# Fermi Surface Symmetric Mass Generation

Da-Chuan Lu  Department of Physics, University of California, San Diego, CA
92093, USA  Meng Zeng  Department of Physics, University of California, San
Diego, CA 92093, USA  Juven Wang  Center of Mathematical Sciences and
Applications, Harvard University, Cambridge, MA 02138, USA  Yi-Zhuang You
Department of Physics, University of California, San Diego, CA 92093, USA

###### Abstract

Symmetric mass generation is a novel mechanism to give gapless fermions a mass
gap by non-perturbative interactions without generating any fermion bilinear
condensation. The previous studies of symmetric mass generation have been
limited to Dirac/Weyl/Majorana fermions with zero Fermi volume in the free
fermion limit. In this work, we generalize the concept of symmetric mass
generation to Fermi liquid with a finite Fermi volume and discuss how to gap
out the Fermi surfaces by interactions without breaking the U(1) loop group
symmetry or developing topological orders. We provide examples of Fermi
surface symmetric mass generation in both (1+1)D and (2+1)D Fermi liquid
systems when several Fermi surfaces together cancel the Fermi surface anomaly.
However, the U(1) loop group symmetry in these cases is still restrictive
enough to rule out all possible fermion bilinear gapping terms, such that a
non-perturbative interaction mechanism is the only way to gap out the Fermi
surfaces. This symmetric Fermi surface reconstruction is in contrast to the
conventional symmetry-breaking mechanism to gap the Fermi surfaces. As a side
product, our model provides a pristine 1D lattice regularization for the
(1+1)D U(1) symmetric chiral fermion model (e.g., the 3-4-5-0 model) by
utilizing a lattice translation symmetry as an emergent U(1) symmetry at low
energy. This opens up the opportunity for efficient numerical simulations of
chiral fermions in their own dimensions _without_ introducing mirror fermions
under the domain wall fermion construction.

##  I Introduction

Fermi liquids are gapless quantum many-body systems of fermions that possess
Fermi surfaces and well-defined quasi-particle excitations at low energy. They
are the models for the most commonly-seen metallic materials in nature. They
are probably also one of the most studied quantum phases of matter in
condensed matter physics since Landau [1, 2]. However, there are still many
novel aspects of Fermi liquids that might not have been well recognized. This
article explores one such aspect: the phenomenon of _symmetric mass
generation_ (SMG, see a recent overview [3] and references therein) in Fermi
liquids.

One intriguing property of the Fermi liquid is the surprising stability of the
Fermi surface under generic local interactions of fermions. Although the
system is gapless with vastly degenerated ground states, local interactions
often do not immediately lift the ground state degeneracy and destabilize the
Fermi liquid towards gapped phases. Early understandings of this property came
from the perturbative renormalization group (RG) analysis, as the Fermi liquid
theory can emerge as a stable RG fix-point of interacting fermion systems [4,
5, 6, 7, 8, 9, 10].

Recently, a modern understanding arose under the name of _Fermi surface
anomaly_ [11, 12], which states that the stability of the Fermi surface can be
viewed as protected by the quantum anomaly of an emergent $\mathrm{LU}(1)$
loop group symmetry at low energy, extending and unifying many related
discussions [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]
about Luttinger’s theorem [29] and Lieb-Schultz-Mattis theorem [30]. Loosely
speaking, the $\mathrm{LU}(1)$ symmetry corresponds to the fermion number
$n_{\bm{k}}$ conservation at each momentum point ${\bm{k}}$ on the Fermi
surface (FS), which is preserved by the Landau Fermi liquid Hamiltonian
$H_{\text{FL}}=\sum_{{\bm{k}}\in\text{FS}}\epsilon_{\bm{k}}n_{\bm{k}}+\sum_{{\bm{k}},{\bm{k^{\prime}}}\in\text{FS}}f_{{\bm{k}}{\bm{k}}^{\prime}}n_{\bm{k}}n_{{\bm{k}}^{\prime}}+\cdots$.
In the presence of the Fermi surface anomaly, the Fermi liquid can only be
gapped by either (i) spontaneously breaking the $\mathrm{LU}(1)$ symmetry or
(ii) spontaneously developing anomalous topological orders (or other non-Fermi
liquid exotic states) that saturate the Fermi surface anomaly. The anomaly
matching is a kinematic constraint, which is non-perturbative and more robust
than the perturbative RG analysis of the Fermi liquid low-energy dynamics.

Over the past decade, the quantum anomaly [31, 32, 33, 34] has been realized
as an important theoretical tool in analyzing the protected gapless boundary
states of interacting topological insulators/superconductors, which belong to
symmetry-protected topological (SPT) phases in a grand scope (see overviews
[35, 36, 37] and references therein). An interesting phenomenon, known as
symmetric mass generation (SMG) [38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48],
was discovered in the study of interacting fermionic SPT states. It was
realized that certain SPT states might look non-trivial at the free-fermion
(non-interacting) level but can be smoothly deformed into a trivial gapped
phase with a unique ground state by fermion interactions. This implies some
integer $\mathbb{Z}$ classification of non-interacting SPT states can be
reduced to a finite abelian elementary order-$n$ group $\mathbb{Z}_{n}$
classification for some interacting SPT states, first emphasized by Fidkowski-
Kitaev [38, 39]. Correspondingly, their gapless boundary states can be gapped
out by (and only by) interaction without breaking the symmetry or developing
the topological order (breaking emergent higher-form symmetry). This provides
a novel mechanism to generate a mass for zero-density relativistic gapless
fermions (e.g., Dirac/Weyl/Majorana fermions occupying only Fermi points with
zero Fermi volumes at the Fermi level, colloquially known as
Dirac/Weyl/Majorana cones) without symmetry breaking. This mechanism is called
SMG, which is distinct from the conventional Higgs mechanism that relies on
symmetry breaking for fermion mass generation.

However, as far as we are aware of the existing literature, the SMG mechanism
has not yet been extended to fermion systems at a finite filling (with a
finite density). The Fermi liquid is one most notable examples of such, which
possesses a Fermi surface enclosing a finite Fermi volume. It is natural to
ask: can SMG happen on the Fermi surface as well, gapping out the Fermi
surface by interaction without breaking the loop group symmetry of interest?
As we will demonstrate in this article, the answer is yes.

Given the spacetime-internal symmetry $G$ of a fermion system, the conditions
[3] for SMG to happen are: (i) the system must be free from $G$-anomaly such
that symmetric gapping (without topological order) becomes possible, and (ii)
the symmetry $G$ must be restricted enough to rule out any symmetric fermion
bilinear gapping term such that the gapping can only be achieved by
interaction. These defining conditions of SMG can be applied to the Fermi
liquid system by considering $G$ as the emergent loop group symmetry on the
Fermi surface. Based on this understanding, we will investigate the Fermi
surface SMG in the presence of the $\mathrm{LU}(1)$ symmetry. The general
feature is that even though a single Fermi surface is anomalous, it is
possible to cancel the Fermi surface anomaly among multiple Fermi surfaces (or
Fermi surfaces with multiple fermion flavors), such that interactions can
drive the transition from the Fermi liquid phase to a symmetric gapped phase.
We shall name this phenomenon as the “Fermi surface Symmetric Mass
Generation”.

The Fermi surface SMG provides us a different possibility to create a gap to
all excitations on the Fermi surface without condensing any fermion bilinear
order parameter, which makes it distinct from the superconducting gap (i.e.,
condensing Cooper pairs) or the density wave gap (i.e., condensing excitons)
that are more familiar in condensed matter physics. Nevertheless, it does
involve condensing some multi-fermion bound states that transform trivially
under the symmetry action. One simplest example is the charge-4e
superconductor [49, 50, 51, 52, 53, 54, 55], which condenses fermion quartets
(four-fermion bound states) that preserves at least the $\mathbb{Z}_{4}$
subgroup of the charge $\mathrm{U}(1)$ symmetry. In this work, we will provide
more carefully designed examples preserving the full $\mathrm{U}(1)$ symmetry
(and other lattice symmetries), but the essential idea of condensing symmetric
multi-fermion operators to generate a many-body excitation gap is the same.
Therefore, the Fermi surface SMG is intrinsically a strong non-perturbative
interaction effect of fermions. The interaction may look irrelevant at the
free-fermion (or the Fermi liquid) fixed-point. However, strong enough
interaction can still drive the gap-opening transition through non-
perturbative effects.

The article will be organized as follows. In Sec. II, we will present a
lattice model of Fermi surface SMG in (1+1)D, as the pristine lattice
regularization of the 3-4-5-0 chiral fermion model, whose phase diagram can be
reliably analyzed by RG approach. In Sec. III, we will extend the discussion
of Fermi surface SMG to (2+1)D in a concrete lattice model, which can be
exactly solved in both the weak and strong interaction limits. Through these
examples, we establish the Fermi surface SMG as a general mechanism to gap out
anomaly-free Fermi surfaces in different dimensions. We summarize our result
and discuss its connection to future directions in Sec. IV.

##  II Fermi Surface SMG in (1+1)D

###  II.1 (1+1)D Fermi Liquid and Fermi Surface Anomaly

In the free-fermion limit, the (1+1)D Fermi liquid can be realized as a system
of fermions occupying a segment of single-particle momentum eigenstates in the
1D momentum space (or Brillouin zone), which can be described by a Hamiltonian
$H=\sum_{k}c_{k}^{\dagger}\epsilon_{k}c_{k}$, where $c_{k}$ (or
$c_{k}^{\dagger}$) is the fermion annihilation (or creation) operator of the
single-particle mode at momentum $k$. For now, we only consider spinless
fermions, such that the $c_{k}$ operator does not carry spin (or any other
internal degrees of freedom). As an example, suppose the band structure is
described by $\epsilon_{k}=(k^{2}-k_{F}^{2})/(2m)$ for non-relativistic
fermions with a finite chemical potential $\mu=k_{F}^{2}/(2m)$. The ground
state of the Hamiltonian $H$ will have fermions occupying the momentum segment
$k\in[-k_{F},k_{F}]$ set by the Fermi momentum $k_{F}$, as illustrated in Fig.
1(a).

Figure 1: (a) A typical single-band Fermi liquid with Fermi surface anomaly.
(b) Two-band model of a Fermi liquid with the Fermi surface anomaly canceled.
Chiral fermions with linearized dispersions around different Fermi points
emerge at low energy.

