)

Then ${\bar{\theta}_{\rm v}}\equiv{\theta_{\rm v}}-{\bf q}_{{\rm
v}}^{\intercal}\cdot\arg({\rm diag}M)$ is invariant under any chiral
transformation for any value of $\upchi$. For this 1+1d model, the Naturalness
says that a generic ${\bar{\theta}_{\rm v}}$ is nonzero. However, if we live
in this 1+1d model and experimentally test that miraculously
${\bar{\theta}_{\rm v}}\simeq 0$, then we encounter the CT or P problem in
this 1+1d model.

In Sec. III, we will provide a solution to this 1+1d CT problem via the
symmetric mass generation. Below we clarify more on the properties of this
1+1d CT problem.

  8. 8.

Let us rephrase the above remark differently. Among the four parameters
$(\upalpha_{L,3},\upalpha_{L,4},\upalpha_{R,5},\upalpha_{R,0})$, we only have
two available degrees of freedom from the anomalous ${\rm U}(1)_{\rm
3rd}\times{\rm U}(1)_{\rm 4th}$ symmetry to rotate the $\theta_{\rm v}$. We
should perform the chiral transformation here as the field redefinition even
before we dynamically gauge ${\rm U}(1)_{\rm 1st}\times{\rm U}(1)_{\rm 2nd}$.
Under the field redefinition $(\upalpha_{\rm 3rd},\upalpha_{\rm 4th})\in{\rm
U}(1)_{\rm 3rd}\times{\rm U}(1)_{\rm 4th}$,

$\displaystyle\theta_{\rm v}$ $\displaystyle\mapsto$ $\displaystyle\theta_{\rm
v}-\upalpha_{\rm 3rd}(q_{{\rm v}}^{\intercal}\cdot K\cdot q_{{\rm
3}})-\upalpha_{\rm 4th}(q_{{\rm v}}^{\intercal}\cdot K\cdot q_{{\rm
4}})=\theta_{\rm v}{-30(\upalpha_{\rm 3rd}+\upalpha_{\rm 4th})}.{}$
$\displaystyle\arg(m_{3,0})\equiv\vartheta_{3,0}$ $\displaystyle\mapsto$
$\displaystyle\vartheta_{3,0}{-3(\upalpha_{\rm 3rd}+\upalpha_{\rm 4th})}.{}$
$\displaystyle\arg(m_{4,5})\equiv\vartheta_{4,5}$ $\displaystyle\mapsto$
$\displaystyle\vartheta_{4,5}{-9(\upalpha_{\rm 3rd}+\upalpha_{\rm 4th})}.{}$
$\displaystyle{\bar{\theta}_{\rm v}}$ $\displaystyle\mapsto$
$\displaystyle{\bar{\theta}_{\rm v}}\equiv{\theta_{\rm v}}-{\bf q}_{{\rm
v}}^{\intercal}\cdot\arg({\rm diag}M)={\theta_{\rm
v}}-\vartheta_{3,0}-3\vartheta_{4,5}.$ (20)

The three complex phases from the $\theta_{\rm v}$, the $\vartheta_{3,0}$ and
$\vartheta_{4,5}$ from the first and second YH term ($m_{3,0}$ and $m_{4,5}$)
terms are generally nonzero. With only a single parameter $(\upalpha_{\rm
3rd}+\upalpha_{\rm 4th})$ available, we cannot rotate all of three generic
complex phases to zero, nor can we rotate the invariant linear combination
${\bar{\theta}_{\rm v}}\equiv{\theta_{\rm v}}-{\bf q}_{{\rm
v}}^{\intercal}\cdot\arg({\rm diag}M)$ to zero. This is precisely the CT or P
problem in the 1+1d model. In short, the generic nonzero $\theta_{\rm v}$,
$\vartheta_{3,0}$, and $\vartheta_{4,5}$ in a generic path integral cannot be
all set to zeros by any field redefinition or any chiral transformation:

$\int[{\cal D}a_{\rm v}][{\cal D}\psi][{\cal D}\bar{\psi}]\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt(S_{\Psi,\rm
gauge}+\int\hskip 1.0pt\mathrm{d}t\hskip 1.0pt\mathrm{d}x(\phi_{H}^{3}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\vartheta_{3,0}}\psi_{L,3}^{\dagger}\psi_{R,0}+\phi_{H}^{\dagger}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\vartheta_{4,5}}\psi_{L,4}^{\dagger}\psi_{R,5}+\text{h.c.}-\mu_{H}|\phi_{H}|^{2}-\lambda_{H}|\phi_{H}|^{4})+\int(\frac{1}{2e^{2}}F^{\rm
v}\wedge\star F^{\rm v}))}\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\int\frac{1}{2\pi}\theta_{\rm v}F^{\rm v}}.$ (21)

  9. 9.

No Peccei-Quinn solution and no axion: One may wonder whether we can imitate
the Peccei-Quinn solution Peccei and Quinn (1977a, b); Weinberg (1978);
Wilczek (1978) to solve this CT problem in 1+1d. However, in 1+1d, there is no
spontaneous continuous symmetry-breaking, no Goldstone bosons and no truly
long-range order, due to the Coleman-Mermin-Wagner-Hohenberg theorem Coleman
(1973); Mermin and Wagner (1966); Hohenberg (1967). So any new scalar field
that we introduce to the model with a modified anomalous U(1) symmetry (known
as the Peccei-Quinn symmetry Peccei and Quinn (1977a, b)) cannot be
spontaneously broken. There will be no corresponding pseudo Goldstone modes or
nearly massless axion in 1+1d, so it is not feasible to solve this 1+1d CT
problem via the conventional Peccei-Quinn solution.

##  III A Solution to the CT or P problem via Symmetric Mass Generation on the
mirror fermions

We provide a new type of solution to the CT problem (21) via Symmetric Mass
Generation (SMG). A schematic spatial configuration of the solution is shown
in Fig. 1 (a). The prevailing physical mechanisms at different energy scales
${\rm E}$ are shown in Fig. 1 (b). We summarize the solution by enumerating
the setup step by step below.

Figure 1: Our model has a chiral fermion $\Psi$ theory and a mirror fermion
$\Psi^{\prime}$ theory sharing the same internal symmetry, placed on two
domain walls, respectively shown on a spatial manifold (e.g., two ends of a
2+1d regularizable manifold ${\cal M}^{2}\times I^{1}_{y}$). The mirror
fermion is gapped by SMG at some energy scale around $\Lambda_{\text{SMG}}$.
The solid-line arrows represent the chiral fermion modes
$3_{L}$-$4_{L}$-$5_{R}$-$0_{R}$, and the dash-line arrows represent the mirror
fermion modes $3_{R}$-$4_{R}$-$5_{L}$-$0_{L}$. The shaded area means to
schematically imply the dash-line mirror fermions are gapped by SMG and become
absent from the low energy IR physics.

(a)  (b)

Figure 2: (a) The relations between the energy scale ${\rm E}$ and the CT
solution ${\bar{\theta}}={\bar{\theta}}^{\prime}=0$ imposed by the parity-
reflection $\mathbb{Z}_{2}^{\rm PR}$ above the SMG scale, are explained in the
main text. (b) The Higgs condensation profile (red curve) becomes dominant at
the energy scale $\Lambda_{\rm H}$ but becomes exponentially small when going
into the bulk ($+\hat{y}$ direction). On the other hand, the SMG strength
becomes dominant at a higher energy scale $\Lambda_{\rm SMG}$, also it becomes
exponentially small when going into the bulk ($-\hat{y}$ direction). The
horizontal axis labels the bulk direction, while the vertical axis labels both
the energy scale and also schematically the strength of (Higgs condensate or
SMG) interaction terms. In principle, it is preferred that the Higgs and SMG
dynamics do not interfere with each other in any spacetime region (i.e., the
red and blue curves do not both have nonzero values at the same region).
However, even if the Higgs condensation and SMG interfere at the same
spacetime region, as long as the SMG does not generate any mean-field mass
like Higgs condensation does, we can still maintain our solution of the 1+1d
CT or P problem to the low energy (IR below the $\Lambda_{\rm H}$ scale).

  1. 1.

1+1d theory on the boundary of a finite-width 2+1d bulk: For the convenience,
the chiral fermion action Eq. (3) can be regarded as the 1+1d boundary theory
on ${{\cal M}^{2}}$ of a 2+1d bulk abelian $K$-matrix Chern-Simons theory
$S_{\rm bulk}$ on ${{\cal M}^{3}}$ (here with a rank-4 matrix $K={\rm
diag}(1,1,-1,-1)$ and ${\rm I,J}\in\\{1,2,3,4\\}$) as an invertible
topological field theory (iTFT) but with the real-valued dynamical gauge field
“${\rm a}$” with a compact ${\rm U}(1)$ gauge group:

$\displaystyle\int[{\cal D}{\rm a}]\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0ptS_{\rm bulk})=\int[{\cal D}{\rm a}]\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\frac{K_{\rm IJ}}{4\pi}\int_{{\cal M}^{3}}{\rm a}_{\rm I}\wedge\hskip
1.0pt\mathrm{d}{\rm a}_{\rm J})=\int[{\cal D}{\rm a}]\exp(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\frac{K_{\rm IJ}}{4\pi}\int_{{\cal M}^{3}}\hskip
1.0pt\mathrm{d}t\hskip 1.0pt\mathrm{d}x\hskip
1.0pt\mathrm{d}y\,\varepsilon^{\mu\nu\rho}{\rm a}_{{\rm
I},\mu}\partial_{\nu}{\rm a}_{{\rm J},\rho}).\;\;\;\;\;\;$ (22)

The 1+1d massless fermion theory (3) on ${{\cal M}^{2}}$ is a specific Lorentz
invariant boundary condition of the 2+1d theory (22) on $\partial{{\cal
M}^{3}}={{\cal M}^{2}}$.444The free massless boundary condition is ${\rm
a}_{{\rm I},t}-K_{\rm IJ^{\prime}}^{-1}V_{\rm J^{\prime}J}{\rm a}_{{\rm
J},x}=0$. We can further choose the velocity matrix $V_{\rm IJ}$ in the
Lorentz invariant relativistic system as $V_{\rm IJ}=c\delta_{\rm IJ}$ with
the speed of light $c=1$. So for each component ${\rm I}$, we have ${\rm
a}_{{\rm I},t}-K_{\rm II}^{-1}{\rm a}_{{\rm I},x}=0$ to give the massless 1+1d
theory (3) on the boundary of a 2+1d bulk. See Appendix A for details. The
bulk theory (22) is the familiar abelian Chern-Simons theory with a ${\rm
U}(1)^{4}$ gauge group and the apparent gauge transformation ${\rm a}_{\rm
I}\to{\rm a}_{\rm I}+\hskip 1.0pt\mathrm{d}\uplambda_{\rm I}$. This 2+1d ${\rm
U}(1)^{4}$ gauge theory also has the extra ${\rm U}(1)^{4}$ global symmetry.
Each of the ${\rm U}(1)^{4}$ global symmetry has its associate ${\rm U}(1)$
global symmetry current that couples to the external background field $A_{\rm
I}$ of the Sec. II. The conserved current of the ${\rm U}(1)$ global symmetry
is $J_{\rm I}^{\rm Bulk}=\star\frac{1}{2\pi}\hskip 1.0pt\mathrm{d}{\rm a}_{\rm
I}$, which obeys the current conservation $\hskip 1.0pt\mathrm{d}\star J_{\rm
I}=0$ thanks to $\hskip 1.0pt\mathrm{d}^{2}=0$ and the Bianchi identity, which
holds strictly when ${{\cal M}^{3}}$ is a closed manifold with no boundary.

We can choose the ${\rm U}(1)^{4}$ global symmetry to be compatible with Table
1, so we redefine $J_{\alpha}^{\rm Bulk}=q_{\alpha,\rm
I}(\star\frac{1}{2\pi}\hskip 1.0pt\mathrm{d}{\rm a}_{\rm I})$ where the ${\rm
I}$ is summed over and the fixed $\alpha\in\\{1,2,3,4\\}$ labels the 1st, 2nd,
3rd, or 4th linear independent ${\rm U}(1)$ symmetry in Table 1. The external
background field $A_{\rm I}$ couples to the bulk current via $A\wedge\star
J_{\alpha}^{\rm Bulk}\sim q_{\alpha,\rm I}\frac{1}{2\pi}A_{\rm I}\wedge\hskip
1.0pt\mathrm{d}{\rm a}_{\rm I}$ for the ${\rm U}(1)_{\alpha{\text{-th}}}$
global symmetry. The bulk-boundary correspondence maps the bulk current
coupling term $A\wedge\star J_{\alpha}^{\rm Bulk}$ to the boundary current
coupling term $A\wedge\star J_{\alpha}$ with the boundary current $J_{\alpha}$
given in (9) and (13).

If ${\rm U}(1)_{\alpha{\text{-th}}}$ is treated as a global symmetry, its
1-connection $q_{\alpha,\rm I}A_{\rm I}$ is just a non-dynamical background
probe gauge field. If instead we promote the non-dynamical $q_{\alpha,\rm
I}A_{\rm I}$ to the dynamical $q_{\alpha,\rm I}a_{\rm I}$, then the path
integral’s current coupling factor has to be adjusted to:

Bulk: $\displaystyle\exp(\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{q_{\alpha,\rm
I}}{2\pi}\int A_{\rm I}\wedge\hskip 1.0pt\mathrm{d}{\rm a}_{\rm
I})\mapsto\int[{\cal D}a]\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\frac{q_{\alpha,\rm I}}{2\pi}\int a_{\rm I}\wedge\hskip
1.0pt\mathrm{d}{\rm a}_{\rm I}).\quad$ (23) Boundary:
$\displaystyle\exp(\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{1}{2\pi}\int\hskip
1.0pt\mathrm{d}t\hskip 1.0pt\mathrm{d}x\;A_{\rm I,\mu}(\Psi_{\rm
I}^{\dagger}q_{{\alpha},\rm I}\tau^{\mu}_{\rm L/R}\Psi_{\rm
I}))\mapsto\int[{\cal D}a]\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\frac{1}{2\pi}\int\hskip 1.0pt\mathrm{d}t\hskip 1.0pt\mathrm{d}x\;a_{\rm
I,\mu}(\Psi_{\rm I}^{\dagger}q_{{\alpha},\rm I}\tau^{\mu}_{\rm L/R}\Psi_{\rm
I})).$ (24)

Due to the mixed anomaly between ${\rm U}(1)_{\rm 1st}\times{\rm U}(1)_{\rm
2nd}$ and ${\rm U}(1)_{\rm 3rd}\times{\rm U}(1)_{\rm 4th}$, we will only gauge
${\rm U}(1)_{\rm 1st}\times{\rm U}(1)_{\rm 2nd}$ to get a consistent 1+1d
theory. These mixed anomalies can be captured by the Laughlin’s style of the
flux insertion thought experiment Laughlin (1981), namely threading a magnetic
flux through the hole of the annulus or cylinder strip to induce the electric
field and the boundary-bulk anomalous current transport Wang and Wen (2013);
Santos and Wang (2014).555After gaining an analytic understanding, later we
will demonstrate the Laughlin’s style of thought experiment together with the
SMG solution to the 1+1d CT or P problem by physical pictures in the
Conclusion Sec. V.

