# Definition and Classification of Fermi Surface Anomalies

Da-Chuan Lu  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

We propose that the Fermi surface anomaly of symmetry group $G$ in any
dimension is universally classified by $G$-symmetric interacting fermionic
symmetry-protected topological (SPT) phases in $(0+1)$-dimensional spacetime.
The argument is based on the perspective that the gapless fermions on the
Fermi surface can be viewed as the topological boundary modes of Chern
insulators in the phase space (position-momentum space). Given the non-
commutative nature of the phase space coordinates, we show that the momentum
space dimensions should be counted as negative dimensions for SPT
classification purposes. Therefore, the classification of phase-space Chern
insulators (or, more generally fermionic SPT phases) always reduces to a
$(0+1)$-dimensional problem, which can then be answered by the cobordism
approach. In addition to the codimension-1 Fermi surface case, we also discuss
the codimension-$p$ Fermi surface case briefly. We provide concrete examples
to demonstrate the validity of our classification scheme, and make connections
to the recent development of Fermi surface symmetric mass generation.

###### Contents

  1. I Introduction
  2. II Effective Descriptions of Fermi Liquids
    1. II.1 Non-Commutative Phase Space Geometry
    2. II.2 Bulk Description: Fermi Sea = Phase-Space Chern Insulator
    3. II.3 Boundary Description: Fermi Surface = Phase-Space Chiral Boundary Fermions
  3. III Definition of Fermi Surface Anomaly
    1. III.1 Emergent Loop Group Symmetry and Perturbative Fermi Surface Anomaly
    2. III.2 Interstitial Defect and Non-Perturbative Fermi Surface Anomaly
  4. IV Classification of Fermi Surface Anomaly
    1. IV.1 Synthetic Dimension Reduction Argument
    2. IV.2 Cobordism Classification Results
    3. IV.3 Examples and Comments
      1. IV.3.1 $G=\mathrm{U}(1)$ and $\mathbb{Z}$ Classification
      2. IV.3.2 $G=\mathrm{U}(n)$ and $\mathbb{Z}$ Classification
      3. IV.3.3 $G=\mathrm{SU}(2n)$ and Trivial Classification
      4. IV.3.4 $G=\mathbb{Z}_{2n}$ and $\mathbb{Z}_{2n}$ Classification
      5. IV.3.5 $G=\mathrm{SU}(2n+1)$ and $\mathbb{Z}_{2}$ Classification
      6. IV.3.6 $G=\mathbb{Z}_{2n+1}$ and $\mathbb{Z}_{4n+2}$ Classification
      7. IV.3.7 $G=\mathrm{U}(1)\rtimes\mathbb{Z}_{2}^{T}$ and $\mathbb{Z}$ Classification
      8. IV.3.8 $G=\mathrm{U}(1)\rtimes_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4}^{TF}$ and $\mathbb{Z}$ Classification
      9. IV.3.9 $G=\mathrm{U}(1)\times\mathbb{Z}_{2}^{T}$ and Trivial Classification
  5. V Summary

##  I Introduction

The Fermi liquid [1, 2] is a conventional and ubiquitous phase of matter in
condensed matter physics, modeling the universal low-energy features of
electrons in metals. Despite its long history of study, there has been renewed
interest in the Fermi liquid, motivated by the quest to understand the
surprising stability [3, 4, 5] of gapless fermions on the Fermi surface. An
emerging paradigm in condensed matter theory is to understand all gapless
quantum phases of matter from the perspective of emergent symmetries and
quantum anomalies [6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. This paradigm has led
to significant progress in understanding the Fermi liquid as a gapless state
of fermions protected by an emergent quantum anomaly known as the _Fermi
surface anomaly_ [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29].

The boundary-bulk correspondence between quantum anomalies and symmetry-
protected topological (SPT) orders has been a key area of study in condensed
matter physics in the past decade [30, 31, 32, 33, 34, 35, 36, 37, 38, 39].
There is a growing consensus [40, 41, 42, 43, 44] that the gapless fermions on
the Fermi surface can be viewed as the topological boundary modes of a bulk
fermionic SPT state and that the Fermi surface anomaly is related to the bulk
SPT order. So what on earth should be the “bulk” of a Fermi surface? The most
honest answer is the Fermi sea — a region in the momentum space enclosed by
the Fermi surface. Then what is “topological” about the Fermi sea? Ref. 43
made a key observation that a $d$-dimensional Fermi sea could be viewed as a
quantum Hall insulator (or, equivalently, a Chern insulator) in the
$2d$-dimensional _phase space_ (i.e. position-momentum space). This sets the
basis for classifying Fermi surface anomaly by classifying topological
insulators in the phase space.

The main goal of this work is to provide a comprehensive and rigorous
classification of the Fermi surface anomaly along the above line of thought.
We will primarily consider codimension-1 Fermi surface [25] (i.e., the Fermi
surface is one dimension less than the momentum space dimension) and comment
on the higher codimension cases in the summary section (Sec. V). Our key
result is that the classification of the Fermi surface anomaly in any
spacetime dimension is universally equivalent to the classification of
interacting fermionic SPT phases in (0+1)-dimensional spacetime. This might
not be too surprising as many thermodynamic and transport properties of Fermi
liquids remain identical across different dimensions already. The proposed
equivalence is established through a careful analysis of the non-commutative
geometry [45, 46, 47] in phase space, the synthetic dimension reduction [48,
49] of a phase-space Dirac fermion field theory, and the use of cobordism
classification [50, 51, 52, 53, 54, 55, 56, 57, 58, 59] for interacting
fermionic SPT states.

We also provide a non-perturbative definition [29] of the Fermi surface
anomaly protected by the internal symmetry $G$ and the translation symmetry.
When $G=\mathrm{U}(1)$, our results match known results such as the Luttinger
theorem [60, 61, 62] for conventional Fermi liquids. When the $\mathrm{U}(1)$
symmetry is broken down to $G=\mathbb{Z}_{4}$ (both contain the fermion parity
symmetry ${\mathbb{Z}_{2}^{F}}$ as a subgroup), we discover non-trivial
examples of Fermi surface symmetric mass generation (SMG) [28], where the
Fermi surface can be gapped out by multi-fermion interactions and deformed to
a trivial product state without breaking any symmetry. These novel gapping
mechanisms may shed light on the understanding of pseudo-gap physics in
cuprates [63, 64].

The article is organized as follows. In Sec. II, we analyze the non-
commutative geometry in the phase space to establish a mathematical foundation
for defining quantum field theory in the phase space. We propose a phase-space
Dirac fermion field theory as the bulk regularization for the Fermi surface
and demonstrate that it reproduces the expected phase space Chern-Simons
response theory of the Fermi liquid, as well as the Fermi surface gapless
modes as topological boundary modes. This sets the stage for our argument. We
then provide a non-perturbative definition of the Fermi surface anomaly and
connect it to the recently proposed emergent loop group anomaly in Sec. III.
Using dimension reduction techniques of synthetic dimensions, we prove our key
result: the equivalence between Fermi surface anomaly and (0+1)-dimensional
fermionic SPT order in Sec. IV. We use cobordism tools to classify a list of
unitary and anti-unitary symmetries and provide physical insights into our
classification results. The article concludes with a summary in Sec. V.

##  II Effective Descriptions of Fermi Liquids

###  II.1 Non-Commutative Phase Space Geometry

Given the spacetime manifold $M_{d}\times\mathbb{R}$ of a $(d+1)$-dimensional
physical system (where $M_{d}$ is the $d$-dimensional spatial manifold and
$\mathbb{R}$ is the time axis), for each position
${\bm{x}}=(x_{1},x_{2},\cdots,x_{d})\in M_{d}$ in the space, the conjugate
momentum ${\bm{k}}=(k_{1},k_{2},\cdots,k_{d})$ generates infinitesimal
translations on the manifold $M_{d}$ in the vicinity of ${\bm{x}}$ and hence
lives in the $d$-dimensional cotangent space $T_{\bm{x}}^{*}M_{d}$. Thus the
phase space is represented by the cotangent bundle
$T^{*}M_{d}:=\\{({\bm{x}},{\bm{k}})|{\bm{x}}\in M_{d},{\bm{k}}\in
T_{\bm{x}}^{*}M_{d}\\}$, equipped with a canonical commutator (setting
$\hbar=1$)

$[x_{i},k_{i}]=\mathrm{i}\quad(i=1,2,\cdots,d),$ (1)

with $\mathrm{i}$ being the imaginary unit. Unlike in a classical space where
all coordinates commute, the phase space coordinates obey non-trivial
commutation relations Eq. (1), which makes the phase space $T^{*}M_{d}$ a non-
commutative manifold.

There are two strategies to deal with the non-commutative phase space
coordinates:

  * (I)

_Phase-space background Berry curvature_. Treat both ${\bm{x}}$ and ${\bm{k}}$
as ordinary commuting coordinates at the price of introducing a uniform
background magnetic field (Berry curvature) in each $(x_{i},k_{i})$-plane,
such that any unit-charged particle moving in such a background magnetic field
will accumulate the same Berry phase as required by the commutation relation
Eq. (1).

  * (II)

_Canonical quantization_. Represent the position operator
${\bm{x}}=\mathrm{i}\partial_{{\bm{k}}}$ as a gradient operator in the
eigenbasis of the momentum operator ${\bm{k}}$, or vice versa
${\bm{k}}=-\mathrm{i}\partial_{{\bm{x}}}$, such that the commutation relation
Eq. (1) is satisfied on the operator level as in quantum mechanics.

The strategy (I) of phase-space background Berry curvature has been used in
many works [43, 22, 23, 25, 26] to formulate the Fermi liquid as a phase-space
quantum Hall insulator. The phase-space Berry curvature is also responsible
for the Berry phase term in Wen’s effective theory of Fermi liquid [24], or
the Wess-Zumino-Witten term in the recently proposed nonlinear bosonization of
Fermi surfaces by the coadjoint orbit method [65]. In this work, we will
explore more of the strategy (II) of canonical quantization and hope to gain
different insights.

For simplicity, we will always restrict our scope to a _translation invariant_
Fermi liquid in the Euclidean position space $M_{d}=\mathbb{R}^{d}$, then the
momentum space is also Euclidean $T_{\bm{x}}^{*}M_{d}=\mathbb{R}^{d}$ and is
identical among all points ${\bm{x}}$. The phase space reduces to a trivial
bundle as a product of the position and the momentum spaces

$T^{*}M_{d}=\mathbb{R}^{d}\Bowtie\mathbb{R}^{d}.$ (2)

We use the symbol $\Bowtie$ instead of $\times$ to indicate the non-
commutative nature between the position and momentum space coordinates.