The low-energy degrees of freedom in the (1+1)D Fermi liquids can be modeled
by the chiral fermions near the 0D Fermi surfaces (namely, Fermi points) at
$\pm k_{F}$, which are described by the following Lagrangian density

$\mathcal{L}=c_{L}^{\dagger}(\mathrm{i}\partial_{t}-v_{F}\mathrm{i}\partial_{x})c_{L}+c_{R}^{\dagger}(\mathrm{i}\partial_{t}+v_{F}\mathrm{i}\partial_{x})c_{R},$
(1)

where $v_{F}=k_{F}/m$ is the Fermi velocity. The operator $c_{L}$ (or $c_{R}$)
annihilates the left (or right) moving fermion modes, defined as

$c_{R/L}(x)=\int_{-\Lambda}^{\Lambda}\mathrm{d}\kappa\;c_{\pm
k_{F}+\kappa}\mathrm{e}^{\mathrm{i}(\pm k_{F}+\kappa)x}$ (2)

around the Fermi points within a small momentum cutoff $\Lambda\ll k_{F}$. The
low-energy effective theory $\mathcal{L}$ in Eq. (1) has an emergent
$\mathrm{U}(1)_{L}\times\mathrm{U}(1)_{R}$ symmetry, corresponding to the
separate charge conservation of the left- and right-moving chiral fermions.
Under the symmetry transformation with the periodic $\phi_{L}$ and $\phi_{R}$
in $[0,2\pi)$,

$\begin{split}\mathrm{U}(1)_{L}&:c_{L}\to\mathrm{e}^{\mathrm{i}\phi_{L}}c_{L},c_{R}\to
c_{R};\\\ \mathrm{U}(1)_{R}&:c_{L}\to
c_{L},c_{R}\to\mathrm{e}^{\mathrm{i}\phi_{R}}c_{R}.\end{split}$ (3)

They can be as well understood as a recombination of the vector
$\mathrm{U}(1)_{V}$ and axial $\mathrm{U}(1)_{A}$ symmetries, by rewriting
$\phi_{L}=\phi-k_{F}\,\delta x$ and $\phi_{R}=\phi+k_{F}\,\delta x$,

$\begin{split}\mathrm{U}(1)_{V}&:c_{k}\to\mathrm{e}^{\mathrm{i}\phi}c_{k}\Rightarrow\left\\{\begin{array}[]{l}c_{L}\to\mathrm{e}^{\mathrm{i}\phi}c_{L},\\\
c_{R}\to\mathrm{e}^{\mathrm{i}\phi}c_{R};\end{array}\right.\\\
\mathrm{U}(1)_{A}&:c_{k}\to\mathrm{e}^{\mathrm{i}k\,\delta
x}c_{k}\Rightarrow\left\\{\begin{array}[]{l}c_{L}\to\mathrm{e}^{-\mathrm{i}k_{F}\,\delta
x}c_{L},\\\ c_{R}\to\mathrm{e}^{+\mathrm{i}k_{F}\,\delta
x}c_{R}.\end{array}\right.\end{split}$ (4)

More precisely, the combined symmetry group should be denoted as
$\mathrm{U}(1)_{V}\times_{\mathbb{Z}_{2}^{F}}\mathrm{U}(1)_{A}\equiv\frac{\mathrm{U}(1)_{V}\times\mathrm{U}(1)_{A}}{\mathbb{Z}_{2}^{F}}$,
because the $\mathrm{U}(1)_{V}$ and $\mathrm{U}(1)_{A}$ symmetries share the
fermion parity $\mathbb{Z}_{2}^{F}$ subgroup (under which $c_{L,R}\to-
c_{L,R}$). The physical meaning of the vector $\mathrm{U}(1)_{V}$ symmetry is
the total $\mathrm{U}(1)$ charge conservation of the fermions, and the axial
$\mathrm{U}(1)_{A}$ symmetry can be considered an effective representation of
the _translation symmetry_ in the infrared (IR) limit (that translates all
fermions by displacement $\delta x$ along the 1D system). Although translation
symmetry is described by a _non-compact_ symmetry group $\mathbb{Z}$ at the
lattice scale, its action on the low-energy chiral fermion fields
$c_{L},c_{R}$ behaves as a _compact_ $\mathrm{U}(1)_{A}$ emergent symmetry
[25, 56].

The stability of the Fermi liquid is protected by the Fermi surface anomaly,
which can be viewed as the mixed anomaly between the $\mathrm{U}(1)_{V}$ and
$\mathrm{U}(1)_{A}$ symmetries. The anomaly index is given by [29, 30, 23]

$1\times k_{F}-1\times(-k_{F})=2k_{F}=2\pi\nu,$ (5)

which can be related to the fermion filling fraction $\nu$. The system is
anomalous if the filling $\nu$ is not an integer. Without breaking the charge
$\mathrm{U}(1)$ and translation symmetries, it is impossible to drive the
Fermi liquid to a trivial gap phase due to the non-vanishing Fermi surface
anomaly. This can be viewed as a consequence of the Lieb-Schultz-Mattis (LSM)
theorem [30]. The situation is also similar to the chiral fermion edge states
on the (1+1)D boundary of a (2+1)D quantum Hall insulator.

###  II.2 Two-Band Model and Anomaly Cancellation

To generate a gap for these low-energy fermions in (1+1)D Fermi liquids, the
Fermi surface anomaly must be canceled. Here we present a two-band toy model
that achieves anomaly cancellation and enables gapping out the Fermi surface
without breaking the charge $\mathrm{U}(1)$ and translation symmetries and
without generating any Fermi bilinear condensation. It will provide a concrete
example of SMG in (1+1)D Fermi liquids.

Consider a 1D lattice (a chain of sites) with two types of fermions $c_{iA}$
and $c_{iB}$ per site. The $A$-type fermion $c_{iA}$ carries charge $q_{A}$
under a global $\mathrm{U}(1)$ symmetry, and the $B$-type fermion $c_{iB}$
carries charge $q_{B}$ under the same $\mathrm{U}(1)$ symmetry. The
Hamiltonian takes the general form of

$\begin{split}H=&-\sum_{ij}(t_{ij}^{A}c_{iA}^{\dagger}c_{jA}+t_{ij}^{B}c_{iB}^{\dagger}c_{jB}+\text{h.c.})\\\
&-\sum_{i}(\mu_{A}c_{iA}^{\dagger}c_{iA}+\mu_{B}c_{iB}^{\dagger}c_{iB})+H_{\text{int}}\end{split}$
(6)

with $H_{\text{int}}$ being some fermion interactions to be specified later in
Eq. (16). The specific details of the hopping coefficients $t_{ij}^{A}$ and
$t_{ij}^{B}$ are not important to our discussion as long as they produce a
band structure that looks like Fig. 1(b) in the Brillouin zone. The $A$-type
fermion forms an electron-like band, and the $B$-type fermion forms a hole-
like band. The two bands overlap in the energy spectrum. This will realize a
two-band Fermi liquid in general. The Hamiltonian $H$ in Eq. (6) has a
$\mathrm{U}(1)\times(\mathbb{Z}\rtimes\mathbb{Z}_{2})$ symmetry (parameterized
by a periodic angle $\phi\in[0,2\pi)$ and an integer $n\in\mathbb{Z}$ as
follows)

$\begin{split}\mathrm{U}(1)&:c_{iA}\to\mathrm{e}^{\mathrm{i}q_{A}\phi}c_{iA},\quad
c_{iB}\to\mathrm{e}^{\mathrm{i}q_{B}\phi}c_{iB};\\\ \mathbb{Z}&:c_{iA}\to
c_{(i+n)A},\quad c_{iB}\to c_{(i+n)B};\\\ \mathbb{Z}_{2}&:c_{iA}\to
c_{(-i)A},\quad c_{iB}\to c_{(-i)B}.\end{split}$ (7)

They correspond to the total charge conservation symmetry $\mathrm{U}(1)$, the
lattice translation symmetry $\mathbb{Z}$, and the lattice reflection symmetry
$\mathbb{Z}_{2}$. The question is whether we can gap the Fermi liquid without
breaking all these symmetries in (1+1)D.

One significant obstruction towards gapping is the Fermi surface anomaly,
which can also be interpreted as a mixed anomaly between the charge
$\mathrm{U}(1)$ and (the IR correspondence of) the translation symmetry. To
cancel the Fermi surface anomaly, we need to fine-tune the chemical potentials
$\mu_{A}$ and $\mu_{B}$ such that the anomaly index vanishes

$q_{A}\nu_{A}+q_{B}\nu_{B}=0\mod 1,$ (8)

where $\nu_{A}$ and $\nu_{B}$ are the filling fractions of the $A$ and $B$
bands (for the hole-like $B$ band, we may assign $\nu_{B}<0$ such that
$|\nu_{B}|$ corresponds to the hole-filling). This is also known as the
_charge compensation_ condition in semiconductor physics.

If the $A$-type and $B$-type fermions carry the same charge as
$q_{A}=q_{B}=1$, the anomaly cancellation condition Eq. (8) simply requires
$\nu_{A}=-\nu_{B}$. In this case, the electron-like Fermi surface of the
$A$-type fermion and the hole-like Fermi surface of the $B$-type fermion are
perfectly nested (with zero nesting momentum). A gap can be opened simply by
tuning on a fermion bilinear term
$\sum_{i}(c_{iA}^{\dagger}c_{iB}+\text{h.c.})$ in the Hamiltonian, which
preserves the full $\mathrm{U}(1)\times(\mathbb{Z}\rtimes\mathbb{Z}_{2})$
symmetry. This is the familiar band hybridization mechanism to open a band gap
in a charge-compensated Fermi liquid, which drives a metal to a band insulator
without breaking symmetry.