In fact, if we choose the 2+1d bulk has only a finite-width (say, along the
$y$ direction so $-w\leq y\leq w$), then the whole theory can be effectively
regarded as a 1+1d system Wang and Wen (2013, 2019); Zeng  _et al._ (2022). We
can just choose the chiral fermion $\Psi$ on one edge on $y=-w$, there are two
possible scenarios on the other edge on $y=w$: (1) In one scenario, we choose
a symmetric gapped boundary condition on the other edge $y=w$; then we shrink
the size of $w$ while lifting the mirror edge energy gapped spectrum from
$y=w$ to a higher energy. (2) In another scenario, we have the mirror fermion
$\Psi^{\prime}$ as the fermion doubling Nielsen and Ninomiya (1981) on the
other edge $y=w$; then we can further introduce the SMG to fully gap the
mirror fermion at some energy scale Wang and Wen (2013, 2019); Zeng  _et al._
(2022). To resolve the 1+1d CT problem of the original chiral fermion theory,
we will start with this second scenario.

  2. 2.

Fermion doubling — chiral fermion and mirror fermion spectra: The 1+1d chiral
fermion theory can be defined nonperturbatively (for example, on a UV
regularization such as a lattice Wang and Wen (2013, 2019); Zeng  _et al._
(2022)). But näively, the 1+1d chiral fermion theory suffers from the Nielsen-
Ninomiya (NN) fermion-doubling problem Nielsen and Ninomiya (1981) — There are
chiral fermions and mirror fermions on two domain walls separated from each
other from a finite-width 2+1d bulk. The chiral and mirror fermions localized
on the two domain walls are related to the lattice domain wall fermion
construction Kaplan (1992). The mirror fermion sector is the generalized
parity (${\cal P}\equiv{\rm P}{\rm R}$) and the generalized time-reversal
(${\cal T}\equiv{\rm T}{\rm R}$) partner of the chiral fermion sector (see the
detail in Remark 3). We can use the subindices $L$ and $R$ to denote the
chirality of Weyl fermions. In the chiral fermion theory side, the four Weyl
fermions $\Psi_{\rm I}\equiv\Big{(}\begin{smallmatrix}\psi_{L,3}\\\
\psi_{L,4}\\\ \psi_{R,5}\\\ \psi_{R,0}\end{smallmatrix}\Big{)}_{\rm I}$ carry
U(1) charges: $3_{L}$, $4_{L}$, and $5_{R}$, $0_{R}$. In the mirror fermion
theory side, the four Weyl fermions $\Psi^{\prime}_{\rm
I}\equiv\Big{(}\begin{smallmatrix}\psi^{\prime}_{R,3}\\\
\psi^{\prime}_{R,4}\\\ \psi^{\prime}_{L,5}\\\
\psi^{\prime}_{L,0}\end{smallmatrix}\Big{)}_{\rm I}$ carry U(1) charges:
$3_{R}$, $4_{R}$, and $5_{L}$, $0_{L}$. On the chiral fermion side, we have
the free fermion action Eq. (3). Similarly, on the mirror fermion side, we
have the free mirror fermion action (due to the fermion doubling):

$\displaystyle S_{\Psi^{\prime},\rm free}=\int\hskip 1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x\big{(}\hskip 1.0pt\mathrm{i}\hskip
1.0pt\psi^{\prime\dagger}_{R,3}(\partial_{t}+\partial_{x})\psi_{R,3}+\hskip
1.0pt\mathrm{i}\hskip
1.0pt\psi^{\prime\dagger}_{R,4}(\partial_{t}+\partial_{x})\psi_{R,4}+\hskip
1.0pt\mathrm{i}\hskip
1.0pt\psi^{\prime\dagger}_{L,5}(\partial_{t}-\partial_{x})\psi_{L,5}+\hskip
1.0pt\mathrm{i}\hskip
1.0pt\psi^{\prime\dagger}_{L,0}(\partial_{t}-\partial_{x})\psi_{L,0}\big{)}.\quad$
(25)

  3. 3.

Discrete R, C, ${\cal P}=$ PR and ${\cal T}=$ TR symmetry transformations:
Here we follow the systematic analysis on the C-P-T symmetries together with
the fermion parity $\mathbb{Z}_{2}^{\rm F}$ in Wang (2021a). In particular, we
generalize the previous C-P-T transformation in (2) to include the full theory
with two domain walls. We focus on the field configurations on the two domain
walls at $y=-w$ and $y=+w$. Other than the chiral fermion $\Psi$ at $y=-w$ and
the mirror fermion $\Psi^{\prime}$ at $y=+w$, there can also be additional
bosonic fields $\Phi$ at $y=-w$ and $\Phi^{\prime}$ at $y=+w$, also the same
set of gauge field $A$ at $y=-w$ and $y=+w$ that can leak to the finite-width
bulk.

Let us spell out the precise symmetry transformations for the _unitary_
reflection R, charge conjugation C, generalized parity ${\cal P}={\rm P}{\rm
R}$, and the _anti-unitary_ generalized time reversal ${\cal T}={\rm T}{\rm
R}$ of this system with chiral and mirror fermions (Fig. 1) in terms of the
active transformations on the multiplet of fermionic fields, bosonic fields,
and gauge field:

$\displaystyle\mathbb{Z}_{2}^{\rm R}$ $\displaystyle:$
$\displaystyle\Psi(t,x,-w)\equiv\Big{(}\begin{smallmatrix}\psi_{L,3}(t,x)\\\
\psi_{L,4}(t,x)\\\ \psi_{R,5}(t,x)\\\
\psi_{R,0}(t,x)\end{smallmatrix}\Big{)}\mapsto\Psi^{\prime}(t,x,w)\equiv\Big{(}\begin{smallmatrix}\psi^{\prime}_{R,3}(t,x)\\\
\psi^{\prime}_{R,4}(t,x)\\\ \psi^{\prime}_{L,5}(t,x)\\\
\psi^{\prime}_{L,0}(t,x)\end{smallmatrix}\Big{)}.\quad\Phi(t,x,-w)\mapsto\Phi^{\prime}(t,x,w).\;\left\\{\begin{array}[]{lcr}A(t,x,-w)&\mapsto&A(t,x,w).\\\
F(t,x,-w)&\mapsto&F(t,x,w).\end{array}\right.\quad\quad$ (28)
$\displaystyle\mathbb{Z}_{2}^{\rm C}$ $\displaystyle:$
$\displaystyle\Psi(t,x,-w)\mapsto(-)^{L}\Psi^{*}(t,x,-w)\equiv\Big{(}\begin{smallmatrix}-\psi^{*}_{L,3}(t,x)\\\
-\psi^{*}_{L,4}(t,x)\\\ \psi^{*}_{R,5}(t,x)\\\
\psi^{*}_{R,0}(t,x)\end{smallmatrix}\Big{)}.\quad\Phi(t,x,-w)\mapsto\Phi^{*}(t,x,-w).\;\left\\{\begin{array}[]{lcr}A(t,x,-w)&\mapsto&-A(t,x,-w).\\\
F(t,x,-w)&\mapsto&-F(t,x,-w).\end{array}\right.\quad$ (31)
$\displaystyle\mathbb{Z}_{2}^{{\cal P}}$ $\displaystyle:$
$\displaystyle\Psi(t,x,-w)\mapsto\Psi^{\prime}(t,-x,w)\equiv\Big{(}\begin{smallmatrix}\psi^{\prime}_{R,3}(t,-x)\\\
\psi^{\prime}_{R,4}(t,-x)\\\ \psi^{\prime}_{L,5}(t,-x)\\\
\psi^{\prime}_{L,0}(t,-x)\end{smallmatrix}\Big{)}.\quad\Phi(t,x,-w)\mapsto\Phi^{\prime}(t,-x,w).\quad\left\\{\begin{array}[]{lcr}A_{0}(t,x,-w)&\mapsto&A_{0}(t,-x,w).\\\
A_{1}(t,x,-w)&\mapsto&-A_{1}(t,-x,w).\\\
F_{01}(t,x,-w)&\mapsto&-F_{01}(t,-x,w).\end{array}\right.\quad\quad$ (35)
$\displaystyle\mathbb{Z}_{2}^{{\cal T}}$ $\displaystyle:$
$\displaystyle\Psi(t,x,-w)\mapsto\Psi^{\prime}(-t,x,w)\equiv\Big{(}\begin{smallmatrix}\psi^{\prime}_{R,3}(-t,x)\\\
\psi^{\prime}_{R,4}(-t,x)\\\ \psi^{\prime}_{L,5}(-t,x)\\\
\psi^{\prime}_{L,0}(-t,x)\end{smallmatrix}\Big{)}.\quad\Phi(t,x,-w)\mapsto\Phi^{\prime}(-t,x,w).\quad\left\\{\begin{array}[]{lcr}A_{0}(t,x,-w)&\mapsto&A_{0}(-t,x,w).\\\
A_{1}(t,x,-w)&\mapsto&-A_{1}(-t,x,w).\\\
F_{01}(t,x,-w)&\mapsto&F_{01}(-t,x,w).\\\ \end{array}\right.\quad\quad$ (39)

The precise $\Phi$ content will soon become clear when we built the full model
including interactions — later $\Phi$ will stand for the multiplet of scalars
including the Higgs or disorder scalars $(\phi_{H},\phi_{1},\phi_{2})$ on the
domain wall $y=-w$, and similarly $\Phi^{\prime}$ stands for
$(\phi^{\prime}_{H},\phi^{\prime}_{1},\phi^{\prime}_{2})$ on the wall $y=w$.
In the above, we write the symmetry transformations of $\Psi$ and $\Phi$;
similarly we have analogous transformations for $\Psi^{\prime}$ and
$\Phi^{\prime}$.

Below let us check what are the symmetries of the chiral fermion theory,
without or with the mirror sector, and without or with the $\theta$ term:

$\bullet$ The 1+1d U(1) symmetric free massless chiral fermion 3-4-5-0 theory
alone is only C symmetric, but it violates the reflection R, and violates the
ordinary P and T symmetries (thus being chiral Wang and Wen (2013)). The
generic $\theta$ term $\frac{\theta}{2\pi}\int F$ violates C, CT, and P
symmetries, but preserves T symmetry. So the 1+1d chiral fermion theory
together with a generic $\theta$ term violates all R, C, P, and T symmetries.

$\bullet$ The parent theory with free massless chiral and mirror fermions
(above the energy ${\rm E}>\Lambda_{\text{SMG}}$) is not only C symmetric but
also ${\cal P}$ and ${\cal T}$ symmetric, however not R symmetric. Notice that
the full parent theory has the generalized ${\cal P}$ and ${\cal T}$
symmetries that contain not merely the ordinary P and T symmetries but also
the R symmetry. Namely, we have already defined the ordinary PR $\equiv{\cal
P}$ as the generalized ${\cal P}$ in (35), and the ordinary TR $\equiv{\cal
T}$ as the generalized ${\cal T}$ in (39).

  4. 4.

Theta term in the parent theory with chiral and mirror fermions: The parent
theory can include a generic $\theta$ term with the field strength $F=\hskip
1.0pt\mathrm{d}A$ for some ${\rm U}(1)$ internal symmetry group (e.g., in our
Table 1, this ${\rm U}(1)$ can be chosen as either ${\rm U}(1)_{\rm 1st}$ or
${\rm U}(1)_{\rm 2nd}$) on two domain walls as

$\displaystyle\frac{\theta}{2\pi}\int_{{\cal
M}^{2}}F(t,x,-w)+\frac{\theta^{\prime}}{2\pi}\int_{{\cal M}^{\prime
2}}F(t,x,w).$ (40)

One domain wall has its 2d spacetime ${{\cal M}^{2}}$, the other has its 2d
spacetime ${{\cal M}^{\prime 2}}$. These 2d spacetime can be general curved
manifolds, where the ${{\cal M}^{2}}\sqcup{{\overline{{\cal
M}^{\prime}}^{2}}}=\partial{\cal M}^{3}$ as the two boundaries of a 2+1d bulk
manifold ${\cal M}^{3}$, with the overline ${\overline{{\cal
M}^{\prime}}^{2}}$ implies the orientation reversal ${{\cal M}^{\prime}}^{2}$.
For the specific configuration in Fig. 1 (a), we can choose ${{\cal
M}^{2}}={{\cal M}^{1}_{\text{space}}}\times{{\cal M}^{1}_{\text{time}}}$ and
${{\cal M}^{\prime 2}}={{\cal M}^{\prime 1}_{\text{space}}}\times{{\cal
M}^{1}_{\text{time}}}$, where the 1d time ${{\cal M}^{1}_{\text{time}}}$ is
shared and the ${{\cal M}^{1}_{\text{space}}}\sqcup{\overline{{\cal
M}}^{1}_{\text{space}}}$ is the boundary of a 2-dimensional strip or a
cylinder. The generic $\theta$ and $\theta^{\prime}$ vacua violate the R, C,
${\cal P}$, and ${\cal T}$ symmetries, if we treat the abelian gauge field on
both sides of domain walls distinctly.

However, the gauge fields on both sides are the same gauge field, so we could
even combine the effects into a combined theta term:

$\displaystyle\frac{\theta+\theta^{\prime}}{2\pi}\int F(t,x).$ (41)

In this case, we have to consider a finite-width strip as Fig. 1 (a) so we set
the electric field strength $F(t,x,-w)=F(t,x,w)$ while ${{\cal M}^{2}}$ is
identified as ${{\cal M}^{\prime 2}}$ in the spacetime integration range. The
$F_{01}(t,x,-w)=F_{01}(t,x,w)=E_{1}$ identified as the same electric field on
two sides of the strip can be precisely derived from the Laughlin’s flux
insertion thought experiment Laughlin (1981).

