# Effective field theories of topological crystalline insulators and
topological crystals

Sheng-Jie Huang  Condensed Matter Theory Center, Department of Physics,
University of Maryland, College Park, Maryland 20742-4111, USA  Joint Quantum
Institute, Department of Physics, University of Maryland, College Park,
Maryland 20742-4111, USA  Chang-Tse Hsieh  Quantum Matter Theory Research
Team, RIKEN CEMS, Wako, Saitama 351-0198, Japan  Jiabin Yu  Condensed Matter
Theory Center, Department of Physics, University of Maryland, College Park,
Maryland 20742-4111, USA

###### Abstract

We present a general approach to obtain effective field theories for
topological crystalline insulators whose low-energy theories are described by
massive Dirac fermions. We show that these phases are characterized by the
responses to spatially dependent mass parameters with interfaces. These mass
interfaces implement the dimensional reduction procedure such that the state
of interest is smoothly deformed into a _topological crystal_ , which serves
as a representative state of a phase in the general classification. Effective
field theories are obtained by integrating out the massive Dirac fermions, and
various quantized topological terms are uncovered. Our approach can be
generalized to other crystalline symmetry protected topological phases and
provides a general strategy to derive effective field theories for such
crystalline topological phases.

##  I Introduction

Topological phases of matter are gapped phases which are characterized by the
patterns of quantum entanglement in the ground stateWen (2019). A ground state
with a short-range quantum entanglement is considered trivial since the state
can be smoothly deformed into a product state of microscopic degrees of
freedom without closing the energy gap. Such deformation might not be possible
in the presence of symmetry, allowing us to define the so-called symmetry
protected topological (SPT) phases Schnyder  _et al._ (2008); Kitaev (2009);
Ryu  _et al._ (2010); Gu and Wen (2009); Pollmann  _et al._ (2010); Fidkowski
and Kitaev (2011); Turner  _et al._ (2011); Chen  _et al._ (2011a, b); Schuch
_et al._ (2011); Chen  _et al._ (2013); Levin and Gu (2012); Kapustin (2014);
Else and Nayak (2014). Explicitly, a SPT phase is a gapped phase of matter
with a unique ground state that cannot be adiabatically deformed to a trivial
state as long as the symmetry is preserved; the adiabatic trivialization is
possible only if the symmetry is allowed to be broken.

For internal symmetries, general understanding of SPT phases has been largely
achieved Freed (2014); Freed and Hopkins (2019a); Xiong (2018); Gaiotto and
Johnson-Freyd (2019). Besides the development of SPT phases with internal
symmetries, there has been a great interest on topological phases with
crystalline point group and space group symmetries Teo and Hughes (2013);
Shiozaki and Sato (2014); Isobe and Fu (2015); Shiozaki  _et al._ (2017); Po
_et al._ (2017); Bradlyn  _et al._ (2017); Kruthoff  _et al._ (2017). Typical
examples include so-called topological crystalline insulators (TCIs), which
are electronic insulators protected, in part, by point group or space group
symmetries Fu (2011); Ando and Fu (2015); Chiu  _et al._ (2016). As a
consequence of crystalline symmetry, higher-order topology may occur in
crystalline SPT (cSPT) phases, where anomalous gapless modes could show up on
the $(d-n)$-dimensional boundary of a $d$-dimensional bulk (with $n>1$)
Schindler  _et al._ (2018); Neupert and Schindler (2018).

There have been two general frameworks which are conjectured to give a general
classification of cSPT phases. The first are dubbed the “topological crystal”
approach Song  _et al._ (2017); Huang  _et al._ (2017); Rasmussen and Lu
(2018); Cheng and Wang (2018); Shiozaki  _et al._ (2018); Xiong and
Alexandradinata (2018); Song  _et al._ (2018, 2019, 2020a); Okuma  _et al._
(2019); Song  _et al._ (2020b); Rasmussen and Lu (2020); Huang and Hsu (2021);
Geier  _et al._ (2021), in which the key idea is to deform a generic cSPT
state into a real-space stacking of “building blocks”, which are lower-
dimensional SPT states with effective internal symmetry. These special kinds
of states are referred to as topological crystals. The classification of cSPT
phases is then given by the deformation classes of the topological crystals.
This approach gives a simple physical picture for understanding cSPT phases.
Interesting physical signatures on the boundaries Huang and Hermele (2018);
Huang (2020) or crystalline defectsGeier  _et al._ (2021) are usually easy to
obtain in this framework.

The second framework is based on the “smooth states”Thorngren and Else (2018).
In this framework, a hypothetical lattice of ancillas (hypothetical degrees of
freedom) is introduced, and the lattice of ancillas has a much smaller lattice
constant than the physical lattice. A smooth state is very smooth in the
lattice of ancillas. However, the radius of the spatial variation of a smooth
state is on the order of the unit cell size of the physical lattice. An
important consequence in this framework is that the classification of cSPT
phases with a crystalline symmetry group G is the same as the classification
of SPT phases with internal symmetry group G, which is known as the
“Crystalline Equivalence Principle” (See also Ref. Jiang and Ran, 2017; Freed
and Hopkins, 2019b; Debray, 2021). The topological crystal and the smooth
state approach are actually equivalent as shown in Ref. Else and Thorngren,
2019.

In principle, SPT phases should be characterized by their response to
background gauge fields. The concept of crystalline gauge fields has been
mathematically defined Thorngren and Else (2018); but the result is rather
formal, which limits the practical usage. Further attempts have been made in
order to turn it into a concrete theoretical tool either in the field theory
frameworkSong  _et al._ (2021); Gioia  _et al._ (2021); Manjunath and
Barkeshli (2021) or on the latticeManjunath and Barkeshli (2021). There has
been an alternative proposal which characterizes cSPT phases through their
response to elastic deformationsNissinen (2020); Else  _et al._ (2021). This
approach is very physical and is well-defined in a continuous field theory.
However, it has only been worked out in some simple examples, and the relation
to the general classification can only be obtained formally. It’s also unclear
how to derive such effective field theories microscopically.

In this paper, we propose a continuous field theoretical description of the
cSPT phases. Focusing on TCIs described by massive Dirac theories, the main
idea is to characterize these phases by the responses to spatially dependent
mass parameters (and to the background gauge field of internal symmetry).
These spatially dependent mass terms implement the dimensional reduction
procedure such that the states trapped at the mass interfaces are precisely
the building blocks in the corresponding topological crystal picture. The
effective field theory is then obtained by integrating out the gapped
fermions. Our approach not only provides well-defined effective field theories
for the cSPT phases, but also gives an explicit connection between the
topological crystal picture and the effective field theories.

The rest of this paper is organized as follows: in Sec. II, we give a review
of the topological crystal approach and summarize our main results. Cellular
cohomology are used throughout this paper, which we review in Appendix A.

In Sec. III, we discuss the effective field theory for 1d atomic insulators
with charge conservation and the lattice translation symmetry as a simple
example to illustrate our approach. Generalizations to 2d and 3d atomic
insulators are also discussed and more details are given in Appendix D. We
point out that the responses described by these topological terms are
generalized Thouless pumps.

In Sec. IV, we apply our approach to various TCIs with point group symmetry
and derive the effective field theories. Sec. IV.1 consider 1d TCIs with
reflection symmetry. Sec. IV.2 generalize the discussion to 2d TCIs with
$C_{N}$ rotational symmetry. Since the form of the topological term for this
case might not be familiar to the readers, we provide a perturbative
derivation in Appendix E. Sec. IV.3 is devoted to 3d 2nd order TCIs protected
by $C_{nv}$ symmetry, where there are gapless hinge modes for appropriate
boundary conditions. The field theories we obtained are essentially the same
as the axion field theory Qi  _et al._ (2008), where the non-trivial
information of the classification is encoded in the theta angle. We also
briefly discuss the physical responses as the a result of the topological
terms in the effective field theories.

Finally, we conclude in Sec. V with a discussion of our results and possible
directions for future work. Other mathematical details are given in Appendix
B, including general discussions on the fundamental domain for crystalline
symmetry, and the construction of the map $f:M\rightarrow BG_{s}$, where $M$
is the real space manifold and the $BG_{s}$ is the classifying space for the
space group $G_{s}$. Appendix C gives a construction of the classifying space
$BG_{s}$ for a space group.

##  II General perspective and summary of the results

###  II.1 Review of the topological crystal approach

Topological crystal approach is a general framework of describing and
classifying cSPT phases. The main idea is that any cSPT phase is adiabatically
connected to a stacking of $d_{b}$-dimensional topological states with
effective internal symmetry arranged in some crystalline pattern in
$d$-dimensional space, where $d_{b}$ ranges from $0$ to $d$. This procedure is
called dimensional reduction and these special kinds of states are referred to
as the topological crystals in Ref. Song  _et al._ , 2019. In order for the
argument to go through, an important assumption is that the correlation length
$\xi$ can be tuned to be arbitrarily small, and, in the presence of
translation symmetry, much smaller than the size of the unit cell, which
requires adding a fine mesh of trivial degrees of freedom.

A systematic way to describe a topological crystal is as follows. We define a
fundamental domain (FD) to be a smallest simply connected closed part of
space, subject to the condition that no two points in the region are related
by a crystalline symmetry. The FD is then copied throughout space using the
crystalline symmetry such that the whole of space is filled. A more formal
definition of the FD is given in Appendix B. This construction gives a
$d$-dimensional space a cell complex structure, where the $d$-cells are copies
of FD. The $(d-1)$-cells lie on faces where two $d$-cells meet, with the
property that no two points in the same $(d-1)$-cell are related by symmetry.
This procedure continues to $0$-cells. As shown in Appendix B, an important
property of this cell complex is that there exists a map $f:M\rightarrow
BG_{s}$, where $BG_{s}$ is the classifying space of the space group $G_{s}$. A
construction of the classifying space $BG_{s}$ is given in Appendix C. As a
result, each $d$-cell can be labeled by a group element in the space group,
and each path connecting a point $\bm{r}$ in a FD to $g\bm{r}$ is also labeled
by an group element $g\in G_{s}$.

With this cell-complex structure, one can understand a topological crystal
state by associating $d_{b}$-dimensional topological phase with each
$d_{b}$-cell. These $d_{b}$-dimensional states are referred to as the
“building blocks” of the topological crystal. When the building blocks
intersect in the bulk, the building blocks must be glued together so as to
eliminate any gapless modes at the junctions while preserving symmetry. An
ordinary crystal is a simple examples of a topological crystal state, which is
formed by periodically arranged atoms as $d_{b}=0$ building blocks.

###  II.2 Characterizing topological crystalline phases by responses to mass
parameters

In this section, we summarize general aspects of our approach on deriving the
effective field theories for cSPT phases. The main idea of this work is to
characterize cSPT phases by its response to spatially dependent mass
parameters. These spatially dependent mass terms have interfaces that
implement the dimensional reduction procedure such that the states trapped at
the mass interfaces are precisely the building blocks in the corresponding
topological crystal picture. The effective field theory is then obtained by
integrating out the gapped fermions. This provides a way to connect the
topological crystal picture to the effective field theories.

One of the simplest example is given in a $1$-dimensional crystal with a
lattice translation symmetry and a $U(1)$ charge conservation symmetry. There
is an integer topological invariant $\nu$ representing the charge per unit
cell. We will show that the topological response of such a system to a
spatially dependent mass parameter background and to a background $U(1)$ gauge
field $A_{\mu}$ is characterized by a quantized topological term

$\frac{\nu}{2\pi}\int\epsilon^{\mu\nu}A_{\mu}\partial_{\nu}\phi(x)d^{2}x,$ (1)

where $\phi(x)$ is a phase variable that parametrizes the winding of the mass
interface. The charge-$\nu$ bound state trapped at the mass interface with a
non-trivial $2\pi\nu$ winding is precisely the $d_{b}=0$ building block in the
topological crystal picture of this phase. Detailed arguments and the
derivation are given in Sec. III.

In order for this approach to make sense, it’s important to consider the
“topological limit”, where one tunes the correlation length $\xi$ to be
arbitrarily small, especially $\xi\ll a$ (where $a$ is the size of a unit
cell) in the presence of lattice translation symmetry. One can imagine that
the system is defined on a much finer lattice with lattice spacing $l\ll a$.
It’s still possible to describe a system in the topological limit by a
continuous field theory—One consider fields that are coarse-grained with
respect to the length scale $l$ so that the fields are smooth on the scale $l$
and could vary on the scale $R$ with $R\gg l$ and $R<a$. Working in the
topological limit, the translation symmetry can not be viewed as an effective
internal symmetry of the field theory—This property emerges only when one goes
to the true “IR limit”, where the fields are smooth even on the scale $a$. In
the topological limit, it thus makes sense to consider a mass interface with a
characteristic length scale $w<a$ so that the dimensional reduction procedure
goes through. In this paper, we will first derive the topological terms in the
topological limit. Once a topological term is obtained, we are free to deform
the field configurations of the mass parameters to be smooth on the scale $a$
while staying in the same classification class. The resulting theory will be
validated in the IR limit. This is the general perspective that we take in
this work.