However, we are more interested in the non-trivial case when the fermions
carry different charges $q_{A}\neq q_{B}$. For example, let us consider the
case of $q_{A}=1$ and $q_{B}=3$, then the anomaly cancellation condition Eq.
(8) requires $\nu_{A}=-3\nu_{B}$, i.e. the electron-like Fermi volume in the
$A$ band must be _three times_ as large as the hole-like Fermi volume in the
$B$ band to cancel the Fermi surface anomaly. Defining the fermion operators
$c_{kA},c_{kB}$ in the momentum space by the Fourier transformation

$c_{kA}=\sum_{i}c_{iA}\mathrm{e}^{-\mathrm{i}ki},\quad
c_{kB}=\sum_{i}c_{iB}\mathrm{e}^{-\mathrm{i}ki},$ (9)

the desired band structure can be effectively described by the following band
Hamiltonian (suppressing the interaction for now)

$H=\sum_{k}(c_{kA}^{\dagger}\epsilon_{kA}c_{kA}+c_{kB}^{\dagger}\epsilon_{kB}c_{kB}),$
(10)

with the band dispersions (see Fig. 1(b))

$\epsilon_{kA}=\frac{k^{2}-(3k_{F})^{2}}{2m_{A}},\quad\epsilon_{kB}=-\frac{k^{2}-k_{F}^{2}}{2m_{B}}.$
(11)

Here we assume $m_{A},m_{B}>0$. The Fermi momentum $k_{F}=|\nu_{B}|\pi$ is set
by the filling $|\nu_{B}|$ which is typically an _irrational_ number (without
fine-tuning). The key feature is that the Fermi momenta of the $A$ and $B$
energy bands must have a $3:1$ ratio that matches the inverse charge ratio
$(q_{A}/q_{B})^{-1}$ precisely. In this case, the energy band hybridization is
forbidden by the charge $\mathrm{U}(1)$ symmetry as the two bands now carry
different charges. Even if the band hybridization is spontaneously generated
at the price of breaking the $\mathrm{U}(1)$ symmetry, it does not gap the
Fermi liquid because the Fermi surfaces of the two bands are no longer nested
at the Fermi level, such that the band hybridization will only create some
avoided energy band crossing below the Fermi level. Then the system remains
metallic because the (upper) hybridized band still crosses the Fermi level.

One can show that it is impossible to symmetrically gap the Fermi liquid by
any fermion _bilinear_ terms in this charge-compensated two-band system with
$q_{A}=1$ and $q_{B}=3$, even if the Fermi surface anomaly has already been
canceled by the charge-compensated filling $\nu_{A}=-3\nu_{B}$. Although the
anomaly vanishes (i.e. there is no obstruction towards gapping in principle),
the symmetry is still restrictive enough to forbid any fermion bilinear
gapping term, such that the only possible gapping mechanism rests on non-
perturbative fermion interaction effects.

To see this, we can single out the low-energy chiral fermions near the four
Fermi points:

$\begin{split}c_{AR}=c_{(3k_{F})A},&\quad c_{BR}=c_{(-k_{F})B},\\\
c_{BL}=c_{(k_{F})B},&\quad c_{AL}=c_{(-3k_{F})A},\end{split}$ (12)

where $A,B$ label the bands that they originated from and $L,R$ label their
chiralities (i.e. left- or right-moving), according to Fig. 1(b). Similar to
Eq. (1), the low-energy effective Lagrangian density reads

$\mathcal{L}=\sum_{a}c_{a}^{\dagger}(\mathrm{i}\partial_{t}+v_{a}\mathrm{i}\partial_{x})c_{a},$
(13)

where the index $a$ sums over the four Fermi point labels $AR$, $BR$, $BL$,
$AL$. Here $v_{a}$ denotes the Fermi velocity near the Fermi point $a$.

The original $\mathrm{U}(1)\times\mathbb{Z}$ symmetry at the lattice fermion
level reduces to the emergent
$\mathrm{U}(1)_{V}\times_{\mathbb{Z}_{2}^{F}}\mathrm{U}(1)_{A}$ symmetry for
the low-energy chiral fermions $c_{a}$ (see Appendix A for more explanations)

$\begin{split}\mathrm{U}(1)\Rightarrow\mathrm{U}(1)_{V}&:c_{a}\to\mathrm{e}^{\mathrm{i}q_{a}^{V}\phi_{V}}c_{a},\\\
\mathbb{Z}\Rightarrow\mathrm{U}(1)_{A}&:c_{a}\to\mathrm{e}^{\mathrm{i}q_{a}^{A}\phi_{A}}c_{a}.\\\
\end{split}$ (14)

Tab. 1 summarizes their charge assignment under $\mathrm{U}(1)_{V}$ and
$\mathrm{U}(1)_{A}$, where the vector $\mathrm{U}(1)_{V}$ symmetry is just the
charge $\mathrm{U}(1)$ symmetry and the axial $\mathrm{U}(1)_{A}$ symmetry is
an emergent symmetry corresponding to the lattice translation symmetry
$\mathbb{Z}$. Alternatively, they can be recombined into the
$\mathrm{U}(1)_{\frac{3V+A}{2}}\times\mathrm{U}(1)_{\frac{3V-A}{2}}$ symmetry,
such that it becomes obvious that all fermion bilinear back-scattering terms
(either the Dirac mass $c_{a}^{\dagger}c_{b}$ or the Majorana mass
$c_{a}c_{b}$ for $a\neq b$ and $a,b\in\\{AR,BR,BL,AL\\}$) are forbidden by the
symmetry because they are all charged non-trivially under the
$\mathrm{U}(1)_{\frac{3V+A}{2}}\times\mathrm{U}(1)_{\frac{3V-A}{2}}$ symmetry
due to the distinct charge assignment to every chiral fermion. Given this
situation, the only hope to gap the Fermi liquid is to evoke the SMG mechanism
that generates the mass for all chiral fermions by non-perturbative multi-
fermion interactions.

Table 1: Charge assignments of low-energy fermions. See also the model in [57] on the same charge assignments. fermion | chirality | $\mathrm{U}(1)_{V}$ | $\mathrm{U}(1)_{A}$ | $\mathrm{U}(1)_{\frac{3V+A}{2}}$ | $\mathrm{U}(1)_{\frac{3V-A}{2}}$
---|---|---|---|---|---
$c_{a}$ | $\operatorname{sgn}v_{a}$ | $q_{a}^{V}$ | $q_{a}^{A}$ | $\frac{1}{2}(3q_{a}^{V}+q_{a}^{A})$ | $\frac{1}{2}(3q_{a}^{V}-q_{a}^{A})$
$c_{AR}$ |  $-1$ (left) | $1$ | $3$ | $3$ | $0$
$c_{BR}$ |  $-1$ (left) | $3$ | $-1$ | $4$ | $5$
$c_{BL}$ |  $+1$ (right) | $3$ | $1$ | $5$ | $4$
$c_{AL}$ |  $+1$ (right) | $1$ | $-3$ | $0$ | $3$

###  II.3 SMG Interaction and RG Analysis

It is worth mentioning that the charge-compensated two-band model with
$q_{A}=1$ and $q_{B}=3$ essentially regularizes the 3-4-5-0 chiral fermion
model [58, 59] on a pristine 1D lattice (without introducing any compact extra
dimensions). The emergent $\mathrm{U}(1)_{\frac{3V\pm A}{2}}$ symmetries act
as the lattice translations decorated by appropriate internal $\mathrm{U}(1)$
rotations, described by the following $\mathbb{Z}$ symmetry groups
(parameterized by integer $n\in\mathbb{Z}$) at the lattice level: (see
Appendix A for derivation)

$\mathbb{Z}\text{ (for {\hbox{\tfrac{3V\pm
A}{2}}})}:\left\\{\begin{array}[]{l}c_{iA}\to\mathrm{e}^{\pm\mathrm{i}3k_{F}n}c_{(i+n)A},\\\
c_{iB}\to\mathrm{e}^{\pm\mathrm{i}9k_{F}n}c_{(i+n)B}.\end{array}\right.$ (15)

The 3-4-5-0 model is a toy model for studying the long-standing problem: the
lattice regularization of the chiral fermion theory in high-energy physics
[60, 61, 62, 63, 64, 65, 66, 67, 68]. Many variants of the model are studied
in the lattice community [69, 70]. This model is anomaly-free — perturbative
local gauge anomaly free within any linear combination of the
$\mathrm{U}(1)_{V}\times_{\mathbb{Z}_{2}^{F}}\mathrm{U}(1)_{A}$ checked by the
Adler-Bell-Jackiw method [71, 72], perturbative local gravitational anomaly
free because of the zero chiral central charge $c_{L}-c_{R}=0$, also
nonperturbative global anomaly free from any gauge or gravitational fields
checked by the cobordism [73]. However, it is known much later that symmetric
gapping can only be achieved by minimally six-fermion interactions among the
four flavors of 3-4-5-0 fermions. The SMG interaction was first proposed by
Wang and Wen [44, 74], which was later discussed by Tong [75] and only
recently verified by the density matrix renormalization group (DMRG) [76, 77]
numerical simulation in Ref. 78.

Given the existing knowledge about the SMG interaction in the 3-4-5-0 chiral
fermion model, we can map the Wang-Wen interaction [44, 74] back to our
lattice model following the correspondence listed in Tab. 1, which gives us
the following SMG interaction (see Appendix B for more details)

$H_{\text{int}}=g\sum_{i}c_{(i-1)B}^{\dagger}c_{(i-1)A}c_{iB}c_{iA}c_{(i+1)B}^{\dagger}c_{(i+1)A}+\text{h.c.}.$
(16)

This is a six-fermion interaction across three adjacent sites on the 1D
lattice. It describes the process that first annihilates both $A$\- and
$B$-type fermions on the center site (which annihilates four units of charges
on the site $i$) and then separately converts $A$-type fermions to $B$-type
fermions on the two adjacent sites (which creates two units of charges on each
of the site $i-1$ and $i+1$), such that the $\mathrm{U}(1)_{V}$ charge is
conserved. The interaction is also manifestly translation and reflection
symmetric, so the full
$\mathrm{U}(1)_{V}\times(\mathbb{Z}\rtimes\mathbb{Z}_{2})$ symmetry is
preserved by the interaction as expected. With this interaction, we claim that
the lattice model Eq. (6) will exhibit an (ersatz) Fermi liquid to SMG
insulator transition when the interaction strength $g$ exceeds a finite
critical value $g_{c}$.