  5. 5.

Impose the $\mathbb{Z}_{2}^{\rm PR}\equiv\mathbb{Z}_{2}^{{\cal P}}$ parity-
reflection symmetry on the theta term: If we impose the PR symmetry on the
parent theory with chiral and mirror fermions and with the
${\theta+\theta^{\prime}}$ term in (41) (above the energy ${\rm
E}>\Lambda_{\text{SMG}}$), then the PR symmetry in (35) demands that the
$\theta$ term on the domain wall on ${{\cal M}^{2}}$ maps to the
$\theta^{\prime}$ term on the domain wall on ${{\cal M}^{\prime 2}}$

$\displaystyle\frac{\theta}{2\pi}\int_{{\cal
M}^{2}}F(t,x,-w)\mapsto\frac{\theta}{2\pi}\int_{{\cal M}^{\prime
2}}-F(t,-x,w)=\frac{-\theta}{2\pi}\int_{{\cal M}^{\prime 2}}F(t,x,w).$ (42)

Note that the right-hand side is the $\frac{\theta^{\prime}}{2\pi}\int_{{\cal
M}^{\prime 2}}F(t,x,w)$ term on ${{\cal M}^{\prime 2}}$. Thus, the PR symmetry
for the full parent theory implies that ${\theta^{\prime}}=-\theta$,

$\displaystyle\theta+\theta^{\prime}=0\quad\Rightarrow\quad\frac{\theta+\theta^{\prime}}{2\pi}\int
F(t,x)=0.$ (43)

Namely (40) and (41) vanishes. Note that (1) the chiral fermions on ${{\cal
M}^{2}}$ and the mirror fermions on ${{\cal M}^{\prime 2}}$ have the opposite
chirality, and (2) their chiral symmetries (with the opposite chiralities on
the two domain walls) coupled to the same U(1) gauge field, the chiral U(1)
symmetry transformation will rotate $\theta\mapsto\theta+\upalpha$ and
$\theta^{\prime}\mapsto\theta^{\prime}-\upalpha$ oppositely, but keeps the
$\theta+\theta^{\prime}\mapsto\theta+\theta^{\prime}$ invariant. Because the
above two reasons, this means that the PR $={\cal P}$ symmetry at the parent
theory solves the zero theta angle problem at a high energy, since
$\theta+\theta^{\prime}=0$ and the chiral transformation with an appropriate
$\upalpha$ allows us to choose both $\theta=0$ and $\theta^{\prime}=0$.

  6. 6.

Solution to the 1+1d CT or P problem by the imposed $\mathbb{Z}_{2}^{\rm
PR}\equiv\mathbb{Z}_{2}^{{\cal P}}$ symmetry at UV and the SMG at an
intermediate energy:

We proceed our solution to the 1+1d CT or P problem by imposing the
$\mathbb{Z}_{2}^{\rm PR}\equiv\mathbb{Z}_{2}^{{\cal P}}$ symmetry further on
the fully interacting parent theory, not just on the free fermions and the
$\theta+\theta^{\prime}$ term, but also on the interaction terms including the
Yukawa-Higgs terms and the SMG multi-fermion terms that we will introduce.

The $\mathbb{Z}_{2}^{\rm R}$ reflection symmetry along the $y=0$ axis can map
the edge theory on one side $y=-w$ to the mirror edge theory on the other side
$y=w$. The $\mathbb{Z}_{2}^{\rm R}$ symmetry transformation sends $y\mapsto-y$
in the passive coordinate transformation, or sends the fields in the active
transformation. Due to the chirality of the chiral fermion at $y=-w$ is
opposite to the mirror fermion at $y=w$, even the full free massless chiral
and mirror fermion theory does not preserve the $\mathbb{Z}_{2}^{\rm R}$
symmetry.

However, there is also a 1+1d parity symmetry $\mathbb{Z}_{2}^{\rm P}$ in (6)
that can flip the chirality of chiral fermions. So we can consider a combined
parity-reflection symmetry $\mathbb{Z}_{2}^{\rm PR}$. The $\mathbb{Z}_{2}^{\rm
PR}$’s passive transformation on the coordinates $x\mapsto-x,\;y\mapsto-y$ is
obvious, but we shall follow the active transformation (35) on the fields in
this work. The full free massless chiral and mirror fermion theory does
preserve the $\mathbb{Z}_{2}^{\rm PR}$ symmetry. In fact, the
$\mathbb{Z}_{2}^{\rm PR}$ symmetry is the $\pi$-rotation in the $x$-$y$ plane
with respect to the 2+1d theory. But here we concern the $\mathbb{Z}_{2}^{\rm
PR}$ symmetry only mapping between the two 1+1d domain walls.

$\bullet$ ${\rm E}>\Lambda_{\text{SMG}}$: We hypothesize that a
$\mathbb{Z}_{2}^{\rm PR}$ reflection symmetry is preserved at some deep
ultraviolet (UV) high energy full parent theory including interaction terms
above some energy scale ${\rm E}>\Lambda_{\text{SMG}}$, while the SMG
dominates below $\Lambda_{\text{SMG}}$. So a full parent field theory with an
action

$\displaystyle{\mathbb{Z}_{2}^{\rm PR}}:S_{\rm UV}=\int\hskip
1.0pt\mathrm{d}t\hskip 1.0pt\mathrm{d}x\big{(}{\cal L}_{\rm UV}(t,x,-w)+{\cal
L}_{\rm UV}(t,x,w)\big{)}\;{\mapsto}\;S_{\rm UV}$ (44)

with its Lagrangian density ${\cal L}_{\rm UV}(x,y,t)$ on $y=-w$ and $y=w$ at
UV respects a $\mathbb{Z}_{2}^{\rm PR}$ symmetry.

This implies that the $\mathbb{Z}_{2}^{\rm PR}$ symmetry is preserved at some
UV scale, for the fermion doubling, all field contents, the kinetic, theta and
interaction terms in the Lagrangian at $y=-w$ and $y=w$, at least
kinematically.

$\bullet$ ${\rm E}\geq M_{\rm P}$ while $M_{\rm P}\gg\Lambda_{\text{SMG}}$: If
the quantum gravity effect is involved at some Planck energy scale $M_{\rm
P}$, then preferably the symmetries (including $\mathbb{Z}_{2}^{\rm PR}$) can
be either dynamically gauged or explicitly broken.

$\bullet$ $\Lambda_{H}<E<\Lambda_{\text{SMG}}$: Below the SMG energy scale
$\Lambda_{\text{SMG}}$ and above the Higgs condensation energy scale
$\Lambda_{H}$, the mirror fermion sector shall be fully gapped by the dominant
SMG effect.

$\bullet$ ${\rm E}<\Lambda_{H}$: Below the Higgs condensation energy scale
$\Lambda_{H}$, the condensed Higgs give the symmetry-breaking mass to the
chiral fermion sector.

Since the energy scale ${\rm E}\sim t^{-1}_{\rm period}$ is inverse
proportional to the time scale $t_{\rm period}$, we can also regard Fig. 1 (b)
as the time evolution of this quantum universe from the beginning (higher
energy ${\rm E}$ and small $t$ at UV) to the late time (lower energy ${\rm E}$
and large $t$ at IR).

  7. 7.

Yukawa-Higgs term for the fermion doubling: To respect the
$\mathbb{Z}_{2}^{\rm PR}$ for a parent theory at the energy ${\rm
E}>\Lambda_{\text{SMG}}$, we introduce the same kind of YH term on the chiral
and mirror fermion sectors:

$\displaystyle S_{{\text{YH-}}\Psi}$ $\displaystyle\equiv$
$\displaystyle\int\hskip 1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x\;(\phi_{H}^{3}\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip
1.0pt\vartheta_{3,0}}\psi_{L,3}^{\dagger}\psi_{R,0}+\phi_{H}^{\dagger}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\vartheta_{4,5}}\psi_{L,4}^{\dagger}\psi_{R,5}+\text{h.c.}+|D_{\mu}\phi_{H}|^{2}-\mu_{H}|\phi_{H}|^{2}-\lambda_{H}|\phi_{H}|^{4}).$
(45) $\displaystyle S_{{\text{YH-}}\Psi^{\prime}}$ $\displaystyle\equiv$
$\displaystyle\int\hskip 1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x\;(\phi^{\prime 3}_{H}\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip
1.0pt\vartheta_{3,0}}{\psi^{\prime\dagger}_{R,3}}\psi^{\prime}_{L,0}+\phi^{\prime\dagger}_{H}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\vartheta_{4,5}}\psi^{\prime\dagger}_{R,4}\psi^{\prime}_{L,5}+\text{h.c.}+|D_{\mu}\phi_{H}|^{2}-\mu^{\prime}_{H}|\phi^{\prime}_{H}|^{2}-\lambda^{\prime}_{H}|\phi^{\prime}_{H}|^{4}).$
(46)

Note that $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\vartheta_{3,0}}$ and $\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\vartheta_{4,5}}$ have the same complex values in
the $S_{{\text{YH-}}\Psi}$ and $S_{{\text{YH-}}\Psi^{\prime}}$ to respect the
$\mathbb{Z}_{2}^{\rm PR}$ symmetry. There are two possible choices of Higgs
profiles. (1) We can choose different Higgs fields $\phi_{H}$ and
$\phi_{H}^{\prime}$ for the chiral and mirror sectors. (2) Alternatively, we
may even choose the same Higgs field on the chiral and mirror sectors (so the
Higgs $\phi_{H}=\phi_{H}^{\prime}$ can propagate from either of two 1+1d
domain walls to the 2+1d finite-width bulk).

The Yukawa-Higgs mass matrices from (45) for the chiral sector and for the
mirror sector are respectively:

$\displaystyle(\psi_{L,3}^{\dagger},\psi_{L,4}^{\dagger})M_{\rm
H}\big{(}\begin{smallmatrix}\psi_{R,0}\\\
\psi_{R,5}\end{smallmatrix}\big{)}+{\rm h.c.}$ $\displaystyle\equiv$
$\displaystyle(\psi_{L,3}^{\dagger},\psi_{L,4}^{\dagger})\begin{pmatrix}\phi_{H}^{3}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\vartheta_{3,0}}&0\\\
0&\phi_{H}^{\dagger}\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\vartheta_{4,5}}\end{pmatrix}\big{(}\begin{smallmatrix}\psi_{R,0}\\\
\psi_{R,5}\end{smallmatrix}\big{)}+{\rm h.c.}$ (47)
$\displaystyle(\psi^{\prime\dagger}_{L,3},\psi^{\prime\dagger}_{L,4})M^{\prime\dagger}_{\rm
H}\big{(}\begin{smallmatrix}\psi^{\prime}_{R,0}\\\
\psi^{\prime}_{R,5}\end{smallmatrix}\big{)}+{\rm h.c.}$ $\displaystyle\equiv$
$\displaystyle(\psi^{\prime\dagger}_{L,3},\psi^{\prime\dagger}_{L,4})\begin{pmatrix}\phi^{\prime\dagger
3}_{H}\hskip 1.0pt\mathrm{e}^{-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\vartheta_{3,0}}&0\\\ 0&\phi^{\prime}_{H}\hskip 1.0pt\mathrm{e}^{-\hskip
1.0pt\mathrm{i}\hskip
1.0pt\vartheta_{4,5}}\end{pmatrix}\big{(}\begin{smallmatrix}\psi^{\prime}_{R,0}\\\
\psi^{\prime}_{R,5}\end{smallmatrix}\big{)}+{\rm h.c.}$ (48)

We shall set $\phi_{H}$ and $\phi^{\prime}_{H}$ to be real-valued, and the
complex phases ($\vartheta_{3,0}$ and $\vartheta_{4,5}$) capture the full
complexity.

  8. 8.

SMG by multi-fermion interactions: We follow the 1+1d SMG by the multi-fermion
interactions Wang and Wen (2013, 2019); Zeng  _et al._ (2022); Tong (2021)
that satisfy a mathematically rigorous gapping condition Haldane (1995);
Kapustin and Saulina (2011); Wang and Wen (2015); Levin (2013) (that defines a
topological gapped boundary condition of the 2+1d bulk Chern-Simons theory).
To respect the $\mathbb{Z}_{2}^{\rm PR}$ symmetry for a parent theory at the
energy ${\rm E}>\Lambda_{\text{SMG}}$, we introduce the same multi-fermion
interactions for the chiral and mirror fermion sectors:

$\displaystyle gS_{\text{multi-}\Psi}$ $\displaystyle\equiv$
$\displaystyle\int\hskip 1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x\;\big{(}{g}_{1}({\psi}_{L,3})({\psi}_{L,4}^{\dagger})^{2}({\psi}_{R,5})({\psi}_{R,0})^{2}+{g}_{2}({\psi}_{L,3})^{2}({\psi}_{L,4})({\psi}_{R,5}^{\dagger})^{2}({\psi}_{R,0})+\text{h.c.}\big{)}.$
(49) $\displaystyle g^{\prime}S_{\text{multi-}\Psi^{\prime}}$
$\displaystyle\equiv$ $\displaystyle\int\hskip 1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x\;\big{(}{g}^{\prime}_{\mathrm{I}}({\psi}^{\prime}_{R,3})(\psi^{\prime\dagger}_{R,4})^{2}(\psi^{\prime}_{L,5})(\psi^{\prime}_{L,0})^{2}+{g}^{\prime}_{2}(\psi^{\prime}_{R,3})^{2}(\psi^{\prime}_{R,4})(\psi^{\prime\dagger}_{L,5})^{2}(\psi^{\prime}_{L,0})+\text{h.c.}\big{)}.$
(50)

The multi-fermion interactions are derived based on the bosonization method
Wang and Wen (2013, 2019). Since the fermions are Grassmann numbers, all
higher power (larger than 1) of any fermion needs to be point-splitted, such
as ${\psi}_{R,0}^{2}\equiv{\psi}_{R,0}(\partial_{t}+\partial_{x}){\psi}_{R,0}$
or ${\psi}_{R,0}(D_{t}+D_{x}){\psi}_{R,0}$ in the covariant derivative form,
etc. On the lattice, we have to split
${\psi}_{R,0}^{2}(x)\equiv{\psi}_{R,0}(x){\psi}_{R,0}(x+\epsilon)$ where
$\epsilon$ is the lattice constant. These terms preserve the Lorentz spacetime
Spin group, and the internal ${\rm U}(1)_{\rm 1st}$ and ${\rm U}(1)_{\rm 2nd}$
symmetries. The coupling $g$ and $g^{\prime}$ control the coupling ${g}_{1}$,
${g}_{2}$, ${g}^{\prime}_{1}$, and ${g}^{\prime}_{2}$ in a dynamical manner —
in the sense that $g$ and $g^{\prime}$ can depend on the time or energy scales
(or the length or momentum scales) illustrated in Fig. 2.