###  II.2 Bulk Description: Fermi Sea = Phase-Space Chern Insulator

A Chern insulator in the phase space $T^{*}M_{d}$ can be formally described by
a low-energy effective Hamiltonian of massive Dirac fermions [43]

$H=\int_{T^{*}M_{d}}\mathrm{d}^{d}{\bm{x}}\mathrm{d}^{d}{\bm{k}}\;\psi^{\dagger}(\mathrm{i}\partial_{{\bm{x}}}\cdot{\bm{\Gamma}}_{x}+\mathrm{i}\partial_{{\bm{k}}}\cdot{\bm{\Gamma}}_{k}+m({\bm{k}})\Gamma^{0})\psi,$
(3)

where $\psi:=\psi({\bm{x}},{\bm{k}})$ is a $2^{d}$-component fermion operator
defined at each “point” of the $2d$-dimensional phase space
$T^{*}M_{d}=\mathbb{R}^{d}\Bowtie\mathbb{R}^{d}$ (let us not worry about the
non-commutativity between ${\bm{x}}$ and ${\bm{k}}$ for now, which will be
resolved later). Let $\Gamma^{\mu}$ (for $\mu=0,1,2,\cdots,2d$) be a set of
$2^{d}\times 2^{d}$ anti-commuting Hermitian matrices, satisfying
$\\{\Gamma^{\mu},\Gamma^{\nu}\\}=2\delta^{\mu\nu}$ and
$\Gamma^{0}=\mathrm{i}^{d}\prod_{\mu=1}^{2d}\Gamma^{\mu}$. These $\Gamma$
matrices can be grouped into the temporal $\Gamma^{0}$, the position spatial
${\bm{\Gamma}}_{x}=(\Gamma^{1},\cdots,\Gamma^{d})$, and the momentum spatial
${\bm{\Gamma}}_{k}=(\Gamma^{d+1},\cdots,\Gamma^{2d})$ components. Here
$\mathrm{i}\partial_{\bm{x}}\cdot{\bm{\Gamma}}_{x}=\sum_{i=1}^{d}\mathrm{i}\partial_{x_{i}}\Gamma^{i}$
denotes the dot product between the differential operator
$\mathrm{i}\partial_{\bm{x}}$ and the set of matrices ${\bm{\Gamma}}_{x}$, and
similarly for $\mathrm{i}\partial_{\bm{k}}\cdot{\bm{\Gamma}}_{k}$. A few
comments on this theory are as follows:

  * •

_Locality_. Without interaction, Eq. (3) looks like a valid local theory of
the fermion field $\psi$ in the phase space. However, once fermion interaction
is introduced, Eq. (3) is no longer a local field theory because the
interaction is generally non-local in the momentum space. Therefore, Eq. (3)
should only be viewed as a “formal” description of the phase-space Chern
insulator. One way to regularize the theory is to evoke the strategy (II) in
Sec. II.1 to resolve the non-commutative phase space geometry by replacing
$\mathrm{i}\partial_{\bm{k}}\to{\bm{x}}$, and rewrite Eq. (3) as

$H=\int_{M_{d}}\mathrm{d}^{d}{\bm{x}}\;\psi^{\dagger}(\mathrm{i}\partial_{{\bm{x}}}\cdot{\bm{\Gamma}}_{x}+{\bm{x}}\cdot{\bm{\Gamma}}_{k}+m(-\mathrm{i}\partial_{\bm{x}})\Gamma^{0})\psi,$
(4)

which is solely defined in the position space and respects the position space
locality such that local interactions can be introduced if needed.

  * •

_Mass profile_. The bulk Dirac mass $m({\bm{k}})$ is supposed to be a
polynomial function of ${\bm{k}}$, which specifies the shape of the Chern
insulator in the phase space. For example, given the Fermi momentum $k_{F}$,
$m({\bm{k}})={\bm{k}}^{2}-k_{F}^{2}$ is one possible choice of the mass
profile. Suppose the Fermi sea occupies a region $\Omega\subset\mathbb{R}^{d}$
in the momentum space enclosed by the $(d-1)$-dimensional Fermi surface
$\partial\Omega$, the Dirac fermion mass profile should satisfy

$m({\bm{k}})\left\\{\begin{array}[]{cc}\leq 0&\text{if }{\bm{k}}\in\Omega,\\\
>0&\text{if }{\bm{k}}\notin\Omega.\end{array}\right.$ (5)

This described a phase-space Chern insulator in the Fermi sea region $\Omega$,
such that the Fermi surface $\partial\Omega$ (as the boundary of the phase-
space Chern insulator) corresponds to the mass domain wall at $m({\bm{k}})=0$.

The fermions are gapped everywhere in the phase space except on the Fermi
surface, where the fermion mass vanishes. This is consistent with the physical
intuition that the gapless fermions on the Fermi surface are the only non-
trivial low-energy feature of the Fermi liquid. We will study these boundary
fermion modes in more detail in Sec. II.3 to show that they travel in the
directions perpendicular to the Fermi surface as expected.

  * •

_Particle-hole symmetry_. Under the particle-hole transformation
$\mathbb{Z}_{2}^{C}$, the inside and outside of the Fermi surface will
interchange, corresponding to flipping the fermion mass
$\mathbb{Z}_{2}^{C}:m\to-m$, or equivalently, conjugating the fermion operator

$\mathbb{Z}_{2}^{C}:\psi\to\mathcal{K}\Gamma^{0}\psi^{*},$ (6)

where $\mathcal{K}$ denotes the complex conjugate operator, such that
$\mathbb{Z}_{2}^{C}:\psi^{\dagger}\Gamma^{0}\psi\to-\psi^{\dagger}\Gamma^{0}\psi$.
Note that $\mathbb{Z}_{2}^{C}$ is _not_ a symmetry of the Hamiltonian $H$ in
Eq. (3), as the mass term $m$ explicitly breaks this symmetry. However, it is
useful in defining the Fermi surface. We propose that the Fermi surface should
be more generally defined as the _particle-hole symmetric_ sub-manifold in the
phase space, specified by the locus of
$\langle\psi^{\dagger}\Gamma^{0}\psi\rangle=0$. This definition applies to the
case of interacting fermions.

  * •

_Phase-space $\mathrm{U}(1)$ symmetry_. The Hamiltonian $H$ in Eq. (3) has a
0-form $\mathrm{U}(1)$ symmetry in the phase space, generated by the charge
operator

$Q=\int_{T^{*}M_{d}}\mathrm{d}^{d}{\bm{x}}\mathrm{d}^{d}{\bm{k}}\;\psi^{\dagger}\psi.$
(7)

The symmetry transformation $\mathrm{e}^{\mathrm{i}\phi Q}$ forms the
$\mathrm{U}(1)$ symmetry group, where $\phi\in[0,2\pi)$ and $Q\in\mathbb{Z}$.
The fermion field transforms as $\psi\to\mathrm{e}^{\mathrm{i}\phi}\psi$ under
the symmetry transformation.

The essential bulk topological response of the Fermi liquid is captured by a
phase-space Chern-Simons theory [43, 22, 23, 25, 26] of the phase-space
$\mathrm{U}(1)$ symmetry. To show that the effective Hamiltonian in Eq. (3)
indeed reproduces the desired topological response, we first gauge the 0-form
$\mathrm{U}(1)$ symmetry of the fermion $\psi$ (under which
$\psi\to\mathrm{e}^{\mathrm{i}\phi}\psi$) by introducing a 1-form gauge field
$A$ in the phase spacetime

$A=A_{0}\mathrm{d}t+{\bm{A}}_{x}\cdot\mathrm{d}{\bm{x}}+{\bm{A}}_{k}\cdot\mathrm{d}{\bm{k}},$
(8)

where $A_{0}$, ${\bm{A}}_{x}=(A_{1},\cdots,A_{d})$,
${\bm{A}}_{k}=(A_{d+1},\cdots,A_{2d})$ are respectively the components of the
$\mathrm{U}(1)$ gauge connection in the time, position, and momentum spaces.
We will treat $A$ as a background gauge field that does not have dynamics. Let
$F:=\mathrm{d}A$ be the $\mathrm{U}(1)$ gauge curvature. Following the
strategy (I) mentioned in Sec. II.1, we must set $F_{i,d+i}=1$ for
$i=1,2,\cdots,d$ to reproduce the position-momentum commutator in Eq. (1).
This background gauge curvature effectively replaces the non-commutative
$2d$-dimensional phase space geometry, and the effective Hamiltonian Eq. (3)
becomes [43]

$H=\int_{T^{*}M_{d}}\mathrm{d}^{d}{\bm{x}}\mathrm{d}^{d}{\bm{k}}\;\psi^{\dagger}(\mathrm{i}D_{\bm{x}}\cdot{\bm{\Gamma}}_{x}+\mathrm{i}D_{\bm{k}}\cdot{\bm{\Gamma}}_{k}+m\Gamma^{0}-A_{0})\psi,$
(9)

where $\mathrm{i}D_{\mu}:=\mathrm{i}\partial_{\mu}-A_{\mu}$ are gauge
covariant derivatives. Now, in Eq. (9), ${\bm{x}}$ and ${\bm{k}}$ are ordinary
_commuting_ coordinates, as the background Berry curvature $F_{i,d+i}=1$ has
been implemented in the $\mathrm{U}(1)$ gauge configuration to resolve the
non-commutativity. Therefore, we can use conventional field theory approaches
to deal with Eq. (9).

Integrating out the fermion field in Eq. (9) generates the following Chern-
Simons action in the $(2d+1)$-dimensional phase spacetime [43, 66] (assuming
the Dirac fermion $\psi$ is such regularized that $m>0$ corresponds to a
trivial insulator)

$S=\frac{1}{(d+1)!(2\pi)^{d}}\int_{T^{*}M_{d}\times\mathbb{R}}\frac{1-\operatorname{sgn}m}{2}A\wedge(\mathrm{d}A)^{\wedge
d}.$ (10)

This is the defining bulk topological field theory whose inflow generates the
Fermi surface anomaly [22, 23, 27]. 111Our discussion here is unrelated to the
previous study of chiral and gravitational anomalies on Fermi surfaces [145],
which is about the non-trivial Berry curvature on Fermi surfaces purely
defined in the momentum space. In particular, if we plug in the phase-space
background gauge configuration $F_{i,d+i}=1$ (i.e.,
$\mathrm{d}A=F=\sum_{i=1}^{d}\mathrm{d}x_{i}\wedge\mathrm{d}k_{i}$), take the
fermion mass profile in Eq. (5), and finish the momentum space integration,
Eq. (10) will reduce to

$S=\frac{1}{(2\pi)^{d}}\int_{M_{d}\times\mathbb{R}}\mathrm{d}t\,\mathrm{d}^{d}{\bm{x}}\;A_{0}\operatorname{vol}\Omega,$
(11)

which indicates that the fermion charge density $\nu$ (filling fraction) is
related to the Fermi volume $\operatorname{vol}\Omega$ by

$\nu=\frac{\delta S}{\delta
A_{0}}=\frac{\operatorname{vol}\Omega}{(2\pi)^{d}},$ (12)

where
$\operatorname{vol}\Omega:=\int\mathrm{d}^{d}{\bm{k}}\;(1-\operatorname{sgn}m({\bm{k}}))/2$
is by-definition the momentum-space volume where $m({\bm{k}})\leq 0$. This is
precisely the Luttinger theorem — a hallmark of the Fermi surface anomaly.
Thus we have confirmed that the effective bulk Hamiltonian Eq. (3) can produce
the correct anomaly inflow to describe a $(d+1)$-dimensional unit-charged
Fermi liquid with a single Fermi surface. The extension to cases of generic
fermion charges and multiple Fermi surfaces is straightforward (see Ref. 22
for example) and will not be elaborated further here.

###  II.3 Boundary Description: Fermi Surface = Phase-Space Chiral Boundary
Fermions

How do we see more explicitly that the effective Hamiltonian Eq. (3)
reproduces the low-energy fermions on a Fermi surface? Since the Fermi surface
is interpreted as the boundary of the phase-space Chern insulator, the gapless
fermions should arise as the topological boundary modes, which can be analyzed
as follows.

Figure 1: Illustration of a point ${\bm{k}}_{F}$ on the Fermi surface
$\partial\Omega$ with the normal vector ${\bm{n}}$ and the tangent vector(s)
${\bm{\tau}}_{j}$, for the case when the Fermi sea $\Omega$ dimension is
$d=2$.