###  II.3 Summary of the results

Ideally we would like to apply this method to obtain the quantized topological
terms for any cSPT phase. However, doing this in full generality is still a
difficult task. In this work, we instead illustrate our approach in physically
relavant systems. We consider a wide range of TCIs in one, two, and three
dimensions with a $U(1)$ charge conservation and a $G_{c}$ crystalline
symmetry. For simplicity, we consider TCIs that can be built from building
blocks with only charge $U(1)$ response, i.e., the building blocks are
$d_{b}$-dimensional topological phases with $U(1)$ symmetry, which transform
trivially under $G_{c}$. These building blocks are characterized by the Chern-
Simons term

$\mathcal{L}_{\text{CS}}^{2s+1}[A]=\frac{1}{(s+1)!}A\wedge(\frac{dA}{2\pi})^{s},$
(2)

where $A$ is the $U(1)$ gauge field. For these examples, we find the quantized
topological terms take the following general form

$S=\int\mathcal{L}_{\text{CS}}^{d-k+1}[A]\wedge\Omega_{k},$ (3)

where $d$ is spatial dimensions and $\Omega_{k}$ is a $k$-form that
corresponds to the codimension-k mass interfaces at which the building blocks
are decorated. We will show that the mass interfaces that implement the
dimensional reduction procedure are classified by $H^{k}(BG_{c},\mathbb{Z})$
with a twisting coefficient when $G_{c}$ contains orientation reversing
elements. The $k$-form $\Omega_{k}$ in the topological term Eq. (3) is given
by $\Omega_{k}=f^{*}\alpha$, where $f$ is the map $f:M\rightarrow BG_{c}$
given by the FDs in crystallography and $\alpha\in H^{k}(BG_{c},\mathbb{Z})$.
In general, the $k$-form $\Omega_{k}$ is determined by a set of integral
relations, which take the following form

$\int_{C_{g}}\Omega_{k}=N_{g},$ (4)

where $C_{g}$ is a $k$-cycle labeled by $g\in G_{c}$, and $N_{g}$ is given by

$N_{g}=\int_{C_{g}}f^{*}\alpha$ (5)

with $\alpha\in H^{k}(BG_{c},\mathbb{Z})$. Here $N_{g}$ is in general an
integer or a $\mathbb{Z}_{n}$ number depending on the element $\alpha$ in the
cohomology group $H^{k}(BG_{c},\mathbb{Z})$. Representative of $\Omega_{k}$
can be obtained by solving these equations. Table 1 summarizes the examples
and the results in this work. From the topological term Eq. (3), one can
obtain electromagnetic responses of TCIs, which will be briefly discussed in
the sections of each case.

Spacetime | Symmetry group $G$ |  $\Omega_{k}$ in Eq. (3) | Integral conditions | Section
---|---|---|---|---
dimensions |  |  |  |
$1+1$D | $U(1)\times\Gamma$ | $\Omega_{1}=E$ | $\int_{C_{t_{1}}}E=\int_{C_{t_{1}}}f^{*}\tau\in\mathbb{Z},\tau\in H^{1}(B\Gamma,\mathbb{Z})$ | Sec. III
$1+1$D | $U(1)\times D_{1}$ | $\Omega_{1}=dP$ | $\int_{C_{g_{r}}}dP=\int_{C_{g_{r}}}f^{*}r\in\mathbb{Z}_{2},r\in H^{1}(BD_{1},\mathbb{Z}^{r})$ | Sec. IV.1
$2+1$D | $U(1)\times\Gamma$ | $\Omega_{2}=\frac{1}{2}\epsilon_{IJ}E^{I}\wedge E^{J}$ | $\int_{C_{t_{I}}}E^{I}=\int_{C_{t_{I}}}f^{*}\tau^{I}\in\mathbb{Z},\tau^{I}\in H^{1}(BT_{I},\mathbb{Z})$ | App. D
$2+1$D | $U(1)\times C_{N}$ | $\Omega_{2}=d\omega_{1}$ | $\int_{D_{u}}d\omega_{1}=\int_{D_{u}}f^{*}b\in\mathbb{Z}_{N},b\in H^{2}(BC_{N},\mathbb{Z})$ | Sec. IV.2
$3+1$D | $U(1)\times\Gamma$ | $\Omega_{3}=\frac{1}{6}\epsilon_{IJK}E^{I}\wedge E^{J}\wedge E^{K}$ | $\int_{C_{t_{I}}}E^{I}=\int_{C_{t_{I}}}f^{*}\tau^{I}\in\mathbb{Z},\tau^{I}\in H^{1}(BT_{I},\mathbb{Z})$ | App. D
$3+1$D | $U(1)\times C_{nv}$ | $\Omega_{1}=dP^{(n)}$ | $\int_{C_{g_{r}}}dP^{(n)}=\int_{C_{g_{r}}}f^{*}r\in\mathbb{Z}_{2},r\in H^{1}(BD_{1},\mathbb{Z}^{r})$ | Sec. IV.3
|  |  | $\int_{C_{u}}dP^{(n)}=\int_{C_{u}}f^{*}a=0,a\in H^{1}(BC_{N},\mathbb{Z})$ |
Table 1: Summary of the results. The second column specified the symmetry
group of the system. $\Gamma$, $D_{1}$, $C_{N}$, $C_{nv}$ denote lattice
translation, reflection, $N$-fold rotation, and $C_{nv}$ point group,
respectively. The third column gives the $k$-form $\Omega_{k}$ that appears in
the topological term Eq. (3). The fourth column lists the integral conditions
that $\Omega_{k}$ needs to be satisfied. Here we use $t_{I}$, $g_{r}$, and $u$
to denote the generator of the lattice translation in the $I$-th direction,
reflection, and the $N$-fold rotations, respectively. A $1$-cycle labeled by
$g$ is denoted by $C_{g}$, and the $2$-cycle labeled by the generator of the
$N$-fold rotation $u$ is denoted by $D_{u}$, of which the boundary is given by
$N$ $1$-cycles $C_{u}$. The last column shows the section of the paper where
each case is discussed.

##  III Warm up: effective field theories of 1d atomic insulators

To illustrate the basic idea, we begin with a simple example: 1d atomic
insulators. The relevant symmetry group is $U(1)\times\Gamma$, where $U(1)$ is
the charge conservation symmetry and $\Gamma\cong\mathbb{Z}$ is a 1d discrete
translation symmetry. Fermion parity is the $\mathbb{Z}_{2}$ subgroup of
$U(1)$. We will focus on such an 1d atomic insulator whose building block
picture has a charge-1 atom per unit cell.

A model of a 1d atomic insulator consisting of spinless fermions on a 1d
lattice with unit cell size $a$ so that there is a unit charge per unit cell.
We then consider a much finer lattice by adding degrees of freedom as ancillas
within each unit cell. Note that these ancillas are not the physical atoms.
This setup is shown in Fig. 1. The new model, which has a lattice spacing
$l\ll a$ 111To simplify the analysis, here we choose $a/l=2m$ with $m\gg 1$
without loss of generality. One is free to choose $a/l$ to be an odd integers,
which will not effect the results., is described by the Hamiltonian:

$H=-t\sum_{x}^{L}(c_{x}c_{x+l}^{\dagger}+h.c.)-\mu\sum_{x}^{L}c_{x}^{\dagger}c_{x}+\cdots\
,$ (6)

where $x$ labels the ancillas. The ellipsis represents various perturbations
consistent with the $U(1)$ and the translation symmetries $\Gamma$ while
preserving the average charge per unit cell. The translation symmetry acts on
the fermions by

$t:c_{x}\rightarrow c_{x+a}.$ (7)

By construction, the new model also has the charge $U(1)$ and translation
symmetry $\Gamma$.

Figure 1: A 1d lattice of ancillas with lattice spacing $l$ much smaller than
the unit cell size $a$. The correlation length $\xi$ is assumed to be much
less than the unit cell $a$. After coarse-graining with respective to the
lattice space $l$, local observables have spatial variation within the unit
cell. The original atoms are represented as the charge density wave (gradiant
blue colors) in the lattice of ancillas.

Now we want to derive the continuum IR limit of the theory where the fields
are coarse-grained with respect to the lattice spacing $l$. To proceed, we
expand the microscopic fermion operator $c_{x}$ in terms of the slowly varying
low-energy fields $\psi_{R/L}$ as

$c_{x}\sim\psi_{R}(x)e^{k_{F}x}+\psi_{L}(x)e^{-k_{F}x},$ (8)

where $k_{F}=\pi/l$. Define $\psi(x)=(\psi_{R},\psi_{L})^{T}$ as the low-
energy fermion field, the continuum IR limit of the theory takes the following
general form

$\mathcal{L}=-i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi+im\bar{\psi}\psi.$ (9)

where $\gamma^{\mu}$ satisfies $\\{\gamma^{\mu},\gamma^{\nu}\\}=2g^{\mu\nu}$,
and $g^{\mu\nu}=$ diag $(-1,1)$ is the Minkowski metric. We choose
$\gamma^{0}=i\sigma_{y}$, $\gamma^{1}=\sigma_{x}$, and
$\gamma^{01}=\sigma_{z}$.

The translation symmetry acts on the low-energy fields by

$t:\psi(x)\rightarrow e^{ik_{F}a\sigma_{z}}\psi(x+a)=\psi(x+a),$ (10)

where we have used the fact that $k_{F}a\in 2\pi\mathbb{Z}$ for integer
filling. Note that, while $\psi(x)$ varies very slowly on the scale of $l$:
$\psi(x)\approx\psi(x+l)$, it’s not the case on the scale of $a$:
$\psi(x)\not\approx\psi(x+a)$, hence that $\psi(x)$ could vary on the scale
$R\gg l$ and $R<a$.

Although this kind of model might not seem like the system one would normally
consider, it has been argued that classifications and the topological
properties of such systems are same as the crystalline phases in general
Thorngren and Else (2018). In other words, all other states belonging to the
same topological crystalline phase are smoothly connected to the ground state
of such models (dubbed as the smooth state in Ref. Thorngren and Else, 2018),
and the smooth state can serve as a representative of the whole phase. In the
next section, we are going to obtain the effective field theory in this
special limit.

###  III.1 The topological term for 1d atomic insulators

As discussed above, the effective theory of an atomic insulator is a $1+1$d
single massive Dirac theory (9). To make contact with the topological crystal
picture, we add a spatially dependent mass term:

$\mathcal{L}_{m}=im_{0}\bar{\psi}e^{i\phi(x)\gamma^{01}}\psi,$ (11)

where $\gamma^{01}=\gamma^{0}\gamma^{1}$, $m_{0}>0$, and we assume $m\gg
m_{0}$. Here the spatial dependence of the mass term is encoded in the
function $\phi(x)$. Similar to the fermion fields, $\phi(x)$ could varies on
the scale $R$. Here we focus on a special configuration of $\phi(x)$ such that
it’s a monotonic function whose value changes abruptly by $2\pi$ at the
location of atoms as shown in Fig. 2. It can be shown that this kind of
spatially dependent mass terms trap a charge-1 bound state with a finite
energy at the interfaces of $\phi(x)$. These charge-1 bound states are
precisely the building blocks in the topological crystal approach and
corresponds to the physical atoms 222More formally, we note that the function
$\phi(x)$ defines a map $\phi:X\rightarrow\mathfrak{M}_{1}$, where
$\mathfrak{M}_{1}$ is the space of $1+1$d fermionic short range entangled
states with $U(1)$ symmetry. The configuration of $\phi(x)$ we choose gives a
noncontractible loop in $\mathfrak{M}_{1}$ every time we go through a unit
cell and the bound state is associated to the winding number
$\pi_{1}(\mathfrak{M}_{1})=\mathbb{Z}$..

Figure 2: A typical configuration of the phase $\phi(x)$ of the mass term.
$\phi(x)$ jump by $2\pi$ at the location of an atom.

We now discuss the classification of such mass interfaces parameterized by
$\phi(x)$. We will show that the mass interfaces is classified by
$H^{1}(B\Gamma,\mathbb{Z})$. To systematically discuss the configurations of
$\phi(x)$, we consider the cell decomposition of $\mathbb{R}$ by the FDs. Here
each one cell $\Sigma_{1,(i)}$ is simply a unit cell and is labeled by a group
element $g$ in the group of discrete translation $\Gamma\cong\mathbb{Z}$. Two
neighboring $1$-cells $\Sigma_{1,(i)}$, $\Sigma_{1,(j)}$ meet at a single
$0$-cell $\Sigma_{0,(ij)}$, which is labeled by the generator $t$ of the
translation group $\Gamma$. This kind of cell decompositions by the FDs
applies to any crystalline symmetry.