  9. 9.

SMG by disorder-scalar interactions: We can also rewrite the multi-fermion
interactions. The disorder scalar term below Zeng  _et al._ (2022) can be
integrated out $\int[{\cal D}\phi_{1}][{\cal D}\phi_{1}^{\prime}][{\cal
D}\phi_{2}][{\cal D}\phi_{2}^{\prime}]$ to derive the multi-fermion
interactions (49):

$\displaystyle gS_{\text{disorder}}\equiv\int\hskip 1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x\;\Big{(}\phi_{1}^{2}\psi_{L,3}\psi_{R,5}+\phi_{1}^{\dagger}\psi_{L,4}^{\dagger}\psi_{R,0}+\phi_{2}^{\dagger}\psi_{L,3}\psi_{R,5}^{\dagger}+\phi_{2}^{2}\psi_{L,4}\psi_{R,0}+\text{h.c.}+\frac{1}{\sqrt{|g_{1}|}}\phi_{1}^{\dagger}\phi_{1}+\frac{1}{\sqrt{|g_{2}|}}\phi_{2}^{\dagger}\phi_{2}\Big{)}.$
(51) $\displaystyle g^{\prime}S_{\text{disorder}^{\prime}}\equiv\int\hskip
1.0pt\mathrm{d}t\hskip 1.0pt\mathrm{d}x\;\Big{(}\phi^{\prime
2}_{1}\psi^{\prime}_{R,3}\psi^{\prime}_{L,5}+\phi^{\prime\dagger}_{1}\psi^{\prime\dagger}_{R,4}\psi^{\prime}_{L,0}+\phi^{\prime\dagger}_{2}\psi^{\prime}_{R,3}\psi^{\prime\dagger}_{L,5}+\phi^{\prime
2}_{2}\psi^{\prime}_{R,4}\psi^{\prime}_{L,0}+\text{h.c.}+\frac{1}{\sqrt{|g_{1}^{\prime}|}}\phi^{\prime\dagger}_{1}\phi^{\prime}_{1}+\frac{1}{\sqrt{|g_{2}^{\prime}|}}\phi^{\prime\dagger}_{2}\phi^{\prime}_{2}\Big{)}.\quad\quad$
(52)

We do not have to introduce explicit kinetic terms for these scalar fields
($\phi_{1},\phi_{1}^{\prime},\phi_{2},\phi_{2}^{\prime}$). We also include any
random configuration of these scalar fields in the spacetime when doing the
path integral $\int[{\cal D}\phi_{1}][{\cal D}\phi_{1}^{\prime}][{\cal
D}\phi_{2}][{\cal D}\phi_{2}^{\prime}]$. Notice that our disorder scalar term
is different from the formulation of Ref. [Chen  _et al._ , 2013] that
includes different and _too many_ types of disorder-scalar interactions (of
Yukawa-Dirac and Yukawa-Majorana terms) that are _not compatible_ with the
topological gapping conditions Haldane (1995); Kapustin and Saulina (2011);
Wang and Wen (2015); Levin (2013). Ref. [Chen  _et al._ , 2013]’s formulation
_cannot_ , but Ref. [Wang and Wen, 2013, 2019; Zeng  _et al._ , 2022]’s
formulation _can_ fully symmetrically gap the chiral fermion theory.

  10. 10.

$\bar{\theta}$ and $\bar{\theta}^{\prime}$ terms that include the theta terms
and complex phases of mass matrices: Now we aim to show that in the full UV
parent theory, the analogous $\bar{\theta}$ and $\bar{\theta}^{\prime}$ terms
like (19) vanish once the $\mathbb{Z}_{2}^{\rm PR}$ is imposed.

There could be theta terms for both ${\rm U}(1)_{\rm 1st}\times{\rm U}(1)_{\rm
2nd}$. Denote the chiral sector’s as $\theta_{1}$ and $\theta_{2}$, and the
mirror sector’s as $\theta_{1}^{\prime}$ and $\theta_{2}^{\prime}$. The
$\mathbb{Z}_{2}^{\rm PR}$ symmetry at some higher energy imposes that (43), so
$\theta_{1}+\theta_{1}^{\prime}=0$ and $\theta_{2}+\theta_{2}^{\prime}=0$.

In the Higgs condensed low energy theory, breaking ${\rm U}(1)_{\rm
1st}\times{\rm U}(1)_{\rm 2nd}$ to ${\rm U}(1)_{\rm vector}$, we can also pay
attention to the $\mathbb{Z}_{2}^{\rm PR}$ symmetry on the unbroken ${\rm
U}(1)_{\rm vector}$’s theta term of the chiral and mirror sector:

$\displaystyle\theta_{v}+\theta_{v}^{\prime}=0.$ (53)

Follow (20) and (46), we have ${\bar{\theta}_{\rm v}}\equiv{\theta_{\rm
v}}-{\bf q}_{{\rm v}}^{\intercal}\cdot\arg({\rm diag}M)={\theta_{\rm
v}}-\vartheta_{3,0}-3\vartheta_{4,5}$ and ${\bar{\theta}^{\prime}_{\rm
v}}\equiv{\theta^{\prime}_{\rm v}}-{\bf q}_{{\rm v}}^{\intercal}\cdot\arg({\rm
diag}M^{\prime\dagger})={\theta^{\prime}_{\rm
v}}+\vartheta_{3,0}+3\vartheta_{4,5}$. Notice that the overall

$\displaystyle{\bar{\theta}_{\rm v}}+{\bar{\theta}^{\prime}_{\rm
v}}\equiv\big{(}{\theta_{\rm v}}-{\bf q}_{{\rm v}}^{\intercal}\cdot\arg({\rm
diag}M)\big{)}+\big{(}{\theta^{\prime}_{\rm v}}-{\bf q}_{{\rm
v}}^{\intercal}\cdot\arg({\rm diag}M^{\prime\dagger})\big{)}={{\theta}_{\rm
v}}+{{\theta}^{\prime}_{\rm v}},$ (54)

the total mass matrix contributions to the complex phases canceled out,
because (1) the chiral fermion mass $M$ and mirror fermion mass $M^{\prime}$
have the opposite pairing on the chiral fermions, and the (2) magnitude of
phases (i.e., here $\vartheta_{3,0}$ and $\vartheta_{4,5}$) imposed by the
$\mathbb{Z}_{2}^{\rm PR}$ symmetry.

So from (53) and (54), we derive that ${\bar{\theta}_{\rm
v}}+{\bar{\theta}^{\prime}_{\rm v}}=0$. Just like Remark 5, because the
${\bar{\theta}_{\rm v}}$ and ${\bar{\theta}^{\prime}_{\rm v}}$ couple to the
same ${\rm U}(1)_{\rm vector}$, we can rotate ${\bar{\theta}_{\rm
v}}\mapsto{\bar{\theta}_{\rm v}}+\delta\upalpha=0$ and
${\bar{\theta}^{\prime}_{\rm v}}\mapsto{\bar{\theta}^{\prime}_{\rm
v}}-\delta\upalpha=0$ by the same symmetry transformation by those anomalous
symmetries that have mixed anomalies with the ${\rm U}(1)_{\rm vector}$
symmetry, with an amount $\delta\upalpha$. This means effectively we have

$\displaystyle{\bar{\theta}_{\rm v}}+{\bar{\theta}^{\prime}_{\rm
v}}=0\;\Rightarrow\;\bar{\theta}_{v}=\bar{\theta}_{v}^{\prime}=0$ (55)

at UV, thanks to the imposed $\mathbb{Z}_{2}^{\rm PR}$ symmetry.

  11. 11.

UV solution to the 1+1d CT or P problem: The full parent theory at UV has the
path integral:

$\displaystyle\int[{\cal D}\psi][{\cal D}\bar{\psi}][{\cal
D}\psi^{\prime}][{\cal D}\bar{\psi}^{\prime}]\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt(S_{\rm bulk}+S_{\Psi,\rm
free}+S_{\Psi^{\prime},\rm
free}+S_{{\text{YH-}}\Psi}+S_{{\text{YH-}}\Psi^{\prime}}+gS_{\text{multi-}\Psi}+g^{\prime}S_{\text{multi-}\Psi^{\prime}})}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\int_{{\cal
M}^{2}}\frac{1}{2\pi}(\theta_{\rm v}+\theta^{\prime}_{\rm v})F^{\rm v}}.$ (56)

In earlier remarks, we have shown that as long as the $\mathbb{Z}_{2}^{\rm
PR}$ symmetry is respected for this full UV theory above some energy scale
$E>\Lambda_{\text{SMG}}$, then $\bar{\theta}_{v}=\bar{\theta}_{v}^{\prime}=0$
solves the 1+1d CT or P problem.

$\bullet$ The multi-fermion interactions are irrelevant operators at IR, but
becomes non-perturbatively important at UV. Thus, the multi-fermion
interactions are _dangerously irrelevant operators_ that can drive the phase
transitions when the coupling strength becomes large above some critical
strength Wang and Wen (2013, 2019); Zeng  _et al._ (2022). This is consistent
with the fact that we set the $\Lambda_{\text{SMG}}$ at a higher UV energy
scale (Fig. 2). The multi-fermion interactions are not renormalizable, however
it is fine to include non-renormalizable terms in the effective field theory
formulation. Moreover, the multi-fermion interactions can be regularized on
the UV scale $\Lambda_{\text{SMG}}$ (e.g., on a lattice Wang and Wen (2013,
2019); Zeng  _et al._ (2022) or by a short distance cutoff in quantum
gravity).

$\bullet$ 1+1d abelian gauge theory exhibits confinement at IR, while shows
asymptotic freedom at UV Schwinger (1962). Namely, the ${\rm U}(1)$ gauge
theory becomes weak coupling at UV. Thus, conceptually, we may regard $A$ as
background gauge field at UV; we regard it as a dynamical gauge field $a$ only
after going below the SMG scale, only then

$\int[{\cal D}a_{\rm v}]\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\int(\frac{1}{2e^{2}}F^{\rm v}\wedge\star F^{\rm v}))$

in the path integral becomes significant. In contrast, in the limit when $A$
is treated as background gauge field, the SMG gapping the mirror fermion
sector while keeping the chiral fermion sector gapless is indeed successfully
numerically verified in Zeng  _et al._ (2022).

  12. 12.

Flow to IR solution to the 1+1d CT or P problem: We also need to demonstrate
that $\bar{\theta}_{v}=\bar{\theta}_{v}^{\prime}=0$ at UV under the
renormalization group (RG) flows to below the Higgs condensation $\Lambda_{\rm
H}$ scale, the model still keeps
$\bar{\theta}_{v}\simeq\bar{\theta}_{v}^{\prime}\simeq 0$ at IR. Let us list
down then rule out probable dangers that can ruin the zero or small IR theta
angles.

$\bullet$ If there is a mass matrix $M=M_{\rm H}+gM_{\rm SMG}$ mixing between
the Higgs condensation mass $M_{\rm H}$ and SMG mass $gM_{\rm SMG}$ at IR,
then the theta angle contribution from (54) gives:

$\displaystyle\arg({\rm diag}(MM^{\prime\dagger}))$ $\displaystyle=$
$\displaystyle\arg({\rm diag}((M_{\rm H}+gM_{\rm SMG})(M^{\prime\dagger}_{\rm
H}+g^{\prime}M^{\prime\dagger}_{\rm SMG}))$ (57) $\displaystyle=$
$\displaystyle\arg({\rm diag}(M_{\rm H}M^{\prime\dagger}_{\rm
H}+gg^{\prime}M_{\rm SMG}M^{\prime\dagger}_{\rm SMG}+gM_{\rm
SMG}M^{\prime\dagger}_{\rm H}+g^{\prime}M_{\rm H}M^{\prime\dagger}_{\rm
SMG}))$ (58)

The $\arg({\rm diag}(M_{\rm H}M^{\prime\dagger}_{\rm H}+gg^{\prime}M_{\rm
SMG}M^{\prime\dagger}_{\rm SMG}))=0$ can hold, because the complex phase
contributions from $M_{\rm H}$ and ${M^{\prime}}_{\rm H}^{\dagger}$ cancel
with each other, and also $M_{\rm SMG}$ and ${M^{\prime}}_{\rm SMG}^{\dagger}$
cancel with each other, when the matrix product are due to the same
mechanisms. But the mixing term $\arg({\rm diag}(gM_{\rm
SMG}M^{\prime\dagger}_{\rm H}+g^{\prime}M_{\rm H}M^{\prime\dagger}_{\rm
SMG}))$ can have a nonzero angle, because we choose the strength in Fig. 2 (b)
as $\frac{|M^{\prime\dagger}_{\rm H}|}{|M_{\rm H}|}\to 0\ll 1$ and
$\frac{|g^{\prime}M^{\prime\dagger}_{\rm SMG}|}{|gM_{\rm SMG}|}\to\infty\gg
1$, so $\frac{|gM_{\rm SMG}M^{\prime\dagger}_{\rm H}|}{|g^{\prime}M_{\rm
H}M^{\prime\dagger}_{\rm SMG}|}\to 0\ll 1$. So the complex phase contribution
of (57) mainly weighs in $\arg({\rm diag}(g^{\prime}M_{\rm
H}M^{\prime\dagger}_{\rm SMG}))$ when the mass gap strength profile is the
Fig. 2 (b) type. In the next remark, we argue that (57) with $\arg({\rm
diag}(MM^{\prime\dagger}))\simeq 0$ solves the CT or P problem at IR.

  13. 13.