As shown in Fig. 1, we consider a point ${\bm{k}}_{F}\in\partial\Omega$ on the
Fermi surface at which the normal vector is specified by ${\bm{n}}$. This
means that the fermion mass will cross zero in the phase space at
${\bm{k}}_{F}$ with a gradient along the ${\bm{n}}$ direction:

$m({\bm{k}}_{F})=0,\quad\partial_{{\bm{k}}}m({\bm{k}}_{F})\propto{\bm{n}}.$
(13)

Such a mass domain wall at ${\bm{k}}_{F}$ will trap gapless fermion modes in
the eigenspace specified by the projection
$P_{0}=(1+\mathrm{i}({\bm{n}}\cdot{\bm{\Gamma}}_{k})\Gamma^{0})/2$. Under this
projection, only those terms that commute with $P_{0}$ can remain, so the
effective Hamiltonian Eq. (3) reduces to

$\begin{split}H&=\int_{\partial\Omega}\mathrm{d}{\bm{k}}_{F}\int_{M_{d}\Bowtie
T_{{\bm{k}}_{F}}\partial\Omega}\mathrm{d}^{d}{\bm{x}}\,\mathrm{d}^{d-1}{\bm{k}}\\\
&\psi^{\dagger}P_{0}\Big{(}\mathrm{i}({\bm{n}}\cdot\partial_{\bm{x}})({\bm{n}}\cdot{\bm{\Gamma}}_{x})+\\\
&\sum_{j=1}^{d-1}\big{(}\mathrm{i}({\bm{\tau}}_{j}\cdot\partial_{\bm{x}})({\bm{\tau}}_{j}\cdot{\bm{\Gamma}}_{x})+\mathrm{i}({\bm{\tau}}_{j}\cdot\partial_{{\bm{k}}})({\bm{\tau}}_{j}\cdot{\bm{\Gamma}}_{k})\big{)}\Big{)}P_{0}\psi,\end{split}$
(14)

where $T_{{\bm{k}}_{F}}\partial\Omega$ denotes the $(d-1)$-dimensional tangent
space of the Fermi surface $\partial\Omega$ at the base point ${\bm{k}}_{F}$,
and ${\bm{\tau}}_{j}$ (for $j=1,2,\cdots,d-1$) denote a set of orthonomal
basis of the tangent space $T_{{\bm{k}}_{F}}\partial\Omega$.

To resolve the non-commutativity between ${\bm{x}}$ and ${\bm{k}}$
coordinates, we evoke the strategy (II) outlined in Sec. II.2. Given that
${\bm{x}}=\mathrm{i}\partial_{\bm{k}}$ resolves the canonical commutation
relation in Eq. (1), we can simply replace the gradient operator
$\mathrm{i}\partial_{\bm{k}}$ by ${\bm{x}}$, and fall back to the standard
quantum mechanical description in the position space $M$ alone. Under this
replacement, Eq. (14) becomes

$\begin{split}H&=\int_{\partial\Omega}\mathrm{d}{\bm{k}}_{F}\int_{M_{d}}\mathrm{d}^{d}{\bm{x}}\;\psi^{\dagger}P_{0}\Big{(}\mathrm{i}({\bm{n}}\cdot\partial_{\bm{x}})({\bm{n}}\cdot{\bm{\Gamma}}_{x})\\\
&\sum_{j=1}^{d-1}\big{(}\mathrm{i}({\bm{\tau}}_{j}\cdot\partial_{\bm{x}})({\bm{\tau}}_{j}\cdot{\bm{\Gamma}}_{x})+({\bm{\tau}}_{j}\cdot{\bm{x}})({\bm{\tau}}_{j}\cdot{\bm{\Gamma}}_{k})\big{)}\Big{)}P_{0}\psi.\end{split}$
(15)

Now the terms
$({\bm{\tau}}_{j}\cdot{\bm{x}})({\bm{\tau}}_{j}\cdot{\bm{\Gamma}}_{k})$ in the
Hamiltonian Eq. (15) can be interpreted as a new set of perpendicular domain
walls of fermion masses (each one is normal to a ${\bm{\tau}}_{j}$ direction).
They will further localize the fermions to the origin in all tangent
directions ${\bm{\tau}}_{j}$ (for $j=1,\cdots,d-1$). The localized fermion
modes are specified by a sequence of further projections
$P_{j}=(1+({\bm{\tau}}_{j}\cdot{\bm{\Gamma}}_{x})({\bm{\tau}}_{j}\cdot{\bm{\Gamma}}_{k}))/2$,
such that the total projection is

$P=P_{0}\prod_{j=1}^{d-1}P_{j}.$ (16)

Under the total projection $P$, only one fermion mode survives. This can be
seen by a simple counting argument: the fermion field $\psi$ has $2^{d}$
components to start with, given $P_{0},\cdots,P_{d-1}$ are $d$ commuting
projectors, each reducing the number of fermion components by half, the
remaining component number is $2^{d}/2^{d}=1$.

The only term in the Hamiltonian that commute with the total projection $P$ is
$\mathrm{i}({\bm{n}}\cdot\partial_{\bm{x}})({\bm{n}}\cdot{\bm{\Gamma}}_{x})$,
which will survive in the low-energy theory. Moreover,
$({\bm{n}}\cdot{\bm{\Gamma}}_{x})$ becomes an identity operator in the
projected subspace, because

$\begin{split}&P({\bm{n}}\cdot{\bm{\Gamma}}_{x})P\\\
=&P({\bm{n}}\cdot{\bm{\Gamma}}_{x})\mathrm{i}({\bm{n}}\cdot{\bm{\Gamma}}_{k})\Gamma^{0}\prod_{j=1}^{d-1}\big{(}\mathrm{i}({\bm{\tau}}_{j}\cdot{\bm{\Gamma}}_{x})({\bm{\tau}}_{j}\cdot{\bm{\Gamma}}_{k})\big{)}P\\\
=&P\Big{(}\mathrm{i}^{d}\prod_{\mu=0}^{2d}\Gamma^{\mu}\Big{)}P=P\mathds{1}P=P.\end{split}$
(17)

The first equality in Eq. (LABEL:eq:_PGP=P) relies on the fact that we can
insert between projection operators $P$ matrices like
$\mathrm{i}({\bm{n}}\cdot{\bm{\Gamma}}_{k})\Gamma^{0}$ or
$\mathrm{i}({\bm{\tau}}_{j}\cdot{\bm{\Gamma}}_{x})({\bm{\tau}}_{j}\cdot{\bm{\Gamma}}_{k})$,
as they all behave like identity operators in the projected subspace. If we
denote the projected fermion mode as $\psi_{{\bm{k}}_{F}}=P\psi$ (the low-
energy fermion localized on the intersection of mass domain walls at the Fermi
momentum ${\bm{k}}_{F}$), the effective Hamiltonian for this fermion mode
reads

$H=\int_{\partial\Omega}\mathrm{d}{\bm{k}}_{F}\int\mathrm{d}({\bm{n}}\cdot{\bm{x}})\;\psi_{{\bm{k}}_{F}}^{\dagger}\mathrm{i}({\bm{n}}\cdot\partial_{\bm{x}})\psi_{{\bm{k}}_{F}},$
(18)

which describes a single chiral fermion moving along the normal direction
${\bm{n}}$ at every momentum ${\bm{k}}_{F}\in\partial\Omega$ on the Fermi
surface, which matches the low-energy physics of Fermi liquid precisely.
Therefore, the phase-space Chern insulator effective Hamiltonian $H$ in Eq.
(3) indeed provides a bulk regularization for the Fermi liquid, reproducing
all the expected low-energy behaviors of gapless fermions on the Fermi
surface. This is an alternative bulk regularization of Fermi liquid compared
to the Weyl fermion regularization proposed by Ma and Wang [25] recently. To
make a comparison between our regularization and that in Ref. 25,

  * •

We use the canonical quantization approach to resolving the non-commutative
phase space geometry, while Ref. 25 uses the phase-space background Berry
curvature approach.

  * •

The low-energy chiral fermions are realized as domain-wall fermions in our
approach, compared to Landau-level Weyl fermions in Ref. 25. The directional
nature of the chiral fermions (i.e., they always move along the normal
direction at each point on the Fermi surface) is more explicit in our
regularization.

##  III Definition of Fermi Surface Anomaly

###  III.1 Emergent Loop Group Symmetry and Perturbative Fermi Surface Anomaly

The chiral boundary fermion effective Hamiltonian Eq. (18) has a rather large
emergent symmetry, described by the loop-$\partial\Omega$ group of
$\mathrm{U}(1)$ [22, 23] or the mapping space from the Fermi surface
$\partial\Omega$ to $\mathrm{U}(1)$, denoted as
$\mathrm{L}_{\partial\Omega}\mathrm{U}(1):=\mathrm{Map}(\partial\Omega,\mathrm{U}(1))$
222For codimension-1 Fermi surface, $\partial\Omega$ is a $(d-1)$-dimensional
closed manifold. In the case that $\partial\Omega$ is diffeomorphic to a
$S^{d-1}$ sphere, the loop group is also denoted as
$\mathrm{L}^{d-1}\mathrm{U}(1)$. Under the group action, fermion operators
transform as

$\mathrm{L}_{\partial\Omega}\mathrm{U}(1):\psi_{{\bm{k}}_{F}}\to\mathrm{e}^{\mathrm{i}\phi({\bm{k}}_{F})}\psi_{{\bm{k}}_{F}}\quad(\forall{\bm{k}}_{F}\in\partial\Omega)$
(19)

where $\phi({\bm{k}}_{F})$ is a continuous function on the Fermi surface
$\partial\Omega$, subject to the equivalence
$\phi({\bm{k}}_{F})\sim\phi({\bm{k}}_{F})+2\pi$. Mathematically, the loop
group $\mathrm{L}_{\partial\Omega}\mathrm{U}(1)$ is the group of all
continuous maps from the closed manifold $\partial\Omega$ to $\mathrm{U}(1)$,
with the group multiplication defined pointwise.

In contrast, for a conventional real-space $\mathrm{U}(1)$-symmetric Chern
insulator, the boundary theory only has the same $\mathrm{U}(1)$ symmetry
inherited from the bulk. In this case, the boundary symmetry is not enlarged
because the gapless fermion mode can propagate (along tangent directions)
throughout the boundary, locking point-wise $\mathrm{U}(1)$ transformations
together into a global $\mathrm{U}(1)$ transformation on the boundary
manifold. However, for the phase-space Chern insulator, due to the non-
commutative nature between the position and momentum coordinates, the boundary
fermion mode is localized in all tangent directions of the Fermi surface and
only propagates along the normal direction ${\bm{n}}$. Therefore, the
$\mathrm{U}(1)$ transformations at different momentum points ${\bm{k}}_{F}$ on
the Fermi surfaces are not locked together, giving rise to the enlarged loop
group symmetry $\mathrm{L}_{\partial\Omega}\mathrm{U}(1)$.

Our argument establishes the loop group symmetry
$\mathrm{L}_{\partial\Omega}\mathrm{U}(1)$ on the Fermi surface as an emergent
symmetry, originated from the $\mathrm{U}(1)$ symmetry in the phase-space
bulk. Therefore, the Fermi surface anomaly, which was proposed [22] to be a
perturbative anomaly of $\mathrm{L}_{\partial\Omega}\mathrm{U}(1)$, can be
described by the bulk topological field theory of a $\mathrm{U}(1)$ connection
$A$ of the $\mathrm{U}(1)$ bundle in the phase spacetime, as derived in Eq.
(10) already,

$S=\frac{k}{(d+1)!(2\pi)^{d}}\int_{M_{d}\times\Omega\times\mathbb{R}}A\wedge(\mathrm{d}A)^{\wedge
d}.$ (20)

Here we have added in the Chern-Simons level $k\in\mathbb{Z}$ for generality,
which should correspond to the multiplicity (degeneracy) of the Fermi surface.
We set $k=1$ for a single Fermi surface. Various physical consequences of this
theory have been discussed in the literature [43, 22, 23, 25, 26], which we
will not reiterate. This description sets the basis to classify the loop group
$\mathrm{L}G$ anomaly on the $(d-1)$-dimensional Fermi surface by the
$G$-symmetric invertible topological phases in the $2d$-dimensional phase
space, which will be our key strategy in Sec. IV.