With this cell decomposition, we can now discuss the configurations of
$\phi(x)$ more systematically. We will first focus on the discontinuous
configurations of $\phi(x)$ since the general structure can be seen more
clearly in this limit. A discontinuous configurations of $\phi(x)$ is modeled
by having a constant function in each $1$-cell $\Sigma_{1,(i)}$. At the
intersecting $0$-cell $\Sigma_{0,(ij)}$, where $\Sigma_{1,(i)}$ and
$\Sigma_{1,(j)}$ meet, we have the relation

$\phi(\Sigma_{1,(j)})=\phi(\Sigma_{1,(i)})+2\pi\tau(\Sigma_{0,(ij)}),\
\tau(\Sigma_{0,(ij)})\in\mathbb{Z}.$ (12)

The integer-valued function $\tau$ satisfies

$\tau(\Sigma_{0,(ij)})+\tau(\Sigma_{0,(jk)})=\tau(\Sigma_{0,(ik)}).$ (13)

Moreover, there is a redundancy since, if we modify the configuration of
$\phi$ as

$\displaystyle\phi(\Sigma_{1,(i)})$ $\displaystyle\rightarrow$
$\displaystyle\phi(\Sigma_{1,(i)})+2\pi h(\Sigma_{1,(i)}),$
$\displaystyle\tau(\Sigma_{0,(ij)})$ $\displaystyle\rightarrow$
$\displaystyle\tau(\Sigma_{0,(ij)})+h(\Sigma_{1,(j)})-h(\Sigma_{1,(i)})$ (14)

with $h(\Sigma_{1,(i)})\in\mathbb{Z}$, we obtain the same configuration of
$\phi$, which means that $\tau$ is a $\mathbb{Z}$-valued cocycle in
$H^{1}(M,\mathbb{Z})$. This, however, does not mean that $\phi(x)$ is
classified by the cellular cohomology $H^{1}(M,\mathbb{Z})$ of the manifold
$M$, and we are not considering arbitrary interfaces. What we are interested
in is the _symmetric deformation classes_ of the interface configurations of
$\phi$ with a charge-1 particle at each interface. Taking the typical
configuration of $\phi(x)$ shown in Fig. 2 as an example, one can see that all
the deformations, which respects the translation symmetry, will not change the
$2\pi$ jumps at the locations of atoms. This structure can be captured by
identifying the integer-valued transition function $\tau$ as the pullback
$\tau=f^{*}\alpha$, where $\alpha\in H^{1}(B\Gamma,\mathbb{Z})$ by the map
$f:M\rightarrow B\Gamma$. The general construction of the map $f$ is given in
Appendix B. In our case the map $f$ is constructed as follows. We note that
for each $1$-cell $\Sigma_{1,(i)}$, there is a dual $0$-cell
$\Sigma^{\vee}_{0,(i)}$. Similarly, for each $0$-cell $\Sigma_{0,(ij)}$, there
is a dual $1$-cell $\Sigma^{\vee}_{1,(ij)}$. The map $f$ is constructed such
that every dual $0$-cell $\Sigma^{\vee}_{0,(i)}$ is mapped to the based point
$*$ in $B\Gamma$, and a dual $1$-cells $\Sigma^{\vee}_{1,(ij)}$ is mapped to
the non-trivial loop labeled by the generator
$t\in\pi_{1}(B\Gamma)=\mathbb{Z}$:

$\displaystyle f:M$ $\displaystyle\rightarrow$ $\displaystyle B\Gamma$
$\displaystyle\Sigma^{\vee}_{0,(i)}$ $\displaystyle\mapsto$ $\displaystyle*$
$\displaystyle\Sigma^{\vee}_{1,(ij)}$ $\displaystyle\mapsto$ $\displaystyle
t\in\pi_{1}(B\Gamma)=\mathbb{Z}.$ (15)

As a result, dual $1$-cells (or the original $0$-cells) are in one-to-one
correspondence with the generator $t$ of the translation group $\Gamma$. This
has a following implication: let $x_{0}$ be a point inside a $1$-cell, a path
connecting $x_{0}$ to $gx_{0}$ is labeled by a group element $g\in\Gamma$.
This property is generally true for any space group $G_{s}$—any such path can
be labeled by a group element $g\in G_{s}$.

Since we have $\tau=f^{*}\alpha$, the redundancy of the mass interfaces Eq.
(14) is restricted as we now discussed. We recall that
$H^{1}(B\Gamma,\mathbb{Z})\cong H^{1}(\Gamma,\mathbb{Z})$ is the quotient of
1-cocycles by 1-coboundaries. The cocycle condition reads

$\alpha(g_{1})+\alpha(g_{2})=\alpha(g_{1}g_{2}).$ (16)

In other words, $\alpha(g)$ is a group homomorphism of $\Gamma$. The
coboundaries in this case are all trivial:

$\displaystyle\delta\mu:\Gamma$ $\displaystyle\rightarrow$
$\displaystyle\mathbb{Z}$ $\displaystyle g$ $\displaystyle\mapsto$
$\displaystyle 0.$ (17)

One can easily see that $H^{1}(B\Gamma,\mathbb{Z})=\mathbb{Z}$ with the
generator given by $\alpha(t)=1$ where $t$ is the generator of the translation
group $\Gamma$. Using our construction of the map $f$, we have
$h(\Sigma_{1,j})=h(\Sigma_{1,i})$ due to Eq. (17). The only remaining
redundancy is given an overall shift by $2\pi$, which will not affect the
value of $\tau$, and hence the $2\pi$ jumps of $\phi(x)$ at the locations of
atoms. This is similar to the redundancy of changing the integer labeling of
the atoms. Therefore, the symmetric deformation classes of the interfaces are
classified by $H^{1}(B\Gamma,\mathbb{Z})=\mathbb{Z}$.

To obtain an effective theory, we couple the fermion to an external background
$U(1)$ gauge field $A_{\mu}$. By integrate out the massive fermion, the
effective action contains the following topological term:

$\displaystyle S_{\text{eff}}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\int\epsilon^{\mu\nu}\phi(x)F_{\mu\nu}d^{2}x$ (18)
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int\epsilon^{\mu\nu}A_{\mu}\partial_{\nu}\phi(x)d^{2}x.$

By requiring the effective action to be gauge invariant under
$A_{\mu}\rightarrow A_{\mu}+\partial_{\mu}\alpha$, we find the current

$J^{\mu}=\frac{1}{2\pi}\epsilon^{\mu\nu}\partial_{\nu}\phi$ (19)

is conserved.

In the limit where $\phi(x)$ is discontinuous, the density is given by

$\rho=\frac{1}{2\pi}\partial_{x}\phi=\sum_{i}\delta(x-x_{i}),$ (20)

where the discrete nature of the density of an atomic insulator is recovered.
Note that, if we define the theory on a finite system with size $L$, the total
charge of the system is

$Q_{\text{tot}}=\int_{0}^{L}\frac{1}{2\pi}\partial_{x}\phi
dx=\frac{1}{2\pi}\left(\phi(L)-\phi(0))\right)=N.$ (21)

This topological term also reveals the “Thouless pump” response for the atomic
insulatorsThouless (1983)—There is a net charge flows through the system when
the phase field $\phi$ winds $2\pi$ in time.

In general, there is a coefficient $\nu$ in front of the topological term Eq.
(18). To show this coefficient is quantized, we proceed with the following
argument. Consider $\phi$ is time-independent and the spatial dependence is
given by Eq. (12) with $\tau=1$. We then integrate along the $x$-direction and
the topological term becomes

$\displaystyle S_{\text{eff}}$ $\displaystyle=$
$\displaystyle\frac{\nu}{2\pi}\int\epsilon^{\mu\nu}A_{\mu}\partial_{\nu}\phi
d^{2}x$ (22) $\displaystyle\sim$ $\displaystyle\nu\sum_{i}\int
A_{0}\delta(x-x_{i})d^{2}x$ $\displaystyle=$ $\displaystyle\sum_{i}\nu\int
A_{0}(x_{i})dt\ ,$

where the gauge transformations of $A_{0}(x_{i})$ can be different for
different $i$. We obtain a sum of $0+1$d effective actions, each of which is
the effective action of an atom couple to the $U(1)$ gauge field. By the gauge
invariance, we see the $\nu$ has to be quantized to integers.

###  III.2 Taking the smooth limit

In the previous section, we take the limit where $\phi(x)$ is discontinuous in
order to make a clear connection to the building block picture. We also see
that these discontinuous configurations of $\phi(x)$ are classified by
$\alpha\in H^{1}(B\Gamma,\mathbb{Z})$. Here we show that it’s possible to take
a limit where $\phi(x)$ is a smooth function such that $\phi(x)$ is still
classified by $H^{1}(B\Gamma,\mathbb{Z})$. Moreover, we are going to take the
limit where $\phi(x)$ is as smooth as possible such that $\partial_{x}\phi(x)$
is uniform. After taking such smooth limit, we will obtain an effective field
theory which works in the usual IR limit, where the correlation length $\xi$
doesn’t have to be much smaller than the unit cell $a$.

Let’s recall that the defining property of an element $\alpha\in
H^{1}(B\Gamma,\mathbb{Z})$ is that the pairing satisfies

$\int_{C_{1}}\alpha\in\mathbb{Z},$ (23)

where $C_{1}$ is the non-trivial $1$-cycle of $B\Gamma$. We pull this back by
using the map $f:M\rightarrow B\Gamma$, and the corresponding $1$-cocycle
$e_{1}\in H^{1}(M,\mathbb{Z})$ satisfies

$\int_{\Sigma_{1}^{\vee}}e_{1}=\int_{\Sigma_{1}^{\vee}}f^{*}\alpha\in\mathbb{Z},$
(24)

where $\Sigma_{1}^{\vee}$ is a dual $1$-cell (unit cell) in the real space.
This integral essentially counts the number of atoms in a unit cell.

We would like to obtain a low-energy effective field theory where all the
fields are smooth. At the same time, we want to preserve the classification of
$\phi(x)$ given by $H^{1}(B\Gamma,\mathbb{Z})$. The way to achieve this is to
consider closed $1$-forms with integral periods 333Recall that a closed
$k$-form $\omega$ on M has integral periods if, for every smooth $k$-cycle $C$
in M, the integral $\int_{C}\omega$ is an integer. Moreover, a closed $k$-form
$\omega$ has integral periods if and only if the de Rham class of $\omega$
lies in the image of the change-of-coefficients map
$H^{k}(M,\mathbb{Z})\rightarrow H^{k}(M,\mathbb{R})\cong H^{k}_{dR}(M),$ (25)
where $H^{k}_{dR}(M)$ denotes the de Rham cohomology of $M$ Simons and
Sullivan (2008). Loosely speaking, a closed $k$-form with an integral period
serves as a differential form representative of an element in
$H^{k}(M,\mathbb{Z})$.. Now we replace the cocycle on the left-hand side of
Eq. (24) by an smooth differential $1$-form $E_{1}$ with integral periods:

$\int_{\Sigma_{1}^{\vee}}E_{1}=N_{t}\in\mathbb{Z},$ (26)

where $E_{1}=d\phi/2\pi$, and $\phi$ is a smooth function. Then we identify
the integer $N_{t}$ with the right-hand side of Eq. (24):

$N_{t}=\int_{\Sigma_{1}^{\vee}}f^{*}\alpha\in\mathbb{Z}.$ (27)

Representatives of the smooth $1$-form $E_{1}$ can be obtained by solving Eq.
(26). We consider a smooth function $\phi(x)$ satisfying

$\int_{x_{0}}^{x_{0}+a}d\phi=(\phi(x_{0}+a)-\phi(x_{0}))=2\pi\tau,$ (28)

where we have restored the unit cell size $a$ for the sake of clarity. An
example of such function, which is as smooth as possible and satisfying Eq.
(28), is given by

$\phi(x)=\frac{2\pi}{a}\tau x=b_{1}\tau x,$ (29)

where $b_{1}=2\pi/a$ is the reciprocal lattice vector (This is essentially the
“labelling” field introduced by Haldane Haldane (1981)). We can then define a
smooth $1$-form

$E_{1}=\frac{1}{2\pi}\partial_{x}\phi(x)dx=\frac{1}{2\pi}\tau b_{1}dx.$ (30)

This is a closed $1$-form with integral period since, if we integrate over a
unit cell, we have

$\int_{\Sigma_{1}^{\vee}}E_{1}=\int_{x_{0}}^{x_{0}+a}\frac{1}{2\pi}\tau
b_{1}dx=\tau\in\mathbb{Z},$ (31)

which is the property that we want.

In general, there will be time-dependence in $\phi$ so that we can define the
time-component of the $1$-form $E_{0}=\partial_{t}\phi dt/2\pi$. Written in
terms of these smooth $1$-form $E$, we have the following topological term:

$\displaystyle\int A\wedge
E=\frac{1}{2\pi}\int\epsilon^{\mu\nu}A_{\mu}b_{\nu}d^{2}x.$ (32)

We have thus reproduce the effective field theory of an atomic insulator in
Ref. Nissinen, 2020; Else  _et al._ , 2021. As one can see from the above
discussion, the $1$-form $E=d\phi/2\pi$ basically tells us where to decorate
the 0d building blocks. Therefore, our approach gives a direct correspondence
between the topological terms and the topological crystal picture.

This discussion can be generalized to atomic insulators in higher dimensions.
For example, in 2d and 3d, we expect there are topological terms of the form:

$\displaystyle S_{\text{eff}}$ $\displaystyle=$
$\displaystyle\frac{\nu}{2}\int\epsilon_{IJ}A\wedge E^{I}\wedge E^{J}.$
$\displaystyle S_{\text{eff}}$ $\displaystyle=$
$\displaystyle\frac{\nu}{6}\int\epsilon_{IJK}A\wedge E^{I}\wedge E^{J}\wedge
E^{K},$ (33)

where, for a translation in the $I$th direction, $E^{I}$ is the differential
form representative of the cocycle in $H^{1}(BT_{I},\mathbb{Z})$. In Appendix
D, we give a detailed derivation of the topological term Eq. (III.2) for the
2d atomic insulators. Generalizing to 3d atomic insulators is straightforward.