To show that the proposed interaction Eq. (16) indeed drives the Fermi liquid
to a gapped interacting insulator, we bosonize [79, 80] the fermion operator
$c_{a}\sim{:\,}\mathrm{e}^{\mathrm{i}\varphi_{a}}{\,:}$ (with
$a\in\\{AR,BR,BL,AL\\}$) and cast the lattice model to an effective Luttinger
liquid theory, described by the following Lagrangian density

$\begin{split}\mathcal{L}=&\frac{1}{4\pi}(\partial_{t}\varphi^{\intercal}K\partial_{x}\varphi-\partial_{x}\varphi^{\intercal}V\partial_{x}\varphi)\\\
&+\sum_{\alpha=1,2}g_{\alpha}\cos(l_{\alpha}^{\intercal}\varphi),\end{split}$
(17)

where
$\varphi=(\varphi_{AR},\varphi_{BR},\varphi_{BL},\varphi_{AL})^{\intercal}$
are compact scalar bosons. The $K$ matrix and the $l_{\alpha}$ vectors are
given by

$K=\left[\begin{smallmatrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&-1&0\\\
0&0&0&-1\end{smallmatrix}\right],\quad l_{1}=\left[\begin{smallmatrix}1\\\
-2\\\ 1\\\ 2\end{smallmatrix}\right],\quad l_{2}=\left[\begin{smallmatrix}2\\\
1\\\ -2\\\ 1\end{smallmatrix}\right].$ (18)

The six-fermion interaction $H_{\text{int}}$ in Eq. (16) translates to the
cosine terms $g_{1}$ and $g_{2}$ in the Luttinger liquid theory in Eq. (17),
with $g_{1}=g_{2}=g$ enforced by the $\mathbb{Z}_{2}$ reflection symmetry. The
RG flow in the log energy scale $\ell=-\ln\Lambda$ is given by [81, 82]

$\frac{\mathrm{d}g}{\mathrm{d}\ell}=(2-\Delta_{\text{int}})g,\quad\frac{\mathrm{d}\Delta_{\text{int}}^{-1}}{\mathrm{d}\ell}=\pi^{2}g^{2},$
(19)

where $\Delta_{\text{int}}$ is the scaling dimension of the SMG interaction.
The RG flow diagram is shown in Fig. 2.

Figure 2: The RG flow of the coupling $g$ and the scaling dimension
$\Delta_{\text{int}}$ of the SMG interaction. The abbreviations stand for the
following terminology: SMG for symmetric mass generation, FL for Fermi liquid,
EFL for ersatz Fermi liquid, LL for Luttinger liquid.

At the Fermi liquid (FL) fixed-point, we have
$\Delta_{\text{int}}=\frac{1}{2}l_{\alpha}^{\intercal}l_{\alpha}=5>2$, meaning
that the SMG interaction is perturbatively irrelevant. If the bare coupling
$g$ (the interaction strength at the lattice scale) is weak ($g<g_{c}$), it
will just flow to zero and disappear in the IR theory. However, the scaling
dimensions of all operators will be renormalized as the coupling $g$ flows
toward zero. Therefore, the FL fixed-point will be deformed into the Luttinger
liquid (LL) fixed-line, along which the fermion quasi-particle is no longer
well-defined, but the system remains gapless. Despite the different dynamical
properties, the LL still preserves all the kinematic properties (e.g. emergent
symmetries and anomalies) as the FL, which can be unified under the concept of
_ersatz Fermi liquid_ (EFL) [11].

If the bare coupling $g$ is strong enough ($g>g_{c}$), the scaling dimension
$\Delta_{\text{int}}$ can be reduced to $\Delta_{\text{int}}<2$ such that the
SMG interaction becomes relevant and flows strong. As the cosine term in Eq.
(17) gets strong, the corresponding vertex operators
$\mathrm{e}^{\mathrm{i}l_{\alpha}^{\intercal}\varphi}$ ($\alpha=1,2$)
condense. Any other operators that braid non-trivially with the condensed
operators will be gapped, which includes all the fermion operators. Therefore
the system enters the SMG insulating phase with all fermion excitations gapped
without breaking the $\mathrm{U}(1)\times(\mathbb{Z}\rtimes\mathbb{Z}_{2})$
symmetry. This has been confirmed by the DMRG simulation in Ref. 78 for a
related model using the domain wall fermion construction, where it has been
verified that the fermion two-point function indeed decays exponentially in
the SMG phase — a direct piece of evidence for the gap generation. On the
lattice level, this corresponds to condensing the six-fermion bound state by
developing the ground state expectation value of $\langle
c_{(i-1)B}^{\dagger}c_{(i-1)A}c_{iB}c_{iA}c_{(i+1)B}^{\dagger}c_{(i+1)A}\rangle\neq
0$. So the gapping is achieved by the multi-fermion condensation (involving
more than two fermions), which is distinct from the fermion bilinear
condensation in the conventional gapping mechanisms of Fermi liquids (such as
the band hybridization or Cooper pairing mechanisms).

The RG analysis also indicates that the ersatz Fermi liquid to SMG insulator
transition (at $g=g_{c}$) is of the Berezinskii-Kosterlitz-Thouless (BKT) [83,
84, 85] transition universality in (1+1)D.

The above analysis established the Fermi surface SMG phenomenon in the lattice
model Eq. (6) (equipped with the gapping interaction Eq. (16)). The
significance of this lattice model is that it provides a pristine 1D lattice
regularization of the 3-4-5-0 chiral fermion model by using lattice
translation to realize the axial $\mathrm{U}(1)_{A}$ symmetry at low energy.
This construction does not require the introduction of a (2+1)D bulk to
realize the chiral fermions as boundary modes, as in the domain wall fermion
constructions [74, 68, 78]. Such pristine 1D lattice regularization is
advantageous for the numerical simulation of chiral fermions, as the model
contains no redundant bulk (or mirror) fermions, such that the computational
resources can be used more efficiently. We will leave the numerical
exploration of this model to future research.

##  III Fermi Surface SMG in (2+1)D

###  III.1 (2+1)D Fermi Liquid and Fermi Surface Anomaly

Given the example of Fermi surface SMG in (1+1)D, we would like to further
explore similar physics in higher dimensions. The most important low-energy
features of a (2+1)D Fermi liquid are the gapless fermions on its 1D Fermi
surface. Suppose we parameterize the 1D Fermi surface
${\bm{k}}_{F}(\theta)\in\partial\mathcal{V}_{F}$ by a continuous and periodic
parameter $\theta$, such that ${\bm{k}}_{F}(\theta+2\pi)={\bm{k}}_{F}(\theta)$
(where we do not require $\theta$ to literally represent the geometrical
angle, as the Fermi surface may not be a perfect circle in general). The
fermions $c_{\theta}$ on the Fermi surface have an emergent symmetry described
by the loop group of $\mathrm{U}(1)$ [11, 12], denoted as $\mathrm{LU}(1)$,
under which

$\mathrm{LU}(1):c_{\theta}\to\mathrm{e}^{\mathrm{i}\phi(\theta)}c_{\theta},$
(20)

where the $\mathrm{U}(1)$ phase factor $\mathrm{e}^{\mathrm{i}\phi(\theta)}$
is a smooth function of $\theta$ with the periodicity
$\mathrm{e}^{\mathrm{i}\phi(\theta+2\pi)}=\mathrm{e}^{\mathrm{i}\phi(\theta)}$.
Both the (global) charge $\mathrm{U}(1)$ and the translation symmetries
$\mathbb{R}^{2}$ are subgroups of $\mathrm{LU}(1)$:

$\mathrm{U}(1):c_{\theta}\to\mathrm{e}^{\mathrm{i}q\phi}c_{\theta},\quad\mathbb{R}^{2}:c_{\theta}\to\mathrm{e}^{\mathrm{i}{\bm{a}}\cdot{\bm{k}}_{F}(\theta)}c_{\theta},$
(21)

assuming the fermions $c_{\theta}$ carry charge $q$ under the global
$\mathrm{U}(1)$ symmetry and are translated by the vector
${\bm{a}}\in\mathbb{R}^{2}$.

The presence of the Fermi surface causes a mixed anomaly between the
$\mathrm{U}(1)$ and translation symmetries [86], which is characterized by the
anomaly index

$\frac{q}{2(2\pi)^{2}}\oint\mathrm{d}\theta({\bm{k}}_{F}\times\partial_{\theta}{\bm{k}}_{F})_{3}=\frac{q\mathcal{V}_{F}}{(2\pi)^{2}}=q\nu,$
(22)

where $\mathcal{V}_{F}$ stands for the Fermi volume in the momentum space, and
$\nu$ is the filling factor. If the Fermi surface anomaly is non-vanishing, it
is impossible to trivially gap out the Fermi liquid without breaking any
symmetry or developing any topological order. The Fermi surface SMG is only
possible if the Fermi liquid system contains multiple Fermi surfaces of
opposite anomaly indices, such that their anomalies cancel as a whole.

###  III.2 Kagome-Triangular Lattice Model

We present a concrete lattice model to demonstrate the Fermi surface SMG in
(2+1)D. Consider two types of _spinless_ fermions labeled by $A$ and $B$ that
are charged under a global $\mathrm{U}(1)$ symmetry with charges $q_{A}=1$ and
$q_{B}=3$, respectively. The $A$-type (or $B$-type) fermion is defined on a
Kagome (or triangular) lattice. As depicted in Fig. 3(a), the Kagome and the
triangular lattices lie on top of each other, with the site $I$ of the
triangular lattice aligned with the upper triangle $\triangle_{I}$ on the
Kagome lattice. We will use the lower-case letters $i,j$ (or the upper-case
letters $I,J$) to label the Kagome (or the triangular) lattice sites.

Figure 3: (a) Overlapping Kagome ($A$) and triangular ($B$) lattices. The
green triangle marks out the unit cell. $A$-type (in blue) and $B$-type (in
red) Fermi surfaces (b) at a general filling such as $\nu_{A}=3/8$ and
$\nu_{B}=7/8(=-1/8)$, or (c) at a special filling $\nu_{A}=\nu_{B}=3/4(=-1/4)$
where the Fermi surfaces coincide.