###  III.2 Interstitial Defect and Non-Perturbative Fermi Surface Anomaly

One drawback of using the phase-space Chern-Simons theory Eq. (20) to
characterize the Fermi surface anomaly is that it is not straightforward to
extend the description to Fermi liquids with a more general symmetry group
$G$, such as $G=\mathbb{Z}_{2n}$. We propose to define the Fermi surface
anomaly in a lattice fermion system by the projective representation of the
internal symmetry $G$ in the presence of an _interstitial defect_ that adds an
extra site to the lattice [69, 29], as illustrated in Fig. 2 (a).

Figure 2: (a) Characterize the Fermi surface anomaly by the projective
representation of the internal symmetry in the presence of an interstitial
defect. (b) On the lattice, an interstitial defect (the red dot) is created by
translating a semi-infinite line of sites along the line direction. (c) In the
phase space, this creates extra Berry curvature (in the shaded plaquettes)
along a line of momenta at the defect position.

Consider a lattice fermion system in $d$-dimensional space with global
internal symmetry $G$ and lattice translation symmetry $\mathbb{Z}^{d}$. Let
$T_{i}$ be the generator of translation symmetry in the $i$-th spatial
direction. In the phase space, the lattice translation symmetry
$\mathbb{Z}^{d}$ acts as an emanant momentum-space dipole symmetry
$\mathrm{U}(1)^{d}$ (i.e. the dipole moment conservation in the momentum
space)

$\psi({\bm{x}},{\bm{k}})\to
T_{i}\psi({\bm{x}},{\bm{k}})T_{i}^{-1}=\mathrm{e}^{\mathrm{i}k_{i}}\psi({\bm{x}},{\bm{k}}).$
(21)

An emanant symmetry [29] is an exact IR symmetry that only acts on low-energy
degrees of freedom. Its action on high-energy degrees of freedom is not well-
defined. However, it arises from a UV symmetry in that any low-energy operator
charged under the emanant symmetry must also be charged under the
corresponding UV symmetry. The momentum-space dipole symmetry
$\mathrm{U}(1)^{d}$ in Eq. (21) emanates from the lattice translation symmetry
$\mathbb{Z}^{d}$ in the sense that any low-energy operator violating the
momentum-space dipole symmetry will also break the lattice translation
symmetry [29, 70], even though there is no group homomorphism between these
two symmetry groups. A similar discussion also appeared in Ref. 24, where the
emanant symmetry was proposed to be $\mathbb{R}^{d}$ (as a non-compact version
of our proposed $\mathrm{U}(1)^{d}$).

An interstitial defect is a point defect that adds one extra site (or unit
cell) to the lattice. It can be created by translating a semi-infinite line of
sites along the line direction as shown in Fig. 2(b) on the lattice level. The
choice of direction for this semi-infinite line does not matter. We may choose
it to be along the positive axis of $x_{1}$. The twist operator
$U_{\mathrm{tw}}$ creates the interstitial defect at the origin ${\bm{x}}=0$,

$U_{\mathrm{tw}}=T_{1}^{\Theta(x_{1})\prod_{i=2}^{d}\delta(x_{i})},$ (22)

where $\Theta$ is the Heaviside step function and $\delta$ is the Kronecker
delta function:

$\Theta(x)=\left\\{\begin{array}[]{ll}1&\text{if }x>0,\\\ 0&\text{if
}x<0,\end{array}\right.\quad\delta(x)=\partial_{x}\Theta(x).$ (23)

They together ensure that the translation is only implemented along the
positive axis of $x_{1}$.

For any field or operator $\mathcal{O}$, we defined the twisted version
$\mathcal{O}_{\mathrm{tw}}$ as
$\mathcal{O}_{\mathrm{tw}}:=U_{\mathrm{tw}}\mathcal{O}U_{\mathrm{tw}}^{-1}$.
In particular, the fermion field is twisted to

$\psi_{\mathrm{tw}}({\bm{x}},{\bm{k}})=\mathrm{e}^{\mathrm{i}k_{1}\Theta(x_{1})\prod_{i=2}^{d}\delta(x_{i})}\psi({\bm{x}},{\bm{k}}).$
(24)

This allows us to define the twisted Hamiltonian $H_{\mathrm{tw}}$ and the
twisted representation of symmetry operation $\rho_{\mathrm{tw}}(g)$ for any
group element $g\in G$ of the internal symmetry group $G$ by replacing all
operators in $H$ or $\rho(g)$ with their twisted version. We say that the
fermion system has a Fermi surface anomaly, if there exists a cyclic subgroup
of $G$ (generated by $g\in G$ and $g^{n}=1$) such that the twisted partition
function accumulates a non-trivial phase $\mathrm{e}^{\mathrm{i}2\pi\nu}\neq
1$ (or equivalently, a non-trivial index $\nu\neq 0\mod 1$) under the cyclic
symmetry action:

$\operatorname{Tr}(\mathrm{e}^{-\beta
H_{\mathrm{tw}}}\rho_{\mathrm{tw}}(g)^{n})=\mathrm{e}^{\mathrm{i}2\pi\nu}\operatorname{Tr}\mathrm{e}^{-\beta
H_{\mathrm{tw}}}.$ (25)

This indicates that the interstitial defect transforms projectively under the
internal symmetry $G$, which provides a non-perturbative definition of the
Fermi surface anomaly. From this perspective, the Fermi surface anomaly may
also be viewed as the mixed anomaly between the internal symmetry $G$ and the
emanant symmetry $\mathrm{U}(1)^{d}$, which is a straightforward
generalization of the mixed $\mathrm{U}(1)\times\mathbb{R}^{d}$ anomaly
proposed by Wen [24].

To demonstrate the validity of the general definition of the Fermi surface
anomaly by Eq. (25), we consider the special case of $G=\mathrm{U}(1)$ and
show that it reproduces the known filling constraints by the Luttinger
theorem. When $G=\mathrm{U}(1)$, for $g=\mathrm{e}^{\mathrm{i}\phi}\in G$, we
have $\rho(g)_{\mathrm{tw}}=\mathrm{e}^{\mathrm{i}\phi Q_{\mathrm{tw}}}$,
where $Q_{\mathrm{tw}}$ is twisted from the charge operator $Q$ in Eq. (7).
The twisted partition function can be defined as

$Z_{\mathrm{tw}}(\beta,\phi)=\operatorname{Tr}(\mathrm{e}^{-\beta
H_{\mathrm{tw}}}\mathrm{e}^{\mathrm{i}\phi Q_{\mathrm{tw}}}).$ (26)

The Fermi surface anomaly is manifested by

$Z_{\mathrm{tw}}(\beta,\phi+2\pi)=\mathrm{e}^{\mathrm{i}2\pi\nu}Z_{\mathrm{tw}}(\beta,\phi),$
(27)

where $\nu$ (mod 1) serves as the anomaly index, and
$\mathrm{e}^{\mathrm{i}2\pi\nu}$ is the same non-trivial phase factor that
appeared in Eq. (25).

To compute the anomaly index $\nu$, we notice that the transformation of the
fermion field in Eq. (24) induces a $\mathrm{U}(1)$ gauge transformation in
the phase space [70, 21], such that

$A_{\mathrm{tw}}=A+\Theta(x_{1})\prod_{i=2}^{d}\delta(x_{i})\,\mathrm{d}k_{1}.$
(28)

This means that the background $\mathrm{U}(1)$ gauge field component
$A_{k_{1}}$ in the phase space is shifted by a uniform amount over the half-
plane of $x_{1}>0$, as shown in Fig. 2(c). As a result, this leads to
additional $\mathrm{U}(1)$ gauge curvature $F:=\mathrm{d}A$ in the phase space
along the interface of $x_{1}=0$,

$F_{\mathrm{tw}}=F+\delta({\bm{x}})\,\mathrm{d}x_{1}\wedge\mathrm{d}k_{1},$
(29)

where $\delta({\bm{x}})=\prod_{i=1}^{d}\delta(x_{i})$. Substitute into the
bulk topological response theory in Eq. (20), and take a phase-space uniform
configuration for the temporal gauge field
$A_{0}({\bm{x}},{\bm{k}})=\varphi\delta(t)$ at the $t=0$ time slice, we have

$S_{\mathrm{tw}}=S+k\varphi\frac{\operatorname{vol}\Omega}{(2\pi)^{d}},$ (30)

hence the twisted charge operator is given by

$Q_{\mathrm{tw}}=\frac{\partial
S_{\mathrm{tw}}}{\partial\varphi}=Q+k\frac{\operatorname{vol}\Omega}{(2\pi)^{d}}.$
(31)

Substitute Eq. (31) to Eq. (26), we can compute the anomalous phase factor in
Eq. (27). Given that the total charge $Q\in\mathbb{Z}$ is quantized, the Fermi
surface anomaly index $\nu$ is associated with the fractional charge of the
global $\mathrm{U}(1)$ symmetry induced by the interstitial defect

$\nu=k\frac{\operatorname{vol}\Omega}{(2\pi)^{d}}\mod 1.$ (32)

The level $k\in\mathbb{Z}$ is integer classified in this case. For generic
Fermi volume $\operatorname{vol}\Omega$ that is not a rational fraction of the
Brillouin zone volume $(2\pi)^{d}$, the Fermi surface anomaly is non-vanishing
as long as the level $k\neq 0$. This reproduces the know results about Fermi
liquid with $\mathrm{U}(1)$ symmetry and demonstrates that our non-
perturbative definition of the Fermi surface anomaly in Eq. (25) falls back to
the perturbative $\mathrm{L}_{\partial\Omega}\mathrm{U}(1)$ anomaly proposed
in Ref. 22, 23 for the case of $G=\mathrm{U}(1)$.

For more general internal symmetry $G$, we proposed that the Fermi surface
anomaly should be defined via Eq. (25), following the general idea of the
twist defect construction by Cheng and Seiberg [29]. The major difference is
that they twist the translation symmetry in time and internal symmetry in
space, while we twist the translation symmetry in space and internal symmetry
in time. This modification allows us to define the Fermi surface anomaly in
general dimensions (beyond $(1+1)$D).

##  IV Classification of Fermi Surface Anomaly

###  IV.1 Synthetic Dimension Reduction Argument

The remaining objective is to classify the Fermi surface anomaly for a general
internal symmetry group $G$. According to Sec. III.2, the anomaly is defined
by the fractionalized representation of $G$ carried by interstitial defects in
the fermionic system, indicating that the anomaly classification can be mapped
to the classification of $(0+1)$-dimensional phase transitions between
$G$-symmetric invertible topological phases of fermions, which is equivalent
to the classification of $(0+1)$-dimensional fermionic SPT states. However, in
Sec. III.1, the bulk topological field theory described by Eq. (20) suggests a
different conclusion that classifying the Fermi surface anomaly of a
$(d+1)$-dimensional Fermi liquid is equivalent to classifying the
$(d+d+1)$-dimensional fermionic SPT states in phase spacetime. This raises a
paradox as the two different counting of dimensions seem to be inconsistent
with each other.