The physical meaning of the topological terms Eq. (III.2) is that there is a
charge-$\nu$ per unit cell as one can see from the effective action of the
mass interface obtained by integrating out the spatial directions. Those terms
also describe higher dimensional analogs of the Thouless pumpKapustin and
Spodyneiko (2020); Hsin  _et al._ (2020). Such kind of topological terms for
atomic insulators are discussed in Ref. Else  _et al._ , 2021 in the context
of topological elasticity theory (see also Ref. Nissinen, 2020), where
$\theta^{I}$ fields are interpreted as the phonon fields. Since the
topological terms we obtained above take essentially the same form, this
suggests that there could be an elasticity interpretation for the spatially
dependent mass terms. Indeed, in the case where the spatially dependent mass
terms are generated from coupling to lattice deformations, our topological
terms are the topological terms in the elasticity theory.

##  IV Topological terms of topological crystalline insulators with point
group symmetry

In this section, we are going to discuss various effective field theories for
TCIs with point group symmetries. We will facus on the reflection, $C_{N}$
rotation, and $C_{nv}$ symmetries for 1d, 2d, and 3d TCIs. The 1d and 2d TCIs
that we are going to discuss are built by placing 0d charges at the high
symmetry points, which do not support protected boundary gapless modes. For 3d
TCIs, we will consider the 2nd-order topological phases with gapless chiral
hinge models, whose building block picture is given by placing 2d IQH states
on high symmetry planes.

###  IV.1 1d insulators with reflection symmetry

We now move on to discuss topological crystalline insulators with reflection
symmetry in 1d. The symmetry group we focus on is $G=U(1)\times D_{1}$, where
the reflection group $D_{1}\cong\mathbb{Z}_{2}$. We are going to focus on the
phase of which the building block picture is given by placing a 0d state
carrying a unit $U(1)$ charge and a trivial irreducible representation of
$D_{1}$ at the reflection center.

The low energy theory of this kind of insulators is given by the following
massive Dirac theory

$\mathcal{L}=-i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-
m_{1}\bar{\psi}\gamma^{01}\psi$ (34)

with the reflection symmetry acting on the fermions by

$g_{r}:\psi(x,t)\rightarrow\gamma_{1}\psi(-x,t).$ (35)

We are going to show that this Dirac theory indeed describes the TCI we are
interested in. We proceed with the dimensional reduction procedure by adding a
spatially varying mass term:

$\mathcal{L}_{m}=-im_{2}\bar{\psi}e^{i\phi(x)\gamma^{01}}\psi,$ (36)

where we require that $m_{1}\gg m_{2}>0$. The reflection symmetry requires
that $\phi(x)=\pi-\phi(-x)$ mod $2\pi$. The phase $\phi(x)$ could wind non-
trivially in space subjecting to the constraint given by the reflection
symmetry. If we consider a configuration of $\phi(x)$ such that it winds
$2\pi$ along a path passing through the reflection center (as shown in Fig.
4), by solving the bound state directly for this mass interface, one can
confirm that there is a charge-1 bound state sitting at the reflection center.
This is precisely the building block picture for this phase, which justifies
the claim that Eq. (34) describes the TCI we are interested in.

We are going to show that the mass interfaces parameterized by $\phi(x)$ are
classified by the cellular cohomology $H^{1}(BD_{1},\mathbb{Z}^{r})$ with a
twisting coefficient $\mathbb{Z}^{r}$, which will be defined below. To
proceed, we first discuss the cell decomposition given by the FDs. As shown in
Fig. 3, there are two 1-cells $\Sigma_{1,(0)}$, $\Sigma_{1,(1)}$ and a single
0-cell $\Sigma_{0,(01)}$ at the reflection center. The dual cell structure is
also shown in Fig. 3, which contains two dual 0-cells $\Sigma^{\vee}_{0,(0)}$,
$\Sigma^{\vee}_{0,(1)}$ and one dual 1-cell $\Sigma^{\vee}_{1,(01)}$. With
this choice of cell-decomposition, there is a map $f:M\rightarrow BD_{1}$.
Written explicitly, the map $f:\Sigma^{\vee}_{0,(0)}\mapsto*$,
$\Sigma^{\vee}_{0,(1)}\mapsto*$, $\Sigma^{\vee}_{1,(01)}\mapsto g_{r}$, where
$g_{r}\in\pi_{1}(BD_{1})\cong D_{1}$ denotes the non-trivial group element in
$D_{1}$.

Figure 3: Reflection symmetric cell-decomposition (above) and the dual-cell-
decomposition (below).

We consider the configurations of $\phi(x)$ such that it’s a constant function
within the two 1-cells $\Sigma_{1,(0)}$, $\Sigma_{1,(1)}$. At the intersecting
0-cell $\Sigma_{0,(01)}$, the constant functions satisfy the relation:

$\phi(\Sigma_{1,(1)})=\phi(\Sigma_{1,(0)})+2\pi r(\Sigma_{0,(01)}),$ (37)

where $r(\Sigma_{0,(01)})\in\mathbb{Z}$. The reflection symmetry gives the
following condition

$\phi(\Sigma_{1,(0)})+\phi(\Sigma_{1,(1)})=\pi\ \text{mod}\ 2\pi$ (38)

as well as a non-trivial action on the integer-valued function $r$

$g_{r}\cdot r=-r.$ (39)

However, there is a redundancy since, if we modify the configuration of $\phi$
as

$\displaystyle\phi(\Sigma_{1,(i)})$ $\displaystyle\rightarrow$
$\displaystyle\phi(\Sigma_{1,(i)})+2\pi h(\Sigma_{1,(i)}),$ $\displaystyle
r(\Sigma_{0,(01)})$ $\displaystyle\rightarrow$ $\displaystyle
r(\Sigma_{0,(01)})+h(\Sigma_{1,(1)})-h(\Sigma_{1,(0)}),$ (40)

we obtain the same configuration of $\phi(x)$. These conditions tell us that
$r$ is a $\mathbb{Z}$-valued cocycle in $H^{1}(M,\mathbb{Z}^{r})$ with a
twisting coefficient given by Eq. (39).

We consider the mass interface such that $r$ is given by the pullback
$r=f^{*}\upsilon$ of $\upsilon\in
H^{1}(BD_{1},\mathbb{Z}^{r})\cong\mathbb{Z}_{2}$. To understand this
$\mathbb{Z}_{2}$ classification, we note that the coboundaries in
$H^{1}(BD_{1},\mathbb{Z}^{r})$ are of the form

$\displaystyle\delta\nu^{0}:D_{1}$ $\displaystyle\rightarrow$
$\displaystyle\mathbb{Z}$ $\displaystyle g_{r}$ $\displaystyle\mapsto$
$\displaystyle 2m$ (41)

for some $m\in\mathbb{Z}$, where $\nu^{0}$ is a function taking the based
point $*\in BD_{1}$ to an integer $\nu^{0}(*)\in\mathbb{Z}$. By using the map
$f:M\rightarrow BD_{1}$, we have

$\displaystyle h(\Sigma_{1,(0)})$ $\displaystyle=$
$\displaystyle\nu^{0}(f(\Sigma_{1,(0)}))=\nu^{0}(*),$ $\displaystyle
h(\Sigma_{1,(1)})$ $\displaystyle=$
$\displaystyle\nu^{0}(f(\Sigma_{1,(1)}))=\nu^{0}(g_{r}\cdot*),$ (42)

where $g\cdot*\sim*$ in the classifying space $BD_{1}$. Using Eq. (41), we
have $h(\Sigma_{1,(1)})-h(\Sigma_{1,(0)})=\delta\nu^{0}=2m$. As a result,
there is an equivalence relation $r\sim r+2m$, and the non-trivial winding of
the phase $\phi(x)$ has a $\mathbb{Z}_{2}$ classification. Pictorially, it’s
easy to see that a interface with a $4\pi$ jump can be deformed into a
configuration with no interface while preserving the reflection symmetry.

Figure 4: A typical configuration of $\phi(x)$ which supports a charge-1 bound
state at the reflection center.

After coupling to the $U(1)$ gauge fields and integrated out the massive
fermions, we obtain an topological term which takes the same form as Eq. (18).
The difference is that $\phi(x)$ has to satisfy Eq. (38) due to the reflection
symmetry. We thus define a 1-form

$dP=\frac{d\phi}{2\pi}.$ (43)

The topological term becomes

$\displaystyle S_{\text{eff}}$ $\displaystyle=$ $\displaystyle k\int A\wedge
dP,$ (44)

where the coefficient $k=1$ in our example. Note that the coefficient $k$ is
$\mathbb{Z}_{2}$ valued since the mass interfaces have $\mathbb{Z}_{2}$
classification. In the next section, we are going to show how to connect the
topological term Eq. (44) to the known result in Ref. Ramamurthy and Hughes,
2015; Nissinen, 2020 by taking the smooth limit for $\phi(x)$.

####  IV.1.1 Smooth limit

Now we would like to consider smooth configurations of $\phi(x)$ such that the
mass interfaces are still classified by $H^{1}(BD_{1},\mathbb{Z}^{r})$. To
proceed, we first rewrite Eq. (37) in terms of the cellular cohomology:

$\int_{\Sigma_{1}^{\vee}}dP^{\prime}_{1}=\int_{\partial\Sigma_{1}^{\vee}}P^{\prime}_{1}=r,$
(45)

where, for the sake of convenience, we have defined
$P^{\prime}_{1}=\phi/2\pi-1/4$ to subtract the constant $\pi/2$ contribution
in $\phi$. $P^{\prime}_{1}$ is now simply odd under refection:
$g_{r}:P^{\prime}_{1}\rightarrow-P^{\prime}_{1}$.

What we are looking for is the smooth version of Eq. (45) such that the
$1$-form $dP_{1}^{\prime}$ is constructed by smooth functions. This is
achieved by considering the following smooth 1-form with integral periods:

$\displaystyle\int_{-x_{0}}^{g_{r}\cdot(-x_{0})}d\tilde{P}_{1}$
$\displaystyle=$ $\displaystyle\int_{-x_{0}}^{x_{0}}d\tilde{P}_{1}$ (46)
$\displaystyle=$ $\displaystyle\tilde{P}_{1}(x_{0})-\tilde{P}_{1}(-x_{0})$
$\displaystyle=$ $\displaystyle r,$

where $|x_{0}|\gg\xi$ and $\tilde{P}$ is smooth. We can thus write the low-
energy effective action in terms of the smooth $1$-form $\tilde{P}$:

$\displaystyle S_{\text{eff}}$ $\displaystyle=$ $\displaystyle k\int A\wedge
d\tilde{P}.$ (47)

Under the reflection symmetry $\tilde{P}\rightarrow-\tilde{P}$. After an
integration by parts, we recognized that Eq. (47) is essentially the effective
action obtained in Ref. Ramamurthy and Hughes, 2015; Nissinen, 2020, and
$\tilde{P}$ can be interpreted as the spatially dependent electric
polarization.

###  IV.2 2d insulators with rotational symmetry

Here we discuss the topological term of fermionic TCIs with $U(1)\times C_{N}$
symmetry 444More precisely, the symmetry is $(U(1)\times
C_{N})/\mathbb{Z}_{2}$, as the $C_{N}$ rotation $U$ defined in Eq. (49) obeys
$U^{N}=(-1)^{N_{f}}$, where $N_{f}$ is the total fermion number under $U(1)$.
. The classification of these systems has been computed in Ref. Cheng and
Wang, 2018 by using the topological crystal approach. For the sake of
simplicity, here we focus on the states with no charge and thermal Hall
conductivity, and with a building block picture given by placing a $0$d state
carrying a unit $U(1)$ charge and a trivial $C_{N}$ charge at the rotational
center.

The low energy field theory of this state is given by a 2+1D massive Dirac
theory:

$\mathcal{L}=-i\bar{\Psi}\gamma^{\mu}\partial_{\mu}\Psi+im_{0}\bar{\Psi}\sigma_{3}\Psi,$
(48)

where $\Psi=(\psi_{1},\psi_{2})$, and $\sigma_{i}$ are Pauli matrices in the
flavor space. The mass term here guarantees that the Chern number is zero. The
$C_{N}$ rotation acts on the fermions by

$u:\Psi(\bm{r})\rightarrow\exp{\left(\frac{i}{2}\frac{2\pi}{N}\gamma^{0}\sigma_{3}\right)}\Psi(R\bm{r}).$
(49)

One can recover the building block state by adding the following spatially
varying mass term:

$\mathcal{L}_{m}=im\bar{\Psi}\left(n^{1}(\bm{r})\sigma_{1}+n^{2}(\bm{r})\sigma_{2}\right)\Psi,$
(50)

where we require that $m_{0}\gg m>0$ and consider the configuration of $n^{1}$
and $n^{2}$ such that there is a bound state carrying a unit $U(1)$ charge at
the origin. An example of such configuration would be a “hedgehog” with a
singularity at the origin. Usually a hedgehog configuration is invariant under
a continuous rotational symmetry, here we only require that it’s invariant
under a discrete $C_{N}$ rotation. We note that Shiozaki shows there there is
an isomorphism between the group of 0d building blocks and the K group of the
Dirac Hamiltonians with the hedgehog-mass potential with a unit winding number
Shiozaki (2019). Starting from a 0d Hamiltonian describing the 0d building
state, one can obtain the massive Dirac theory Eq. (48) by using his general
construction.

