# Universal topological marker

Wei Chen  Department of Physics, PUC-Rio, 22451-900 Rio de Janeiro, Brazil

###### Abstract

We elaborate that for topological insulators and topological superconductors
described by Dirac models in any dimension and symmetry class, the topological
order can be mapped to lattice sites by a universal topological marker.
Deriving from a recently discovered momentum-space universal topological
invariant, we introduce a topological operator that consists of alternating
projectors to filled and empty lattice eigenstates and the position operators,
multiplied by the Dirac matrices that are omitted in the Hamiltonian. The
topological operator projected to lattice sites yields the topological marker,
whose form is explicitly constructed for every topologically nontrivial
symmetry class from 1D to 3D. The off-diagonal elements of the topological
operator yields a nonlocal topological marker, which decays with a correlation
length that diverges at topological phase transitions, and represents a
Wannier state correlation function. Various prototype examples, including Su-
Schrieffer-Heeger model, Majorana chain, Chern insulators, Bernevig-Hughes-
Zhang model, 2D chiral and helical $p$-wave superconductors, lattice model of
3He B-phase, and 3D time-reversal symmetric topological insulators, etc, are
employed to demonstrate the ubiquity of our formalism.

##  I Introduction

The celebrated topological order in topological insulators (TIs) and
superconductors (TSCs) has been recognized as the principle behind various
seemingly unrelated phenomena in these materialsHasan and Kane (2010); Qi and
Zhang (2011). The topological phases of these materials are characterized by
topological invariants that are derived from the Bloch state of the electrons
or quasiparticles in momentum space, which have been well-understood within
the context of symmetry classification that classifies the Dirac Hamiltonians
of these materials according to their symmetries and dimensionsSchnyder  _et
al._ (2008); Ryu  _et al._ (2010); Kitaev (2009); Chiu  _et al._ (2016). On
the other hand, it has been pointed out that several kinds of topological
invariants can be expressed as real space quantities completely defined from
lattice eigenstates, giving rise to the notion of topological markers. The
earliest and most widely investigated example is the Chern marker in 2D time-
reversal (TR) symmetry-breaking systems, where the corresponding topological
invariant is the Chern number calculated from the momentum-integration of
Berry curvatureBianco and Resta (2011); Prodan  _et al._ (2010); Prodan (2010,
2011). Through rewriting the Berry curvature into projectors to the valence
and conduction bands, which can further be expressed in terms of projectors to
the filled and empty states of the lattice Hamiltonian after a momentum
integration, the diagonal element of the resulting Chern operator on lattice
site ${\bf r}$ yields the correct Chern numberBianco and Resta (2011). Since
the discovery of Chern marker, various topological markers have been proposed
to generalize this concept to other dimensions and symmetry classesLoring and
Hastings (2010); Bianco and Resta (2013); Mondragon-Shem  _et al._ (2014);
Marrazzo and Resta (2017); Cardano  _et al._ (2017); Meier  _et al._ (2018);
Huang and Liu (2018a, b); Focassio  _et al._ (2021); Sykes and Barnett (2021);
Jezequel  _et al._ (2022); Wang  _et al._ (2022); Hannukainen  _et al._
(2022), which have been proved to be a powerful tool to investigate how the
real space inhomogeneity, such as disorder and interfaces, can influence the
topological order locally and globally. In addition, several theoretical
proposals suggest that some topological markers may be measured by real space
experimentsMolignini  _et al._ (2022); Chen (2022).

In this paper, we address two important issues that naturally arises along the
development of topological marker theories: (1) Firstly, is it possible to
formulate a universal topological marker that can be ubiquitously applied to
lattice models of TIs and TSCs in any dimension and symmetry class? This
question is raised because recently, a wrapping number has been proposed as
the universal momentum space topological invariant in any dimension and
symmetry classvon Gersdorff  _et al._ (2021), which has the physical meaning
as the number of times that the Brillouin zone (BZ) torus $T^{D}$ wraps around
the target sphere $S^{D}$ of the Dirac Hamiltonian. Similar to the derivation
of Chern marker from rewriting the Chern number into real spaceBianco and
Resta (2011), we demonstrate that the wrapping number can always be expressed
in real space as the trace of an object that we call the topological operator.
The diagonal element of the topological operator at lattice site ${\bf r}$
then corresponds in a universal topological marker. (2) Secondly, can
topological phase transitions (TPTs) also be detected in real space by some
universal quantity valid for any dimension and symmetry class? This question
arises because the integrand of the wrapping number, which plays the role of
the Jacobian of the aforementioned $T^{D}\rightarrow S^{D}$ map, has a
universal critical behavior in Dirac models, namely it narrows and flips sign
at the gap-closing high symmetry point ${\bf k}_{0}$ as the system crosses
TPTsChen (2016); Chen  _et al._ (2016, 2017); Chen and Sigrist (2019); Chen
and Schnyder (2019). We show that this critical behavior can be detected
ubiquitously by the $({\bf r+R,r})$-th off-diagonal element of the topological
operator that we call the nonlocal topological marker, which is equivalently
the Fourier transform of the Jacobian that narrows and flips sign, and thus
decays in real space with a decay length that diverges at TPTs.

The structure of the paper is organized in the following manner. In Sec. II,
we introduce the formalism that rewrites the wrapping number into projectors
to valence and conduction band states, and how it can further be expressed in
terms of filled and empty lattice eigenstates, yielding the topological
operator. The local and nonlocal topological markers are further introduced as
the diagonal and off-diagonal elements of the topological operator,
respectively, and their interpretations in terms of Wannier states are given.
In Sec. III, IV, and V, we explicitly construct the topological operator for
the topologically nontrivial symmetry classes in 3D, 2D, and 1D, and examine
various prototype lattice models to demonstrate the universal features of the
local and nonlocal topological markers. Section VI summarizes the results, and
lists a numbers of open questions that remain to be explored.

##  II General formalism in any dimension and symmetry class

###  II.1 Topological operators

Our aim is to formulate a real space topological marker for TIs and TSCs in
$D$-dimension described by Dirac Hamiltonian $H={\bf d}({\bf
k})\cdot{\bm{\Gamma}}$, where
$\Gamma_{i}=(\Gamma_{0},\Gamma_{1}...\Gamma_{2n})$ are the $n$-th order Dirac
matrices of dimension $2^{n}\times 2^{n}$ that satisfy
$\left\\{\Gamma_{i},\Gamma_{j}\right\\}=2\delta_{ij}$, and ${\bf d}({\bf
k})=(d_{0},d_{1}...d_{D})$ characterizes the momentum dependence of the
HamiltonianSchnyder  _et al._ (2008); Ryu  _et al._ (2010); Chiu  _et al._
(2016). It is often more convenient to work on the spectrally flattened Dirac
Hamiltonian $\tilde{Q}={\bf n}({\bf k})\cdot{\bm{\Gamma}}$ at momentum ${\bf
k}$, where ${\bf n}={\bf d}/|{\bf d}|$ is a momentum-dependent unit vector.
The precise form of the $\Gamma$-matrices depends on the dimension and
symmetry class of the system at hand. Nevertheless, it has recently been
discovered that all the dimensions and symmetry classes can be described by a
universal topological invariant calculated from momentum-integration of the
cyclic derivative of the ${\bf d}$-vector or ${\bf n}$-vector