The paradox can be resolved by considering the non-trivial dimension counting
in the phase space. Because position and momentum are non-commuting
coordinates, their dimensions should not be simply added together. Instead,
the correct classification should consider the momentum dimensions as
“negative” spatial dimensions [48], effectively defining the bulk SPT phase in
a $(d-d+1)=(0+1)$-dimensional spacetime, aligning with the view from the
interstitial defect.

To understand this unusual dimension counting, we revisit the effective bulk
Hamiltonian in Eq. (3), which describes a phase-space Chern insulator (or,
equivalently, a $d$-dimensional Fermi sea). Following the strategy (II) of
canonical quantization to regularize the bulk Hamiltonian by replacing
$\mathrm{i}\partial_{\bm{k}}\to{\bm{x}}$ as Eq. (4), we have

$H=\int_{M_{d}}\mathrm{d}^{d}{\bm{x}}\;\psi^{\dagger}(\mathrm{i}\partial_{{\bm{x}}}\cdot{\bm{\Gamma}}_{x}+{\bm{x}}\cdot{\bm{\Gamma}}_{k}+m\Gamma^{0})\psi.$
(33)

This describes a series of perpendicular mass domain walls (one in each
independent direction) that intersect at ${\bm{x}}=0$, trapping a single
fermion mode at the intersection point, which is described by the following
effective Hamiltonian:

$H=m(\psi^{\dagger}\psi-1/2),$ (34)

where $m$ plays the role of the chemical potential, and the $2^{d}$-component
spinor is projected to $1$-dimensional spinor $\psi$. This single fermion mode
can also be understood as the topological zero mode of the Dirac operator
$\mathrm{i}D=\mathrm{i}D_{\bm{x}}\cdot{\bm{\Gamma}}_{x}+\mathrm{i}D_{\bm{k}}\cdot{\bm{\Gamma}}_{k}$
in the phase space $T^{*}M_{d}$, as required by the index theorem (assuming
$M_{d}=\mathbb{R}^{d}$ is Euclidean):

$\text{index}(D)=\int_{T^{*}M_{d}}\text{ch}(D)=\frac{1}{d!}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}\Big{(}\frac{\mathrm{d}A}{2\pi}\Big{)}^{d}=1.$
(35)

Therefore, regardless of the spatial dimension $d$ of a Fermi sea, its
corresponding bulk description as a phase-space Chern insulator is always
equivalent to a $(0+1)$-dimensional fermion mode at low energy under dimension
reduction. As a result, the classification of the Fermi surface anomaly for a
Fermi liquid in $(d+1)$-dimensional spacetime is equivalent to the
classification of fermionic SPT phases in $(0+1)$-dimensional spacetime.

The above statement holds true even in the presence of fermion _interactions_.
It was originally realized by Teo and Kane [48] that momentum space (or
parameter space) dimensions should be treated as negative dimensions in
classifying topological defects in free fermion SPT states. The argument is
recently generalized by Jian and Xu [49] to classify interacting fermionic SPT
phases with synthetic dimensions, which is relevant to our discussion here as
the Hamiltonian in Eq. (33) precisely describes a fermionic SPT system with
physical dimension $d$ and synthetic dimension $\delta=d$. According to Ref.
49, the key criterion to distinguish the physical and synthetic dimensions
relies on the locality of fermion interactions: the interactions must be local
in the physical coordinate space and the synthetic momentum space, while non-
local in the physical momentum space and the synthetic coordinate space. Based
on this principle, the physical momentum is equivalent to the synthetic
coordinate in dimension counting; thus, the momentum space dimension should be
treated as the synthetic dimension. The main result of Ref. 49 is that the
classification of interacting fermionic SPT states in $(d,\delta)$ physical-
synthetic dimension is the same as that in $d_{\text{eff}}$-dimensional
physical space with

$d_{\text{eff}}=d-\delta.$ (36)

Applying this result to our case, we conclude that the classification of
interacting phase-space Chern insulators (or phase-space fermionic SPT states
more generally) in any spatial dimension $d$ is equivalent to the
classification of real-space interacting fermionic SPT states in
$(d-\delta)=(d-d)=0$-dimensional space (or, correspondingly, in
$(0+1)$-dimensional spacetime).

###  IV.2 Cobordism Classification Results

Using the cobordism classification [50, 51, 52, 53, 54, 55, 56, 57, 58, 59] of
interacting fermionic SPT states, we propose:

> The classification of the Fermi surface anomaly associated with the loop
> group symmetry $\mathrm{L}G$ is equivalent to the classification of
> $(0+1)$-dimensional interacting fermionic SPT phases with symmetry $G$,
> which is given by $\text{TP}_{1}(\mathrm{Spin}\ltimes G)$.

Here $G$ is the global internal symmetry group, and $\mathrm{Spin}\ltimes G$
denotes the total spacetime-internal symmetry group given by the extension
$1\to G\to\mathrm{Spin}\ltimes G\to\mathrm{Spin}\to 1$, with $\mathrm{Spin}$
being the spin group of the spacetime manifold. Although we start with the
Dirac fermion theory Eq. (3) in the $2d$-dimensional phase space, the
effective Euclidean spacetime manifold is only $(0+1)$-dimensional after the
synthetic dimension reduction, so the Euclidean spacetime rotation symmetry of
the fermionic spinor field is described by the $\mathrm{Spin}(1)$ group. In
the presence of time-reversal symmetry, the $\mathrm{Spin}$ structure can be
further extended to $\mathrm{Pin}^{\pm}$ structures [58]. The Fermi surface
$\partial\Omega$ with symmetry $G$ can have an emergent loop-$\partial\Omega$
group of $G$ symmetry denoted as $\mathrm{L}G$ in general. The notion of loop
group symmetry is more subtle when $G$ is discrete, which will be discussed
case by case later.

Table 1: Cobordism classification of the Fermi surface anomaly of the loop group symmetry $\mathrm{L}G$ by $\text{TP}_{1}(\mathrm{Spin}\ltimes G)$. In the table, $n\in\mathbb{N}$ stands for any natural number, and $\mathrm{Spin}\times_{H}G:=(\mathrm{Spin}\times G)/H$ denotes the quotient of the group product by their shared normal subgroup. $\mathbb{Z}_{2}^{F}$ denotes the Fermion parity symmetry. $\mathrm{L}G$ | $G$ | $\mathrm{Spin}\ltimes G$ | $\text{TP}_{1}$
---|---|---|---
$\mathrm{L}\mathrm{U}(1)$ | $\mathrm{U}(1)$ | $\mathrm{Spin}^{c}$ | $\mathbb{Z}$
$\mathrm{L}\mathrm{U}(n)$ | $\mathrm{U}(n)$ | $\mathrm{Spin}\times_{\mathbb{Z}_{2}^{F}}\mathrm{U}(n)$ | $\mathbb{Z}$
$\mathrm{L}\mathrm{SU}(2n)$ | $\mathrm{SU}(2n)$ | $\mathrm{Spin}\times_{\mathbb{Z}_{2}^{F}}\mathrm{SU}(2n)$ | $0$
$\tilde{\mathrm{L}}\mathrm{U}(1)\times\mathbb{Z}_{2n}$ | $\mathbb{Z}_{2n}$ | $\mathrm{Spin}\times_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{2n}$ | $\mathbb{Z}_{2n}$
$\mathrm{L}\mathrm{SU}(2n+1)$ | $\mathrm{SU}(2n+1)$ | $\mathrm{Spin}\times\mathrm{SU}(2n+1)$ | $\mathbb{Z}_{2}$
$\tilde{\mathrm{L}}\mathrm{U}(1)\times\mathbb{Z}_{2n+1}$ | $\mathbb{Z}_{2n+1}$ | $\mathrm{Spin}\times\mathbb{Z}_{2n+1}$ | $\mathbb{Z}_{4n+2}$
$\mathrm{LU}(1)\rtimes\mathbb{Z}_{2}^{T}$ | $\mathrm{U}(1)\rtimes\mathbb{Z}_{2}^{T}$ | $\mathrm{Pin}^{-}\ltimes_{\mathbb{Z}_{2}^{F}}\mathrm{U}(1)$ | $\mathbb{Z}$
$\mathrm{LU}(1)\rtimes_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4}^{TF}$ | $\mathrm{U}(1)\rtimes_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4}^{TF}$ | $\mathrm{Pin}^{+}\ltimes_{\mathbb{Z}_{2}^{F}}\mathrm{U}(1)$ | $\mathbb{Z}$
$\mathrm{LU}(1)\times\mathbb{Z}_{2}^{T}$ | $\mathrm{U}(1)\times\mathbb{Z}_{2}^{T}$ | $\mathrm{Pin}^{c}$ | 0

In $(0+1)$-dimensional spacetime, SPT phases protected by the total symmetry
$\mathrm{Spin}\ltimes G$ are classified by the cobordism group
$\text{TP}_{1}(\mathrm{Spin}\ltimes G)$ [53] and their topological invariants
are given by the cobordism group generators (i.e., the cobordism invariants).
Here TP is shorthand for the topological phase [53, 54, 55, 58]. Tab. 1
summarizes a few examples of the cobordism classification of Fermi surface
anomalies. The cobordism group element $k\in\text{TP}_{1}(\mathrm{Spin}\ltimes
G)$ is always an integer index given by

$k=\pm qN,$ (37)

where $q$ is the symmetry charge carried by the fermion, $N$ is the
multiplicity (flavor degeneracy) of the Fermi surface and the sign depends on
whether the Fermi surface is electron-like ($+$) or hole-like ($-$). If there
are multiple Fermi surfaces in the system, each one can have an independent
integer-valued cobordism index
$k_{\alpha}\in\text{TP}_{1}(\mathrm{Spin}\ltimes G)$. The total Fermi surface
anomaly is characterized by a $\mathrm{U}(1)$-valued index $\nu$,

$\nu=\sum_{\alpha}k_{\alpha}\frac{\operatorname{vol}\Omega_{\alpha}}{(2\pi)^{d}}\mod
1.$ (38)

Each cobordism index $k_{\alpha}$ is multiplied by the fraction of Fermi
volume $\operatorname{vol}\Omega_{\alpha}$ in the Brillouin zone. The Fermi
surface anomaly can vanish in the following cases:

  * •

$\operatorname{vol}\Omega_{\alpha}/(2\pi)^{d}\in\mathbb{Z}$. The Fermi volume
is an integer multiple of the Brillouin zone volume, i.e., the fermion filling
is an integer per unit cell for every fermion flavor. In this case, there is
no Fermi surface anomaly regardless of the cobordism index $k$.

  * •

$k_{\alpha}\sim 0$ (meaning $k_{\alpha}=0$ when $k\in\mathbb{Z}$ or
$k_{\alpha}=0\mod 2n$ when $k\in\mathbb{Z}_{2n}$). When the cobordism index
$k_{\alpha}$ is trivial, there is no Fermi surface anomaly, regardless of the
filling. This scenario becomes particularly noteworthy when the cobordism
group is $\mathbb{Z}_{2n}$, as in this case $k_{\alpha}=2n$ multiples of the
(unit-charged) Fermi surface can collectively cancel the anomaly and become
deformable to a symmetric product state.

  * •

Multiple Fermi surfaces of different cobordism indices $k_{\alpha}$ and Fermi
volumes $\operatorname{vol}\Omega_{\alpha}$ can cancel the anomaly
collectively, if $\nu$ adds up to an integer. Examples of such have been
recently studied in the context of Fermi surface symmetric mass generation
(SMG) [28].