The lattice model is described by following the Hamiltonian

$\begin{split}H&=H_{A}+H_{B}+H_{\text{int, CF}},\\\ H_{A}&=-t_{A}\sum_{\langle
ij\rangle}(c_{i}^{\dagger}c_{j}+\text{h.c.})-\mu_{A}\sum_{i}c_{i}^{\dagger}c_{i},\\\
H_{B}&=-t_{B}\sum_{\langle
IJ\rangle}(c_{I}^{\dagger}c_{J}+\text{h.c.})-\mu_{B}\sum_{I}c_{I}^{\dagger}c_{I},\\\
H_{\text{int,
CF}}&=-g\sum_{I|ijk\in\triangle_{I}}(c_{I}^{\dagger}c_{i}c_{j}c_{k}+\text{h.c.}),\end{split}$
(23)

where $\langle ij\rangle$ (or $\langle IJ\rangle$) denotes the nearest
neighboring link on the $A$ (or $B$) lattices and $ijk\in\triangle_{I}$ stands
for the three $A$-sites $i,j,k$ at the vertices of the upper triangle
surrounding the $B$-site labeled by $I$. The model has a $\mathrm{U}(1)$
symmetry that acts as

$\mathrm{U}(1):c_{i}\to\mathrm{e}^{\mathrm{i}\phi}c_{i},\quad
c_{I}\to\mathrm{e}^{\mathrm{i}3\phi}c_{I}.$ (24)

The Hamiltonian in Eq. (23) preserves the internal $\mathrm{U}(1)$ symmetry
and all symmetries of the Kagome-triangular lattice (most importantly, the
lattice translation symmetry).

The model Eq. (23) describes the two types of fermions hopping separately on
their corresponding lattices. Because every unit cell contains four sites
(three from the Kagome lattice and one from the triangle lattice), the hopping
model will give rise to four energy bands (three bands for $A$-type fermions
and one band for $B$-type fermions). The chemical potentials $\mu_{A}$ and
$\mu_{B}$ are adjusted to ensure the desired filling of these fermions. We
will focus on a simple case when only the lowest $A$-type (Kagome lattice)
bands and the $B$-type (triangular lattice) bands are filled by filling
fractions $\nu_{A}$ and $\nu_{B}$ respectively, such that the Fermi surface
only involves _two_ of the four bands.

The $A$-type and $B$-type fermions are coupled together only through a four-
fermion interaction $H_{\text{int, CF}}$ in Eq. (23) that fuses three $A$-type
(charge-1) fermions to one $B$-type (charge-3) fermions (and vice versa)
within each unit cell. We will call it a _charge fusion_ (CF) interaction. The
CF interaction breaks the separate $U(1)$ charge conservation laws for
$A$-type and $B$-type fermions in the hopping model to a joint $\mathrm{U}(1)$
charge conservation, associated with the symmetry action in Eq. (24). Similar
interactions also appear in a recent study [87] of quantum breakdown.

Without interaction ($g=0$), the system is in a Fermi liquid phase. According
to Eq. (22), the Fermi surface anomaly cancellation condition requires

$q_{A}\nu_{A}+q_{B}\nu_{B}=0\mod 1.$ (25)

Given the charge assignment of $q_{A}=1$ and $q_{B}=3$, it requires
$\nu_{A}=-3\nu_{B}$. There is no further requirement on the choice of
$\nu_{A}$ itself. With a generic choice of filling (assuming $\nu_{A}<3/4$) as
in Fig. 3(b), the $A$-type fermions (on the Kagome lattice) will form an
electron-like Fermi surface, whose Fermi volume is three times as large as
that of the hole-like Fermi surfaces formed by the $B$-type fermions (on the
triangular lattice). Although the Fermi liquid has a vanishing Fermi surface
anomaly, the charge $\mathrm{U}(1)$ and the lattice translation symmetries are
still restrictive enough to forbid any gap opening on the free-fermion level.
For example, any pairing (charge-2e superconducting) gap will break the
$\mathrm{U}(1)$ symmetry. The only possibility to gap the Fermi liquid relies
on the fermion interaction.

We claim that the charge fusion interaction $H_{\text{int, CF}}$ in Eq. (23)
is a valid SMG interaction that drives the Fermi liquid into a trivially gaped
insulator without breaking symmetry (or developing any topological order). To
see this, we go to the strong coupling limit by taking $g\to\infty$. Of
course, the chemical potentials $\mu_{A},\mu_{B}$ must increase
correspondingly to keep the fermion fillings fixed. The model Hamiltonian
decouples to each unit cell in the strong coupling limit

$\begin{split}H=\sum_{I|ijk\in\triangle_{I}}&-\mu_{A}(n_{i}+n_{j}+n_{k})-\mu_{B}n_{I}\\\
&-g(c_{I}^{\dagger}c_{i}c_{j}c_{k}+\text{h.c.}),\end{split}$ (26)

where $n_{i}=c_{i}^{\dagger}c_{i}$ (and $n_{I}=c_{I}^{\dagger}c_{I}$) denotes
the fermion number operator. Within each unit cell, there are only two
relevant states $|1110\rangle$ and $|0001\rangle$ (in the Fock state basis
$|n_{i}n_{j}n_{k}n_{I}\rangle$) acted by the Hamiltonian. Their hybridization
will produce the ground state in each unit cell. The full-system ground state
will be the following direct product state

$|\text{SMG}\rangle=\bigotimes_{I}\big{(}\sqrt{p}|1110\rangle+\sqrt{1-p}|0001\rangle\big{)}_{I},$
(27)

where
$p=\frac{1}{2}\big{(}1+\frac{-3\mu_{A}+\mu_{B}}{\sqrt{(-3\mu_{A}+\mu_{B})^{2}+4g^{2}}}\big{)}$
is the probability to observe the $|1110\rangle$ state in the unit cell, which
is tunable by adjusting $\mu_{A},\mu_{B}$ relative to $g$. The fermion
fillings (per unit cell) in the ground state $|\text{SMG}\rangle$ will be

$\nu_{A}=3p,\quad\nu_{B}=1-p=-p\mod 1,$ (28)

which automatically satisfies the anomaly cancellation condition
$\nu_{A}=-3\nu_{B}$ (as it should be). The ground state $|\text{SMG}\rangle$
is non-degenerated and gapped from all excited states (with a gap of the order
$g$). It also preserves the charge $\mathrm{U}(1)$ and all the lattice
symmetries and does not have topological order. Therefore, we have explicitly
shown that the system ends up in the SMG insulator phase as $g\to\infty$. As a
gapped phase, we expect it to be stable against perturbations (such as the
hopping terms $t_{A},t_{B}$) over a finite region in the parameter space. The
SMG phase is a strongly interacting insulating phase, which has no
correspondence in the free-fermion picture.

Established the Fermi liquid (metallic) phase at $g=0$ and the SMG insulator
phase at $g\to\infty$, there must be an SMG transition (an interaction-driven
metal-insulator transition) at some intermediate coupling strength $g_{c}$.
However, the nature of the transition is still an open question, which we will
leave for future numerical study. In the following, we will only analyze the
SMG transition at a special filling: $\nu_{A}=\nu_{B}=3/4$, where the Fermi
surfaces coincide precisely and take the perfect hexagon shapes as shown in
Fig. 3(c). This allows us to gain some analytic control of the problem.

###  III.3 RG Analysis of the SMG Transition

In this subsection, we analyze the interaction effect in Eq. (23) when the
filling is $\nu_{A}=\nu_{B}=3/4$. In this case, the Fermi surface contains
three Van Hove singularities (VHSs) (or three hot spots) at three distinct $M$
points as shown in Fig. 3(c). We can investigate the interaction effects using
the hot-spot renormalization group (RG) method at the one-loop level [88, 89,
90, 91, 92, 93, 94, 95]. The hot-spot RG assumes that the low-energy physics
emerges from the correlated effects of the fermions near the VHSs, because the
density of states diverges near the VHSs, leading to the highest instability
towards ordering.

Under RG, the charge fusion interaction $H_{\text{int, CF}}$ will generate two
types of density-density interactions at the one-loop level, namely,
$H_{\text{int, AA}}=\sum_{i,j}n_{i}n_{j}$ and $H_{\text{int,
AB}}=\sum_{I,i}n_{I}n_{i}$ as well as other (less important) exchange
interactions. These density-density interactions are more important in the
sense that they will in turn contribute to the correction of $H_{\text{int,
CF}}$. Therefore, we should include $H_{\text{int, CF}},H_{\text{int,
AA}},H_{\text{int, AB}}$ altogether in the RG analysis and study the RG flow
jointly.

To proceed, we transform the interactions into the momentum space. The fermion
operators are labeled by the flavor index $S=A,B$ and the hot-spot index
$\alpha=1,2,3$ (referring to the three different VHSs). We note that
$H_{\text{int, CF}}$ would vanish if it is naively restricted to the hot spots
because the momentum conservation requires multiple $A$-type fermion operators
to appear on the same hot spot, which violates the Pauli exclusion principle
of fermions. So we need to introduce point splitting in the momentum space
around each hot spot. Our strategy is to further split the $A$-type fermion
into three modes $A_{s}$ labeled by $s=1,2,3$, and define the interaction,

$\displaystyle H_{\text{int,
CF}}=g_{\text{rs}}\sum_{\alpha}\epsilon^{ijk}c_{B\alpha}^{\dagger}c_{A_{i}\alpha}c_{A_{j}\alpha}c_{A_{k}\alpha}$
$\displaystyle+g_{\text{rt}}\sum_{\alpha\neq\beta}\epsilon^{ijk}c_{B\alpha}^{\dagger}c_{A_{i}\alpha}c_{A_{j}\beta}c_{A_{k}\beta}+\text{h.c.}$
(29)

$g_{\text{rs}}$ and $g_{\text{rt}}$ are the CF interaction decomposed into
different momentum transfer channels: the intra-hot-spot scattering
$g_{\text{rs}}$ and the inter-hot-spot scattering $g_{\text{rt}}$.

These CF interactions receive corrections from the following density-density
interactions at the one-loop level,

$\displaystyle H_{\text{int, AA}}+H_{\text{int,
AB}}=g_{\text{as}}\sum_{\alpha,st}n_{A_{s}\alpha}n_{A_{t}\alpha}+(A_{s}\leftrightarrow
A_{t})$
$\displaystyle+g_{\text{bt}}\sum_{\alpha\neq\beta,s}n_{B\alpha}n_{A_{s}\beta}+(A_{s}\leftrightarrow
B)+\text{h.c.}+...$ (30)

where $...$ refers to the other interactions that are decoupled from
$g_{\text{rs}},g_{\text{rt}},g_{\text{as}},g_{\text{bt}}$ in the RG equations.
The scattering processes of these four important interactions are illustrated
in Fig. 4. The complete set of all possible interactions is presented in
Appendix C.