$\displaystyle{\rm deg}[{\bf n}]$ $\displaystyle=$
$\displaystyle\frac{1}{V_{D}}\int d^{D}{\bf
k}\,\varepsilon_{i_{0}...i_{D}}\frac{1}{|{\bf
d}|^{D+1}}d^{i_{0}}\partial_{1}d^{i_{1}}...\partial_{D}d^{i_{D}}$ (1)
$\displaystyle=$ $\displaystyle\frac{1}{V_{D}}\int d^{D}{\bf
k}\,\varepsilon_{i_{0}...i_{D}}n^{i_{0}}\partial_{1}n^{i_{1}}...\partial_{D}n^{i_{D}},$

which has been referred to as the wrapping number or degree of the map that
counts the number of times the $T^{D}$ BZ wraps around the unit sphere $S^{D}$
that the ${\bf n}$-vector forms, and the integrand $J_{\bf
k}=\varepsilon_{i_{0}...i_{D}}n^{i_{0}}\partial_{1}n^{i_{1}}...\partial_{D}n^{i_{D}}$
is the Jacobian of the mapvon Gersdorff  _et al._ (2021). Here
$V_{D}=2\pi^{(D+1)/2}/\Gamma(\frac{D+1}{2})$ is the volume of the $D$-sphere
of unit radius, and $\partial_{j}\equiv\partial/\partial k_{j}$. The true
topological invariant is either ${\rm deg}[{\bf n}]$ if the system belongs to
the so-called primary or complex series, $2\,{\rm deg}[{\bf n}]$ for the even
series, and $(-1)^{{\rm deg}[{\bf n}]}$ for the first and second descendants.

We now elaborate that the ${\rm deg}[{\bf n}]$ in Eq. (1) can be expressed as
the momentum integration of the trace ${\rm Tr}[W\tilde{Q}(d\tilde{Q})^{D}]$,
where

$\displaystyle\tilde{Q}(d\tilde{Q})^{D}\equiv\tilde{Q}\partial_{1}\tilde{Q}\partial_{2}\tilde{Q}...\partial_{D}\tilde{Q}.$
(2)

and $W$ is the product of Dirac matrices that are omitted in the Dirac
Hamiltonian for the system at hand, or the identity matrix $W=I$ if all the
Dirac matrices are used. This connection is made because the trace of the
product of all the $n$-th order $\Gamma$-matrices is a constant

$\displaystyle{\rm Tr}\left[\Gamma_{0}\Gamma_{1}...\Gamma_{2n}\right]=2^{n}c,$
(3)

where the prefactor $c=\left\\{1,-1,i,-i\right\\}$ depends on the
representation of the $\Gamma$-matrices for the system at hand. Now suppose
for a specific TI or TSC, the Dirac Hamiltonian uses only
$\left\\{\Gamma_{0},\Gamma_{1},...\Gamma_{D}\right\\}$, leaving
$\left\\{\Gamma_{D+1},\Gamma_{D+2},...\Gamma_{2n}\right\\}$ unused. If we
define the product of all the unused ones to be
$W=\Gamma_{D+1}\Gamma_{D+2}...\Gamma_{2n}$, it then follows that (repeating
indices are summed)

$\displaystyle{\rm Tr}[W\tilde{Q}(d\tilde{Q})^{D}]$ $\displaystyle={\rm
Tr}\left[\Gamma_{D+1}\Gamma_{D+2}...\Gamma_{2n}\Gamma_{i_{0}}\Gamma_{i_{1}}...\Gamma_{i_{D}}\right]n^{i_{0}}\partial_{1}n^{i_{1}}...\partial_{D}n^{i_{D}}$
$\displaystyle=2^{n}c\,\varepsilon_{i_{0}...i_{D}}n^{i_{0}}\partial_{1}n^{i_{1}}...\partial_{D}n^{i_{D}},$
(4)

where the permutation $\varepsilon_{i_{0}...i_{D}}$ comes from the fact that
the $\Gamma$-matrices anticommute, and the trace is nonzero and given by Eq.
(3) only if every $\Gamma$-matrix appears once and only once. Taking a
momentum integration and comparing with Eq. (1), we obtain

$\displaystyle{\rm deg}[{\bf
n}]=\frac{(2\pi)^{D}}{2^{n}c\,V_{D}}\int\frac{d^{D}{\bf k}}{(2\pi)^{D}}{\rm
Tr}[W\tilde{Q}(d\tilde{Q})^{D}].$ (5)

This quantity $\int d^{D}{\bf k}\,{\rm Tr}[W\tilde{Q}(d\tilde{Q})^{D}]$ is our
bridge to a real space topological marker, because it allows to adopt the
projector algebra that originally derives the Chern markerBianco and Resta
(2011). To see this, we observe that the spectrally flattened Hamiltonian
$\tilde{Q}$ can be separated into the projector $p$ into the valence band
states $|n({\bf k})\rangle$ and the projector $q$ into the conduction band
states $|m({\bf k})\rangle$,

$\displaystyle\tilde{Q}=q-p,\;\;\;p=\sum_{n}|n\rangle\langle
n|,\;\;\;q=\sum_{m}|m\rangle\langle m|.$ (6)

and $q+p=I$. As a result, the derivative of $\tilde{Q}$ over a certain
component of momentum $\partial/\partial k_{j}\equiv\partial_{j}$ is
equivalently $\partial_{i}\tilde{Q}=2\partial_{i}q=-2\partial_{i}p$.
Consequently, we can write $\tilde{Q}(d\tilde{Q})^{D}$ into the form that
consists of alternating derivatives of $p$ and $q$.

Consider first $D=$ odd, in which case the integrand of Eq. (5) reads

$\displaystyle W\tilde{Q}(d\tilde{Q})^{D}|_{D\in odd}=2^{D}(-1)^{(D+1)/2}$
$\displaystyle\times
W\left\\{q\,\partial_{1}p\,\partial_{2}q...\partial_{D}p+p\,\partial_{1}q\,\partial_{2}p...\partial_{D}q\right\\}$
$\displaystyle=2^{D}(-1)^{(D+1)/2}\sum_{m_{1}\sim m_{(D+1)/2}}\sum_{n_{1}\sim
n_{(D+1)/2}}$ $\displaystyle W\left\\{|m_{1}\rangle\langle
m_{1}|\partial_{i_{1}}n_{1}\rangle\langle
n_{1}|\partial_{i_{2}}m_{2}\rangle...\right.$ $\displaystyle\left....\langle
m_{(D+1)/2}|\partial_{i_{D}}n_{(D+1)/2}\rangle\langle
n_{(D+1)/2}|+(m\leftrightarrow n)\right\\}.$ (7)

The motivation to rewrite it into this form of alternating $p$ and $q$ is to
use the identityBianco and Resta (2011)