More generally, the Fermi surface SMG refers to the phenomenon that the Fermi
surface anomaly vanishes $\nu\sim 0$. Still, no symmetric fermion bilinear
operator can gap out the Fermi surface into a symmetric product state. Then
the symmetric gapping of the Fermi surface can only be achieved through non-
trivial interaction effects. It generalizes the concepts of the interaction-
reduced SPT classification [71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82,
83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96] and symmetric mass
generation [97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110,
111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125] to
the case of finite fermion filling. We will explore more examples of such in
the next subsection.

###  IV.3 Examples and Comments

In the following, we will provide some physical understanding of the cobordism
classifications in several different cases. To focus our discussion on the
_discrete_ aspect of the Fermi surface anomaly (as characterized by the
integer-valued cobordism index $k$), we will restrict our scope to a unit-
charged ($q=1$) single Fermi surface of multiplicity $N$ (such that the
cobordism index is $k=N$) with a generic Fermi volume
$\operatorname{vol}\Omega$ (e.g. $\operatorname{vol}\Omega$ is some irrational
fraction of the Brillouin zone volume), such that the Fermi surface anomaly
index $\nu=k\operatorname{vol}\Omega/(2\pi)^{d}$ is only trivialized when the
cobordism index $k\sim 0$ belongs to the trivial class.

Our starting point will be the dimension-reduced $(0+1)$-dimensional effective
bulk theory of the Fermi liquid, as described by the single-mode fermion
Hamiltonian Eq. (34). The objective is to understand the interacting fermionic
SPT classification in this $(0+1)$-dimensional quantum system and make
connections to the classification of Fermi surface anomaly. After the case by
case discussions, we summarize the anomaly-free condition in Tab. 2.

####  IV.3.1 $G=\mathrm{U}(1)$ and $\mathbb{Z}$ Classification

The $G=\mathrm{U}(1)$ is the most common symmetry in the conventional
discussion of Fermi liquids, under which the fermion operator $\psi$
transforms as $\psi\to\mathrm{e}^{\mathrm{i}\phi}\psi$ for $\phi\in[0,2\pi)$.
The dimension-reduced bulk effective Hamiltonian $H=m(\psi^{\dagger}\psi-1/2)$
has only two eigenstates: $|n_{\psi}=0\rangle$ and $|n_{\psi}=1\rangle$,
labeled by the two distinct eigenvalues of the fermion number operator
$n_{\psi}:=\psi^{\dagger}\psi$. The excitation gap closes at $m=0$ as the
ground state switches from one to another, which is also the point where the
particle-hole symmetry $\mathbb{Z}_{2}^{C}$ is restored. The gap closing
signifies a “quantum phase transition” in the $(0+1)$-dimensional system.
Therefore, $m<0$ and $m>0$ should be identified as two different SPT phases.
If there are many copies of such system, each copy can undergo the SPT
transition separately, leading to $\mathbb{Z}$-classified SPT phases.

In the presence of the $\mathrm{U}(1)$ symmetry, this gap closing can not be
avoided even under interaction. Because the $\mathrm{U}(1)$ symmetry enforces
that the interaction can only take the form of a polynomial of $n_{\psi}$,
which does not change the fact that $|n_{\psi}=0\rangle$ and
$|n_{\psi}=1\rangle$ are still eigenstates of the interacting Hamiltonian.
Then the two states have to degenerate on the locus of $\langle
n_{\psi}\rangle=1/2$ where the particle-hole symmetry $\mathbb{Z}_{2}^{C}$ is
restored, resulting in the unavoidable gap closing. So the $\mathbb{Z}$
classification is robust against fermion interaction, confirming the cobordism
calculation.

As discussed previously in Sec. II.2, the Fermi surface should be defined as
the particle-hole symmetric sub-manifold in the phase space. Tuning the mass
parameter $m$ across 0 in the effective theory can be viewed as going across
the Fermi surface in the momentum space. The inevitable gap closing at $m=0$
(or at the particle-hole symmetric point) corresponds to the protected gapless
fermions on the Fermi surface. The cobordism index $k\in\mathbb{Z}$ labels the
number of gapless fermion modes (assuming fermions are unit-charged under
$\mathrm{U}(1)$) both at the SPT transition in the effective theory and on the
Fermi surface in the Fermi liquid system.

In this case, the emergent symmetry on the Fermi surface is
$\mathrm{L}_{\partial\Omega}\mathrm{U}(1):\psi_{{\bm{k}}_{F}}\to\mathrm{e}^{\mathrm{i}\phi({\bm{k}}_{F})}\psi_{{\bm{k}}_{F}}$,
which is defined for any smooth phase function
$\mathrm{e}^{\mathrm{i}\phi({\bm{k}}_{F})}$ on the Fermi surface
$\partial\Omega$. There is no further constraint on the choice of the function
$\phi({\bm{k}}_{F})$. The loop group symmetry is denoted as
$\mathrm{L}\mathrm{U}(1)$ for short in Tab. 1.

####  IV.3.2 $G=\mathrm{U}(n)$ and $\mathbb{Z}$ Classification

Apart from carrying $\mathrm{U}(1)$ charge, the fermions may also have
internal degrees of freedom. For example, electrons also carry the
$\mathrm{SU}(2)$ spin freedom, such that for electronic Fermi liquid in a
metal, the total internal symmetry is
$\mathrm{U}(1)\times_{\mathbb{Z}_{2}^{F}}\mathrm{SU}(2)=\mathrm{U}(2)$. More
generally, we may consider a $\mathrm{U}(n)$ symmetry, under which an
$n$-component fermion field $\psi$ transforms as $\psi_{a}\to U_{ab}\psi_{b}$
for $U\in\mathrm{U}(n)$. The classification of Fermi surface anomaly for
$G=\mathrm{U}(n)$ is the same as that of $G=\mathrm{U}(1)$, which is
$\mathbb{Z}$, because the protecting symmetry is only the
$\mathrm{U}(1)=\mathrm{U}(n)/\mathrm{SU}(n)$ quotient group. In this case, the
emergent symmetry on the Fermi surface is
$\mathrm{L}_{\partial\Omega}\mathrm{U}(n):\psi_{{\bm{k}}_{F}}\to
U({\bm{k}}_{F})\psi_{{\bm{k}}_{F}}$ with $U({\bm{k}}_{F})\in\mathrm{U}(n)$,
denoted as $\mathrm{LU}(n)$ in Tab. 1.

####  IV.3.3 $G=\mathrm{SU}(2n)$ and Trivial Classification

However, once the internal symmetry is reduced from $\mathrm{U}(2n)$ to
$\mathrm{SU}(2n)$, the classification collapses, and there is no Fermi surface
anomaly for any Fermi volume. From the perspective of the $(0+1)$-dimensional
effective theory, a $\mathrm{SU}(2n)$ fundamental fermion $\psi$ (which
contains $2n$ flavor components $\psi_{a}$ for $a=1,2\cdots,2n$) can always be
gapped by the following multi-fermion interaction,

$H_{\text{int}}=\prod_{a=1}^{2n}\psi_{a}+\text{h.c.}.$ (39)

This interaction always stabilizes a unique $\mathrm{SU}(2n)$ singlet ground
state. In the presence of this interaction, the $m<0$ and $m>0$ phases can be
smoothly tuned to each other without gap closing. Therefore, the
$(0+1)$-dimensional interacting fermionic SPT states have only a trivial class
under the $\mathrm{SU}(2n)$ symmetry.

The vanishing Fermi surface anomaly implies that the $\mathrm{SU}(2n)$
symmetric Fermi liquid at any filling level (of any Fermi volume) can always
be deformed into a gapped product state without breaking the $\mathrm{SU}(2n)$
and translation symmetry. For $n=1$, this gapping term is simply the $s$-wave
spin-singlet pairing. For $n>1$, the gapping will be achieved by uniform
$\mathrm{SU}(2n)$-singlet multi-fermion condensation. Such multi-fermion
condensation can happen independently on each site (or in each unit cell),
resulting in a gapped symmetric product state.

####  IV.3.4 $G=\mathbb{Z}_{2n}$ and $\mathbb{Z}_{2n}$ Classification

When reducing the symmetry from $\mathrm{U}(2n)$ to $\mathrm{SU}(2n)$, what
essentially happens is that the $\mathrm{U}(1)=\mathrm{U}(2n)/\mathrm{SU}(2n)$
quotient group is broken to its $\mathbb{Z}_{2n}$ subgroup, which is also the
$\mathbb{Z}_{2n}$ center of $\mathrm{SU}(2n)$. In fact, we only need to keep
this essential $\mathbb{Z}_{2n}$ center symmetry, under which the fermion
operator transforms as $\psi\to\mathrm{e}^{\frac{2\pi\mathrm{i}}{2n}m}\psi$
for $m=0,1,\cdots,2n$. The multi-fermion condensation interaction Eq. (39) is
still the gapping interaction to trivialize the SPT phase (or to gap out the
SPT phase transition). However, since the $\mathbb{Z}_{2n}$ group has only
1-dimensional representations, the fermionic SPT root state (the generator
state) only contains one fermion flavor. Therefore, the trivialization is
achieved at $2n$ copies of the root state so that the classification is
$\mathbb{Z}_{2n}$.

In particular, for $n=1$, a $\mathbb{Z}_{2}$ symmetric Fermi liquid allows the
opening of a pairing gap by superconductivity. In this case, the
$\mathbb{Z}_{2}$ classification of the Fermi surface anomaly indicates that
for generic Fermi volume, the deformation of the Fermi liquid to a symmetric
product state is only achievable when there are two fermion flavors (like
spin-1/2 electrons) with the cobordism index $k=2\sim 0$, which enables the
$s$-wave spin-singlet pairing. One may wonder, even when the fermion flavor
number is one (like spinless fermions in condensed matter language) with the
cobordism index $k=1$, it is still possible to fully gap the Fermi surface by
$p_{x}+\mathrm{i}p_{y}$ pairing in $(2+1)$D, although the Fermi surface
anomaly is not canceled for general Fermi volume. However, one should note
that the $p_{x}+\mathrm{i}p_{y}$ superconductor is not a trivial gaped state,
as it is not deformable to a product state due to its chiral edge mode. The
non-vanishing Fermi surface anomaly at $k=1$ enforces the non-trivial
invertible topological order in the gapped state. This is related to many
discussions about filling-enforced SPT states in the literature [126, 18,
127].

Another case worth discussing is the $n=2$ case, which is the simplest case
where Fermi surface SMG [28] can occur. In this case, the fermions have a
$\mathbb{Z}_{4}$ internal symmetry that forbids any pairing gap from opening
on the fermion bilinear level. The $\mathbb{Z}_{4}$ classification indicates
that every four copies of the Fermi surface (with generic Fermi volume) can be
deformed to a gapped product state by interaction. A simple lattice model to
demonstrate this phenomenon is described by the following Hamiltonian,

$H=\sum_{a=1}^{4}\sum_{ij}t_{ij}\psi_{ia}^{\dagger}\psi_{ja}+g\sum_{i}\psi_{i1}\psi_{i2}\psi_{i3}\psi_{i4}+\text{h.c.}.$
(40)

There are four fermion modes $\psi_{ia}$ ($a=1,2,3,4$) on each site $i$. The
$t_{ij}$ term describes a generic fermion hopping model on the lattice.
Without fine-tuning the chemical potential, the fermion system will generally
fall in the Fermi liquid phase with a generic Fermi volume. Gapping of the
Fermi surface can be achieved by the $\mathbb{Z}_{4}$-symmetric interaction
$g$, which drives four-fermion condensation on each site, leading to a gapped
symmetric product state in the $g\to\infty$ limit. This gapping mechanism
applies to lattice fermions in any spatial dimension. So the
$\mathbb{Z}_{4}$-symmetric Fermi liquid is universally $\mathbb{Z}_{4}$
classified in any dimension.