Since the system is rotational invariant, it’s convenient to parametrize the
mass term by

$n^{1}=n_{r}\cos{n_{\theta}},\ n^{2}=n_{r}\sin{n_{\theta}}.$ (51)

where $n_{r}$ can be taken to be a constant almost everywhere and
$n_{r}\rightarrow 0$ as $r\rightarrow 0$. Following from the $C_{N}$
transformation on the fermions Eq. (49), $n_{\theta}$ must satisfy

$n_{\theta}(\theta+\frac{2\pi}{N})=n_{\theta}(\theta)+\frac{2\pi}{N}\
\text{mod}\ 2\pi.$ (52)

In general, the singularity at the origin is described by a one-form
$\omega_{1}$ such that
$d\omega_{1}=(\partial_{i}\partial_{j}-\partial_{j}\partial_{i})n_{\theta}\neq
0$. As we will discuss in detail below, this one-form $\omega_{1}$ is
classified by $H^{1}(BC_{N},\mathbb{Z}_{N})$.

After coupling the fermion to a $U(1)$ gauge field and integrate out the
massive fermions, the effective theory contains the following topological
term:

$\displaystyle S_{\text{eff}}=\int A\wedge\omega_{2}$ (53)

where the 2-form $\omega_{2}$ is the Euler class in $H^{2}(M,\mathbb{Z})$. A
perturbative derivation of this topological term is given in Appendix E. In
general, we have $\omega_{2}=n^{*}\tau_{2}$, where $\tau_{2}$ is $2$-form in
the space of the mass parameters. However, we are going to show that the
topological crystal picture tells us that the $2$-form $\omega_{2}$ should be
identified with the pullback $f^{*}\alpha_{2}$, where $\alpha_{2}\in
H^{2}(BC_{N},\mathbb{Z})$ and the map $f:M\rightarrow BC_{N}$ is the map from
the manifold $M$ to the classifying space $BC_{N}$.

The appearance of the Euler class $H^{2}(M,\mathbb{Z})$ is very natural (see
Ref. Else and Thorngren, 2019; Hason  _et al._ , 2020 for encountering the
Euler class in similar situations). Note that, since $n^{1}$ and $n^{2}$
transform as the regular representation of the $C_{N}$ rotational symmetry, it
makes sense to view them as a vector field $\bm{n}=(n^{1},n^{2})$. Formally,
we have a $\mathbb{R}^{2}$ real vector bundle $V$ over the manifold $M$ (We
will always assume the manifold $M$ to be a euclidean space $\mathbb{E}^{2}$),
and the vector field $\bm{n}$ is the section of this bundle: $n:M\rightarrow
V$ such that $\pi\circ n=id$, where $\pi$ is the projection $\pi:V\rightarrow
M$. The singular configurations of $\bm{n}$ corresponds to the zero sections
of the vector bundle $V$, and it’s well-known that the Euler class $e(V)\in
H^{2}({M,\mathbb{Z}})$ counts the number of zero sectionsBott and Tu (1982).

Figure 5: (a) The cell decomposition for a system with $C_{4}$ symmetry, and
(b) the corresponding dual-cell decomposition.

Now we discuss how to construct the $2$-form $\omega_{2}$ explicitly. We begin
with the the cell decomposition of $\mathbb{R}^{2}$ given by the FDs. Each
$2$-cell $\Sigma_{2,(i)}$ is labeled by a group element $g\in C_{N}$. Two
neighboring two cells $\Sigma_{2,(i)}$, $\Sigma_{2,(j)}$ meet at a single
$1$-cell $\Sigma_{1,(ij)}$, which is labeled by the generator $u\in C_{N}$.
There is a unique $0$-cell $\Sigma_{0}$ sitting at the rotational center. An
example of the $C_{4}$ symmetric cell-decomposition is shown in Fig. 5(a), and
the corresponding dual-cell decomposition is shown in Fig. 5(b).

Figure 6: A typical configuration of $n_{\theta}$ for systems with $C_{4}$
symmetry.

We now discuss the configurations of the mass term parameterized by Eq. (51)
from a more general perspective. Since we only require that the mass term to
be invariant under a discrete $C_{N}$ rotation, $n_{\theta}$ can be chosen to
be a constant in each $2$-cell $\Sigma_{2,(i)}$. At the intersections of
$2$-cells, $n_{\theta}$ could jump abruptly. We proceed with the following
systematically discussion. At the $1$-cell $\Sigma_{1,(ij)}$ where
$\Sigma_{2,(i)}$ and $\Sigma_{2,(j)}$ meet, we have the relation

$n_{\theta}(\Sigma_{2,(j)})=n_{\theta}(\Sigma_{2,(i)})+\frac{2\pi}{N}c(\Sigma_{1,(ij)}),\
c(\Sigma_{1,(ij)})\in\mathbb{Z}.$ (54)

A typical configuration of $n_{\theta}$ for $C_{4}$ symmetric system is shown
in Fig. 6. The integer-valued function $c$ satisfies

$c(\Sigma_{1,(ij)})+c(\Sigma_{1,(jk)})=c(\Sigma_{1,(ik)}).$ (55)

There is a redundancy of the function $c(\Sigma_{1,(ij)})$ since we obtain the
same configuration of $n_{\theta}$ after we modify the configuration as

$\displaystyle n_{\theta}(\Sigma_{2,(i)})$ $\displaystyle\rightarrow$
$\displaystyle n_{\theta}(\Sigma_{2,(i)})+2\pi h(\Sigma_{2,(i)}),$
$\displaystyle c(\Sigma_{1,(ij)})$ $\displaystyle\rightarrow$ $\displaystyle
c(\Sigma_{1,(ij)})+h(\Sigma_{2,(j)})-h(\Sigma_{2,(i)}),$ (56)

, where $h(\Sigma_{2,(i)})\in\mathbb{Z}$. Formally, $c$ is a
$\mathbb{Z}$-valued cocycle in $H^{1}(M,\mathbb{Z})$. Here we consider the
interface such that $c=f^{*}\alpha_{1}$, where $\alpha_{1}\in
H^{1}(BC_{N},\mathbb{Z}_{N})$, which is lifted to $C^{1}(BC_{N},\mathbb{Z})$
by the embedding $\mathbb{Z}_{N}\sim[0,N-1)\subset\mathbb{Z}$. The deformation
classes of the $n_{\theta}$ interfaces are classified by
$H^{1}(BC_{N},\mathbb{Z}_{N})=\mathbb{Z}_{N}$. If we choose
$c(\Sigma_{1,(ij)})=1$, we obtain a class of interfaces of $n_{\theta}$
satisfying Eq. (52). Such choice can always be made since we can always choose
$\alpha_{1}(u)=1$, where $u$ is the generator of the $C_{N}$ rotation and
$\alpha(u)\in H^{1}(BC_{N},\mathbb{Z}_{N})$. We then have

$c(\Sigma_{1,(ij)})=\alpha_{1}(f(\Sigma_{1,(ij)}))=\alpha_{1}(u)=1,$ (57)

where we have used the fact that $f:\Sigma_{1,(ij)}\mapsto u$ for every
$1$-cell $\Sigma_{1,(ij)}$.

We now use the cocycle $c\in H^{1}(M,\mathbb{Z})$ to construct the desire
$2$-form in the topological term. At the level of cellular cohomology, we
would like to have some 2-cocycle $w=\delta c$ satisfying the following
property:

$\displaystyle\frac{1}{N}\int_{\Sigma_{2}^{\vee}}w=\frac{1}{N}\int_{\Sigma_{2}^{\vee}}\delta
c=\frac{1}{N}\int_{\partial\Sigma_{2}^{\vee}}c=1.$ (58)

Physically, this means that $n_{\theta}$ rotates by $2\pi$ as it goes around
the origin along the $1$-cells, which, intuitively, corresponds to having a
vortex configuration of $\bm{n}$ with strength $1$. There will be a localized
charge-1 bound state trapped at the vortex core, which reproduces the building
block picture. We claim the property Eq. (58) is satisfied if $\delta
c=f^{*}\delta\alpha_{1}$. Using Eq. (57), we can explicitly check Eq, (58) is
satisfied:

$\displaystyle\frac{1}{N}\int_{\Sigma_{2}^{\vee}}w$ $\displaystyle=$
$\displaystyle\frac{1}{N}\int_{\partial\Sigma_{2}^{\vee}}c$ (59)
$\displaystyle=$ $\displaystyle\frac{1}{N}\sum_{i=1}^{N}c(\Sigma_{1,(i,i+1)})$
$\displaystyle=$ $\displaystyle\frac{1}{N}\sum_{i=1}^{N}\alpha_{1}(g)$
$\displaystyle=$ $\displaystyle 1.$

The two form $\omega_{2}$ we are looking for is the differential form
representative of the $2$-cocycle $w$, which is a $2$-form with integral
periods. Now we discuss an explicit construction of the $2$-form $\omega_{2}$.
First we need to use two patches $U_{0}$ and $U_{1}$ to cover the real space
$\mathbb{R}^{2}$ such that the singularity of the section $\bm{n}$ (at the
origin) is contain entirely in $U_{0}$, and that $U_{1}$ covers
$\mathbb{R}^{2}-D_{0}$, where $D_{0}$ is the disc covered by $U_{0}$. Let
$\rho_{0}(r)$ and $\rho_{1}(r)$ be the partition of unity satisfying
$\rho_{0}(r)+\rho_{1}(r)=1$ subordinate to $U_{0}$ and $U_{1}$. In the patch
$U_{1}$, we use the configuration of $n_{\theta}$ discussed above and
partition of unity to define a 1-form

$\omega_{1}^{(1)}=-\rho_{0}\frac{dn_{\theta}}{2\pi}=\rho_{0}\sum_{i=1}^{N}\frac{c_{i,i+1}}{N}\delta\left(\theta-\frac{2\pi
c_{i,i+1}}{N}\right)d\theta,$ (60)

where we have used a short hand notation $c_{i,j}$ to denote
$c(\Sigma_{1,ij})$ and $c_{N,N+1}=c_{N,1}$. In the patch $U_{0}$, we define
the $1$-form $\omega_{1}^{(0)}=\rho_{1}dn_{\theta}/2\pi$.

Finally, we construct the two form

$\omega_{2}=d\omega_{1}^{(1)}=-\frac{1}{2\pi}d\rho_{0}\wedge dn_{\theta}$ (61)

such that it has an integral period:

$\displaystyle\int_{M}\omega_{2}$ $\displaystyle=$ $\displaystyle 1.$ (62)

From the property of the partition of unity, $\omega_{2}$ has support near the
intersection $U_{0}\cap U_{1}$. One can show that
$d\omega_{1}^{(0)}=d\omega_{1}^{(1)}$ so that they piece together to a well-
defined $2$-form $\omega_{2}$. We will then drop the superscript when it’s not
relevant to the context. Eq. (62) can be checked explicitly as follows:

$\displaystyle\int_{M}\omega_{2}$ $\displaystyle=$
$\displaystyle\int_{M}d\omega_{1}^{(1)}$ (63) $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\int_{M}d\rho_{0}\wedge dn_{\theta}$
$\displaystyle=$
$\displaystyle-\frac{1}{2\pi}(\rho_{0}(\infty)-\rho_{0}(0))\int dn_{\theta}$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int\sum_{i=1}^{N}\frac{c_{i,i+1}}{N}\delta\left(\theta-\frac{2\pi
c_{i,i+1}}{N}\right)d\theta$ $\displaystyle=$ $\displaystyle 1$

Explicit form of the topological term is given by substituting this $2$-form
$\omega_{2}=d\omega_{1}$ into Eq. (53).

In this example, the coefficient in front of the topological term is $1$ by
construction. In general, there is a coefficient $\kappa$ and the topological
term takes the form:

$\kappa\int A\wedge d\omega_{1}.$ (64)

To show that the coefficient $\kappa$ is quantized, we integrate out the real
space. By construction, we have

$\kappa\int_{M\times S^{1}}A\wedge d\omega_{1}=\kappa\int_{S^{1}}A_{0}dt,$
(65)

which is precisely the effective action of a $0$d particle carrying
charge-$\kappa$. Gauge invariance requires that $\kappa$ is quantized.
Moreover, $\kappa$ is a mod $N$ integer as we now show. From the block
equivalence relation in the topological crystal picture, a state with $0$
charge is equivalent to a state with $N$ charge at the rotational center. This
can be understood as the following deformation process. Starting from a state
with no charge, we bring in $N$ additional charge-$1$ particles to the
rotational center while preserving the $C_{N}$ symmetry (and sending $N$
additional charge-$(-1)$ particles to infinite, which are not relevant to the
bulk property). At the level of field theory, if there are $N$ charge-$1$
particles at the rotational center, the interface is described by a $2$-form
$d\omega_{1}^{\prime}$, in which $c^{\prime}_{i,i+1}=N$. However, this
$2$-form is trivial as one can check as follows:

$\displaystyle\int_{M}d\omega^{\prime}_{1}{}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\sum_{i=1}^{N}\frac{c^{\prime}_{i,i+1}}{N},$ (66)
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\sum_{i=1}^{N}\frac{c^{\prime}_{i,i+1}-dh_{i,i+1}}{N},$
$\displaystyle=$ $\displaystyle 0$

where, in the first equality, we have integrated over the $r$ and $\theta$
directions, and, in the second equality, we have shifted the cocycle
$c^{\prime}_{i,i+1}$ by a coboundary $dh_{i,i+1}=N$. This implies that
$\kappa$ is a mod $N$ integer.