$\displaystyle\langle
m|\partial_{i}|n\rangle=-i\langle\psi_{m}|{\hat{i}}|\psi_{n}\rangle,$ (8)

provided $n\neq m$, where the $|\psi_{n}\rangle=|\psi_{n}({\bf k})\rangle$ is
the full wave function that satisfies $\langle{\bf r}|\psi_{n}({\bf
k})\rangle=\psi_{n{\bf k}}({\bf r})=u_{n{\bf k}}({\bf r})e^{i{\bf k\cdot
r}}=\langle{\bf r}|n({\bf k})\rangle e^{i{\bf k\cdot r}}$ with $u_{n{\bf
k}}({\bf r})$ the Bloch periodic part of the wave function. Here ${\hat{i}}$
is the position operator, which is a diagonal matrix where all the $2^{n}$
internal degrees of freedom (spin, orbit, particle-hole, etc) within a unit
cell located at the Bravais lattice vector ${\bf r}=(x,y,z...)$ are assigned
with the same ${\bf r}$. This identity allows us to write

$\displaystyle\int\frac{d^{D}{\bf k}}{(2\pi)^{D}}{\rm
Tr}[W\tilde{Q}(d\tilde{Q})^{D}]_{D\in odd}$
$\displaystyle=2^{D}i\int\frac{d^{D}{\bf
k}}{(2\pi)^{D}}\varepsilon^{i_{1}i_{2}...i_{D}}\sum_{m_{1}\sim
m_{(D+1)/2}}\sum_{n_{1}\sim n_{(D+1)/2}}$ $\displaystyle{\rm
Tr}\left\\{W|\psi_{m_{1}}\rangle\langle\psi_{m_{1}}|{\hat{i}_{1}}|\psi_{n_{1}}\rangle\langle\psi_{n_{1}}|...\right.$
$\displaystyle\left....\langle\psi_{m_{(D+1)/2}}|{\hat{i}_{D}}|\psi_{n_{(D+1)/2}}\rangle\langle\psi_{n_{(D+1)/2}}|+(m\leftrightarrow
n)\right\\}$ $\displaystyle=2^{D}i{\rm Tr}\left[W\int\frac{d^{D}{\bf
k}_{m_{1}}}{(2\pi)^{D}}\sum_{m_{1}}|\psi_{m_{1}}\rangle\langle\psi_{m_{1}}|{\hat{i}_{1}}\right.$
$\displaystyle\times\int\frac{d^{D}{\bf
k}_{n_{1}}}{(2\pi)^{D}}\sum_{n_{1}}|\psi_{n_{1}}\rangle\langle\psi_{n_{1}}|{\hat{i}_{2}}...$
$\displaystyle...\int\frac{d^{D}{\bf
k}_{m_{(D+1)/2}}}{(2\pi)^{D}}\sum_{m_{(D+1)/2}}|\psi_{m_{(D+1)/2}}\rangle\langle\psi_{m_{(D+1)/2}}|{\hat{i}_{D}}$
$\displaystyle\times\int\frac{d^{D}{\bf
k}_{n_{(D+1)/2}}}{(2\pi)^{D}}\sum_{n_{(D+1)/2}}|\psi_{n_{(D+1)/2}}\rangle\langle\psi_{n_{(D+1)/2}}|$
$\displaystyle\left.+(n\leftrightarrow m)\right].$ (9)

In the last step, we identity the projector to the valence bands integrated
over momentum as the projector to the filled band states $|E_{n}\rangle$ of a
lattice Hamiltonian, and likewisely the projector to the conduction bands
integrated over momentum as the projector to the empty band states
$|E_{m}\rangle$

$\displaystyle\int\frac{d^{D}{\bf
k}_{m_{j}}}{(2\pi)^{D}}\sum_{m_{j}}|\psi_{m_{j}}\rangle\langle\psi_{m_{j}}|=\sum_{m}|E_{m}\rangle\langle
E_{m}|\equiv Q,$ $\displaystyle\int\frac{d^{D}{\bf
k}_{n_{j}}}{(2\pi)^{D}}\sum_{n_{j}}|\psi_{n_{j}}\rangle\langle\psi_{n_{j}}|=\sum_{n}|E_{n}\rangle\langle
E_{n}|\equiv P,$ (10)

which are $\left(L^{D}2^{n}\right)\times\left(L^{D}2^{n}\right)$ matrices,
where $L^{D}$ is the total number of unit cells in the lattice, and each unit
cell contains $2^{n}$ degrees of freedom. Equation (9) then becomes

$\displaystyle\int\frac{d^{D}{\bf k}}{(2\pi)^{D}}{\rm
Tr}[W\tilde{Q}(d\tilde{Q})^{D}]_{D\in odd}$ $\displaystyle=2^{D}i{\rm
Tr}\left[WQ\,{\hat{i}_{1}}P\,{\hat{i}_{2}}...Q\,{\hat{i}_{D}}P+WP\,{\hat{i}_{1}}Q\,{\hat{i}_{2}}...P\,{\hat{i}_{D}}Q\right],$
(11)

where in the second line we have enlarged $W\rightarrow W\otimes
I_{L^{D}\times L^{D}}$. In this way we have written the momentum space
topological invariant into a object that is completely defined from the
eigenstates of a lattice Hamiltonian.

In $D=$ even dimensions, the construction is similar. We again seek to write
the integrand $W\tilde{Q}(d\tilde{Q})^{D}$ into alternating derivatives of $p$
and $q$, which in even dimensions becomes

$\displaystyle W\tilde{Q}(d\tilde{Q})^{D}|_{D\in even}=2^{D}(-1)^{D/2}$
$\displaystyle\times
W\left\\{q\,\partial_{i_{1}}p\,\partial_{i_{2}}q...\partial_{i_{D}}q-p\,\partial_{i_{1}}q\,\partial_{i_{2}}p...\partial_{i_{D}}p\right\\}$
$\displaystyle=2^{D}(-1)^{D/2}\sum_{m_{1}\sim m_{D/2+1}}\sum_{n_{1}\sim
n_{D/2}}$ $\displaystyle W\left\\{|m_{1}\rangle\langle
m_{1}|\partial_{i_{1}}n_{1}\rangle\langle
n_{1}|\partial_{i_{2}}m_{2}\rangle...\right.$ $\displaystyle\left....\langle
n_{D/2}|\partial_{i_{D}}m_{D/2+1}\rangle\langle m_{D/2+1}|-(m\leftrightarrow
n)\right\\},$ (12)

whose momentum integration becomes

$\displaystyle\int\frac{d^{D}{\bf k}}{(2\pi)^{D}}{\rm
Tr}[W\tilde{Q}(d\tilde{Q})^{D}]_{D\in even}$
$\displaystyle=2^{D}\int\frac{d^{D}{\bf k}}{(2\pi)^{D}}\sum_{m_{1}\sim
m_{D/2+1}}\sum_{n_{1}\sim n_{D/2}}$ $\displaystyle{\rm
Tr}\left\\{W|\psi_{m_{1}}\rangle\langle\psi_{m_{1}}|{\hat{i}_{1}}|\psi_{n_{1}}\rangle\langle\psi_{n_{1}}|...\right.$
$\displaystyle\left....\langle\psi_{n_{D/2}}|{\hat{i}_{D}}|\psi_{m_{D/2+1}}\rangle\langle\psi_{m_{D/2+1}}|-(m\leftrightarrow
n)\right\\}$ $\displaystyle=2^{D}{\rm Tr}\left[W\int\frac{d^{D}{\bf
k}_{m_{1}}}{(2\pi)^{D}}\sum_{m_{1}}|\psi_{m_{1}}\rangle\langle\psi_{m_{1}}|{\hat{i}_{1}}\right.$
$\displaystyle\times\int\frac{d^{D}{\bf
k}_{n_{1}}}{(2\pi)^{D}}\sum_{n_{1}}|\psi_{n_{1}}\rangle\langle\psi_{n_{1}}|{\hat{i}_{2}}...$
$\displaystyle...\int\frac{d^{D}{\bf
k}_{n_{D/2}}}{(2\pi)^{D}}\sum_{n_{D/2}}|\psi_{n_{D/2}}\rangle\langle\psi_{n_{D/2}}|{\hat{i}_{D}}$
$\displaystyle\times\int\frac{d^{D}{\bf
k}_{m_{D/2+1}}}{(2\pi)^{D}}\sum_{m_{D/2+1}}|\psi_{m_{D/2+1}}\rangle\langle\psi_{m_{D/2+1}}|$
$\displaystyle-\left.(n\leftrightarrow m)\right].$ (13)