Non-Mean-Field SMG Conditions to solve the CT or P problem at IR: Yukawa-Higgs
mass matrices in (45) and (47) are the mean-field mass, so $\langle M_{\rm
H}\rangle$ and $\langle M^{\prime}_{\rm H}\rangle$ are generally non-zeros. In
comparison, we show that as long as the SMG mass in (49) does not contribute
any _mean-field quadratic mass matrix_ form, then $M_{\rm SMG}=0$ and
$M^{\prime\dagger}_{\rm SMG}=0$, which gives zero IR complex phase in (57). We
list down the conditions for $M_{\rm SMG}=M^{\prime\dagger}_{\rm SMG}=0$ for
the inserted operators (49) for the mirror fermion $\Psi^{\prime}$ must obey
that:

$\displaystyle{\cal
O}^{\prime}_{\iota\upsilon}\psi^{\prime}_{R,\iota}\psi^{\prime}_{L,\upsilon}+{\rm
h.c.}\text{ requires }\langle{\cal O}^{\prime}_{\iota\upsilon}\rangle=0,$ (59)
$\displaystyle\tilde{{\cal
O}}^{\prime}_{\iota\upsilon}\psi^{\prime\dagger}_{R,\iota}\psi^{\prime}_{L,\upsilon}+{\rm
h.c.}\text{ requires }\langle\tilde{{\cal
O}}^{\prime}_{\iota\upsilon}\rangle=0,$ (60)

where $\psi^{\prime}_{R,\iota}$ represents ${\rm U}(1)_{\rm 1st}$ charge 3 or
charge 4 Weyl fermions, while $\psi^{\prime}_{L,\upsilon}$ represents ${\rm
U}(1)_{\rm 1st}$ charge 5 or charge 0 Weyl fermions. Similar conditions hold
for the inserted operators (49) for the chiral fermion $\Psi$:

$\displaystyle{\cal O}_{\iota\upsilon}\psi_{L,\iota}\psi_{R,\upsilon}+{\rm
h.c.}\text{ requires }\langle{\cal O}_{\iota\upsilon}\rangle=0,$ (61)
$\displaystyle\tilde{{\cal
O}}_{\iota\upsilon}\psi^{\dagger}_{L,\iota}\psi_{R,\upsilon}+{\rm h.c.}\text{
requires }\langle\tilde{{\cal O}}_{\iota\upsilon}\rangle=0.$ (62)

We know that the above conditions in (59) and in (61) are indeed true for the
SMG interactions (49) — because that if any of these expectation values
$\langle{\cal O}^{\prime}_{\iota\upsilon}\rangle,\langle\tilde{{\cal
O}}^{\prime}_{\iota\upsilon}\rangle,\langle{\cal O}_{\iota\upsilon}\rangle,$
and $\langle\tilde{{\cal O}}_{\iota\upsilon}\rangle$ are nonzero, they would
break the ${\rm U}(1)_{\rm 1st}$ and ${\rm U}(1)_{\rm 2nd}$ symmetries of
Table 1. But the SMG exactly demands that those anomaly-free symmetries must
be preserved, so $\langle{\cal
O}^{\prime}_{\iota\upsilon}\rangle=\langle\tilde{{\cal
O}}^{\prime}_{\iota\upsilon}\rangle=\langle{\cal
O}_{\iota\upsilon}\rangle=\langle\tilde{{\cal O}}_{\iota\upsilon}\rangle=0$.

The $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt(S_{\text{multi-}\Psi}+S_{\text{multi-}\Psi^{\prime}})}$ can also be
rewritten as the disorder scalar $\int[{\cal D}\phi_{1}][{\cal
D}\phi_{1}^{\prime}][{\cal D}\phi_{2}][{\cal D}\phi_{2}^{\prime}]$ $\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt(gS_{\text{disorder}}+g^{\prime}S_{\text{disorder}^{\prime}})}$
formulation in (51). In that case, the above no-mean-field-mass conditions
(59) and (61) become

$\displaystyle\langle\phi^{\prime}_{1}\rangle=\langle\phi^{\prime}_{2}\rangle=\langle\phi^{\prime
2}_{1}\rangle=\langle\phi^{\prime 2}_{2}\rangle=0,\text{ and
}\langle\phi_{1}\rangle=\langle\phi_{2}\rangle=\langle\phi^{2}_{1}\rangle=\langle\phi^{2}_{2}\rangle=0.$
(63)

These above conditions guarantee that $M_{\rm SMG}=M^{\prime\dagger}_{\rm
SMG}=0$, so (57) becomes $\arg({\rm diag}(MM^{\prime\dagger}))|_{\rm
UV}=\arg({\rm diag}(M_{\rm H}M^{\prime\dagger}_{\rm H}))|_{\rm UV}=0$, which
is zero at UV when the UV theory (56) respects the $\mathbb{Z}_{2}^{\rm PR}$
symmetry. At IR, only the Higgs condensate develops at the chiral fermion side
so $M_{\rm H}|_{\rm IR}\neq 0$ but $M^{\prime}_{\rm H}|_{\rm IR}=0$, we expect
that our $\arg({\rm diag}(MM^{\prime\dagger}))|_{\rm IR}=\arg({\rm
diag}(M_{\rm H}M^{\prime\dagger}_{\rm H}))|_{\rm IR}\approx 0$ strictly.

In summary, our solution has a smaller or zero $\bar{\theta}$ at IR, even
better than the higher-loop calculation arguments given in Babu and Mohapatra
(1989, 1990); Barr  _et al._ (1991). Because Ref. [Babu and Mohapatra, 1989,
1990; Barr  _et al._ , 1991] require only the Higgs mechanism, thus their
Higgs mixing on both the chiral and mirror sectors can generate complex phase
at the higher-order quantum corrections beyond the tree level semiclassical
analysis. Here instead we have two mechanisms in our solution: Higgs mechanism
dominates on the chiral sector and the SMG dominates on the mirror sector, and
there is no mean-field mass matrix mixing to generate quantum corrections to
$\bar{\theta}\approx 0$.

  14. 14.

Low-energy dynamics: Here we comment on the IR dynamics of the theory with
$\bar{\theta}_{v}=\bar{\theta}_{v}^{\prime}=0$.

$\bullet$ Without ${\rm U}(1)$ gauge fields, the multi-fermion interaction
drives from the gapless phase to the gapped phase via the Berezinskii-
Kosterlitz-Thouless (BKT) universality class transition Zeng  _et al._ (2022).
In terms of the mirror fermion theory in (49), this means that
$|g^{\prime}|<|g_{c}^{\prime}|$ for the gapless phase, and
$|g^{\prime}|>|g_{c}^{\prime}|$ for the SMG phase, for some nonperturbative
critical strength Zeng  _et al._ (2022).

$\bullet$ With ${\rm U}(1)$ gauge fields, in the perturbative field theory
analysis around the free field Gaussian fixed point, the gauge coupling
$[e]=1$ has an energy scaling dimension 1, the Anderson-Higgs mass of fermion
$[m_{\Psi}]=1$ (proportional to some power of the Higgs condensate
$\langle\phi_{H}\rangle$) and the Higgs quadratic mass coefficient
$[\mu_{H}]=1$ in (45) all have an energy scaling dimension 1. We can consider
the phase diagram parametrized by $\frac{m_{\Psi}^{2}}{e^{2}}$ and also
$\frac{\mu_{H}}{e^{2}}$ similar to the analysis in Tong (2021).

$\bullet$ With ${\rm U}(1)$ gauge fields, but suppose we remove the chiral
fermions out of the theory (e.g. $|\frac{m_{\Psi}^{2}}{e^{2}}|\gg 1$): In the
1+1d U(1) gauge theory phase diagram, the $\frac{\mu_{H}}{e^{2}}\gg 1$ gives
the Coulomb phase, and $\frac{\mu_{H}}{e^{2}}\ll-1$ gives the Higgs
phase.666More precisely, in our model with Table 1, the
$\frac{\mu_{H}}{e^{2}}\gg 1$ gives the Coulomb phase for both ${\rm U}(1)_{\rm
1st}$ and ${\rm U}(1)_{\rm 2nd}$. The $\frac{\mu_{H}}{e^{2}}\ll-1$ Higgs down
${\rm U}(1)_{\rm 1st}$ and ${\rm U}(1)_{\rm 2nd}$ to the Coulomb phase of
${\rm U}(1)_{\rm vector}$. But for 1+1d U(1) gauge theory with
$\bar{\theta}_{v}=\bar{\theta}_{v}^{\prime}=0$, the Coulomb phase and Higgs
phase all are confined phases Callan  _et al._ (1977). How to understand this
confinement? In the pure electrodynamics Coulomb phase, the U(1) electric
field $F_{01}=E_{1}$ has a constant profile solution in the spacetime under
the equation of motion. The U(1) electric field goes along the only available
spatial direction $x$ in 1+1d, and the energy stored in two U(1) unit charged
$\pm e$ particles with a distance $l$ apart is $\frac{1}{2}e^{2}l$. This is a
linear potential proportional to the stretched distance $l$ for the
confinement. In the Higgs phase, the low energy theory is a gas of vortices
and anti-vortices, but it also shows the confinement.

$\bullet$ With ${\rm U}(1)$ gauge fields, also we include the chiral fermions
into the theory in the bare massless limit $m_{\Psi}=0$ or
$\frac{m_{\Psi}^{2}}{e^{2}}\to 0$: This also means that no Higgs condensate
$\langle\phi_{H}\rangle=0$, or the Higgs scalar becomes massive and decouples
$\frac{\mu_{H}}{e^{2}}\gg 1$. The 1+1d ${\rm U}(1)$ gauge theory is in the
Coulomb phase which is a confined phase.

We are able to regard one term of multi-fermion interactions (49) as the gauge
singlet Dirac fermion mass term of ${\rm U}(1)_{\rm 1st}$ for
$q_{1}=(3,4,5,0)$; namely,
${g}_{2}\big{(}({\psi}_{L,3})^{2}({\psi}_{L,4})({\psi}_{R,5}^{\dagger})^{2}\big{)}({\psi}_{R,0})+\text{h.c.}\sim{g}_{2}{\psi}_{L,0}^{\dagger}{\psi}_{R,0}+\text{h.c.}$
and
${g}^{\prime}_{2}\big{(}(\psi^{\prime}_{R,3})^{2}(\psi^{\prime}_{R,4})(\psi^{\prime\dagger}_{L,5})^{2}\big{)}(\psi^{\prime}_{L,0})+\text{h.c.}\sim{g}^{\prime}_{2}\psi^{\prime\dagger}_{R,0}\psi^{\prime}_{L,0}+\text{h.c.}$
The ${\psi}_{L,0}$ and ${\psi}^{\prime}_{R,0}$ may be regarded as the gauge
singlet composite fermion due to the ${\rm U}(1)_{\rm 1st}$ confinement Tong
(2021).

We are also able to regard the other term of multi-fermion interactions as the
gauge singlet of another Dirac fermion mass term of ${\rm U}(1)_{\rm 2nd}$ for
$q_{2}=(0,5,4,3)$; namely,
${g}_{1}({\psi}_{L,3})({\psi}_{L,4}^{\dagger})^{2}({\psi}_{R,5})({\psi}_{R,0})^{2}+\text{h.c.}\sim{g}_{1}{\psi}_{L,3}{\psi}_{R,3}^{\dagger}+\text{h.c.}$
and
${g}^{\prime}_{1}({\psi}^{\prime}_{R,3})(\psi^{\prime\dagger}_{R,4})^{2}(\psi^{\prime}_{L,5})(\psi^{\prime}_{L,0})^{2}+\text{h.c.}\sim{g}^{\prime}_{1}{\psi}^{\prime}_{R,3}{\psi}^{\prime\dagger}_{L,3}+\text{h.c.}$
The ${\psi}_{R,3}$ and ${\psi}^{\prime}_{L,3}$ may be regarded as the gauge
singlet composite fermion due to the ${\rm U}(1)_{\rm 2nd}$ confinement.

However, there are no gauge singlet mean-field mass term deformations to
express all multi-fermion interactions to preserve both ${\rm U}(1)_{\rm 1st}$
and ${\rm U}(1)_{\rm 2nd}$. Thus, this suggests that the multi-fermion SMG
interactions (49) by preserving all anomaly-free U(1) (both ${\rm U}(1)_{\rm
1st}$ and ${\rm U}(1)_{\rm 2nd}$) are beyond the mean-field descriptions even
at IR. Similarly, the disorder scalar interactions (51) are also beyond the
mean-field descriptions.

$\bullet$ With ${\rm U}(1)$ gauge fields, also we include the chiral fermions
paired with nonzero mass $m_{\Psi}\neq 0$ via the Higgs condensate
$\langle\phi_{H}\rangle\neq 0$: The 1+1d gauge theory is Higgs down to ${\rm
U}(1)_{\rm vec}$ but still in a confine phase. The fermion theory becomes
vector $q_{\rm v}=(1,3,3,1)$ charged under the ${\rm U}(1)_{\rm vec}$ gauge
field. The fermions pair up with nonzero vector mass
$\langle\phi_{H}^{3}\rangle\psi_{L,3}^{\dagger}\psi_{R,0}+\langle\phi_{H}^{\dagger}\rangle\psi_{L,4}^{\dagger}\psi_{R,5}+\text{h.c.}$
preserving the ${\rm U}(1)_{\rm vec}$. The low energy property of this
resulting theory is controlled by the energy scales of the fermion mass
$m_{\Psi}$ and the ${\rm U}(1)_{\rm vec}$ confinement energy gap.

##  IV Another CT or P solution via replacing the Higgs phase to SMG for the
chiral fermions

Earlier we shape the analogous “Strong CP problem” as the CT or P problem in
1+1d in (21) with the Yukawa-Higgs interaction and Anderson-Higgs mechanism to
gap the chiral fermion sector. This 1+1d Higgs field is meant to mimic the
phenomena of the 3+1d Standard Model’s Higgs field with vacuum expectation
values giving masses to the Standard Model’s confirmed fermions including
quarks and leptons.

However, suppose we live in this 1+1d world and perform the experiment to only
confirm that the fermions are massive and gapped, but we do not yet confirm
the mass generating mechanism. Instead, we may alternatively assume the chiral
fermions in the chiral sector are gapped with some energy gap by SMG (not by
the Higgs mechanism). Namely, the path integral of our current interest is not
(15) but instead this new one:

$\displaystyle Z=\int[{\cal D}\psi][{\cal D}\bar{\psi}]\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt(S_{\Psi,\rm
free}+gS_{\text{multi-}\Psi}+\int(\frac{1}{2e^{2}}F^{\rm 1st}\wedge\star
F^{\rm 1st})+\int(\frac{1}{2e^{2}}F^{\rm 2nd}\wedge\star F^{\rm 2nd}))}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\int\frac{1}{2\pi}(\theta_{\rm 1st}F^{\rm 1st}+\theta_{\rm 2nd}F^{\rm
2nd})}.$ (64)

Below we only need the chiral fermion sector and do not require the mirror
fermion sector anymore.