A key feature of our dimension counting argument is that the classification of
the Fermi surface anomaly does not depend on the spacetime dimension. Instead,
if we naively considered $(d+1)$D Fermi liquid as a quantum Hall insulator in
the $2d$-dimensional phase space, we might mistakenly classify the Fermi
surface anomaly by fermionic SPT states in $(2d+1)$-dimensional spacetime. The
problem may not be exposed if the symmetry is $\mathrm{U}(1)$ because the
classification is always $\mathbb{Z}$ and never gets reduced by the
interaction effect. So we would not tell any difference. However, once the
$\mathrm{U}(1)$ symmetry is broken to its $\mathbb{Z}_{4}$ subgroup, the
discrepancy will be manifest. Take $d=2$ for example, the phase space is a
4-dimensional space, and the $\mathbb{Z}_{4}$-symmetric fermionic SPT states
in $(4+1)$-dimensional spacetime is $\mathbb{Z}_{16}$ classified, which
clearly deviates from the $\mathbb{Z}_{4}$-classified Fermi surface anomaly
predicted by our theory. We know that $\mathbb{Z}_{4}$ should be the correct
answer because the lattice model Eq. (40) explicitly trivialized the Fermi
surface in multiples of four (not sixteen). This speaks for the correctness of
our dimension counting approach that the momentum space should be treated as
negative dimensions, and Fermi liquids in any dimension are topologically
equivalent to $(0+1)$-dimensional fermionic SPT states (with boundaries).

Finally, we would like to comment on the emergent loop group symmetry on the
Fermi surface when the $\mathrm{U}(1)$ symmetry is broken to
$\mathbb{Z}_{2n}$. With the multi-fermion condensation term $g$, the low-
energy theory takes the form of

$\begin{split}&H=\sum_{{\bm{k}}_{F}\in\partial\Omega}\epsilon_{{\bm{k}}_{F}}\psi_{{\bm{k}}_{F}}^{\dagger}\psi_{{\bm{k}}_{F}}+\cdots\\\
&+g\sum_{\\{{\bm{k}}_{F}^{(a)}\\}\in\partial\Omega}\delta_{\sum_{a=1}^{2n}{\bm{k}}_{F}^{(a)}}\prod_{a=1}^{2n}\psi_{{\bm{k}}_{F}^{(a)}}+\text{h.c.},\end{split}$
(41)

which is symmetric under

$\psi_{{\bm{k}}_{F}}\to\mathrm{e}^{\mathrm{i}\frac{2\pi
p}{2n}}\mathrm{e}^{\mathrm{i}\phi({\bm{k}}_{F})}\psi_{{\bm{k}}_{F}},$ (42)

with $p=0,1,\cdots,2n$ labeling a $\mathbb{Z}_{2n}$ group element and
$\phi({\bm{k}}_{F})\sim\phi({\bm{k}}_{F})+2\pi$ being a smooth function of
${\bm{k}}_{F}$ subject to the following constraint:

$\forall\sum_{a=1}^{2n}{\bm{k}}_{F}^{(a)}=0:\sum_{a=1}^{2n}\phi({\bm{k}}_{F}^{(a)})=0\mod
2\pi.$ (43)

All the $\mathrm{U}(1)$ functions $\mathrm{e}^{\mathrm{i}\phi({\bm{k}}_{F})}$
satisfying the constraint in Eq. (43) form a group under pointwise
multiplication. We denoted this constrained loop group as
$\tilde{\mathrm{L}}_{\partial\Omega}\mathrm{U}(1)$. Then the emergent symmetry
on the Fermi surface is
$\tilde{\mathrm{L}}_{\partial\Omega}\mathrm{U}(1)\times\mathbb{Z}_{2n}$, or
short-handed as $\tilde{\mathrm{L}}\mathrm{U}(1)\times\mathbb{Z}_{2n}$ in Tab.
1.

####  IV.3.5 $G=\mathrm{SU}(2n+1)$ and $\mathbb{Z}_{2}$ Classification

We have discussed the case of $\mathrm{SU}(2n)$ flavor symmetry with an even
number of fermion flavors. Now we turn to the case when the fermion flavor
number is odd and the flavor symmetry is $\mathrm{SU}(2n+1)$. The major
difference here is that the $\mathrm{SU}(2n+1)$ flavor symmetry group no
longer contains the $\mathbb{Z}_{2}^{F}$ fermion parity symmetry as a
subgroup. In this case, the Fermi surface anomaly is $\mathbb{Z}_{2}$
classified. The physical argument is that with a single copy of the
$\mathrm{SU}(2n+1)$ fundamental fermion $\psi$ (with contains $2n+1$ flavor
components $\psi_{a}$ for $a=1,2,\cdots,2n+1$), it is no longer possible to
write down the $\mathrm{SU}(2n+1)$-singlet multi-fermion gapping term of the
form $\prod_{a=1}^{2n+1}\psi_{a}+\text{h.c.}$ in the $(0+1)$-dimensional
effective theory, because such a term contains an odd number of fermion
operators and does not respect the $\mathbb{Z}_{2}^{F}$ fermion parity
symmetry. Therefore, one has to double the system and introduce two
$\mathrm{SU}(2n+1)$ fundamental fermions $\psi_{1}$ and $\psi_{2}$, such that
the following gapping interaction becomes possible

$H_{\text{int}}=\prod_{a=1}^{2n}\psi_{1a}\psi_{2a}+\text{h.c.}.$ (44)

Similar multi-fermion interaction is applicable to gap out the Fermi surface
at a generic Fermi volume if there are two copies of $\mathrm{SU}(2n+1)$
fundamental fermions on the Fermi surface, which explains the $\mathbb{Z}_{2}$
classification. This is also an example of the Fermi surface SMG.

####  IV.3.6 $G=\mathbb{Z}_{2n+1}$ and $\mathbb{Z}_{4n+2}$ Classification

If the $\mathrm{SU}(2n+1)$ flavor symmetry is broken to its center
$\mathbb{Z}_{2n+1}$ symmetry group, under which the fermion operator
transforms as $\psi\to\mathrm{e}^{\frac{2\pi\mathrm{i}}{2n+1}m}\psi$ for
$m=0,1,\cdots,2n+1$, the Fermi surface anomaly classification will be
$\mathbb{Z}_{4n+2}$. The physics is essentially the same as the
$G=\mathrm{SU}(2n+1)$ case, which relies on the same multi-fermion interaction
Eq. (44) to drive the SMG in the $(0+1)$-dimensional effective theory. Similar
interaction also drives Fermi surface SMG. The SMG gapping mechanism only
works when the fermion flavor number is $4n+2$, which is consistent with the
$\mathbb{Z}_{4n+2}$ classification.

####  IV.3.7 $G=\mathrm{U}(1)\rtimes\mathbb{Z}_{2}^{T}$ and $\mathbb{Z}$
Classification

We can extend our discussion to anti-unitary symmetries [128, 129], which will
be generally denoted as time-reversal symmetries $\mathbb{Z}_{2}^{T}$. There
are different ways that an anti-unitary symmetry can be combined with the
$\mathrm{U}(1)$ charge conservation symmetry of the fermion. Let us first
consider the case of $G=\mathrm{U}(1)\rtimes\mathbb{Z}_{2}^{T}$, where the
$\mathrm{U}(1)$ rotation does not commute with the anti-unitary symmetry
action $\mathcal{T}\in\mathbb{Z}_{2}^{T}$ and $\mathcal{T}^{2}=+1$. More
specifically, the fermion operator transforms as

$\begin{split}\mathrm{U}(1)&:\psi\to\mathrm{e}^{\mathrm{i}\phi}\psi,\\\
\mathbb{Z}_{2}^{T}&:\psi\to\mathcal{K}\psi,\psi^{\dagger}\to\mathcal{K}\psi^{\dagger},\end{split}$
(45)

where $\mathcal{K}\mathrm{i}\mathcal{K}^{-1}=-\mathrm{i}$ denotes the complex
conjugation operator that acts on all complex coefficients in the operator
algebra.

In this scenario, the presence of the anti-unitary symmetry does not alter the
anomaly classification. The $(0+1)$-dimensional effective theory,
characterized by the Hamiltonian $H=m(\psi^{\dagger}\psi-1/2)$, still includes
the mass term $m$ which is symmetric under $\mathbb{Z}_{2}^{T}$. As the anti-
unitary symmetry does not impose additional restrictions on the Hamiltonian,
the SPT classification remains unchanged from the case with $G=\mathrm{U}(1)$,
which is $\mathbb{Z}$. As a result, the Fermi surface anomaly is still
classified as $\mathbb{Z}$.

####  IV.3.8 $G=\mathrm{U}(1)\rtimes_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4}^{TF}$
and $\mathbb{Z}$ Classification

Another way to combine the anti-unitary symmetry with $\mathrm{U}(1)$ is to
consider $G=\mathrm{U}(1)\rtimes_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4}^{TF}$,
meaning that the $\mathrm{U}(1)$ rotation does not commute with the generator
$\mathcal{T}\in\mathbb{Z}_{4}^{TF}$ of the anti-unitary symmetry, but
$\mathcal{T}^{2}=-1$ (or more precisely, $\mathcal{T}$ squares to the fermion
parity operator, hence the anti-unitary symmetry is four-fold and sharing the
$\mathbb{Z}_{2}^{F}$ subgroup with $\mathrm{U}(1)$). This is actually the
standard time-reversal symmetry of electrons that enforces a Kramers doublet
[130]. The fermion operator
$\psi=(\psi_{\uparrow},\psi_{\downarrow})^{\intercal}$ is a doublet, which
transforms under the symmetry as

$\begin{split}\mathrm{U}(1)&:\psi\to\mathrm{e}^{\mathrm{i}\phi}\psi,\\\
\mathbb{Z}_{4}^{TF}&:\psi_{\uparrow}\to\mathcal{K}\psi_{\downarrow},\quad\psi_{\downarrow}\to\mathcal{K}\psi_{\uparrow}.\end{split}$
(46)

The time-reversal symmetry is denoted as a $\mathbb{Z}_{4}$ group because its
two-fold action is non-trivial and corresponds to the fermion parity operation
($\psi\to-\psi$) that falls in the $\mathbb{Z}_{2}^{F}$ subgroup of
$\mathrm{U}(1)$.

The mass term $H=m(\psi^{\dagger}\psi-1/2)$ is still allowed in the effective
Hamiltonian under the $\mathbb{Z}_{4}^{TF}$ symmetry. As the anti-unitary
symmetry does not introduce new restrictions, the SPT classification remains
the same as the $G=\mathrm{U}(1)$ case, which is $\mathbb{Z}$. Therefore, the
Fermi surface anomaly is also $\mathbb{Z}$ classified in this case.

####  IV.3.9 $G=\mathrm{U}(1)\times\mathbb{Z}_{2}^{T}$ and Trivial
Classification

We further consider $G=\mathrm{U}(1)\times\mathbb{Z}_{2}^{T}$ where the
$\mathbb{Z}_{2}^{T}$ anti-unitary symmetry operation commutes with the
$\mathrm{U}(1)$ symmetry operation. The symmetry action can be realized on the
fermion operator as

$\begin{split}\mathrm{U}(1)&:\psi\to\mathrm{e}^{\mathrm{i}\phi}\psi,\\\
\mathbb{Z}_{2}^{T}&:\psi\to\mathcal{K}\psi^{\dagger},\psi^{\dagger}\to\mathcal{K}\psi.\end{split}$
(47)

The anti-unitary symmetry $\mathbb{Z}_{2}^{T}$ here should be interpreted as a
particle-hole symmetry, which maps $\psi$ and $\psi^{\dagger}$ to each other.