Using the projectors in Eq. (10), we arrive at

$\displaystyle\int\frac{d^{D}{\bf k}}{(2\pi)^{D}}{\rm
Tr}[W\tilde{Q}(d\tilde{Q})^{D}]_{D\in even}$ $\displaystyle=2^{D}{\rm
Tr}\left[WQ\,{\hat{i}_{1}}P\,{\hat{i}_{2}}...P\,{\hat{i}_{D}}Q-WP\,{\hat{i}_{1}}Q\,{\hat{i}_{2}}...Q\,{\hat{i}_{D}}P\right].$
(14)

Once again we have written the topological invariant into a form that is
completely defined from the lattice eigenstates.

Equations (11) and (14) suggest a universal topological operator of the form

$\displaystyle{\hat{\cal
C}}=N_{D}W\left[Q\,{\hat{i}_{1}}P\,{\hat{i}_{2}}...\,{\hat{i}_{D}}{\cal
O}+(-1)^{D+1}P\,{\hat{i}_{1}}Q\,{\hat{i}_{2}}...{\hat{i}_{D}}{\overline{\cal
O}}\right],$ (15)

where the last operators $\left\\{{\cal O},\overline{\cal
O}\right\\}=\left\\{P,Q\right\\}$ if $D=$ odd, and $\left\\{{\cal
O},\overline{\cal O}\right\\}=\left\\{Q,P\right\\}$ if $D=$ even owing to the
alternating order of the projectors $Q$ and $P$. Using Eqs. (5), (11), and
(14), one sees that the wrapping number in Eq. (1) is equal to the trace of
this operator

$\displaystyle{\rm deg}[{\bf n}]=\frac{1}{L^{D}}{\rm Tr}[\,{\hat{\cal C}}\,].$
(16)

where $L^{D}$ is the total number of unit cells, and ${\rm Tr}[...]$
represents the trace over all the lattice sites. The normalization factor
$N_{D}$ in Eq. (15) has the expression

$\displaystyle N_{D}=\frac{i^{D}2^{2D-n}\pi^{D}}{c\,V_{D}},$ (17)

which depends on the dimension $D$, the volume
$V_{D}=\left\\{V_{1},V_{2},V_{3}...\right\\}=\left\\{2\pi,4\pi,2\pi^{2}...\right\\}$,
the order $n$ and the prefactor $c={\rm
Tr}\left[\Gamma_{0}\Gamma_{1}...\Gamma_{2n}\right]/2^{n}=\left\\{1,-1,i,-i\right\\}$
of the representation of $\Gamma$-matrices for the system under question.
Equations (15) to (17) are the central results of this work, and we will
demonstrate their validity using concrete models in the following sections.

###  II.2 Local and nonlocal topological markers

Similar to the original construction of Chern marker as the diagonal elements
of Chern operatorBianco and Resta (2011), the correspondence between the
wrapping number and the trace of the topological operator in Eq. (16) suggests
to define the (local) topological marker on a lattice site ${\bf r}$ by

$\displaystyle{\cal C}({\bf r})=\langle{\bf r}|{\hat{\cal C}}|{\bf
r}\rangle=\sum_{\sigma}\langle{\bf r}\sigma|{\hat{\cal C}}|{\bf
r}\sigma\rangle,$ (18)

i.e., the marker is the diagonal element of topological operator at ${\bf r}$.
Here $\sum_{\sigma}$ represents the summation over all the $2^{n}$ internal
degrees of freedom inside the unit cell at ${\bf r}$, such as spin, orbital,
particle-hole, etc.

We further introduce a nonlocal topological marker as the $({\bf r+R,r})$-th
off-diagonal matrix element of the topological operatorMolignini  _et al._
(2022); Chen (2022)

$\displaystyle{\cal C}({\bf r+R,r})=\langle{\bf r+R}|{\hat{\cal C}}|{\bf
r}\rangle,$ (19)

where ${\bf R}$ is a Bravais lattice vector. For a homogeneous lattice model
in the thermodynamic limit, this nonlocal marker is independent of ${\bf r}$,
and is equivalently the Fourier transform of the integrand of the wrapping
number in Eq. (1)

$\displaystyle{\cal C}({\bf r+R,r})=\tilde{F}({\bf R})$
$\displaystyle\equiv\frac{1}{V_{D}}\int d^{D}{\bf
k}\,\varepsilon_{i_{0}...i_{D}}n^{i_{0}}\partial_{1}n^{i_{1}}...\partial_{D}n^{i_{D}}e^{i{\bf
k\cdot R}}.$ (20)

that has been previously denoted by $\tilde{F}({\bf R})$Chen  _et al._ (2017);
Chen and Schnyder (2019); Chen and Sigrist (2019), and has the physical
meaning as a correlation function that measures the overlap between Wannier
states that are a distance ${\bf R}$ apart, as we shall see in Sec. II.3. The
identification ${\cal C}({\bf r+R,r})=\tilde{F}({\bf R})$ can be seen by
considering equivalently the Fourer transform of ${\rm
Tr}[W\tilde{Q}(d\tilde{Q})^{D}]$, which contains the projection $\langle{\bf
r}|W|\psi_{m_{1}}\rangle e^{i{\bf k\cdot R}}$ of the first ket state of Eqs.
(9) and (13) that is equal to $\langle{\bf r+R}|W|\psi_{m_{1}}\rangle$

$\displaystyle\langle{\bf r}|W|\psi_{m_{1}}\rangle e^{i{\bf k\cdot
R}}=e^{i{\bf k(r+R)}}W\,u_{m_{1}{\bf k}}({\bf r+R})$
$\displaystyle=W\,\psi_{m_{1}{\bf k}}({\bf r+R})=\langle{\bf
r+R}|W|\psi_{m_{1}}\rangle,$ (21)

owing to the cell periodicity of the Bloch state $u_{m_{1}{\bf k}}({\bf
r+R})=u_{m_{1}{\bf k}}({\bf r})$. The correspondence between the momentum
space topological invariants and the real space topological markers is
summarized schematically in Fig. 1.