How could we distinguish the Anderson-Higgs mass mechanism (8) vs SMG
mechanism (64) in this world of the 1+1d chiral fermion sector? We can
distinguish them by examining their masses, their symmetry properties, and the
energy-momentum dispersion relations — these are falsifiable signatures by
numerics or experiments:

$\bullet$ The Anderson-Higgs mass from
$\langle\phi_{H}^{3}\rangle\psi_{L,3}^{\dagger}\psi_{R,0}+\langle\phi_{H}^{\dagger}\rangle\psi_{L,4}^{\dagger}\psi_{R,5}+\text{h.c.}$
demands that the $\psi_{L,3}$ and $\psi_{R,0}$ paired up to be Dirac fermion
and get the Dirac mass. It also demands that the $\psi_{L,4}$ and $\psi_{R,5}$
paired up to be another Dirac fermion and get another Dirac mass. So in this
scenario, $\psi_{L,3}$ mass $m_{3}$ and $\psi_{R,0}$ mass $m_{0}$ are the
same: $m_{3}=m_{0}\equiv m_{3,0}$, while $\psi_{L,4}$ mass $m_{4}$ and
$\psi_{R,5}$ mass $m_{5}$ are the same: $m_{4}=m_{5}\equiv m_{4,5}$. The Higgs
condensation also breaks ${\rm U}(1)_{\rm 1st}\times{\rm U}(1)_{\rm 2nd}$ down
to ${\rm U}(1)_{\rm vec}$. The dispersion relations above the mass gap are
${\rm E}(p_{x})=\sqrt{m_{3,0}^{2}+p_{x}^{2}}$ for the $\psi_{L,3}$ and
$\psi_{R,0}$’s momentum $p_{x}$, and ${\rm
E}(p_{x})=\sqrt{m_{4,5}^{2}+p_{x}^{2}}$ for the $\psi_{L,4}$ and
$\psi_{R,5}$’s momentum $p_{x}$.

$\bullet$ The SMG (by the multi-fermion (49) and the disorder scalar (51))
actually demands that the disorder scalar $\phi_{1}$ pairing up $\psi_{L,3}$
and $\psi_{R,5}$ while the disorder scalar $\phi_{2}$ pairing up $\psi_{L,4}$
and $\psi_{R,0}$. It can be numerically verified that $\psi_{L,3}$ and
$\psi_{R,5}$ have the same mass $m_{3}=m_{5}\equiv m_{3,5}$, while
$\psi_{L,4}$ and $\psi_{R,0}$ have another mass $m_{4}=m_{0}\equiv m_{4,0}$
Zeng  _et al._ (2022). The SMG of course preserve the full ${\rm U}(1)_{\rm
1st}\times{\rm U}(1)_{\rm 2nd}$. The SMG dispersion relations above the energy
gap suggest that ${\rm E}(p_{x})=\sqrt{m_{3,5}^{2}+p_{x}^{2}}$ for the
$\psi_{L,3}$ and $\psi_{R,5}$’s momentum $p_{x}$, and ${\rm
E}(p_{x})=\sqrt{m_{4,0}^{2}+p_{x}^{2}}$ for the $\psi_{L,4}$ and
$\psi_{R,0}$’s momentum $p_{x}$. Surprisingly, the SMG dispersion relations
above the energy gap still suggest the same form like the mean-field mass gap
Zeng  _et al._ (2022). But SMG and the mean-field Anderson-Higgs mass show
different mass gap structures.

By assuming the gapped chiral fermions (64) is due to SMG, the variation of
(64) under the ${\rm U}(1)_{\rm 3rd}\times{\rm U}(1)_{\rm 4th}$ symmetry
transformations, $\Psi_{\rm I}\mapsto\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0ptq_{{\rm 3},\rm I}\upalpha_{\rm 3rd}}\Psi_{\rm I}$
and $\Psi_{\rm I}\mapsto\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0ptq_{{\rm 4},\rm I}\upalpha_{\rm 4th}}\Psi_{\rm I}$ lead to

$\displaystyle Z$ $\displaystyle\mapsto$ $\displaystyle\int[{\cal
D}\psi][{\cal D}\bar{\psi}]\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt(S_{\Psi,\rm
free}+gS^{(\upalpha)}_{\text{multi-}\Psi}+\int(\frac{1}{2e^{2}}F^{\rm
1st}\wedge\star F^{\rm 1st})+\int(\frac{1}{2e^{2}}F^{\rm 2nd}\wedge\star
F^{\rm 2nd})+\int(\upalpha_{\rm 3rd}\hskip 1.0pt\mathrm{d}\star J_{\rm
3rd}+\upalpha_{\rm 4th}\hskip 1.0pt\mathrm{d}\star J_{\rm 4th}))}$ (66)
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\int\frac{1}{2\pi}\big{(}(\theta_{\rm 1st}-\upalpha_{\rm 3rd}(q_{{\rm
1}}^{\intercal}\cdot K\cdot q_{{\rm 3}})-\upalpha_{\rm 4th}(q_{{\rm
1}}^{\intercal}\cdot K\cdot q_{{\rm 4}}))F^{\rm 1st}+(\theta_{\rm
2nd}-\upalpha_{\rm 3rd}(q_{{\rm 2}}^{\intercal}\cdot K\cdot q_{{\rm
3}})-\upalpha_{\rm 4th}(q_{{\rm 2}}^{\intercal}\cdot K\cdot q_{{\rm
4}}))F^{\rm 2nd}\big{)}}.$

Of course, this ${\rm U}(1)_{\rm 1st}\times{\rm U}(1)_{\rm 2nd}$ anomaly-free
theory can be dynamically gauged by summing over the U(1) gauge connections
and bundles in $\int[{\cal D}a_{1}][{\cal D}a_{2}]$. What makes the difference
between the previous path integral (15) and our present path integral (66)?

$\bullet$ Previously, the Yukawa-Higgs term has a mean-field mass $S_{\rm
YH}^{(\upalpha)}$, thus varying $\upalpha$ to absorb $\theta$ to 0 would
result in a compensating complex phase $\upalpha$ gained in the mean-field
mass. Thus the 1+1d CT or P problem _cannot_ yet be solved by varying
$\upalpha$ in (15) in the chiral sector. We had provided one solution by
introducing the mirror fermion sector in Sec. III.

$\bullet$ Now we only have the multi-fermion
$gS^{(\upalpha)}_{\text{multi-}\Psi}$ or the equivalent disorder scalar
$gS^{(\upalpha)}_{\text{disorder}}$ formulation with all configuration summed
$\int[{\cal D}\phi_{1}][{\cal D}\phi_{2}]$. The chiral fermions do not have
any mean-field mass, but only have an interacting energy gap. Varying
$\upalpha$ to absorb $\theta$ to 0 would result in a compensating complex
phase $\upalpha$ in the $gS^{(\upalpha)}_{\text{multi-}\Psi}$ and
$gS^{(\upalpha)}_{\text{disorder}}$. But this is just a shift of $\upalpha$ in
the random disorder sample of disorder scalars. This shift of $\upalpha$ has
no mean-field consequence. It is easier to explain in the disorder scalar
formulation, where $gS_{\text{disorder}}$ in (51) maps to

$\mapsto\int\hskip 1.0pt\mathrm{d}t\hskip 1.0pt\mathrm{d}x\;g\;\Big{(}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt(-2\upalpha_{\rm
3rd}-4\upalpha_{\rm 4th})}\phi_{1}^{2}\psi_{L,3}\psi_{R,5}+\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt(-4\upalpha_{\rm
3rd}-8\upalpha_{\rm 4th})}\phi_{1}^{\dagger}\psi_{L,4}^{\dagger}\psi_{R,0}\\\
+\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt(8\upalpha_{\rm
3rd}+4\upalpha_{\rm
4th})}\phi_{2}^{\dagger}\psi_{L,3}\psi_{R,5}^{\dagger}+\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt(4\upalpha_{\rm
3rd}+2\upalpha_{\rm
4th})}\phi_{2}^{2}\psi_{L,4}\psi_{R,0}+\text{h.c.}+\phi_{1}^{\dagger}\phi_{1}+\phi_{2}^{\dagger}\phi_{2}\Big{)}.$
(67)

The SMG no-mean field mass condition requires
$\langle\phi_{1}\rangle=\langle\phi_{2}\rangle=\langle\phi^{2}_{1}\rangle=\langle\phi^{2}_{2}\rangle=0$,
so does

$\displaystyle\langle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt(-2\upalpha_{\rm 3rd}-4\upalpha_{\rm 4th})}\phi_{1}^{2}\rangle=\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt(-2\upalpha_{\rm
3rd}-4\upalpha_{\rm 4th})}\langle\phi_{1}^{2}\rangle=0.$
$\displaystyle\quad\quad\langle\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt(-4\upalpha_{\rm 3rd}-8\upalpha_{\rm
4th})}\phi_{1}^{\dagger}\rangle=\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt(-4\upalpha_{\rm 3rd}-8\upalpha_{\rm
4th})}\langle\phi_{1}^{\dagger}\rangle=0.$ (68) $\displaystyle\langle\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt(8\upalpha_{\rm
3rd}+4\upalpha_{\rm 4th})}\phi_{2}^{\dagger}\rangle=\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt(8\upalpha_{\rm
3rd}+4\upalpha_{\rm 4th})}\langle\phi_{2}^{\dagger}\rangle=0.$
$\displaystyle\quad\quad\langle\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt(4\upalpha_{\rm 3rd}+2\upalpha_{\rm
4th})}\phi_{2}^{2}\rangle=\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt(4\upalpha_{\rm 3rd}+2\upalpha_{\rm 4th})}\langle\phi_{2}^{2}\rangle=0.$
(69)

Thus, we can use $\upalpha_{\rm 3rd}$ and $\upalpha_{\rm 4th}$ to rotate
$\theta_{\rm 1st}$ and $\theta_{\rm 2nd}$ to zeros in (66), while leave those
$\upalpha$ undetectable in the disorder non-mean field mass. Namely,
$\bar{\theta}_{\rm 1st}=\theta_{\rm 1st}=0$ and $\bar{\theta}_{\rm
2nd}=\theta_{\rm 2nd}=0$. This solves the 1+1d CT or P problem without
introducing the mirror fermion sector, which concludes Sec. IV as another
solution.

##  V Conclusion

The 1+1d toy model formulated in this work is certainly far from a realistic
model for the 3+1d Standard Model world. However, inspired by many previous
milestone works Gross and Neveu (1974); Callan  _et al._ (1977) studied 1+1d
toy models as simplified models of the 3+1d world, we believe that the
phenomena demonstrated in the 1+1d toy model indicates some important features
that we can anticipate in the more realistic 3+1d model to solve the Strong CP
problem Hamada and Wang .

###  V.1 Thought Experiment

Let us summarize Sec. III’s solution to the 1+1d CT or P problem via SMG on
the mirror fermions in terms of Laughlin’s style of the flux insertion thought
experiment Laughlin (1981) (see also Wang and Wen (2013); Santos and Wang
(2014)) in Fig. 3.

(a)  (b)

Figure 3: (a) The time-dependent magnetic flux from an extra-dimension (here
2+1d) threading through the hole of the cylinder strip can induce the 1+1d
electric field (say of ${\rm U}(1)_{\alpha{\text{-th}}}$) along the $x$
direction, which follows from $\frac{\hskip 1.0pt\mathrm{d}\Phi_{B,y}}{\hskip
1.0pt\mathrm{d}t}=-\oint E_{x}\cdot\hskip 1.0pt\mathrm{d}l$ and the Bianchi
identity of the field strength $\hskip 1.0pt\mathrm{d}F=0$. The electric field
can further cause current transport (say of ${\rm U}(1)_{\beta{\text{-th}}}$)
from one edge of chiral fermion to the other edge of mirror fermion, if and
only if the anomaly-free coefficient $q^{\intercal}_{\alpha}Kq_{\beta}$ and
the Hall conductance
$\sigma_{xy}=\frac{e^{2}}{2\pi}q^{\intercal}_{\alpha}Kq_{\beta}$ is nonzero.
(b). Because the 1+1d fermion is anomaly-free within the symmetry $G$, the
2+1d bulk invertible TQFT is a trivial vacuum and thus is detachable. The 1+1d
mirror fermion is also SMG gappable without symmetry-breaking in $G$.

$\bullet$ ${\rm E}>\Lambda_{\text{SMG}}$: We hypothesize that a
$\mathbb{Z}_{2}^{\rm PR}$ reflection symmetry mapping between the chiral and
mirror fermion is preserved at some deep ultraviolet (UV) high energy full
parent theory including interaction terms above some energy scale. The
$\theta$ in the chiral sector and the $\theta^{\prime}=-\theta$ in the mirror
sector are cancelled. The
$\bar{\theta}=\bar{\theta}^{\prime}=\theta=\theta^{\prime}=0$ restricts the
induced electric field to be zero, because the expectation value of the
electric field $\langle F_{01}\rangle={e^{2}}{n-\frac{\theta}{2\pi}}+\dots$
with $n\in\mathbb{Z}$ and $\dots$ are the contribution from the matter
contents (e.g., bosonized phase fields of fermions). The restricts theta also
restricts the Laughlin flux insertion types of the transport shown in Fig. 3
(a).

$\bullet$ $\Lambda_{H}<E<\Lambda_{\text{SMG}}$: Below the SMG energy scale
$\Lambda_{\text{SMG}}$ but above the Higgs condensation energy scale
$\Lambda_{H}$, the mirror fermion sector shall be fully gapped out by the SMG,
shown in Fig. 3 (b).

$\bullet$ ${\rm E}<\Lambda_{H}$: Below the Higgs condensation energy scale
$\Lambda_{H}$, the condensed Higgs give the symmetry-breaking mass to the
chiral fermion sector.

###  V.2 Massless Fermion vs Mean-Field Order Mass vs Interacting or Disorder
Mass Fermion

We can also cross compare the massless fermion solution of ’t Hooft ’t Hooft
(1976b) (Fig. 4 (a)), the Peccei-Quinn symmetry-breaking mass solution with
axions Peccei and Quinn (1977a, b); Weinberg (1978); Wilczek (1978) (Fig. 4
(b)), and Sec. IV’s interacting SMG mass solution (Fig. 4 (c)), schematically
shown in Fig. 4.