####  IV.2.1 Smooth limit

Here we show that it’s possible to deform the function $n_{\theta}$ to a
smooth function such that the 2-form $\omega_{2}$ is smooth. We begin with the
$1$-form $\omega_{1}$. To go to the smooth limit, the key is to consider the
smooth $1$-form $d\tilde{n}_{\theta}$ of the angular variable
$\tilde{n}_{\theta}$ such that

$\displaystyle\int_{\Sigma_{1}^{\vee}}d\tilde{n}_{\theta}$ $\displaystyle=$
$\displaystyle\int_{\theta_{0}}^{\theta_{0}+2\pi/N}d\tilde{n}_{\theta}$ (67)
$\displaystyle=$
$\displaystyle\tilde{n}_{\theta}\left(\theta_{0}+\frac{2\pi}{N}\right)-\tilde{n}_{\theta}\left(\theta_{0}\right)$
$\displaystyle=$ $\displaystyle\frac{2\pi}{N}\tilde{c},$

where $\tilde{c}\in\mathbb{Z}$ is one-to-one correspondent to $c\in
H^{1}(M,\mathbb{Z})$ in Eq. (54), which is determined by the pullback
$f^{*}\alpha_{1}$ with $\alpha_{1}\in H^{1}(BC_{N},\mathbb{Z}_{N})$. The
smooth $2$-form $\tilde{\omega}_{2}$ is constructed in same way as in the
discontinuous case:

$\tilde{\omega}_{2}=d\tilde{\omega_{1}},$ (68)

where $\tilde{\omega}_{1}=-\rho_{0}d\tilde{n}_{\theta}/2\pi$ with
$\tilde{n}_{\theta}$ satisfying Eq. (67). Substituting this 2-form into Eq.
(64), we obtain the topological term in the smooth limit.

###  IV.3 3d topological crystalline insulators

Here we discuss the effective theories of the 3d topological crystalline
insulators with $U(1)\times G_{pg}$, where $G_{pg}$ is a point group symmetry.
The non-interacting classifications have been obtained in Ref. Okuma  _et al._
, 2019 by computing the Atiyah-Hirzebruch spectral sequence for $K$-homology,
which is the rigorous mathematical framework of the topological crystal
approach for free fermion systems. We will be focusing on one type of the 2nd
order topological insulators Benalcazar  _et al._ (2017); Schindler  _et al._
(2018); Varnava and Vanderbilt (2018); Ahn and Yang (2019), which break time-
reversal symmetry and host protected gapless chiral hinge modes in an open
geometry. To illustrate the basic idea, we will consider $G_{pg}=C_{nv}$ for
$n=1,2,3,4,6$ but our approach can be generalized to other point groups. Such
2nd order phases can be described by the topological crystal pictures of
having some 2d integer quantum hall (IQH) states placing at some high symmetry
planes. With proper open geometry, the protected chiral hinge modes are
directly given by the gapless edge modes of the 2d IQH states Schindler  _et
al._ (2018); Ahn and Yang (2019). It was known that such a type of 2nd order
topological insolators can be well-described by the effective axion field
proposed in Ref. [Qi  _et al._ , 2008] (see also Ref. Fu  _et al._ , 2021). It
coincides with the fact that the 2nd order topological insulators that we
considered can be well-described by the effective axion field, and the chiral
hinge modes are the domain-wall modes between two gapped surface with opposite
half quantum anomalous Hall effect Schindler  _et al._ (2018). In this
section, we will show that the same topological terms can be reproduced by our
approach.

The topological crystal states that we are going to focus on are shown in Fig.
7. Those states are obtained by placing a IQH state at each reflection planes
with the requirement that all the gapless modes at the rotational axis are
gapped out while preserving the symmetry.

Figure 7: The top surface view of the 3d topological crystals built from 2d
IQH states for $C_{nv}$ point groups. Solid lines represent an IQH state on
each mirror plane. The arrows represent the edge chiralities of the IQH
states.

It has been shown in Ref. Okuma  _et al._ , 2019 that the bulk of the 2nd
order phases can be described by the following $3+1$D massive Dirac theory:

$\mathcal{L}=-i\bar{\psi}\gamma^{\mu}\partial_{\mu}\psi-im_{0}\bar{\psi}\psi,$
(69)

where we use the following convention for the gamma matrices:

$\displaystyle\gamma^{0}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}\mathbb{1}&0\\\ 0&-\mathbb{1}\\\
\end{pmatrix}=\tau^{3},$ $\displaystyle\gamma^{i}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&\sigma^{i}\\\ -\sigma^{i}&0\\\
\end{pmatrix}=i\sigma^{i}\tau^{2},$ $\displaystyle\gamma^{5}$ $\displaystyle=$
$\displaystyle\begin{pmatrix}0&\mathbb{1}\\\ \mathbb{1}&0\\\
\end{pmatrix}=\tau^{1}.$ (70)

The reflection and the rotation symmetry in the $C_{nv}$ symmetry acts on the
fermions by

$\displaystyle g_{r}^{y}:\psi(\bm{r})$ $\displaystyle\rightarrow$
$\displaystyle i\sigma^{2}\tau^{3}\psi(g_{r}^{y}\bm{r}),$ $\displaystyle
R:\psi(\bm{r})$ $\displaystyle\rightarrow$ $\displaystyle
e^{\frac{i}{2}\theta\sigma^{3}}\psi(R\bm{r}).$ (71)

Other reflection symmetries in the $C_{nv}$ group can be generated by the
combination of the $y$-reflection $g_{y}$ and the rotation $R$.

To connect the massive Dirac theory to the topological crystal states, we add
the following spatially dependent mass term:

$\mathcal{L}_{m}=-im\bar{\psi}e^{i\phi(\bm{r})\gamma^{5}\tau^{2}}\psi.$ (72)

The reflection and the rotational symmetry acts on the phase variable $\phi$
by

$\displaystyle g_{r}^{y}:\phi(\bm{r})$ $\displaystyle\rightarrow$
$\displaystyle-\phi(g_{r}^{y}\bm{r}).$ $\displaystyle R:\phi(\bm{r})$
$\displaystyle\rightarrow$ $\displaystyle\phi(R\bm{r}).$ (73)

In order for Eq. (72) to be invariant under the symmetry, we must have

$\phi(\bm{r})=-\phi(g_{r}^{y}\bm{r})$ (74)

and

$\phi(\bm{r})=\phi(R\bm{r}).$ (75)

We are going to consider the interface configurations of $\phi$ such that
there is an IQH state at each reflection plane. To simplified the discuss,
let’s first focus on $G_{pg}=C_{1v}$ in which there is only one reflection
symmetry $g_{r}^{y}$. We are going to show that such interfaces of $\phi$ are
classified by $H^{1}(BC_{1v},\mathbb{Z}^{r})$.

Figure 8: (a) The cell-decomposition for 3d systems with reflection symmetry,
and (b) the corresponding dual-cell decomposition.

The cell decomposition given by the fundamental domains is the shown in Fig.
8. There are two $3$-cells $\Sigma_{3,(0)}$, $\Sigma_{3,(1)}$, and one
$2$-cell $\Sigma_{2,(01)}$. Similar to the previous discussion, the map
$f:M\rightarrow BC_{1v}$ maps the dual $0$-cells to the based point of
$BC_{1v}$, and it maps the dual $1$-cell to the non-trivial loop in $BC_{1v}$
labeled by $g_{r}^{y}$.

We consider the configurations of $\phi(\bm{r})$ such that it’s a constant
function within the two $3$-cells $\Sigma_{3,(0)}$, $\Sigma_{3,(1)}$. At the
intersecting $2$-cell $\Sigma_{2,(01)}$, we have the following relation:

$\phi(\Sigma_{3,(1)})=\phi(\Sigma_{3,(0)})+2\pi r(\Sigma_{2,(01)}),$ (76)

where $r(\Sigma_{2,(01)})\in\mathbb{Z}$. The reflection symmetry gives the
following condition on $\phi$:

$\phi(\Sigma_{3,(0)})=-\phi(\Sigma_{3,(1)})$ (77)

There is also a non-trivial symmetry action on the integer-valued function $r$

$g_{r}^{y}\cdot r=-r.$ (78)

However, there is a redundancy since, if we modify the configuration of
$\phi(\bm{r})$ as

$\displaystyle\phi(\Sigma_{3,(i)})$ $\displaystyle\rightarrow$
$\displaystyle\phi(\Sigma_{3,(i)})+2\pi h(\Sigma_{3,(i)}),$ $\displaystyle
r(\Sigma_{2,(01)})$ $\displaystyle\rightarrow$ $\displaystyle
r(\Sigma_{2,(01)})+h(\Sigma_{3,(1)})-h(\Sigma_{3,(0)}),$ (79)

we obtain the same configuration. From these conditions, we see that $r$ is a
$\mathbb{Z}$-valued cocycle in $H^{1}(M,\mathbb{Z}^{r})$ with a twisting
coefficient due to the non-trivial action of the reflection Eq. (78). A
typical configuration of $\phi$ is shown in Fig. 9. Similar to the discussion
in Sec. IV.1, we identify $r$ to be the pull back of $c$: $r=f^{*}c$, where
$c\in H^{1}(BC_{1v},\mathbb{Z}^{r})\cong\mathbb{Z}_{2}$. The deformation class
of such $\phi(\bm{r})$ interfaces is classified by $\mathbb{Z}_{2}$.

Figure 9: A typical configuration of $\phi(\bm{r})$ as a function of $y$.
$\phi(\bm{r})$ has no spatial variation along the $x$-, and $z$-directions.

With this systematic discussion on the interfaces, we are now ready to
discussion the effective field theory. After coupling the fermions to the
$U(1)$ gauge field and integrate out the massive fermions, the effective
theory contains the following topological term

$\displaystyle S=\frac{1}{4\pi}\int AdA\wedge dP,$ (80)

where we have defined $P=\phi/2\pi$. If we consider the non-trivial interface
configuration of $\phi$ given by $r=1$, it’s easy to see that, at the
interface, we have

$S_{\text{Interface}}=\frac{1}{4\pi}\int AdA,$ (81)

as we expected that there is an IQH state at the interface. In Ref. Okuma  _et
al._ , 2019, the explicit solution of the low-energy states that are localized
at the interface is obtained, as well as an effective 2d massive Dirac
Hamiltonian that describes an IQH state with a unit Chern number.

####  IV.3.1 Smooth limit

To go to the smooth limit, we simply replace the discontinuous 1-form $dP$ by
the following smooth $1$-form $d\tilde{P}$ with integral periods satisfying

$\int_{\bm{r}_{0}}^{g_{r}^{y}\bm{r}_{0}}d\tilde{P}=r,$ (82)

where $|\bm{r}_{0}|$ is much larger than the correlation length $\xi$.

The generalization to other $C_{nv}$ symmetry for $n=2,3,4,6$ is
straightforward. In the smooth limit, we have a $1$-form $d\tilde{P}^{(n)}$
with integral periods satisfying Eq. (82) and

$\int_{\bm{r}_{0}}^{R\bm{r}_{0}}d\tilde{P}^{(n)}=0.$ (83)

This comes from the requirement that the gapless modes at the rotational axis
should be gapped out in order for the IQH states to be glued together. The
resulting effective theory is given by

$\displaystyle S=\frac{1}{4\pi}\int AdA\wedge d\tilde{P}^{(n)}.$ (84)

The above action is essentially the same as the action for effective axion
field proposed in Ref. [Qi  _et al._ , 2008], where $\tilde{P}^{(n)}$ serves
as the effective axion field. Therefore, our method indeed reproduce the axion
effective action for the 2nd order topological insulators. Moreover, our
approach shows that the 1-form $d\tilde{P}^{(n)}$ constructed from the axion
field $\tilde{P}^{(n)}$ should be classified by
$H^{1}(BG_{pg},\mathbb{Z}^{r})$ for the 2nd order topological insulators.

####  IV.3.2 Physical responses

For the sake of completeness, we briefly discuss the magnetoelectric responses
for the 2nd order topological insulators governed by $\tilde{P}^{(n)}$. When
$\tilde{P}^{(n)}$ is a constant, the integrand of Eq. (84) is a total
derivative, and thus Eq. (84) cannot have any response. Therefore, we need
certain spatial or temporal dependence in $\tilde{P}^{(n)}$ to get any
responses. Here the spatial dependence in $\tilde{P}^{(n)}$ come from the mass
interfaces in the bulk as discussed in the previous section. In particular, we
focus on the case where $\tilde{P}^{(n)}$ is smooth, and the space is infinite
in order to avoid subtle issues that could appear on the boundary. The
responses we discuss below are essentially the same as in Ref. [Qi  _et al._ ,
2008].