We derive the RG equations based on the systematic approach developed in Ref.
96. Since we are interested in the flow of $H_{\text{int, CF}}$, the relevant
part of the RG equations reads,

$\displaystyle\frac{dg_{\text{bt}}}{d\ell}=2d_{0}d_{\text{AB}}g_{\text{bt}}^{2},\quad\frac{dg_{\text{as}}}{d\ell}=-2g_{\text{as}}^{2},$
(31)
$\displaystyle\frac{dg_{\text{rs}}}{d\ell}=-6g_{\text{as}}g_{\text{rs}},\quad\frac{dg_{\text{rt}}}{d\ell}=4d_{0}d_{\text{AB}}g_{\text{bt}}g_{\text{rt}}-2g_{\text{as}}g_{\text{rt}}.$

where the RG parameter is defined by the Cooper-pairing susceptibility of
$A$-type fermions
$\ell=\chi_{pp,\text{AA}}({\bm{k}}=0,E)\sim\nu_{0}\log^{2}(\Lambda/E)$, in
which $\nu_{0}\log(\Lambda/E)$ is the diverging density of states at the VHS,
$E$ is the running energy scale and $\Lambda$ is the high energy cutoff.
$d_{0}=d\chi_{ph,\text{AA}}({\bm{Q}})/d\ell\leq 1$ is the nesting parameter of
$A$-type fermions, which saturates to one in the perfectly nested limit
($d_{0}\to 1$). In our case, different VHSs are half-nested (only one of the
two crossing Fermi surfaces is perfectly nested between every pair of
different VHSs), so $d_{0}=1/2$ is a suitable estimation. Similarly, we define
$d_{\text{AB}}=d\chi_{pp,\text{AB}}({\bm{0}})/d\ell$, which depends on the
energies of $A$ and $B$-type fermions near the VHS. The full RG equations and
details are listed in Appendix C.

Figure 4: Scattering of fermions between Van Hove singularities (VHSs) by (a)
density-density interactions $g_{\text{bt}}$ (red), $g_{\text{as}}$ (blue) and
(b) non-vanishing processes $g_{\text{rs}}$ (yellow), $g_{\text{rt}}$ (green)
of $H_{\text{int, CF}}$. Thin (or thick) arrows correspond to $A$-type (or
$B$-type) fermions.

According to the one-loop RG equations, if the density-density interactions
$g_{\text{bt}},g_{\text{as}}$ are initially zero, then the CF interactions
$g_{\text{rs}},g_{\text{rt}}$ remain marginal along the RG flow. However, if
we turn on small density-density interactions $g_{\text{bt}},g_{\text{as}}$
with correct signs ($g_{\text{bt}}>0$ or $g_{\text{as}}<0$), the charge fusion
interactions $g_{\text{rs}},g_{\text{rt}}$ will be marginally relevant. The
solutions of RG equations Eq. (31) are

$\displaystyle
g_{\text{bt}}(\ell)=\frac{g_{\text{bt}}(0)}{1-2d_{0}d_{\text{AB}}g_{\text{bt}}(0)\ell},\quad
g_{\text{as}}(\ell)=\frac{g_{\text{as}}(0)}{1+2g_{\text{as}}(0)\ell},$
$\displaystyle
g_{\text{rs}}(\ell)=\frac{g_{\text{rs}}(0)}{(1+2g_{\text{as}}(0)\ell)^{3}},$
$\displaystyle
g_{\text{rt}}(\ell)=\frac{g_{\text{rt}}(0)}{(1+2g_{\text{as}}(0)\ell)(1-2d_{0}d_{\text{AB}}g_{\text{bt}}(0)\ell)^{2}}.$
(32)

As the RG parameter $\ell$ increases under the RG flow, the coupling strengths
can diverge at some critical scale $\ell_{c}$, when any of the denominators in
Eq. (III.3) vanish. The critical scale is set by the bare density-density
interaction strengths $g_{\text{bt}}(0)$ and $g_{\text{as}}(0)$, but the CF
interaction strengths $g_{\text{rs}},g_{\text{rt}}$ diverge faster than the
density-density interactions as the critical scale is approached. Therefore,
the RG fixed points are characterized by the behavior of
$g_{\text{rs}},g_{\text{rt}}$.

Figure 5: The RG phase diagram with respect to the density-density
interactions $g_{\text{as}},g_{\text{bt}}$. In the Fermi liquid (FL) phase,
the gapping interaction flows to zero. In the symmetric mass generation (SMG)
phase, the gapping interaction flows to infinity.

Depending on the bare density-density interaction strengths $g_{\text{as}}(0)$
and $g_{\text{bt}}(0)$, the system can flow towards different RG fixed points
as shown Fig. 5. When $g_{\text{as}}(0)>0$ and $g_{\text{bt}}(0)<0$, all
interactions flow to zero, which corresponds to the Fermi liquid fixed point.
When $g_{\text{as}}(0)<\min(0,-d_{0}d_{\text{AB}}g_{\text{bt}}(0))$, both
charge fusion interactions $g_{\text{rs}},g_{\text{rt}}$ flow to infinity,
which should correspond to the SMG phase according to the previous lattice
model analysis. However, we also find a region in the phase diagram, described
by $g_{\text{bt}}>\max(0,-g_{\text{as}}/d_{0}d_{\text{AB}})$, where
$g_{\text{rs}}\to 0$ and $g_{\text{rt}}\to\infty$, i.e. flowing towards
different limits. We are not sure how to interpret the physical meaning of
this RG fixed point. It might still be in the SMG phase as one interaction
still flows strong. But it could as well end up in a spontaneous symmetry
breaking (SSB) phase that breaks the $\mathrm{LU}(1)$ symmetry since the
$A$-type and $B$-type Fermi surfaces have pretty strong nesting instability.
This might also be an artifact of the hot-spot RG method, as it does not fully
capture all low-energy fermionic degrees of freedom of the Fermi surface.

Functional RG [97, 98, 99, 100] might provide a better resolution of the Fermi
surface and remove the uncertainty in the phase diagram Fig. 5. A recent study
[101] has demonstrated the functional RG method in a triangle lattice model
with spinless fermions. The same technique might apply to our model as well.
However, we will leave such study for future research.

By tuning $g_{\text{as}}(0)$ across zero on the $g_{\text{bt}}(0)<0$ side, one
can drive a FL to SMG transition. The gapping interaction is marginally
relevant at the transition point. According to the solution of the RG
equations in Eq. (III.3), the coupling diverges at the critical scale
$\ell_{c}\sim\nu_{0}\log^{2}(\Lambda/\Delta_{\text{SMG}})$ when the
denominator $(1+2g_{\text{as}}(0)l_{c})$ vanishes. This implies that the SMG
gap $\Delta_{\text{SMG}}$ (the energy gap between the ground state and the
first excited state) opens up as [102, 103]

$\Delta_{\text{SMG}}\sim\Lambda\exp(-c/\sqrt{g_{\text{as}}(0)\nu_{0}}),$ (33)

where $\Lambda$ is the UV cutoff energy scale, $\nu_{0}$ is the coefficient in
front of the diverging density of state at the VHS, and $c$ is some non-
universal constant.

##  IV Summary and Discossion

In this work, we propose the concept of Fermi surface SMG: a mechanism to gap
out Fermi surfaces by non-perturbative interaction effects without breaking
the $\mathrm{LU}(1)$ symmetry. This phenomenon can only happen when the Fermi
surface anomaly is canceled out in the fermion system. We present (1+1)D and
(2+1)D examples of Fermi surface SMG. We expect that the mechanism can
generally occur in all dimensions.

Fermi surface SMG belongs to a broader class of phenomena, called the
symmetric Fermi surface reconstruction (SFSR), as summarised in Fig. 6. The
SFSR is in contrast to the more conventional symmetry-breaking Fermi surface
reconstruction, where the Fermi surface is reconstructed (or gapped) by
developing spontaneous symmetry-breaking orders. Depending on the cancellation
of the Fermi surface anomaly, the SFSR further splits into two classes: the
Fermi surface symmetric mass generation (SMG) if the anomaly vanishes, or the
Fermi surface topological mass generation (TMG) if the anomaly does not
vanish. The former class, the Fermi surface SMG, is the focus of this work.
The latter class, the Fermi surface TMG, is also discussed in the literature,
where the non-vanishing Fermi surface anomaly is absorbed by an anomalous
topological quantum field theory (TQFT), such that the SFSR is achieved by
developing the corresponding topological order. This gives rise to
deconfined/fractionalized Fermi liquid (FL∗) [104, 105, 106] or orthogonal
metal [107, 108, 109]. _Symmetry extension_ [110] has provided a unified
framework, to understand TMG and SMG for bosons or fermions of zero Fermi
volume [111, 112, 113, 114, 115, 116], where the symmetric gapping can be
achieved by extending the symmetry group to lift any gapping obstruction that
was otherwise imposed by the symmetry. Similar constructions can be applied to
understand SFSR more generally.

Figure 6: Classification of Fermi surface reconstruction mechanisms, based on
the $\mathrm{LU}(1)$ loop group symmetry.

Fermi surface SMG deforms an anomaly-free (charge-compensated) Fermi liquid
state to a fully gapped product state. Although the resulting SMG gapped state
does not have non-trivial features like topological order, the SMG transition
from the Fermi liquid phase to the SMG phase can still be quite exotic. The
SMG transition of relativistic fermions has been proposed to be a deconfined
quantum critical point [117, 118], where the physical fermion fractionalizes
to bosonic and fermionic partons with emergent gauge fluctuations at and only
at the critical point. It is conceivable that similar scenarios might apply to
the Fermi surface SMG transition as well, where deconfined Fermi liquid
(orthogonal metal) could emerge at the critical point. The lattice models
presented in this study lay the ground for future theoretical and numerical
studies of the exotic SMG transition in these models.

It is also known that the fermion single-particle Green’s function has
symmetry-protected zeros at zero frequency in the SMG phase [119, 120, 121,
122]. It will be interesting to investigate further the Green’s function
structure in the Fermi surface SMG phase. Whether or not the SMG interaction
will replace the original Fermi surface (a loop of poles) with a loop of zeros
in the Green’s function is still an open question to explore.