Figure 4: (a) The massless fermion is drawn as no mass field (such as the
Higgs scalar $\phi_{H}$). The dashed circle means the target space of the mass
field. The origin means the mass ordering is zero. (b) The mean-field or
symmetry-breaking order mass is shown as the mass field ordered in a fixed
oriented arrow direction in the target space, while small continuous
fluctuations in the gray wedge region around the arrow direction gives
Goldstone modes or axions. This solution is unavailable in 1+1d. (c) The
interacting SMG or disorder mass gives mass even without symmetry-breaking
within an anomaly-free symmetry. The mass field is disordered so its arrow is
not oriented in a fixed direction in the target space. Instead the disordered
arrow in target space is shown as the full gray sphere in the whole target
space.

$\bullet$ The massless fermion solution ’t Hooft (1976b) says that if any of
the quark $\psi$ (say up quark) has no mean-field nor Higgs mass, then we can
do the chiral transformation on this quark $\psi$ alone to rotate the $\theta$
away but without gaining a complex phase in the mass matrix since there is no
mean-field mass for $\psi$. This sets $\bar{\theta}=0$ thus solve the the 1+1d
CT or P problem (or Strong CP problem in 3+1d).

$\bullet$ The mean-field symmetry-breaking order mass and Peccei-Quinn
solution: The arbitrariness of $\bar{\theta}$ can be relaxed by the symmetry
breaking dynamics to $\bar{\theta}\simeq 0$, The symmetry breaking here is the
Peccei-Quinn symmetry that also uses the mean-field Yukawa-Higgs symmetry
breaking mass term. The fluctuation around a fixed $\bar{\theta}=0$ gives a
low-energy Goldstone mode or an axion. However, this solution is unavailable
in 1+1d. There are no spontaneous continuous symmetry breaking, no truly long-
range order, and no Goldstone modes in 1+1d Coleman (1973); Mermin and Wagner
(1966); Hohenberg (1967). So we do not have a Peccei-Quinn solution Peccei and
Quinn (1977a, b); Weinberg (1978); Wilczek (1978), no corresponding pseudo
Goldstone modes or nearly massless axion in 1+1d.

$\bullet$ Interacting SMG or disorder mass: In contrast, Sec. IV solution says
that if all the fermions are anomaly-free under a $G$-symmetry then they can
gain the mass via SMG to preserve $G$. The SMG mass can be viewed as an
interacting mass from multi-fermion interactions, or the disorder mass from
disorder interactions. But they have no mean-field nor Higgs mass. Therefore,
we can also do the chiral transformation on any fermion $\psi$ to rotate the
$\theta$ away but without gaining a complex phase in the mass matrix since
there is no mean-field mass for $\psi$. This also sets $\bar{\theta}=0$ thus
also solves the the 1+1d CT or P problem.

_Acknowledgments_ — JW thanks Yuta Hamada, Daniel Jafferis, Matthew Reece, Yi-
Zhuang You, and Meng Zheng for the encouraging conversations. JW especially
thanks Yuta Hamada for the collaboration on Hamada and Wang . JW is supported
by Harvard University CMSA.

##  Appendix A Fermion to Boson Field Theory

###  A.1 Fermions

The original free massless fermion theory action is

$\displaystyle S_{\Psi,\rm free}=\int\hskip 1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x{\cal L}_{\Psi,\rm free}=\int\hskip 1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x\hskip 1.0pt\mathrm{i}\hskip 1.0pt\Psi_{\rm
I}^{\dagger}(\delta_{\rm IJ}\partial_{t}-K_{\rm IJ^{\prime}}^{-1}V_{\rm
J^{\prime}J}\partial_{x})\Psi_{\rm J}=\int\hskip 1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x\hskip 1.0pt\mathrm{i}\hskip 1.0pt\Psi_{\rm
I}^{\dagger}(\partial_{t}-K_{\rm II}^{-1}\partial_{x})\Psi_{\rm I}.$ (70)

The velocity matrix $V_{\rm IJ}$ in the Lorentz invariant relativistic system
can be chosen as $V_{\rm IJ}=c\delta_{\rm IJ}$, here we set the speed of light
$c=1$. Given the ${\rm U}(1)$ charge vector $q$ (like in Table 1), the ${\rm
U}(1)$ symmetry transformation acts on $\psi_{I}$ as:

$\displaystyle\Psi_{\rm I}\mapsto\exp(\hskip 1.0pt\mathrm{i}\hskip 1.0ptq_{\rm
I}\vartheta)\Psi_{\rm I},\quad\vartheta\in[0,2\pi)\cong{\rm U}(1).$ (71)

For example, for $q=q_{1}$ as the first charge vector ${\rm U}(1)_{\rm 1st}$
in Table 1, then $q_{\rm I}=q_{1,\rm I}=(3,4,5,0)$ where the $\rm I$ specifies
the component of the $q$ vector. The symmetry transformation on the 0d point-
like _charged object_ $\Psi_{\rm I}$ implies that there is a corresponding co-
dimension-1 _charge operator_ $U$ also known as the _symmetry generator_. The
symmetry generator $U$ is also a topological operator meaning that the
deformation cannot change the symmetry measurement — as long as the
topological configuration (i.e. the link between the charged object and charge
operator) is not altered under the deformation Gaiotto  _et al._ (2015).

More precisely for this specific 1+1d example, the co-dimension-1 charge
operator $U$ is 1d,

$\displaystyle U$ $\displaystyle=$ $\displaystyle\exp(-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\theta\int\hskip
1.0pt\mathrm{d}x^{\prime}J^{0}(x^{\prime}))\equiv\exp(-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\theta\int\hskip 1.0pt\mathrm{d}x^{\prime}\Psi_{\rm
I}^{\dagger}{q_{\rm I}}\Psi_{\rm I}).$ (72) $\displaystyle U\Psi_{\rm
I}U^{-1}$ $\displaystyle=$ $\displaystyle\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0ptq_{\rm I}\vartheta)\Psi_{\rm I},\quad\vartheta\in[0,2\pi)\cong{\rm
U}(1).$ (73)

More generally, $J^{\mu}\equiv\Psi_{\rm I}^{\dagger}{q_{\rm I}}\tau^{\mu}_{\rm
L/R}\Psi_{\rm I}$ with $\tau^{\mu}_{\rm L/R}$ given in the main text (12). In
the above, the first line has the repeated indices summed over (namely
$\sum_{\rm I}$); the second line has the fixed index true for each ${\rm I}$.
Here we also define ${Q_{\rm I}}\equiv{K_{\rm IJ}q_{\rm J}}$. To derive (73),
we use the fact that the $\Psi_{\rm I}$ and the conjugate momentum
$\frac{\delta{{\cal L}_{\Psi,\rm free}}}{\delta(\partial_{t}\Psi_{\rm
I})}=\hskip 1.0pt\mathrm{i}\hskip 1.0pt\Psi_{\rm I}^{\dagger}$ satisfies the
canonical quantization relation

$\displaystyle\\{\Psi_{\rm I}(x),\hskip 1.0pt\mathrm{i}\hskip 1.0pt\Psi_{\rm
J}^{\dagger}(x^{\prime})\\}=\hskip 1.0pt\mathrm{i}\hskip 1.0pt\delta_{\rm
IJ}\delta(x-x^{\prime})\Rightarrow\\{\Psi_{\rm I}(x),\Psi_{\rm
J}^{\dagger}(x^{\prime})\\}=\delta_{\rm IJ}\delta(x-x^{\prime}).$ (74)

The anti-commutator for each 1+1d one-component Weyl fermion $\Psi_{\rm I}$ is
in agreement with the anti-commutator of the 1+1d two-component Dirac fermion
$\Psi_{\rm D}=\Big{(}\begin{smallmatrix}\psi_{L}\\\
\psi_{R}\end{smallmatrix}\Big{)}$, which says $\\{\Psi_{\rm D,a}(x),\Psi_{\rm
D,b}^{\dagger}(x^{\prime})\\}=\delta(x-x^{\prime})\delta_{\rm ab}$ where ${\rm
a,b}$ are the 2-component indices.

###  A.2 Bosonization

It is helpful to compare with the bosonization language Coleman (1975);
Mandelstam (1975) of this 1+1d chiral fermion theory. For the convenience to
compare with a familiar generic $K$-matrix formulation of a bosonized theory,
we rewrite the above fermionic action (70) by rescaling with the unimodular
$K$ matrix as:

$\displaystyle\int\hskip 1.0pt\mathrm{d}t\hskip 1.0pt\mathrm{d}x{\cal
L}_{\Psi,\rm free}=\int\hskip 1.0pt\mathrm{d}t\hskip 1.0pt\mathrm{d}x\hskip
1.0pt\mathrm{i}\hskip 1.0pt\Psi_{\rm I}^{\dagger}(K_{\rm
IJ}\partial_{t}-V_{\rm IJ}\partial_{x})\Psi_{\rm J}.$ (75)

Define ${Q_{\rm I}}\equiv{K_{\rm IJ}q_{\rm J}}$, the co-dimension-1 charge
operator $U$ in (76) is modified to

$\displaystyle U$ $\displaystyle=$ $\displaystyle\exp(-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\theta\int\hskip
1.0pt\mathrm{d}x^{\prime}J^{0}(x^{\prime}))\equiv\exp(-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\theta\int\hskip 1.0pt\mathrm{d}x^{\prime}\Psi_{\rm
I}^{\dagger}{K_{\rm IJ}q_{\rm J}}\Psi_{\rm I})\equiv\exp(-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\theta\int\hskip 1.0pt\mathrm{d}x^{\prime}\Psi_{\rm
I}^{\dagger}Q_{\rm I}\Psi_{\rm I}).$ (76) $\displaystyle U\Psi_{\rm I}U^{-1}$
$\displaystyle=$ $\displaystyle\exp(\hskip 1.0pt\mathrm{i}\hskip 1.0ptq_{\rm
I}\vartheta)\Psi_{\rm I},\quad\vartheta\in[0,2\pi)\cong{\rm U}(1).$ (77)

In the above, the first line has the repeated indices summed over (namely
$\sum_{\rm I,J}$ and $\sum_{\rm I}$); the second line has the fixed index true
for each ${\rm I}$. To derive (77), we also use the fact that the $\Psi_{\rm
I}$ and the conjugate momentum $\frac{\delta{{\cal L}_{\Psi,\rm
free}}}{\delta(\partial_{t}\Psi_{\rm I})}=\hskip 1.0pt\mathrm{i}\hskip
1.0ptK_{\rm IJ}\Psi_{\rm J}^{\dagger}$ satisfies the canonical quantization
relation

$\displaystyle\\{\Psi_{\rm I}(x),\hskip 1.0pt\mathrm{i}\hskip 1.0ptK_{\rm
IJ}\Psi_{\rm J}^{\dagger}(x^{\prime})\\}=\hskip 1.0pt\mathrm{i}\hskip
1.0pt\delta(x-x^{\prime})\Rightarrow\\{\Psi_{\rm I}(x),\Psi_{\rm
J}^{\dagger}(x^{\prime})\\}=K^{-1}_{\rm IJ}\delta(x-x^{\prime}).$ (78)

Because the action (70) is modified to (75), the commutation relation is
modified from (74) to (78).

This free massless chiral fermion theory action (75) can be bosonized to a
multiplet compact chiral boson $\varphi_{\rm I}$ theory Floreanini and Jackiw
(1987), here each $\varphi_{\rm I}\in[0,2\pi)$, with a bosonized action:

$\displaystyle S_{\varphi,\rm free}=\int\hskip 1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x{\cal L}_{\varphi,\rm free}=\frac{1}{4\pi}\int\hskip
1.0pt\mathrm{d}t\hskip 1.0pt\mathrm{d}x(K_{\rm IJ}\partial_{t}\varphi_{\rm
I}\partial_{x}\varphi_{\rm J}-V_{\rm IJ}\partial_{x}\varphi_{\rm
I}\partial_{x}\varphi_{\rm J}).$ (79)

Strictly speaking, the bosonic action (79) is only equivalent to the fermionic
action (75) when the fermion parity $\mathbb{Z}_{2}^{{\rm F}}$ is gauged.
Again we choose the velocity matrix $V_{\rm IJ}$ as $V_{\rm IJ}=c\delta_{\rm
IJ}$ with the speed of light $c=1$. Although most of the discussions here are
true for any general symmetric $K$ matrix, we consider in particular the
diagonal $K_{\rm IJ}=K_{\rm II}\delta_{\rm IJ}$ for this 1+1d chiral fermion
theory, such that $K_{\rm II}=+1$ for the left-moving mode and $K_{\rm II}=-1$
for the right-moving mode. Here $K_{\rm II}$ is the ${\rm I}$-th diagonal
element _without_ summing over ${\rm I}$.