The general form of the magnetoelectric effect is given by the conserved
current

$J^{\mu}=\frac{1}{2\pi}\epsilon^{\mu\nu\lambda\delta}\partial_{\nu}\tilde{P}^{(n)}\partial_{\lambda}A_{\delta}.$
(85)

Specifically, when $\tilde{P}^{(n)}$ is static, the electric current can be
induced by an applied electric field as

$J^{i}=\sigma^{ij}E_{j},$ (86)

with the $3$d Hall conductivity

$\sigma^{ij}=\frac{1}{2\pi}\epsilon^{ijk}\partial_{k}\tilde{P}^{(n)}.$ (87)

We note that the conductivity $\sigma^{ij}$ is not quantized by itself in
general. The quantized quantity is given by

$\frac{1}{2}\int_{\bm{r}_{0}}^{g_{r}^{y}\bm{r}_{0}}\epsilon_{ijk}\sigma^{jk}dx^{i}=\frac{1}{2\pi}\int_{\bm{r}_{0}}^{g_{r}^{y}\bm{r}_{0}}\partial_{i}\tilde{P}^{(n)}dx^{i}=\frac{1}{2\pi}r,$
(88)

which is quantized in the unit of $e^{2}/2\pi h$ if we restore the proper
unit. When $\tilde{P}^{(n)}$ is dynamic yet homogeneous, the electric current
can be induced by an applied magnetic field as

$J^{i}=\frac{1}{2\pi}\partial_{t}\tilde{P}^{(n)}B^{i}\ .$ (89)

##  V Discussion and outlook

In this work, we have proposed a general approach to characterize cSPT phases
by its response to spatially dependent mass parameters with interfaces
configurations. These mass interfaces implement the dimensional reduction
procedure such that the bound states trapped at the interfaces are precisely
the building blocks in the topological crystal picture. To illustrate the main
idea, we have focused on the TCIs with both $U(1)$ charge conservation and the
crystalline symmetry. We have shown that such mass interfaces with codimension
$k$ are classified by $H^{k}(BG_{s},\mathbb{Z})$, and discussed the
corresponding topological terms generated by integrating out the massive
fermions.

One physical correspondence of the spatially dependent mass terms in TCIs is
non-homogeneous lattice distortions or strain Ilan  _et al._ (2020); Yu and
Liu (2021). In the case when the couplings between electrons and the lattice
distortions take the same form as the mass terms in this work, the topological
terms will have a interpretations in the elasticity theory. This point of view
provides a guiding principal on characterizing TCIs through lattice
distortions.

The topological terms discussed in this paper is by no means an exhaustive
list. In particular, these terms are incapable of describing the building
blocks that transform non-trivially under the crystalline symmetry. It will be
desirable to study this kind of terms in the future and matching with the
formal classifications.

When we consider more general topological crystalline phases, it might be the
case that there is no local Lagrangian description for the building block of
interest. A simple example is given by a weak topological crystalline
superconductors protected by the translation symmetry. It can be think of as a
stacking of 1d Kitaev chains. It’s has been known that the Kitaev chain is
characterized by the Arf invariant, which can not be expressed by a local
differential form. One can nevertheless apply the dimensional reduction
procedure by adding a mass term to a $2+1$D Dirac theory with the interfaces
configurations with respective to the translation symmetry. We expect such
mass interfaces are classified by $\tau\in H^{1}(B\mathbb{Z},\mathbb{Z})$,
which can be bullback by the map $f:M\rightarrow B\mathbb{Z}$, giving
$x=f^{*}\tau\in H^{1}(M,\mathbb{Z})$. The result is that there is a 1+1D Dirac
theory describing the Kitaev chain at each mass interface. After integrating
out the fermions, we can write the topological term in the following schematic
form

$\int x\cup\text{Arf}\equiv\text{Arf}(\text{PD}(x)),$ (90)

where $\text{PD}(x)$ denotes a collection of codimension $1$ submanifolds,
which is Poincaré dual to $x=f^{*}\tau\in H^{1}(M,\mathbb{Z})$.
$\text{Arf}(\text{PD}(x))$ denotes the Arf invariant defined on the
submanifolds $\text{PD}(x)$. The codimension $1$ submanifolds $\text{PD}(x)$
are precisely the location of the mass interfaces. These kinds of topological
terms have been considered in Ref. Guo  _et al._ , 2020 from a more formal
point of view. Applying our approach also leads to these kinds of topological
terms naturally, and it will be interesting to study these kinds of terms more
systematically in the future. Finally, we point out that our approach can be
applied to non-invertible topological crystalline phases as well, which
provides a way to study these phases in the continuous field theory framework
while keeping a clear physical picture.

###### Acknowledgements.

S.-J.H. is grateful to Dominic Else, Abhinav Prem, and Andrey Gromov for
related collaborations which inspires this work. S.-J.H. acknowledges support
from a JQI postdoctoral fellowship and the Laboratory for Physical Sciences.
C.-T.H. is supported by JSPS KAKENHI Grant No.19K14608. J. Y. is supported by
the Laboratory for Physical Sciences.

##  Appendix A Review of the cellular cohomology

Let $X$ be a space with a cell decomposition in terms of CW-complexes. A
cellular $k$-chain is a formal linear combination of oriented k-cells with
integer coefficient $\mathbb{Z}$. These generate an abelian group
$C_{k}(X,\mathbb{Z})$. The cellular $k$-cochain is defined to be a map

$\alpha:C_{k}(X,\mathbb{Z})\rightarrow A$ (91)

and these form a group denoted as $C^{k}(X,A)$. The pairing of a $k$-cochain
$\alpha\in C^{k}(X,A)$ and a $k$-cycle $\Gamma\in C_{k}(X,\mathbb{Z})$ is a
map $C^{k}(X,A)\otimes C_{k}(X,\mathbb{Z})\rightarrow\mathbb{R}$, which we
denote by

$\int_{\Gamma}\alpha.$ (92)

The cellular coboundary map

$\delta:C^{k}(X,A)\rightarrow C^{k+1}(X,A)$ (93)

is defined as

$\int_{\Gamma}\delta\alpha=\int_{\partial\Gamma}\alpha.$ (94)

One can show that $\delta^{2}=0$. We denote the kernel of $\delta$ as
$Z^{k}(X,A)$, whose elements are cellular $k$-cocycles, and the image of
$\delta$ in $Z^{k}(X,A)$ as $B^{k}(X,A)$, the group of exact cellular
k-cocycles. The $k$th cellular cohomology of $X$ with coefficients in $A$ is
defined as

$H^{k}(X,A)=Z^{k}(X,A)/B^{k}(X,A).$ (95)

##  Appendix B Construction of the map $f:M\rightarrow BG_{s}$ for general
space groups

Here we give a construction of the map $f:M\rightarrow BG_{s}$. We begin by
the following cell-decomposition of the real-space $M$, which is assumed to be
the euclidean space $\mathbb{E}^{d}$.

Given a space group $G_{s}$, we can partition the euclidean space
$\mathbb{E}^{d}$ into fundamental domains accordingly. A _fundamental domain_
(FD), also know as an _asymmetric unit_ (AU) in crystallography, is a smallest
simply connected closed part of space from which, by application of all
symmetry operations of the space group, the whole of space is filled.
Formally, the partition is written as

$\mathbb{E}^{d}=\bigcup_{g\in G_{s}}g\mathcal{F},$ (96)

where $\mathcal{F}$ is a fundamental domain and $g\mathcal{F}$ its image under
the action of $g\in G_{s}$. If $g$ is not the identity of space group $G_{s}$,
then by definition $\mathcal{F}$ and $g\mathcal{F}$ only intersect in their
surfaces at most. The choice of fundamental domain is often not unique; a
regular choice of fundamental domain for each wallpaper group and 3D space
group is available in the International Tables for Crystallography.

This construction gives the euclidean space $\mathbb{E}^{d}$ a cell
decomposition $\Sigma$. For example, in three dimensions, the 3-cells are the
individual (non-overlapping) copies of FDs. The 2-cells lie on faces where two
3-cells meet, with the property that no two distinct points in the same 2-cell
are related by symmetry. Similarly, 1-cells are edges where two or more faces
meet, and 0-cells are points where edges meet.

The construction of the map $f:M\rightarrow BG_{s}$ is based on the dual cell
decomposition of the one given above. In particular, there is a one-to-one
correspondence between $k$-cells $\Sigma_{k}$ and dual $(d-k)$-cells
$\Sigma^{\vee}_{d-k}$ such that they intersect at a single point. Each
fundamental domain $\mathcal{F}$ then corresponds to a dual $0$-cell
$\Sigma^{\vee}_{0}$ and is labeled by a group element in $G_{s}$. Moreover, a
dual 1-cell $\Sigma^{\vee}_{1}$ connecting a dual $0$-cell $\Sigma^{\vee}_{0}$
(associated to $\mathcal{F}$) to $g\Sigma^{\vee}_{0}$ (associated to
$g\mathcal{F}$) is also labeled by a group element $g\in G_{s}$. The map $f$
is then constructed such that it maps these dual $0$-cells $\Sigma^{\vee}_{0}$
to the base point $\\{*\\}$ in $BG_{s}$, and maps a dual $1$-cell
$\Sigma^{\vee}_{1}$ labeled by $g$ to a link in $BG_{s}$ labeled by the same
$g\in\pi_{1}(BG_{s})$ and so on.

##  Appendix C Classifying space of space groups

Let $\Gamma$ be the translation group in $\mathbb{R}^{d}$ and $P$ the point
group. The $d$-dimensional space group $G_{s}$ fits into a short exact
sequence,

$1\rightarrow\Gamma\rightarrow G\rightarrow P\rightarrow 1.$ (97)

In general, $G_{s}$ is a subgroup of $\mathbb{R}^{d}\rtimes O(d)$. We can
write an element of $G_{s}$ as $(v,P)$ with $v\in\Gamma$ and $p\in P$.

Following Ref. Xiong, 2019, the classifying space of $G_{s}$ can be
constructed as follows. First we note that the classifying space $B\Gamma$ of
$\Gamma$ is the $d$-torus $T^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$. The point
group $P$ has a non-trivial action on $B\Gamma$ by

$p\cdot[v]=[a(p)+pv],$ (98)

where $[v]$ denotes an element in $B\Gamma$ with the representative
$v\in\mathbb{R}^{d}$, and $a$ is a lift

$a:P\rightarrow\mathbb{R}^{d}$ (99)

such that $(a(p),p)\in G_{s}$. For symmorphic space groups, $a$ can be chosen
to be trivial. For nonsymmorphic space groups, $a$ has to be nontrivial. We
have the usual point group action on the contractible universal cover $EP$ of
the classifying space $BP$ of $P$. The classifying space $BG_{s}$ of $G_{s}$
can be construed as

$BG_{s}=B\Gamma\times_{P}EP,$ (100)

where $B\Gamma\times_{P}EP$ denotes the quotient space $(B\Gamma\times EP)/P$.
One can show that this space is the same as

$BG_{s}=E\Gamma\times_{G_{s}}EP,$ (101)

where $G_{s}$ acts on $E\Gamma=\mathbb{R}^{d}$ according to the space-group
action and on $EP$ by first projecting $G_{s}$ to $P$. The space
$E\Gamma\times EP$ is the universal cover since it’s contractable.

In this paper, we only consider the symmetry group of the from $G=G_{s}\times
G_{\text{int}}$, where $G_{\text{int}}$ is the internal symmetry group. Since
$G_{\text{int}}$ has trivial action on $\mathbb{R}^{d}$, the classifying space
of $G$ splits as

$BG=BG_{s}\times BG_{\text{int}}.$ (102)

##  Appendix D The topological term of atomic insulators in two-dimensions

Here we consider the topological terms of higher dimensional atomic
insulators. We will illustrate the main idea in two dimensions, and
generalization to higher-dimensions is straightforward.

The symmetry group of a 2d atomic insulators is $U(1)\times\Gamma$, where
$\Gamma=T_{x}\times T_{y}\cong\mathbb{Z}^{2}$, and the fermion parity is the
$\mathbb{Z}_{2}$ subgroup of $U(1)$. It’s known that the general
classification of is $\mathbb{Z}\times\mathbb{Z}$. One of the $\mathbb{Z}$
factor is the integer quantum hall state, which is not our focus. We will
focus on the other $\mathbb{Z}$ factor, which is the classification of 2d
atomic insulators. The building block picture is having an atom carrying
$U(1)$ charges per unit cell. We will focus on the charge-1 case. We assume
that we have added the ancillas and performing the coarse-graining with
respected to the ancillas lattice. The unit cell size is much larger than the
lattice space of the ancillas lattice. The minimal low energy field theory is
given by a two flavor massive Dirac theory:

$\mathcal{L}=-i\bar{\Psi}\gamma^{\mu}\partial_{\mu}\Psi+im_{0}\bar{\Psi}\sigma_{3}\Psi,$
(103)

where $\Psi=(\psi_{1},\psi_{2})$, and $\sigma^{i}$ are Pauli matrices in the
flavor space. The mass term here guarantees that there is no Chern number.