The spatial profile of the nonlocal marker ${\cal C}({\bf r+R,r})$ decays with
${\bf R}$, with a decay length $\xi$ that diverges at TPTs, thereby serving as
a faithful quantity to identify TPTs. This can be seen by considering the
linear Dirac model $d^{0}=M$, $d^{i\neq 0}=vk_{i}$, which is a low energy
effective model near the gap-closing momentum that describes the majority of
TPTs. The integrand of Eq. (1) has a Lorentzian shape in this model

$\displaystyle\varepsilon_{i_{0}...i_{D}}\frac{1}{|{\bf
d}|^{D+1}}d^{i_{0}}\partial_{1}d^{i_{1}}...\partial_{D}d^{i_{D}}=\frac{M}{\left[M^{2}+v^{2}k^{2}\right]^{(D+1)/2}}$
$\displaystyle\approx\frac{{\rm Sgn}(M)|M|^{-D}}{1+\xi^{2}k^{2}},$ (22)

implying that its Fourier transform ${\cal C}({\bf r+R,r})$ decays with a
correlation length $\xi\sim|M|^{-\nu}$ that diverges at the critical point
$M_{c}=0$, with the critical exponent $\nu=1$. In addition, owing to the
relation between the integrand and the quantum metric of the valence band
stateProvost and Vallee (1980), a relation that has been called the metric-
curvature correspondencevon Gersdorff and Chen (2021), the integrand at ${\bf
k}=0$ that scales like $\sim|M|^{-D}$ has the meaning as the fidelity
susceptibility near TPTsPanahiyan  _et al._ (2020), and hence has been
assigned with the exponent $\gamma=D$.

Figure 1: Summary of the correspondence between the topological order in
momentum space calculated from the Jacobian $J_{\bf
k}=\varepsilon_{i_{0}...i_{D}}n^{i_{0}}\partial_{1}n^{i_{1}}...\partial_{D}n^{i_{D}}$,
where $n^{i}$ characterizes the spectrally flattened Dirac Hamiltonian
$\tilde{Q}={\bf n}\cdot{\bm{\Gamma}}$, and that in real space calculated from
the topological operator ${\cal C}$ constructed from projectors and position
operators. For homogeneous systems in the thermodynamic limit, the wrapping
number is equal to the topological marker ${\rm deg}[{\bf n}]={\cal C}({\bf
r})$, and the Wannier state correlation function is equivalently the nonlocal
topological marker $\tilde{F}({\bf R})={\cal C}({\bf r+R,r})$.

###  II.3 Wannier state representations

We proceed to elaborate that in the homogeneous and thermodynamic limit, both
the local and nonlocal topological markers can be expressed in terms of
overlap of Wannier states. Given the Bloch states $|\ell\rangle=|\ell_{\bf
k}\rangle$ of either the valence $\ell=n$ or conduction $\ell=m$ band states,
we can introduce the Wannier state $|{\bf R}\ell\rangle$ by

$\displaystyle|\ell_{{\bf k}}\rangle=\sum_{{\bf R}}e^{-i{\bf k}\cdot({\hat{\bf
r}}-{\bf R})}|{\bf R}\ell\rangle,\;\;\;|{\bf R}\ell\rangle=\sum_{\bf
k}e^{i{\bf k}\cdot({\hat{\bf r}}-{\bf R})}|\ell_{{\bf
k}}\rangle.\;\;\;\;\;\;\;$ (23)

Inserting these definitions into Eqs. (9) and (13) yields an expression for
the local topological marker

$\displaystyle{\cal C}({\bf r})|_{D\in odd}\propto\int\frac{d^{D}{\bf
k}}{(2\pi)^{D}}{\rm Tr}[W\tilde{Q}(d\tilde{Q})^{D}]_{D\in odd}$
$\displaystyle=2^{D}i\varepsilon^{i_{1}...i_{D}}\sum_{R_{1}\sim
R_{2D+1}}\sum_{m_{1}\sim m_{(D+1)/2}}\sum_{n_{1}\sim n_{(D+1)/2}}$
$\displaystyle\langle{\bf R}_{1}n_{(D+1)/2}|W|{\bf
R}_{2}m_{1}\rangle\langle{\bf R}_{3}m_{1}|{\hat{i}}_{1}|{\bf
R}_{4}n_{1}\rangle...$ $\displaystyle\langle{\bf
R}_{2D+1}m_{(D+1)/2}|{\hat{i}}_{D}|({\bf R}_{1}-{\bf R}_{2}+...+{\bf
R}_{2D+1})n_{(D+1)/2}\rangle$ $\displaystyle+(n\leftrightarrow m),$
$\displaystyle{\cal C}({\bf r})|_{D\in even}\propto\int\frac{d^{D}{\bf
k}}{(2\pi)^{D}}{\rm Tr}[W\tilde{Q}(d\tilde{Q})^{D}]_{D\in even}$
$\displaystyle=2^{D}\varepsilon^{i_{1}...i_{D}}\sum_{R_{1}\sim
R_{2D+1}}\sum_{m_{1}\sim m_{D/2+1}}\sum_{n_{1}\sim n_{D/2}}$
$\displaystyle\langle{\bf R}_{1}m_{D/2+1}|W|{\bf R}_{2}m_{1}\rangle\langle{\bf
R}_{3}m_{1}|{\hat{i}}_{1}|{\bf R}_{4}n_{1}\rangle...$
$\displaystyle\langle{\bf R}_{2D+1}n_{D/2}|{\hat{i}}_{D}|({\bf R}_{1}-{\bf
R}_{2}+{\bf R}_{3}...+{\bf R}_{2D+1})m_{D/2+1}\rangle$
$\displaystyle-(n\leftrightarrow m).$ (24)

Likewisely, the nonlocal topological markers ${\cal C}({\bf r+R,r})$ in the
homogeneous limit in either even or odd dimensions can also be expressed in
terms of Wannier states, which is simply given by the results in Eq. (24) with
the last position argument replaced by $({\bf R}_{1}-{\bf R}_{2}+{\bf
R}_{3}...+{\bf R}_{2D+1})\rightarrow({\bf R}_{1}-{\bf R}_{2}+{\bf
R}_{3}...+{\bf R}_{2D+1}-{\bf R})$. As a result, the nonlocal marker ${\cal
C}({\bf r+R,r})$ has the physical meaning as the measure of the overlap of
Wannier states weighted by the position operators, which decays with ${\bf R}$
according to the argument after Eq. (22). Moreover, the decay length $\xi$
diverges at TPTs, and hence ${\cal C}({\bf r+R,r})$ serves as a faithful
correlator that characterizes the quantum criticality near TPTs.

###  II.4 Applications to lattice models from 1D to 3D

The TIs and TSCs can be classified according to the TR, particle-hole (PH) and
chiral symmetries of the single particle Hamiltonian defined bySchnyder  _et
al._ (2008); Ryu  _et al._ (2010); Kitaev (2009); Chiu  _et al._ (2016)

$\displaystyle TH({\bf k})T^{-1}=H(-{\bf k}),$ $\displaystyle CH({\bf
k})C^{-1}=-H(-{\bf k}),$ $\displaystyle SH({\bf k})S^{-1}=-H({\bf k}),$ (25)

yielding a total of 10 symmetry classes. The result of the classification
gives 5 topologically nontrivial symmetry classes in each spatial dimension
$D$. For practical reasons, in the following sections, we explicitly construct
the topological operators in Eq. (15) for all the 15 nontrivial symmetry
classes from 1D to 3D. Moreover, for those classes described by $2\times 2$
and $4\times 4$ Dirac matrices, which cover 13 out of the 15 nontrivial
classes, we will use cubic lattice models to explicitly demonstrate the
validity of the local and nonlocal topological markers. The two cases left
unexamined are classes CI and CII in 3D described by $8\times 8$ Dirac
matrices, which are less explored in the literature and will be left for
future investigations.