Each complex chiral Weyl fermionic field $\Psi_{\rm I}$ is bosonized to the
vertex operator of a compact chiral bosonic field

$\displaystyle{\Psi_{\rm I}\equiv:\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0ptK_{\rm II}\varphi_{\rm I}):=:\exp(\pm\hskip 1.0pt\mathrm{i}\hskip
1.0pt\varphi_{\rm I}):}$ (80)

with the proper normal ordering $::$ that moves the annihilation operators to
the right and the creation operators to the left. The $\pm$ sign in the
exponent of (80) depends on the chirality of fermions (e.g., left-moving
$\psi_{L}$ for $K_{\rm II}=+1$, right-moving $\psi_{R}$ for $K_{\rm II}=-1$).
Under the (80) map, the multi-fermion interaction (49) such as
$gS_{\text{multi-}\Psi}$ and $g^{\prime}S_{\text{multi-}\Psi^{\prime}}$ on the
chiral and mirror sectors become:

$\displaystyle\tilde{g}S_{\cos{\text{-}}\varphi}$ $\displaystyle\equiv$
$\displaystyle\int\hskip 1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x\;\big{(}\tilde{g}_{1}\cos({\varphi}_{L,3}-2{\varphi}_{L,4}+\varphi_{R,5}+2\varphi_{R,0})+\tilde{g}_{2}\cos(2{\varphi}_{L,3}+{\varphi}_{L,4}-2\varphi_{R,5}+\varphi_{R,0})\big{)},$
(81) $\displaystyle\tilde{g}^{\prime}S_{\cos{\text{-}}\varphi^{\prime}}$
$\displaystyle\equiv$ $\displaystyle\int\hskip 1.0pt\mathrm{d}t\hskip
1.0pt\mathrm{d}x\;\big{(}\tilde{g}^{\prime}_{2}\cos({\varphi}^{\prime}_{R,3}-2{\varphi}^{\prime}_{R,4}+\varphi^{\prime}_{L,5}+2\varphi^{\prime}_{L,0})+\tilde{g}^{\prime}_{2}\cos(2{\varphi}^{\prime}_{R,3}+{\varphi}^{\prime}_{R,4}-2\varphi^{\prime}_{L,5}+\varphi^{\prime}_{L,0})\big{)}.$
(82)

The U(1) symmetry transformation on the fermion $\psi_{\rm I}$ in (71) becomes
the following U(1) symmetry transformation on the compact bosonized field
$\varphi_{\rm I}$:

$\displaystyle\varphi_{\rm I}\mapsto\varphi_{\rm I}+q_{\rm
I}\vartheta,\quad\varphi_{\rm I},\vartheta\in[0,2\pi)\cong{\rm U}(1),\quad
q_{\rm I}\in\mathbb{Z}.$ (83)

The co-dimension-1 charge operator $U$ as a 1d topological operator in the
bosonized field is,

$\displaystyle U$ $\displaystyle=$ $\displaystyle\exp(-\hskip
1.0pt\mathrm{i}\hskip 1.0pt\theta\int\hskip
1.0pt\mathrm{d}x^{\prime}J^{0}(x^{\prime}))\equiv\exp(\hskip
1.0pt\mathrm{i}\hskip 1.0pt\theta\frac{K_{\rm I^{\prime}J^{\prime}}q_{\rm
J^{\prime}}}{2\pi}\int\hskip
1.0pt\mathrm{d}x^{\prime}\partial_{x^{\prime}}\varphi_{\rm
I^{\prime}})\equiv\exp(\hskip 1.0pt\mathrm{i}\hskip 1.0pt\theta\frac{Q_{\rm
I^{\prime}}}{2\pi}\int\hskip
1.0pt\mathrm{d}x^{\prime}\partial_{x^{\prime}}\varphi_{\rm I^{\prime}}).$ (84)
$\displaystyle U\varphi_{\rm I}U^{-1}$ $\displaystyle=$
$\displaystyle\varphi_{\rm I}-\hskip 1.0pt\mathrm{i}\hskip
1.0pt\theta\frac{K_{\rm I^{\prime}J^{\prime}}q_{\rm
J^{\prime}}}{2\pi}\int\hskip 1.0pt\mathrm{d}x^{\prime}[\varphi_{\rm
I},\partial_{x^{\prime}}\varphi_{\rm I^{\prime}}]=\varphi_{\rm I}+q_{\rm
I}\vartheta.$ (85)

Again we define ${Q_{\rm I}}\equiv{K_{\rm IJ}q_{\rm J}}$. If we consider the
case that the 1d space is extended along the real line
$x^{\prime}\in\mathbb{R}=(-\infty,\infty)$, then the integration range is
$\int_{-\infty}^{\infty}$. Alternatively for the 1d circular space
$x^{\prime}\in[0,2\pi R)$ of the radius size $R$, the integration range needs
to be appropriately modified to $\int_{0}^{2\pi R}$ if we choose the reference
point at $x^{\prime}=0$. In (85), we also use the fact that the $\varphi_{\rm
I}$ and the conjugate momentum $\Pi_{\rm I}\equiv\frac{\delta{{\cal
L}_{\varphi,\rm free}}}{\delta(\partial_{t}\varphi_{\rm
I})}=\frac{1}{4\pi}K_{\rm IJ}\partial_{x}\varphi_{\rm J}$ satisfies the
canonical quantization relation

$\displaystyle[\varphi_{\rm I}(x),\Pi_{\rm J}(x^{\prime})]=\frac{1}{2}\hskip
1.0pt\mathrm{i}\hskip 1.0pt\delta_{\rm
IJ}\delta(x-x^{\prime})\Rightarrow[\varphi_{\rm
I}(x),\partial_{x^{\prime}}\varphi_{\rm J}(x^{\prime})]=\hskip
1.0pt\mathrm{i}\hskip 1.0pt2\pi K^{-1}_{\rm IJ}\delta(x-x^{\prime}).$ (86)

Here the commutator gives a $\frac{1}{2}$ factor because we have each
$\varphi_{\rm I}$ mode as a 1+1d chiral boson.

More generally, we can place the 1d charge operator $U$ along any 1d curve in
the 1+1d spacetime, while $U$ is obtained by the exponential of integrating
$J^{\mu}$ current along this 1d curve. There is a correspondence between the
operators from fermionic $\Psi$ and bosonic $\varphi$ fields:

$\displaystyle J^{\mu}$ $\displaystyle\equiv$ $\displaystyle\Psi_{\rm
I^{\prime}}^{\dagger}{K_{\rm I^{\prime}J^{\prime}}q_{\rm
J^{\prime}}}\tau^{\mu}_{\rm L/R}\Psi_{\rm I^{\prime}}=-\frac{1}{2\pi}{K_{\rm
I^{\prime}J^{\prime}}q_{\rm
J^{\prime}}}\epsilon^{\mu\nu}\partial_{\nu}\varphi_{\rm
I^{\prime}}=-\frac{1}{2\pi}{Q_{\rm
I^{\prime}}}\epsilon^{\mu\nu}\partial_{\nu}\varphi_{\rm I^{\prime}}.$ (87)
$\displaystyle J$ $\displaystyle\equiv$ $\displaystyle J^{\mu}\hskip
1.0pt\mathrm{d}x_{\mu}.$ (88) $\displaystyle U$ $\displaystyle\equiv$
$\displaystyle\exp(-\hskip 1.0pt\mathrm{i}\hskip 1.0pt\theta{\int(\hskip
1.0pt\mathrm{d}x^{\prime}_{\mu})^{\perp}J^{\mu}}).$ (89)

Here we denote $(\hskip 1.0pt\mathrm{d}x^{\prime}_{\mu})^{\perp}$ as the co-
dimension volume form of the current direction $x^{\prime}_{\mu}$. For
example, for $\mu=0$, the density $J^{0}$ has a corresponding volume $(\hskip
1.0pt\mathrm{d}x^{\prime}_{0})^{\perp}=\hskip 1.0pt\mathrm{d}x$ along the
$\mu=1$ spatial direction that we integrate over.

###  A.3 Light Sectors vs Dark Sector: Anomaly-free Condition vs Topological
Gapping Condition

The bosonized interaction cosine terms (81) (and their fermionized multi-
fermion interaction counterpart (49)) in the dark sector (gapped mirror
sector) are carefully designed to satisfy the topological gapping condition
Haldane (1995); Kapustin and Saulina (2011); Wang and Wen (2015); Levin
(2013), which are tightly constrained by the
$\mathbb{anomaly}$-$\mathbb{free}$ $\mathbb{condition}$ in the
$\mathbb{light}$ sector (nearly gapless chiral sector). In fact, it is proven
in Wang and Wen (2013) (also reviewed in Wang and You (2022)) that the
anomaly-free condition can be mapped to topological gapping condition; thus
these conditions are equivalent (if and only if) or dual to each other,
schematically shown in Fig. 5.

(a)
(b)

Figure 5: (a) The $\mathbb{light}$ sector is the chiral fermion sector, which
is also light nearly massless/gapless. The dark sector is the mirror fermion
sector, which is also heavy and gapped. (b) The anomaly-free condition (90)’s
$q^{\intercal}_{\alpha}Kq_{\beta}=0$ and the topological gapping condition
(92)’s $\ell^{\intercal}_{\alpha}K^{-1}\ell_{\beta}$ can be mapped to each
other. It is proven in Wang and Wen (2013) (also reviewed in Wang and You
(2022)) that one condition holds if and only if the other condition holds.
Thus, the anomaly-free condition in the chiral fermion sector guarantees the
design of the topological gapping condition in the mirror fermion sector.

To explain Fig. 5 and its use in our 1+1d toy model of $N$-left moving and
$N$-right moving Weyl fermions with zero central charge $c_{-}=0$ and with an
internal symmetry ${\rm U}(N)_{L}\times{\rm U}(N)_{R}\supset{\rm U}(1)^{2N}$,
we recall that the anomaly-free condition for the ${\rm U}(1)$ charges
$q_{\alpha}$ and $q_{\beta}$ is the anomaly coefficient (7) vanishes:

$\displaystyle q^{\intercal}_{\alpha}Kq_{\beta}\equiv\sum_{\rm
I,J}q_{\alpha,\rm I}K_{\rm IJ}q_{\beta,\rm J}=0,$ (90)

where $K$ can be a general rank-$2N$ symmetric bilinear integer-valued matrix
(with $2N\times 2N$ integer entries), and ${\rm I,J}\in\\{1,\dots,2N\\}$. Note
at most ${\alpha,\beta}\in\\{1,\dots,N\\}$, because the largest anomaly-free
subgroup of ${\rm U}(1)^{2N}$ is ${\rm U}(1)^{N}$. In particular, for the
3-4-5-0 model, we take $\alpha,\beta\in\\{1,2\\}$ for ${\rm U}(1)_{\rm
1st}\times{\rm U}(1)_{\rm 2nd}$ in Table 1, for $N=2$ and $2N=4$.

On the other hand, the topological gapping condition Haldane (1995); Kapustin
and Saulina (2011); Wang and Wen (2015); Levin (2013) demands to find a set of
$N$ cosine terms, $\sum_{\alpha=1}^{N}g_{\alpha}\cos(\ell_{\alpha,{\rm
I}}\varphi_{\rm I}),$ to gap the theory. To design the cosine terms preserving
the ${\rm U}(1)$ symmetry $\varphi_{\rm I}\to\varphi_{\rm I}+{q_{\beta,{\rm
I}}\upalpha}$, we require that

$\displaystyle\ell_{\alpha}^{\intercal}q_{\beta}\equiv\sum_{\rm
I}\ell_{\alpha,{\rm I}}q_{\beta,{\rm I}}=0.$ (91)

The large $g$ coupling limit of the cosine term causes the condensation of the
vertex operator $\langle\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\ell_{\alpha,{\rm I}}\varphi_{\rm I})\rangle\neq 0$ in the 1+1d. The 1+1d
gapping condition can be regarded as the topological boundary condition
specified by the line operator $A$ in a 2+1d bulk of $K$ matrix Chern-Simons
theory. The topological boundary condition is the Dirichlet boundary condition
$\sum_{\rm I}\ell_{\alpha,{\rm I}}A_{\rm I}\big{|}_{{\cal M}^{2}}=0$ of line
operator with $\alpha\in\\{1,\dots,N\\}$ and ${\rm I,J}\in\\{1,\dots,2N\\}$.
There are $N$ different Dirichlet boundary conditions among
$\alpha\in\\{1,\dots,N\\}$, and they must be compatible such that the link
invariant of any linked loop in the 2+1d bulk must give a trivial phase:
$\exp(\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\ell^{\intercal}_{\alpha}K^{-1}\ell_{\alpha})=1$ and $\exp(\hskip
1.0pt\mathrm{i}\hskip
1.0pt2\pi\ell^{\intercal}_{\alpha}K^{-1}\ell_{\beta})=1$. This phase can be
computed as the ratio of path integral $Z[\text{Hopf link};S^{3}]/Z[S^{3}]$
Putrov  _et al._ (2017), with two line operators labeled by $\ell_{\alpha}$
and $\ell_{\beta}$ linked as a Hopf link say in a 3-sphere $S^{3}$. Thus the
topological boundary condition is now related to the trivial Berry phase or
statistical phase between the bulk lines, that projects to the boundary
constraint:

$\displaystyle\ell^{\intercal}_{\alpha}K^{-1}\ell_{\beta}\equiv\sum_{\rm
I,J}\ell_{\alpha,\rm I}K^{-1}_{\rm IJ}\ell_{\beta,\rm J}=0.$ (92)

We may call the symmetry condition (91) and the topological gapping condition
(92) together as the SMG gapping condition.

To explain that the SMG gapping condition holds implies that the anomaly-free
condition also holds, Ref. [Wang and Wen, 2013] shows that given the set of
$\ell_{\alpha}$ satisfying (91) and (92), one can find the set of $q_{\alpha}$
simultaneously satisfying (90). This is true because that given
$\ell_{\alpha}$, we can choose $q_{\alpha}=K^{-1}\ell_{\alpha}$.

To explain that the anomaly-free condition holds implies that the SMG gapping
condition also holds, Ref. [Wang and Wen, 2013] shows that given the set of
$q_{\alpha}$ satisfying (90), one can find the set of $l_{\alpha}$
simultaneously satisfying (91) and (92). This is true because that given
$q_{\alpha}$, we can choose $\ell_{\alpha}=Kq_{\alpha}$.

In Table 1, we choose the two mutual anomaly-free ${\rm U}(1)_{\rm
1st}\times{\rm U}(1)_{\rm 2nd}$ charge $q_{{1}}=(3,4,5,0)$ and
$q_{{2}}=(0,5,4,3)$. So the two corresponding gapping terms can be
$\ell_{1}=Kq_{1}=q_{3}=(3,4,-5,0)$ and $\ell_{2}=Kq_{2}=q_{4}=(0,5,-4,-3)$.
Note that they are also the other mutual anomaly-free ${\rm U}(1)_{\rm
3rd}\times{\rm U}(1)_{\rm 4th}$ charge $q_{3}$ and $q_{4}$. In Table 1, we
choose a different linear combination, we redefine
$\frac{1}{3}(\ell_{1}-2\ell_{2})=(1,-2,1,2)$ as $\ell_{1}$ and redefine
$\frac{1}{3}(2\ell_{1}-\ell_{2})=(2,1,-2,1)$ as $\ell_{2}$, to minimize the
scaling dimensions of the cosine terms. These data are compatible with the
conditions of (90), (91), and (92). The underlying structure is also known as
the Narain lattice of the compact boson theory Narain  _et al._ (1987).

This concludes the demonstration of the map “$\Leftrightarrow$” (namely, if
and only if conditions) in Fig. 5. Because this proof works for a large class
of rank-$2N$ symmetric bilinear integer-valued $K$ matrix, we can generalize
the 1+1d 3-4-5-0 toy model to a general 1+1d U(1)N symmetry fermion or gauge
theory. This means that the 1+1d CT or P solution provided in this work can be
straightforwardly generalized to all U(1)N-anomaly-free models.

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