To obtain the building block picture, we add the following spatially varying
mass term:

$\mathcal{L}=im\bar{\Psi}(n^{1}\sigma_{1}+n^{2}\sigma_{2})\Psi,$ (104)

where $n^{1}$, $n^{2}$ have spatial dependence. Translation symmetries $T_{x}$
and $T_{y}$ require that $n^{i}(\bm{r})=n^{i}(\bm{r}+\bm{a})$ for $i=1,2$. We
would like to choose configurations of $n^{1}$ and $n^{2}$ such that there is
a charge-$1$ bound state at original lattice site. It turns out that it’s
enough to consider $n^{1}$ depends only on $x$ and $n^{2}$ on $y$. The
conditions on $n^{1}$ and $n^{2}$ from the translation symmetries become

$n^{1}(x)=n^{1}(x+1),\;n^{2}(y)=n^{2}(y+1).$ (105)

The generic forms of $n^{1}$ and $n^{2}$ satisfying Eq. (105) are given by

$\displaystyle n^{1}(x)$ $\displaystyle=$ $\displaystyle
x-\frac{1}{2\pi}\theta^{1}(x),$ $\displaystyle n^{2}(y)$ $\displaystyle=$
$\displaystyle y-\frac{1}{2\pi}\theta^{2}(y),$ (106)

where $\theta^{1}(x)$, $\theta^{2}(y)$ are $\mathbb{R}/2\pi\mathbb{Z}$-valued
functions and their values jump by $2\pi$ at the location of atoms. In other
words, under that translations, we have

$\displaystyle\theta^{1}(x+1)$ $\displaystyle=$
$\displaystyle\theta^{1}(x)+2\pi N^{1},\ N^{1}\in\mathbb{Z}\ ,$
$\displaystyle\theta^{2}(y+1)$ $\displaystyle=$
$\displaystyle\theta^{2}(y)+2\pi N^{2},\ N^{2}\in\mathbb{Z}\ .$ (107)

Eq, (107) implies that the parameter space is a 2-torus $T^{2}=S^{1}\times
S^{1}$. If we consider the system with periodic boundary condition, we can
think of the field $\theta$ as a map $\theta:S^{1}\times S^{1}\rightarrow
S^{1}\times S^{1}$.

We now discuss the general classification of the interface configurations of
$\theta^{1}$ and $\theta^{2}$. Let $\Sigma_{2,(x,y)}$ and $\Sigma_{2,(x+1,y)}$
be two neighboring 2-cells that cover the two neighboring unit cell in real
space, related by a translation in the $x$-direction, and similarly,
$\Sigma_{2,(x,y)}$ and $\Sigma_{2,(x,y+1)}$ are two neighboring 2-cells
related by a translation in the $y$-direction. According to Eq, (107), we have

$\displaystyle\theta^{1}(\Sigma_{2,(x+1,y)})$ $\displaystyle=$
$\displaystyle\theta^{1}(\Sigma_{2,(x,y)})+2\pi N^{1}(\Sigma_{1,(x+1,x)}),$
$\displaystyle\theta^{2}(\Sigma_{2,(x,y+1)})$ $\displaystyle=$
$\displaystyle\theta^{2}(\Sigma_{2,(x,y)})+2\pi N^{2}(\Sigma_{1,(y+1,y)}),$
(108)

where $N^{1}\in\mathbb{Z}$, $N^{2}\in\mathbb{Z}$, and $\Sigma_{1,(x+1,x)}$ is
the 1-cell where the two neighboring 2-cells $\Sigma_{2,x+1}$ and
$\Sigma_{2,x}$ meet, and similarly for $\Sigma_{1,(y+1,y)}$. It’s easy to see
that $N^{1}$ and $N^{2}$ satisfy

$\displaystyle
N^{1}(\Sigma_{1,(x_{i},x_{j})})+N^{1}(\Sigma_{1,(x_{j},x_{k})})$
$\displaystyle=$ $\displaystyle N^{1}(\Sigma_{1,(x_{i},x_{k})}),$
$\displaystyle
N^{2}(\Sigma_{1,(x_{i},x_{j})})+N^{2}(\Sigma_{1,(x_{j},x_{k})})$
$\displaystyle=$ $\displaystyle N^{2}(\Sigma_{1,(x_{i},x_{k})}).$ (109)

There is a redundancy since, if we modify the configuration of $\theta^{1}$ as

$\displaystyle\theta^{1}(\Sigma_{2,(x_{i},x_{j})})$ $\displaystyle\rightarrow$
$\displaystyle\theta^{1}(\Sigma_{2,(x_{i},x_{j})})+2\pi
h^{1}(\Sigma_{2,(x_{i},x_{j})}),\
h^{1}(\Sigma_{2,(x_{i},x_{j})})\in\mathbb{Z}$ $\displaystyle
N^{1}(\Sigma_{1,(x_{i}+1,x_{i})})$ $\displaystyle\rightarrow$ $\displaystyle
N^{1}(\Sigma_{1,(x_{i}+1,x_{i})})+h^{1}(\Sigma_{2,(x_{i}+1,x_{j})})-h^{1}(\Sigma_{2,(x_{i},x_{j})})$
(110)

we obtain the same configuration of $\theta^{1}$. There is a similar
redundancy for $N^{2}$ as well. Therefore, we see that $N^{1}$ and $N^{2}$ are
$\mathbb{Z}$-valued cocycles in $H^{1}(M,\mathbb{Z})$. Moreover,
$N^{1}=f^{*}\alpha$ and $N^{2}=f^{*}\beta$ are the pull back of the cocycle
$\alpha\in H^{1}(BT_{x},\mathbb{Z})$ and $\beta\in H^{1}(BT_{y},\mathbb{Z})$.

To obtain the effective field theory, we couple Eq, (103) to a background
$U(1)$ gauge field and integrate out the massive Dirac fermions. According to
Ref. Hsin  _et al._ , 2020, we have the following topological term

$\displaystyle S_{\text{eff}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int\epsilon^{\mu\nu\lambda}A_{\mu}\partial_{\nu}n^{I}\partial_{\lambda}n^{J}\tau_{2,IJ}(n)d^{3}x,$
(111) $\displaystyle=$ $\displaystyle\int A\wedge n^{*}\tau_{2}$

where $\tau_{2}=\frac{1}{2}\tau_{2,IJ}dn^{I}\wedge dn^{J}$ is a $2$-form on
the parameter space. Here, since our parameter space is a 2-torus $T^{2}$
parametrized by $\theta^{I}$, the $2$-form $\tau_{2}$ should be proportional
to the volume form of the $2$-torus:

$\tau_{2}=\frac{1}{8\pi^{2}}\epsilon_{IJ}d\theta^{I}\wedge d\theta^{J}.$ (112)

The topological term becomes

$S_{\text{eff}}=\frac{1}{8\pi^{2}}\int\epsilon^{\mu\nu\lambda}\epsilon_{IJ}A_{\mu}\partial_{\nu}\theta^{I}\partial_{\lambda}\theta^{J}d^{3}x.$
(113)

Requiring that Eq. (113) to be gauge invariant under $A_{\mu}\rightarrow
A_{\mu}+\partial_{\mu}f$, we find that the following current has to be
conserved

$J^{\mu}=\frac{1}{2}\epsilon^{\mu\nu\lambda}\epsilon_{IJ}\partial_{\nu}\theta^{I}\partial_{\lambda}\theta^{J}.$
(114)

To gain more intuition, we focus on the charge density

$\rho=J^{0}=\frac{1}{2}\epsilon^{kl}\epsilon_{IJ}\partial_{k}\theta^{I}\partial_{l}\theta^{J}.$
(115)

When the interface is discontinuous, we have

$\rho=\sum_{i}\delta(x-x_{i})\delta(y-y_{i}).$ (116)

We thus recover the discrete nature of the charge density of an atomic
insulator.

###  D.1 Smooth limit

Now we discuss how to take the smooth limit for the $\theta^{I}$ fields. To
preserve the information of $H^{1}(B\Gamma,\mathbb{Z})$, we consider the
following smooth 1-forms with integral periods:

$\displaystyle\int_{x_{0}}^{x_{0}+1}E^{1}dx$ $\displaystyle=$ $\displaystyle
N^{1}\in\mathbb{Z},$ $\displaystyle\int_{y_{0}}^{y_{0}+1}E^{2}dy$
$\displaystyle=$ $\displaystyle N^{2}\in\mathbb{Z},$ (117)

where we have defined $E^{I}=d\theta^{I}/2\pi$. The smooth limit of the two
form $\tau_{2}$ then takes the form

$\tau_{2}=\frac{1}{2}\epsilon_{IJ}E^{I}\wedge E^{J}.$ (118)

Written in terms of the smooth differential forms, the effective action
becomes

$S_{\text{eff}}=\frac{1}{2}\int\epsilon_{IJ}A\wedge E^{I}\wedge E^{J}.$ (119)

The simplest configurations of $\theta^{I}$ (and hence $E^{I}$) that satisfy
Eq. (D.1) and (117) are given by

$\displaystyle\theta^{1}(x)$ $\displaystyle=$ $\displaystyle 2\pi
N^{1}x=b_{1}N^{1}x,$ $\displaystyle\theta^{2}(y)$ $\displaystyle=$
$\displaystyle 2\pi N^{2}y=b_{2}N^{2}y,$ (120)

where $b_{1}$ and $b_{2}$ are the reciprocal lattice vector of $x$\- and
$y$-directions. This discussion can be straightforwardly generalized to higher
dimensions.

##  Appendix E The topological term of rotation-invariant insulators in two
dimensions

In this section, we show how to derive the topological term of rotation-
invariant insulators in two-dimensions by integrating out fermions. We start
from the following Lagrangian

$\mathcal{L}=i\bar{\psi}\gamma^{\mu}D_{\mu}\psi-
m_{0}\bar{\psi}\psi-\bar{\psi}m\bm{n}\cdot(\gamma^{3},-\mathrm{i}\gamma^{5})\psi,$
(121)

where $D_{\mu}=\partial_{\mu}+\mathrm{i}A_{\mu}$,
$\bm{n}(\bm{r})=(n^{1},n^{2})=n_{r}(\cos n_{\theta},\sin n_{\theta})$, $m_{0}$
is a constant, and the gamma matrices are

$\displaystyle\begin{split}&\gamma^{0}=\tau_{z}\sigma_{z}\ ,\
\gamma^{1}=\mathrm{i}\tau_{y}\sigma_{z}\ ,\
\gamma^{2}=-\mathrm{i}\tau_{x}\sigma_{z}\ ,\
\gamma^{3}=\mathrm{i}\tau_{0}\sigma_{y}\ ,\\\
&\gamma^{5}=\mathrm{i}\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}\ .\end{split}$
(122)

Here the expressions of the gamma matrices differs from those in the main text
by a factor $\mathrm{i}$, but this difference has no influence on the
resultant effective field theory.

Figure 10: The Feynman diagrams for Eq. (LABEL:eq:2D_rotation_Seff). The solid
lines stand for the fermion field.

Performing the Fourier transformation gives the action in the momentum space
as

$S=-\int_{k}\bar{\psi}_{k}G^{-1}(k)\psi_{k}-\int_{k}\int_{q}\bar{\psi}_{k+\frac{q}{2}}\left[A^{\mu}(q)\gamma_{\mu}+m\bm{n}(q)\cdot(\gamma^{3},-\mathrm{i}\gamma^{5})\right]\psi_{k-\frac{q}{2}}\
,$ (123)

where $\mu=0,1,2$, $k=(\omega,\bm{k})$, $\int_{k}=\int d^{3}k/(2\pi)^{3}$, and
$G^{-1}(k)=k_{\mu}\gamma^{\mu}+m_{0}$. Here we use the same Fourier
transformation rule for $\psi$, $A$, and $n$ as

$\psi_{x}=\int_{k}\ e^{\mathrm{i}kx}\psi_{k}\ ,$ (124)

and $kx=k_{\mu}x^{\mu}$. Integrating out the fermions, the $n^{1}n^{2}A$ term
is given by the two diagrams in Fig. 10, and reads

$\displaystyle\begin{split}&-\mathrm{i}\int_{k,q_{1},q_{2}}m^{2}n^{1}(q_{1})n^{2}(-q_{1}-q_{2})A_{\mu}(q_{2})\left(\hbox{$\mathrm{Tr}$}\,[G(k)\gamma^{3}G(k-q_{1})\gamma^{5}G(k+q_{2})\gamma^{\mu}]+\hbox{$\mathrm{Tr}$}\,[G(k-q_{2})\gamma^{5}G(k+q_{1})\gamma^{3}G(k)\gamma^{\mu}]\right)\\\
&=-\int_{k,q_{1},q_{2}}m^{2}n^{1}(q_{1})n^{2}(-q_{1}-q_{2})A_{\mu}(q_{2})8m_{0}\epsilon^{\mu\nu\rho}\frac{q_{1\nu}q_{2\rho}}{(k^{2}+m_{0}^{2})^{3}}+O(q^{3})\\\
&=\mathrm{i}\frac{1}{4\pi}\frac{m^{2}}{m_{0}^{2}}\int
d^{3}x\epsilon^{\mu\nu\rho}\partial_{\nu}n_{1}n_{2}\partial_{\rho}A_{\mu}+...\
.\end{split}$ (125)

Then, the leading order contribution to the corresponding effective action
reads

$S_{eff}=-\frac{1}{4\pi}\frac{m^{2}}{m_{0}^{2}}\int
d^{3}x\epsilon^{\mu\nu\rho}\partial_{\nu}n_{1}\partial_{\rho}n_{2}A_{\mu}\ .$
(126)

Substituting $\bm{n}=n_{r}(\cos(n_{\theta}),\sin(n_{\theta}))$ and
$\kappa=-m^{2}/(4m_{0}^{2})$ into the equation, we arrive at

$\frac{\kappa}{2\pi}\int
d^{3}x\epsilon^{\mu\nu\rho}A_{\mu}\partial_{\nu}n_{r}^{2}\partial_{\rho}n_{\theta}\
.$ (127)

The above expression is Eq. (64) with

$d\omega_{1}=\frac{1}{2\pi}\partial_{\nu}n_{r}^{2}\partial_{\rho}n_{\theta}dx^{\nu}\wedge
dx^{\rho}\ .$ (128)

The quantization of $\kappa$ can be derived from Eq. (65).

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