To elaborate the ubiquity of the topological operators and markers, in each
lattice model, we choose periodic boundary condition in all spatial
directions, focus on one specific critical point of the mass term $M_{c}$, and
examine four parameters of $M$ denoted by

$\displaystyle M_{1}:\;\;\;{\rm nontrivial\;phase\;far\;from\;}M_{c}\;({\rm
red})$ $\displaystyle M_{2}:\;\;\;{\rm
nontrivial\;phase\;close\;to\;}M_{c}\;({\rm green})$ $\displaystyle
M_{3}:\;\;\;{\rm trivial\;phase\;close\;to\;}M_{c}\;({\rm blue})$
$\displaystyle M_{4}:\;\;\;{\rm trivial\;phase\;far\;from\;}M_{c}\;({\rm
orange})$ (26)

The purpose of examining these 4 parameters is to elaborate that the behavior
of local and nonlocal topological markers is the same in any dimension and
symmetry class (see Fig. 2, 3, and 4 in the following sections, with the same
color code indicated in Eq. (26)): Deep inside the bulk, $M_{1}$ and $M_{2}$
have the same integer-valued topological marker ${\cal C}({\bf r})$ since they
are in the same topological phase, but the nonlocal topological marker ${\cal
C}({\bf r+R,r})$ of $M_{2}$ has a longer decay length than $M_{1}$ since it is
closer to the critical point according to the discussion after Eq. (22).
Likewisely, $M_{3}$ and $M_{4}$ have the same ${\cal C}({\bf r})$ deep inside
the bulk, but $M_{3}$ has a longer decay length of ${\cal C}({\bf r+R,r})$
than $M_{4}$ since it is closer to the critical point. We also remark that
numerically, we find that the local marker at the boundary sites deviates from
the bulk value even if periodic boundary condition is employed. This is a
feature well-known for this type of constructionBianco and Resta (2011), since
the position operators $\hat{i}$ in Eq. (15) do not respect translational
invariance. Such an anomaly may be fixed by exponentiating the position
operatorProdan  _et al._ (2010); Prodan (2011), which shall be explored
elsewhere.

Figure 2: Local (left column) and nonlocal (right column) topological markers
for lattice models in three out of the five topologically nontrivial symmetry
classes in 3D, including (a) a regularized lattice model for class AIII, (b) a
lattice model of 3He B-phase in class DIII, and (c) the prototype 3D TR-
symmetric TIs in class AII.

##  III Topological markers in three dimensions

In 3D TIs and TSCs, the topological operator has the following general form

$\displaystyle{\hat{\cal
C}}_{3D}=N_{D}W\left[Q\hat{x}P\hat{y}Q\hat{z}P+P\hat{x}Q\hat{y}P\hat{z}Q\right].$
(27)

The $W$ matrix has different interpretations in different classes. The
numerical results for classes AIII, DIII, and AII, are presented in Fig. 2
using prototype cubic lattice models simulated on a 3D lattice of dimension
$L_{x}\times L_{y}\times L_{z}=20\times 8\times 8$. We use a lattice that is
elongated in the ${\hat{\bf x}}$-direction and plot the results along this
direction for the sake of increasing numerical accuracy. The details of each
symmetry class is described below.

###  III.1 3D class AIII

In 3D classes AIII, the $\Gamma$-matrices are given byRyu  _et al._ (2010)

$\displaystyle\Gamma_{1\sim
5}=\left(\alpha_{x},\alpha_{y},\alpha_{z},\beta,-i\beta\gamma^{5}\right),$
$\displaystyle\alpha_{i}=\left(\begin{array}[]{cc}0&\sigma_{i}\\\
\sigma_{i}&0\end{array}\right),\;\;\beta=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right),\;\;\gamma^{5}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right).\;\;\;$ (34)

The chiral operator $S=\beta$ demands $d^{4}({\bf k})=0$, so for a linear
Dirac model one chooses $d_{1}=Ak_{x}$, $d_{2}=Ak_{y}$, $d_{3}=Ak_{z}$, and
$d_{5}=M+B\sum_{\ell=1}^{3}k_{\ell}^{2}$ as the mass term that contains a
quadratic term to avoid fermion doubling. The spinor of 3D class AIII contains
only annihilation operators, which we name them generically as $\psi=(c_{{\bf
k}1},c_{{\bf k}2},c_{{\bf k}3},c_{{\bf k}4})$, where $c_{{\bf k}J}$ is the
$J$-th degree of freedom. Regularizing the linear Dirac model in the whole BZ
by

$\displaystyle A\,k_{\ell}\rightarrow A\sin k_{\ell},$ $\displaystyle
M\rightarrow M+B\sum_{\ell=1}^{D}k_{\ell}^{2}\rightarrow
M+2DB-2B\sum_{\ell=1}^{D}\cos k_{\ell},\;\;\;$ (35)

with $D=3$ in this case, we may further construct a cubic lattice Hamiltonian
by performing a Fourier transform

$\displaystyle\sum_{k}\cos k_{\ell}c_{{\bf k}I}^{{\dagger}}c_{{\bf
k}J}\rightarrow\frac{1}{2}\sum_{i}\left\\{c_{iI}^{{\dagger}}c_{i+\ell
J}+c_{i+\ell I}^{{\dagger}}c_{iJ}\right\\},$ $\displaystyle\sum_{k}i\sin
k_{\ell}c_{{\bf k}I}^{{\dagger}}c_{{\bf
k}J}\rightarrow\frac{1}{2}\sum_{i}\left\\{c_{iI}^{{\dagger}}c_{i+\ell
J}-c_{i+\ell I}^{{\dagger}}c_{iJ}\right\\}.$ (36)

Denoting $t=A/2$ and $t^{\prime}=B$, the resulting lattice model is

$\displaystyle
H=\sum_{i}t\left\\{-ic_{i1}^{{\dagger}}c_{i+x4}+ic_{i+x1}^{{\dagger}}c_{i4}-ic_{i2}^{{\dagger}}c_{i+x3}+ic_{i+x2}^{{\dagger}}c_{i3}\right\\}$
$\displaystyle+\sum_{i}t\left\\{-c_{i1}^{{\dagger}}c_{i+y4}+c_{i+y1}^{{\dagger}}c_{i4}+c_{i2}^{{\dagger}}c_{i+y3}-c_{i+y2}^{{\dagger}}c_{i3}\right\\}$
$\displaystyle+\sum_{i}t\left\\{-ic_{i1}^{{\dagger}}c_{i+z3}+ic_{i+z1}^{{\dagger}}c_{i3}+ic_{i2}^{{\dagger}}c_{i+z4}-ic_{i+z2}^{{\dagger}}c_{i4}\right\\}$
$\displaystyle+\sum_{i\delta}it^{\prime}\left\\{c_{i1}^{{\dagger}}c_{i+\delta
3}+c_{i+\delta 1}^{{\dagger}}c_{i3}+c_{i2}^{{\dagger}}c_{i+\delta
4}+c_{i+\delta 2}^{{\dagger}}c_{i4}\right\\}$
$\displaystyle+\sum_{i}(-iM-i6t^{\prime})\left\\{c_{i1}^{{\dagger}}c_{i3}+c_{i2}^{{\dagger}}c_{i4}\right\\}+h.c.$
(37)

The omitted Dirac matrix is the chiral operator $W=\beta=\Gamma_{4}=S$, and
the normalization factor is $N_{D}=-8\pi i$. The parameters $t=t^{\prime}=0.5$
and
$\left\\{M_{1},M_{2},M_{3},M_{4}\right\\}=\left\\{-0.5,-0.2,0.2,0.5\right\\}$
are used in the numerical simulation.