In the presence of this symmetry, the original mass term
$H=m(\psi^{\dagger}\psi-1/2)$ is forbidden in the effective Hamiltonian. A
symmetry-allowed mass term can only be realized in the doubled system, where
the fermion operator $\psi=(\psi_{+},\psi_{-})^{\intercal}$ must contain two
components, and the symmetry $G=\mathrm{U}(1)\times\mathbb{Z}_{2}^{T}$ acts as

$\begin{split}\mathrm{U}(1)&:\psi_{\pm}\to\mathrm{e}^{\mathrm{i}\phi}\psi_{\pm},\\\
\mathbb{Z}_{2}^{T}&:\psi_{\pm}\to\mathcal{K}\psi_{\mp}^{\dagger},\psi_{\pm}^{\dagger}\to\mathcal{K}\psi_{\mp},\end{split}$
(48)

such that two anti-commuting mass terms are allowed

$H=m(\psi_{+}^{\dagger}\psi_{+}-\psi_{-}^{\dagger}\psi_{-})+m^{\prime}(\mathrm{i}\psi_{-}^{\dagger}\psi_{+}+\text{h.c.}).$
(49)

It is possible to tune smoothly from $m<0$ to $m>0$ without closing the
excitation gap of this $(0+1)$-dimensional system in the presence of
$m^{\prime}\neq 0$. Therefore, all gapped state belongs to the same SPT phase
and the SPT classification is trivial.

Mapping to the Fermi surface, imposing the particle-hole symmetry enforces the
Fermi surface to be perfectly nested [131]. Tuning $m$ from the inside ($m<0$)
to the outside ($m>0$) of the Fermi surface, two bands cross at the Fermi
level. In this case, a band hybridization term (similar to $m^{\prime}$) is
sufficient to gap out the Fermi surface fully without symmetry breaking (note
that the nesting momentum is already zero in this case). Therefore, the system
is free of Fermi surface anomaly, consistent with the trivial classification.

##  V Summary

In this work, we propose an approach to classify the Fermi surface anomaly by
leveraging the correspondence between the Fermi liquid and the Chern insulator
in the phase space. Specifically, we suggest using the classification of
interacting fermionic symmetry-protected topological (SPT) states in the phase
space to determine the Fermi surface anomaly. The non-commutative geometry of
the phase space implies that the phase-space SPT states follow unusual
dimension counting, where the momentum space dimensions are treated as
negative dimensions. As a result, the effective spacetime dimension for the
classification problem is reduced to $(0+1)$D. To support our argument, we
analyze a phase-space Dirac fermion field theory of fermionic SPT states and
apply the dimension reduction technique after resolving the non-commutative
geometry. Our proposed approach offers a comprehensive and rigorous way to
classify the Fermi surface anomaly, providing valuable insights into the
universal low-energy features of electrons in metals.

Table 2: The summary of the Fermi surface anomaly-free condition. The system with a certain number of copies can be symmetrically gapped. The anomaly is free if and only if $\nu=k\frac{\operatorname{vol}\Omega}{(2\pi)^{d}}=0\mod 1$. The integer-valued index $k$ is classified by cobordism in Tab. 1. The case that the normalized Fermi volume $\frac{\operatorname{vol}\Omega}{(2\pi)^{d}}$ is irrational is discussed in Sec. IV.3 and summarized in the fourth column. The condition for the normalized Fermi volume being rational number $p/q$ with $p,q\in\mathbb{Z}$ is summarized in the third column.  $\mathrm{L}G$ | integer | Number of copies to trivialize
---|---|---
index $k$ | $\frac{\operatorname{vol}\Omega}{(2\pi)^{d}}=p/q$ |  $\frac{\operatorname{vol}\Omega}{(2\pi)^{d}}$ is irrational
$\mathrm{L}\mathrm{U}(1)$ | $\mathbb{Z}$ | $q$ | Never
$\mathrm{L}\mathrm{U}(n)$ | $\mathbb{Z}$ | $q$ | Never
$\mathrm{L}\mathrm{SU}(2n)$ | $0$ | $1$ | 1
$\tilde{\mathrm{L}}\mathrm{U}(1)\times\mathbb{Z}_{2n}$ | $\mathbb{Z}_{2n}$ | $\gcd(q,2n)$ | $2n$
$\mathrm{L}\mathrm{SU}(2n+1)$ | $\mathbb{Z}_{2}$ | $\gcd(q,2)$ | 2
$\tilde{\mathrm{L}}\mathrm{U}(1)\times\mathbb{Z}_{2n+1}$ | $\mathbb{Z}_{4n+2}$ | $\gcd(q,4n+2)$ | $4n+2$
$\mathrm{LU}(1)\rtimes\mathbb{Z}_{2}^{T}$ | $\mathbb{Z}$ | $q$ | Never
$\mathrm{LU}(1)\rtimes_{\mathbb{Z}_{2}^{F}}\mathbb{Z}_{4}^{TF}$ | $\mathbb{Z}$ | $q$ | Never
$\mathrm{LU}(1)\times\mathbb{Z}_{2}^{T}$ | 0 | $1$ | $1$

To summarize, the Fermi surface anomaly can be defined by the projective
representation of the internal symmetry $G$ on the interstitial defect in the
fermion system. It is characterized by a $\mathrm{U}(1)$-valued anomaly index

$\nu=\sum_{\alpha}k_{\alpha}\frac{\operatorname{vol}\Omega_{\alpha}}{(2\pi)^{d}}\mod
1,$ (50)

which is a sum of contributions from each Fermi surface labeled by $\alpha$.
Each term in the summation contains an integer-valued index $k_{\alpha}$
multiplied with a real-valued fraction
$\operatorname{vol}\Omega_{\alpha}/(2\pi)^{d}$. The ratio
$\operatorname{vol}\Omega_{\alpha}/(2\pi)^{d}$ describes the fraction of Fermi
volume $\operatorname{vol}\Omega_{\alpha}$ in the Brillouin zone. The integer
$k_{\alpha}=\pm q_{\alpha}N_{\alpha}$ is given by the fermion charge
$q_{\alpha}$ and multiplicity (flavor degeneracy) $N_{\alpha}$ of the Fermi
surface and classified by the cobordism group
$\text{TP}_{1}(\mathrm{Spin}\ltimes G)$. Assuming a generic Fermi volume for
each Fermi surface (i.e. $\operatorname{vol}\Omega_{\alpha}/(2\pi)^{d}$ is not
a rational number), the Fermi surface anomaly is determined by the cobordism
index $k_{\alpha}\in\text{TP}_{1}(\mathrm{Spin}\ltimes G)$. The classification
result for a list of internal symmetries $G$ is shown in Tab. 1.

The complete gapping of the Fermi surface into a product state is feasible if
and only if the Fermi surface anomaly vanishes, i.e. $\nu\sim 0$. This can
occur through the opening of a superconducting gap (when $G=\mathbb{Z}_{2}$)
or a perfect-nested band hybridization gap (when
$G=\mathrm{U}(1)\times\mathbb{Z}_{2}^{T}$) at the free fermion level, when the
fermion flavor number falls in the trivial cobordism class. Nevertheless,
unconventional mechanisms exist for gapping, referred to as the Fermi surface
symmetric mass generation (SMG) [28], that can solely be realized via
interaction effects when the Fermi surface anomaly vanishes but no fermion
bilinear gapping term is allowed due to symmetry constraints. One informative
example of such is the quartet (charge-4e) fermion condensation [132, 133,
134, 135, 136, 137, 138, 139, 140, 141], on Fermi surfaces with internal
$G=\mathbb{Z}_{4}$ symmetry, where the Fermi surface anomaly is
$\mathbb{Z}_{4}$ classified. In this scenario, every four multiples of Fermi
surfaces can be collectively gapped via four-fermion interactions. The fact
that this gapping mechanism is feasible in all dimensions aligns with our
assertion that the Fermi surface anomaly is universally categorized by
$(0+1)$-dimensional fermionic SPT phases. More cases of Fermi surface
trivialization are summarized in Tab. 2.

It is worth mentioning that we have only focused on the codimension-1 Fermi
surface in this work. However, the synthetic dimension reduction argument in
Eq. (36) applies to more general codimension-$p$ Fermi surfaces. Assuming the
spatial dimension is $d$, a codimension-$p$ Fermi surface will be a
$(d-p)$-dimensional closed manifold in the momentum space, which is the
boundary of a $\delta=(d-p+1)$-dimensional Fermi sea. As the momentum space
(Fermi sea) dimension $\delta$ should be considered as negative dimension, the
effective spatial dimension $d_{\text{eff}}$ for SPT classification is
$d_{\text{eff}}=d-\delta=p-1$, and the corresponding effective spacetime
dimension is $d_{\text{eff}}+1=p$. Therefore, we propose:

> The codimension-$p$ Fermi surface anomaly with the loop group symmetry $LG$
> is classified by $G$-symmetric interacting fermionic SPT phases in
> $p$-dimensional spacetime, which is given by
> $\mathrm{TP}_{p}(\mathrm{Spin}\ltimes G)$.

For example, consider $\mathrm{LU}(1)$ symmetric generalized Fermi surfaces in
$d=3$ dimension, classifications of Fermi surface anomalies are summaried in
Tab. 3. The results are consistent with the understanding that Fermi rings are
topologically unprotected with $\mathrm{U}(1)$ symmetry only, but Weyl points
and Fermi surfaces are topologically stable.

Table 3: Classification of codimension-$p$ Fermi surface anomaly with $\mathrm{LU}(1)$ symmetry in $(3+1)$D spacetime. | Fermi surface | Fermi sea |  |
---|---|---|---|---
| codim $p$ | dim | dim | $d_{\text{eff}}+1$ | $\mathrm{TP}_{p}$
Weyl points | $3$ | $0$ | $1$ | $2+1$ | $\mathbb{Z}\times\mathbb{Z}$
Fermi rings | $2$ | $1$ | $2$ | $1+1$ | $0$
Fermi surfaces | $1$ | $2$ | $3$ | $0+1$ | $\mathbb{Z}$

The classification of Fermi surface anomalies can help us understand the
possible ways a Fermi surface can be gapped and the role of interactions in
this process. The recent proposal of the ancilla qubit approach [63, 64] for
pseudo-gap physics draws a connection between the pseudo-gap metal to Fermi
liquid transition with the Fermi surface SMG transition in the ancilla layers,
as both transitions are described by field theories of fermionic deconfined
quantum critical points [109, 113, 142, 143, 144]. The Fermi surface anomaly
constrains the dynamical behavior of such field theories and can potentially
shed light on the open problem of pseudo-gap transition in correlated
materials.

###### Acknowledgements.

We acknowledge the discussions with Xiao-Liang Qi, Cenke Xu, Chao-Ming Jian,
Chong Wang, Meng Cheng, Nathan Seiberg, Dominic Else, Ryan Thorngren, Zhen Bi,
Umang Mehta, Ashvin Vishwanath, Charles Kane, Ya-Hui Zhang, Subir Sachdev,
John McGreevy. DCL and YZY are supported by the National Science Foundation
(NSF) Grant DMR-2238360 “Theoretical and Numerical Investigation of Symmetric
Mass Generation”. 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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