###### Acknowledgements.

We thank David Tong, Tin Sulejmanpasic, Max Metlitski, Xiao-Liang Qi, Cenke
Xu, Subir Sachdev, John Preskill for inspiring discussions. We acknowledge the
workshop “Paths to Quantum Field Theory 2022” at Durham University, where the
discussion with David Tong and Tin Sulejmanpasic motivates the authors to
think about pristine lattice regularizations of the 3-4-5-0 chiral fermion
model. DCL, MZ, and YZY are supported by a startup fund at UCSD. JW is
supported by the Center for Mathematical Sciences and Applications at Harvard
University and NSF Grant DMS-1607871 “Analysis, Geometry and Mathematical
Physics.”

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##  Appendix A Emergent $\mathrm{U}(1)$ Symmetries in the (1+1)D Two-Band
Model

Start from the definition of charge $\mathrm{U}(1)$ (parameterized by a
periodic angle $\phi\in[0,2\pi)$) and lattice translation $\mathbb{Z}$
(parameterized by an integer $n\in\mathbb{Z}$) symmetries as defined in Eq.
(7)

$\begin{split}\mathrm{U}(1)&:c_{iA}\to\mathrm{e}^{\mathrm{i}q_{A}\phi}c_{iA},\quad
c_{iB}\to\mathrm{e}^{\mathrm{i}q_{B}\phi}c_{iB};\\\ \mathbb{Z}&:c_{iA}\to
c_{(i+n)A},\quad c_{iB}\to c_{(i+n)B}.\end{split}$ (34)

Follow the definition Eq. (9) of the fermion operators in the momentum space

$c_{kA}=\sum_{i}c_{iA}\mathrm{e}^{-\mathrm{i}ki},\quad
c_{kB}=\sum_{i}c_{iB}\mathrm{e}^{-\mathrm{i}ki},$ (35)

where the wave number $k\in[-\pi,\pi)$ is a dimensionless periodic variable
defined in the first Brillouin zone. (Note: the _dimensionful_ momentum $p$
should be related to the _dimensionless_ wave number $k$ by $p=\hbar k/a$ with
$a$ being the lattice constant and the site coordinate $x\in\mathbb{R}$ is
related to the site index $i\in\mathbb{Z}$ by $x=ai$, such that the Fourier
factor $\mathrm{e}^{-\mathrm{i}px/\hbar}=\mathrm{e}^{-\mathrm{i}ki}$ is
consistent with the quantum mechanics convention.) It is straightforward to
show that the $\mathrm{U}(1)\times\mathbb{Z}$ symmetry acts in the momentum
space as

$\begin{split}\mathrm{U}(1)&:c_{kA}\to e^{\mathrm{i}q_{A}\phi}c_{kA},c_{kB}\to
e^{\mathrm{i}q_{B}\phi}c_{kB};\\\ \mathbb{Z}&:c_{kA}\to
e^{\mathrm{i}kn}c_{kA},c_{kB}\to e^{\mathrm{i}kn}c_{kB}.\\\ \end{split}$ (36)

Apply these transformations to the low-energy fermion near the four Fermi
points. According to Eq. (12)

$\begin{split}c_{AR}&=c_{(3k_{F})A},\\\ c_{BR}&=c_{(-k_{F})B},\\\
c_{BL}&=c_{(k_{F})B},\\\ c_{AL}&=c_{(-3k_{F})A},\end{split}$ (37)

Eq. (36) becomes

$\begin{split}\mathrm{U}(1)&:\left\\{\begin{array}[]{rcl}c_{AR}&\to&\mathrm{e}^{\mathrm{i}q_{A}\phi}c_{AR},\\\
c_{BR}&\to&\mathrm{e}^{\mathrm{i}q_{B}\phi}c_{BR},\\\
c_{BL}&\to&\mathrm{e}^{\mathrm{i}q_{B}\phi}c_{BL},\\\
c_{AL}&\to&\mathrm{e}^{\mathrm{i}q_{A}\phi}c_{AL};\\\ \end{array}\right.\\\
\mathbb{Z}&:\left\\{\begin{array}[]{rcl}c_{AR}&\to&\mathrm{e}^{3\mathrm{i}k_{F}n}c_{AR},\\\
c_{BR}&\to&\mathrm{e}^{-\mathrm{i}k_{F}n}c_{BR},\\\
c_{BL}&\to&\mathrm{e}^{\mathrm{i}k_{F}n}c_{BL},\\\
c_{AL}&\to&\mathrm{e}^{-3\mathrm{i}k_{F}n}c_{AL}.\\\
\end{array}\right.\end{split}$ (38)

Because $k_{F}=|\nu_{B}|\pi$ is _almost always_ (i.e., with probability 1) a
irrational multiple of $\pi$ (because $|\nu_{B}|$ is almost always an
irrational number without fine tuning), $k_{F}n\mod 2\pi$ can approach any
angle in $[0,2\pi)$ (with $2\pi$ periodicity) as close as we want (given
$n\in\mathbb{Z}$). This allows us to define two angular variables $\phi_{V}$
and $\phi_{A}$, both are periodic in $[0,2\pi)$,

$\phi_{V}=\phi,\quad\phi_{A}=k_{F}n\mod 2\pi,$ (39)

then Eq. (38) can be compactly written as

$\begin{split}\text{UV symmetry}&\Rightarrow\text{IR symmetry}\\\
\mathrm{U}(1)&\Rightarrow\mathrm{U}(1)_{V}:c_{a}\to\mathrm{e}^{\mathrm{i}q_{a}^{V}\phi_{V}}c_{a},\\\
\mathbb{Z}&\Rightarrow\mathrm{U}(1)_{A}:c_{a}\to
e^{\mathrm{i}q_{a}^{A}\phi_{A}}c_{a},\end{split}$ (40)

for $a=AR,BR,BL,AL$ enumerating over the four Fermi point labels, together
with the charge vectors (given that $q_{A}=1$ and $q_{B}=3$):

${\bm{q}}^{V}=\left[\begin{matrix}q_{A}\\\ q_{B}\\\ q_{B}\\\
q_{A}\end{matrix}\right]=\left[\begin{matrix}1\\\ 3\\\ 3\\\
1\end{matrix}\right],\quad{\bm{q}}^{A}=\left[\begin{matrix}3\\\ -1\\\ 1\\\
-3\end{matrix}\right].$ (41)

Therefore, the global charge $\mathrm{U}(1)$ symmetry is simply reinterpreted
as the $\mathrm{U}(1)_{V}$ vector symmetry, and the translation symmetry
(described by a non-compact $\mathbb{Z}$ group) in the UV becomes an emergent
$\mathrm{U}(1)_{A}$ axial symmetry (described by a compact $\mathrm{U}(1)$
group) in the IR. The symmetry transformation in Eq. (40) precisely matches
Eq. (14) in the main text with the correct charge assignment as listed in Tab.
1.

Recombining the charge vectors of $\mathrm{U}(1)_{V}$ and $\mathrm{U}(1)_{A}$,
we can define two alternative emergent $\mathrm{U}(1)$ symmetries, denoted as
$\mathrm{U}(1)_{\frac{3V\pm A}{2}}$ with the charge vectors

${\bm{q}}^{\frac{3V\pm A}{2}}=\frac{1}{2}(3{\bm{q}}^{V}\pm{\bm{q}}^{A}),$ (42)

as their names implied. More explicitly, the charge vectors match the chiral
charge assignements for the 3-4-5-0 fermions:

${\bm{q}}^{\frac{3V+A}{2}}=\left[\begin{matrix}3\\\ 4\\\ 5\\\
0\end{matrix}\right],\quad{\bm{q}}^{\frac{3V-A}{2}}=\left[\begin{matrix}0\\\
5\\\ 4\\\ 3\end{matrix}\right].$ (43)

The fermions are expected to transform under $\mathrm{U}(1)_{\frac{3V\pm
A}{2}}$ as (parameterized by the periodic angles $\phi_{\pm}\in[0,2\pi)$)

$\mathrm{U}(1)_{\frac{3V\pm
A}{2}}:c_{a}\to\mathrm{e}^{\mathrm{i}\frac{1}{2}(3q_{a}^{V}\pm
q_{a}^{A})\phi_{\pm}}c_{a}.$ (44)

This can be viewed as the combined transformation of $\mathrm{U}(1)_{V}$ and
$\mathrm{U}(1)_{A}$ with the vector rotation angle $\phi_{V}$ and the axial
rotation angle $\phi_{A}$ given by

$\phi_{V}=\frac{3}{2}\phi_{\pm},\quad\phi_{A}=\pm\frac{1}{2}\phi_{\pm},$ (45)

as can be verified by comparing Eq. (44) with Eq. (40). Now we can connect
these rotation angle back to the original $\mathrm{U}(1)\times\mathbb{Z}$
symmetry of the lattice fermions using the relation Eq. (39),

$\phi=\frac{3}{2}\phi_{\pm},\quad\pm\frac{1}{2}\phi_{\pm}=k_{F}n\mod 2\pi.$
(46)

Eliminate $\phi_{\pm}$ from the equations, we obtain the relation

$\phi=\pm 3k_{F}n\mod 2\pi,$ (47)

for the $\mathrm{U}(1)_{\frac{3V\pm A}{2}}$ symmetries. Therefore, in order to
reproduce the IR emergent $\mathrm{U}(1)_{\frac{3V\pm A}{2}}$ symmetries, the
corresponding UV symmetries (at the lattice level) must be such implemented
that every $n$-step translation should be followed by a charge $\mathrm{U}(1)$
rotation with the rotation angle $\phi=\pm 3k_{F}n$. So we establish the
following correspondence between the IR and UV symmetries

$\begin{split}\text{IR symmetry}&\Rightarrow\text{UV symmetry}\\\
\mathrm{U}(1)_{\frac{3V\pm A}{2}}&\Rightarrow\mathbb{Z}(\tfrac{3V\pm
A}{2}):\left\\{\begin{array}[]{l}c_{iA}\to\mathrm{e}^{\pm
3\mathrm{i}q_{A}k_{F}n}c_{(i+n)A},\\\ c_{iB}\to\mathrm{e}^{\pm
3\mathrm{i}q_{B}k_{F}n}c_{(i+n)B}.\end{array}\right.\end{split}$ (48)

Here the compact $\mathrm{U}(1)$ symmetries in the IR get mapped to the non-
compact symmetries $\mathbb{Z}$ in the UV, because the UV symmetries are
parameterized by the integer variable $n\in\mathbb{Z}$. Given that $q_{A}=1$
and $q_{B}=3$, Eq. (48) becomes Eq. (15), as claimed in the main text.
Therefore the 3-4-5-0 chiral fermion model is indeed realized in by the (1+1)D
two-band lattice model at low energy.