###  III.2 3D class DIII

A concrete example of 3D class DIII is the B-phase of superfluid 3HeBalian and
Werthamer (1963); Volovik (2009). For the purpose of discussing the 2D class
DIII case after a dimensional reduction, which will be addressed in Sec. IV.5,
we use the representation of $\Gamma$-matrices in the Bernevig-Hughes-Zhang
(BHZ) modelBernevig  _et al._ (2006); König  _et al._ (2007)

$\displaystyle\Gamma^{1\sim 5}=\left\\{s_{x}\otimes\sigma_{z},s_{y}\otimes
I,s_{z}\otimes
I,s_{x}\otimes\sigma_{x},s_{x}\otimes\sigma_{y}\right\\}.\;\;\;$ (38)

The TR, PH, and chiral operators in this basis are $T=-iI\otimes\sigma_{y}K$,
$C=s_{x}\otimes IK$, and $S=\Gamma^{5}$. Since we aim to demonstrate the
topological marker on a lattice, we regularize the pairing terms of the
B-phase of 3He on a square lattice by $\Delta k_{i}\rightarrow\Delta\sin
k_{i}$, and likewisely the kinetic terms, and arrange our spinor according to
Eq. (38) by $\eta_{\bf k}^{{\dagger}}=(c_{\bf
k\uparrow}^{{\dagger}},c_{\bf-k\uparrow},c_{\bf
k\downarrow}^{{\dagger}},c_{\bf-k\downarrow})$. The leads to the
parametrization of the Hamiltonian

$\displaystyle H=\sum_{\bf k}\eta_{\bf
k}^{{\dagger}}\left(\sum_{i=1}^{4}d_{i}\Gamma^{i}\right)\eta_{\bf
k},\;\;\;d_{1}=\Delta\sin k_{x},\;\;\;d_{2}=\Delta\sin k_{y},$ $\displaystyle
d_{3}=2t\left(\cos k_{x}+\cos k_{y}+\cos
k_{z}\right)-\mu,\;\;\;d_{4}=-\Delta\sin k_{z},$ (39)

so the unused Dirac matrix is the chiral operator $W=S=\Gamma^{5}$, and the
normalization factor is $N_{D}=-8\pi i$. We then construct a square lattice
model in a similar manner as Eq. (36), yielding

$\displaystyle
H=\sum_{i\sigma\delta}-t\left(c_{i\sigma}^{{\dagger}}c_{i+\delta\sigma}+c_{i+\delta\sigma}^{{\dagger}}c_{i\sigma}\right)-\mu\sum_{i\sigma}c_{i\sigma}^{{\dagger}}c_{i\sigma}$
$\displaystyle+\sum_{i}\Delta\left(-ic_{i\uparrow}c_{i+x\uparrow}+ic_{i+x\uparrow}^{{\dagger}}c_{i\uparrow}^{{\dagger}}+c_{i\uparrow}c_{i+y\uparrow}+c_{i+y\uparrow}^{{\dagger}}c_{i\uparrow}^{{\dagger}}\right)$
$\displaystyle+\sum_{i}\Delta\left(ic_{i\downarrow}c_{i+x\downarrow}-ic_{i+x\downarrow}^{{\dagger}}c_{i\downarrow}^{{\dagger}}+c_{i\downarrow}c_{i+y\downarrow}+c_{i+y\downarrow}^{{\dagger}}c_{i\downarrow}^{{\dagger}}\right)$
$\displaystyle+\sum_{i}\Delta\left(ic_{i\uparrow}c_{i+z\downarrow}-ic_{i+z\downarrow}^{{\dagger}}c_{i\uparrow}^{{\dagger}}+ic_{i\downarrow}c_{i+z\uparrow}-ic_{i+z\uparrow}^{{\dagger}}c_{i\downarrow}^{{\dagger}}\right).$
(40)

where $\delta=\left\\{x,y,z\right\\}$. We use $t=\Delta=0.5$ and
$\left\\{\mu_{1},\mu_{2},\mu_{3},\mu_{4}\right\\}=\left\\{4,3.5,2.5,2\right\\}$
in the numerical calculation.

###  III.3 3D class AII

The 3D class AII is relevant to prototype TIs such as Bi2Se3 and Bi2Te3. To
draw relevance to real materials, we will use the model for the low energy
sector described by the Dirac matricesZhang  _et al._ (2009); Liu  _et al._
(2010)

$\displaystyle\Gamma_{1\sim
5}=\left\\{\sigma^{x}\otimes\tau^{x},\sigma^{y}\otimes\tau^{x},\sigma^{z}\otimes\tau^{x},I_{\sigma}\otimes\tau^{y},I_{\sigma}\otimes\tau^{z}\right\\}.$

The spinor is $\psi_{\bf k}=\left(c_{{\bf k}s\uparrow},c_{{\bf
k}p\uparrow},c_{{\bf k}s\downarrow},c_{{\bf k}p\downarrow}\right)^{T}$, where
$s$ and $p$ stand for the $P1_{-}^{+}$ and $P2_{+}^{-}$ orbitals in real
materials. The low energy Hamiltonian given by the lowest order term in the
${\bf k\cdot p}$ theory

$\displaystyle\hat{H}=\left(M+M_{1}k_{z}^{2}+M_{2}k_{x}^{2}+M_{2}k_{y}^{2}\right)\Gamma_{5}+B_{0}\Gamma_{4}k_{z}$
$\displaystyle+A_{0}\left(\Gamma_{1}k_{y}-\Gamma_{2}k_{x}\right),$ (42)

can be regularized on a cubic lattice, yieldingChen (2020)

$\displaystyle H=-\sum_{iI\sigma}\mu
c_{iI\sigma}^{{\dagger}}c_{iI\sigma}+\sum_{i\sigma}\tilde{M}\left\\{c_{is\sigma}^{{\dagger}}c_{is\sigma}-c_{ip\sigma}^{{\dagger}}c_{ip\sigma}\right\\}$
$\displaystyle+\sum_{iI}t_{\parallel}\left\\{c_{iI\uparrow}^{{\dagger}}c_{i+a\overline{I}\downarrow}-c_{i+aI\uparrow}^{{\dagger}}c_{i\overline{I}\downarrow}\right.$
$\displaystyle\left.-ic_{iI\uparrow}^{{\dagger}}c_{i+b\overline{I}\downarrow}+ic_{i+bI\uparrow}^{{\dagger}}c_{i\overline{I}\downarrow}+h.c.\right\\}$
$\displaystyle+\sum_{i\sigma}t_{\perp}\left\\{-c_{is\sigma}^{{\dagger}}c_{i+cp\sigma}+c_{i+cs\sigma}^{{\dagger}}c_{ip\sigma}+h.c.\right\\}$
$\displaystyle-\sum_{i\sigma}M_{1}\left\\{c_{is\sigma}^{{\dagger}}c_{i+cs\sigma}-c_{ip\sigma}^{{\dagger}}c_{i+cp\sigma}+h.c.\right\\}$
$\displaystyle-\sum_{i\delta\sigma}M_{2}\left\\{c_{is\sigma}^{{\dagger}}c_{i+\delta
s\sigma}-c_{ip\sigma}^{{\dagger}}c_{i+\delta p\sigma}+h.c.\right\\},$ (43)