##  Appendix B Wang-Wen Interaction

In the bonsonization language, the Wang-Wen interaction is described by

$\mathcal{L}_{\text{int}}=\sum_{\alpha=1,2}g_{\alpha}\cos(l_{\alpha}^{\intercal}\varphi)$
(49)

with
$\varphi=(\varphi_{AR},\varphi_{BR},\varphi_{BL},\varphi_{AL})^{\intercal}$
and the interaction vectors given by

$l_{1}=\left[\begin{smallmatrix}1\\\ -2\\\ 1\\\
2\end{smallmatrix}\right],\quad l_{2}=\left[\begin{smallmatrix}2\\\ 1\\\ -2\\\
1\end{smallmatrix}\right].$ (50)

Mapping back to the chiral fermions by the correspondence
$c_{a}\sim{:\,}\mathrm{e}^{\mathrm{i}\varphi_{a}}{\,:}$, the interaction reads

$\begin{split}H_{\text{int}}&=\frac{g_{1}}{2}(c_{AR}c_{BL})(c_{BR}^{\dagger}c_{AL})^{2}+\text{h.c.}\\\
&+\frac{g_{2}}{2}(c_{BR}c_{AL})(c_{AR}c_{BL}^{\dagger})^{2}+\text{h.c.}.\end{split}$
(51)

According to Eq. (12) and use the inverse Fourier transformation,

$\begin{split}c_{AR}&=c_{(3k_{F})A}=\sum_{i}c_{iA}\mathrm{e}^{3\mathrm{i}k_{F}i},\\\
c_{BR}&=c_{(-k_{F})B}=\sum_{i}c_{iB}\mathrm{e}^{-\mathrm{i}k_{F}i},\\\
c_{BL}&=c_{(k_{F})B}=\sum_{i}c_{iB}\mathrm{e}^{\mathrm{i}k_{F}i},\\\
c_{AL}&=c_{(-3k_{F})A}=\sum_{i}c_{iA}\mathrm{e}^{-3\mathrm{i}k_{F}i}.\end{split}$
(52)

Plugging Eq. (52) into Eq. (51), the interaction becomes

$H_{\text{int}}=\sum_{i_{1},\cdots,i_{6}}g_{i_{1}\cdots
i_{6}}(c_{i_{1}B}^{\dagger}c_{i_{2}A})(c_{i_{3}B}c_{i_{4}A})(c_{i_{5}B}^{\dagger}c_{i_{6}A})+\text{h.c.},$
(53)

with

$\begin{split}g_{i_{1}\cdots
i_{6}}&=\frac{g_{1}}{2}\mathrm{e}^{\mathrm{i}k_{F}(i_{1}-3i_{2}+i_{3}+3i_{4}+i_{5}-3i_{6})}\\\
&+\frac{g_{2}}{2}\mathrm{e}^{\mathrm{i}k_{F}(-i_{1}+3i_{2}-i_{3}-3i_{4}-i_{5}+3i_{6})}.\end{split}$
(54)

Notice that under lattice reflection symmetry $\mathbb{Z}_{2}:c_{iA}\to
c_{(-i)A},c_{iB}\to c_{(-i)B}$, $g_{1}$ and $g_{2}$ maps to each other. To
simplify, we can impose the reflection symmetry and requires $g_{1}=g_{2}=g$,
then the coupling coefficient is

$\begin{split}g_{i_{1}\cdots
i_{6}}=g\cos\big{(}k_{F}(i_{1}-3i_{2}+i_{3}+3i_{4}+i_{5}-3i_{6})\big{)}.\end{split}$
(55)

The dominant $s$-wave interaction is given by

$i_{1}-3i_{2}+i_{3}+3i_{4}+i_{5}-3i_{6}=0,$ (56)

such that $g_{i_{1}\cdots i_{6}}=g$ is uniform. We seek a local interaction
that has minimal span on the lattice. The optimal solution of Eq. (56) is
given by

$i_{1}=i_{2}=i-1,i_{3}=i_{4}=i,i_{5}=i_{6}=i+1,$ (57)

for any choice of $i$. With this solution Eq. (57), Eq. (53) reduces to

$H_{\text{int}}=g\sum_{i}c_{(i-1)B}^{\dagger}c_{(i-1)A}c_{iB}c_{iA}c_{(i+1)B}^{\dagger}c_{(i+1)A}+\text{h.c.}.$
(58)

which is the SMG interaction Eq. (16) proposed in the main text.

##  Appendix C Full Renormalization Group Equations

We start with the interaction $H_{\text{int, CF}}$

$\displaystyle H_{\text{int,
CF}}=g_{\text{rs}}\sum_{\alpha}\epsilon^{ijk}c_{B\alpha}^{\dagger}c_{A_{i}\alpha}c_{A_{j}\alpha}c_{A_{k}\alpha}$
$\displaystyle+g_{\text{rt}}\sum_{\alpha\neq\beta}\epsilon^{ijk}c_{B\alpha}^{\dagger}c_{A_{i}\alpha}c_{A_{j}\beta}c_{A_{k}\beta}+\text{h.c.}$
(59)

Under RG, the following the density-density and exchange interactions will be
generated

$\displaystyle H_{\text{int,
AA}}=g_{\text{as}}\sum_{\alpha,st}n_{A_{s}\alpha}n_{A_{t}\alpha}+g_{\text{at}}\sum_{\alpha\neq\beta,st}n_{A_{s}\alpha}n_{A_{t}\beta}$
$\displaystyle+g_{\text{ae}}\sum_{\alpha\neq\beta,st}c_{A_{s}\alpha}^{\dagger}c_{A_{s}\beta}c_{A_{t}\beta}^{\dagger}c_{A_{t}\alpha}+(A_{s}\leftrightarrow
A_{t})+\text{h.c.}$ (60)

and

$\displaystyle H_{\text{int,
AB}}=g_{\text{bs}}\sum_{\alpha,s}n_{B\alpha}n_{A_{s}\alpha}+g_{\text{bt}}\sum_{\alpha\neq\beta,s}n_{B\alpha}n_{A_{s}\beta}$
$\displaystyle+g_{\text{be}}\sum_{\alpha\neq\beta,s}c_{B\alpha}^{\dagger}c_{B\beta}c_{A_{s}\beta}^{\dagger}c_{A_{s}\alpha}+(A_{s}\leftrightarrow
B)+\text{h.c.}$ (61)

There is an additional density-density interaction that will correct
$H_{\text{int, AA}},H_{\text{int, AB}}$,

$H_{\text{int,
BB}}=g_{\text{bb}}\sum_{\alpha\beta}n_{B\alpha}n_{B\beta}-c_{B\alpha}^{\dagger}c_{B\beta}c_{B\beta}^{\dagger}c_{B\alpha}.$
(62)

Putting all interactions together, the full RG equations are

$\displaystyle\frac{dg_{\text{bb}}}{d\ell}=4d_{0}d_{\text{BB}}g_{\text{bb}}^{2}+3d_{0}g_{\text{be}}^{2}$
$\displaystyle\frac{dg_{\text{bs}}}{d\ell}=-2d_{\text{AB}}g_{\text{bs}}^{2}+\frac{9g_{\text{rs}}^{2}}{2}+g_{\text{rt}}^{2}$
$\displaystyle\frac{dg_{\text{bt}}}{d\ell}=2d_{0}d_{\text{AB}}g_{\text{bt}}^{2}$
$\displaystyle\frac{dg_{\text{be}}}{d\ell}=-6d_{0}g_{\text{ae}}g_{\text{be}}+2d_{0}g_{\text{at}}g_{\text{be}}+4d_{0}d_{\text{BB}}g_{\text{bb}}g_{\text{be}}+3g_{\text{rs}}g_{\text{rt}}+\frac{g_{\text{rt}}^{2}}{2}$
$\displaystyle\frac{dg_{\text{as}}}{d\ell}=-2g_{\text{as}}^{2}$
$\displaystyle\frac{dg_{\text{at}}}{d\ell}=2d_{0}g_{\text{at}}^{2}-d_{0}d_{\text{AB}}g_{\text{rt}}^{2}$
$\displaystyle\frac{dg_{\text{ae}}}{d\ell}=-d_{0}d_{\text{AB}}g_{\text{rt}}^{2}+4d_{0}g_{\text{ae}}g_{\text{at}}-6d_{0}g_{\text{ae}}^{2}-2d_{0}d_{\text{BB}}g_{\text{be}}^{2}$
$\displaystyle\frac{dg_{\text{rs}}}{d\ell}=-6g_{\text{as}}g_{\text{rs}}$
$\displaystyle\frac{dg_{\text{rt}}}{d\ell}=4d_{0}d_{\text{AB}}g_{\text{bt}}g_{\text{rt}}-2g_{\text{as}}g_{\text{rt}}$

where
$d_{\text{AB}}=d\chi_{pp,\text{AB}}({\bm{0}})/d\ell,d_{\text{BB}}=d\chi_{pp,\text{BB}}({\bm{0}})/d\ell$.
These ratios depend on the energies of $A$ and $B$-type fermions near the
VHSs. The two types of fermions have similar band structures, which can be
approximated as $E_{\bm{k}}^{A,B}=\epsilon^{A,B}f({\bm{k}})$. The ratios are
then given by
$d_{\text{AB}}=\frac{2|\epsilon^{A}|}{|\epsilon^{A}|+|\epsilon^{B}|}$ and
$d_{\text{BB}}=\frac{|\epsilon^{A}|}{|\epsilon^{B}|}$. If $A$ and $B$-type
fermions have the same band structure, then $d_{\text{AB}}=d_{\text{BB}}=1$.