where $\tilde{M}=M+2M_{1}+4M_{2}$, $t_{\parallel}=A_{0}/2$,
$t_{\perp}=B_{0}/2$, $I=\left\\{s,p\right\\}$ and
$\overline{I}=\left\\{p,s\right\\}$ are the orbital indices,
$\delta=\left\\{a,b,c\right\\}$ denotes the lattice constants, and
$\sigma=\left\\{\uparrow,\downarrow\right\\}$ is the spin index. From Eq.
(42), it is clear that the Dirac matrix that has not been used is
$W=\Gamma_{3}$, and the normalization factor is $N_{D}=-8\pi i$. In the
numerical calculation, we use the parameters
$t_{\parallel}=t_{\perp}=M_{1}=M_{2}=1$, and four values
$\left\\{-2,-1,1,2\right\\}$ for the mass term $M$ to capture the critical
behavior near $M_{c}=0$.

###  III.4 3D class CII

The minimal model of 3D class CII is a $8\times 8$ Dirac modelSchnyder  _et
al._ (2008), where the seven $\Gamma$-matrices are given byRyu  _et al._
(2010)

$\displaystyle\Gamma^{a}=\Gamma_{4\times 4}^{a}\otimes\eta_{x}\;,\;\;\;{\rm
for}\;a=1\sim 4\;,$ $\displaystyle\Gamma^{5}=I_{4\times
4}\otimes\eta_{y}\;,\;\;\;\Gamma^{6}=I_{4\times 4}\otimes\eta_{z}\;,$
$\displaystyle\Gamma^{7}=(-i)^{3}\Gamma^{1}\Gamma^{2}...\Gamma^{6}\;,$ (44)

where $\Gamma_{4\times 4}^{a}$ are those in Eq. (34). The chiral symmetry is
implemented by $\Gamma^{6}=S$. The Hamiltonian expressed in terms of the other
six $\Gamma$-matrices has a block-off-diagonal form

$\displaystyle H({\bf
k})=\sum_{i=1,2,3,4,5,7}d_{i}\Gamma^{i}=\left(\begin{array}[]{cccc}&&D_{11}&D_{12}\\\
&&D_{21}&D_{22}\\\ D_{11}^{\ast}&D_{12}&&\\\
D_{21}&D_{22}^{\ast}&&\end{array}\right)\;,$ (49) $\displaystyle
D_{11}=\left(\begin{array}[]{cc}h&\\\
&h\end{array}\right)=-D_{22}^{\ast}\;,\;\;\;D_{12}=\left(\begin{array}[]{cc}g&f\\\
f^{\ast}&-g^{\ast}\end{array}\right)=D_{21}^{{\dagger}}.$ (54)

Within linear Dirac model, we consider

$\displaystyle
f=d_{1}-id_{2}=Ak_{x}-iAk_{y},\;\;\;g=d_{3}+id_{7}=d_{3}=Ak_{z},$
$\displaystyle h=d_{4}-id_{5}=d_{4}=M.$ (56)

The Dirac matrices that are omitted in the Hamiltonian are
$\left\\{\Gamma^{5},\Gamma^{6},\Gamma^{7}\right\\}$, so we choose
$W=\Gamma^{5}\Gamma^{6}\Gamma^{7}$, and the normalization factor is
$N_{D}=-4\pi i/c$. Since this symmetry class is less explored in the
literature, and given that complexity involved in this $8\times 8$ model, the
examination of the lattice model is left for future investigations.

###  III.5 3D class CI

For 3D class CI, we resort to the $8\times 8$ Hamiltonian expanded by 4 out of
the 7 Dirac matricesSchnyder  _et al._ (2008)

$\displaystyle H=\sum_{i=1}^{4}d_{i}\Gamma_{i}=\left(\begin{array}[]{cc}&D\\\
D^{{\dagger}}&\end{array}\right),$ (59) $\displaystyle
D=\left(\begin{array}[]{cc}&D_{12}\\\
D_{21}&\end{array}\right)=\left(\begin{array}[]{cccc}&&f^{\ast}&-g\\\
&&-g^{\ast}&-f\\\ -f^{\ast}&g&&\\\ g^{\ast}&f&&\end{array}\right)\;,$ (66)

with $f=d_{1}-id_{2}$ and $g=d_{3}+id_{4}$. The $8\times 8$ TR and PH
operators are $T=I\otimes I\otimes\sigma_{x}K$ and $C=I\otimes
I\otimes(-i\sigma_{y})K$, which require $d_{1}\sim d_{3}$ to be odd in
momentum, and $d_{4}$ to be the mass term that is even in momentum. Note that
we do not need to know the explicit form of the unused
$\Gamma_{5}\sim\Gamma_{7}$ matrices to calculate their product
$W=\Gamma_{5}\Gamma_{6}\Gamma_{7}$, since we know that

$\displaystyle\Gamma_{1}\Gamma_{2}\Gamma_{3}\Gamma_{4}\Gamma_{5}\Gamma_{6}\Gamma_{7}=\Gamma_{1}\Gamma_{2}\Gamma_{3}\Gamma_{4}W=c\,I_{8\times
8}.$ (67)

Because $\Gamma_{1}\Gamma_{2}\Gamma_{3}\Gamma_{4}={\rm diag}(-1,1)_{8\times
8}$, one sees that $W={\rm diag}(-c,c)_{8\times 8}=-cS$ is given by the chiral
operator, and $N_{D}=-4\pi i/c$. The exploration of the lattice model
corresponding to this $8\times 8$ Hamiltonian will be left for further
investigations.

##  IV Topological markers in two dimensions

For 2D TIs and TSCs, the topological operator reads

$\displaystyle{\hat{\cal
C}}_{2D}=N_{D}W\left[Q{\hat{x}}P{\hat{y}}Q-P{\hat{x}}Q{\hat{y}}P\right].$ (68)

We find that there are only two kinds of topological markers in 2D: For
classes A, C, and D that break TR symmetry, the topological marker is the
Chern markerBianco and Resta (2011) described by $W\propto I$. On the other
hand, the TR-symmetric classes AII and DIII are described by the spin Chern
number, yielding a spin Chern marker $W\propto\sigma_{z}$ that counts the
difference between the spin up and down channels. The numerical calculation
using $20\times 20$ lattices for all the 5 nontrivial symmetry classes is
presented in Fig. 3, as detailed below.

Figure 3: Local (left column) and nonlocal (right column) topological markers
for prototype square lattice models in the five topologically nontrivial
symmetry classes in 2D, including (a) the Chern insulator in class A and the
BHZ model in class AII that have identical results, (b) the chiral $p$-wave SC
in class D and helical $p$-wave SC in class DIII that have identical results,
(c) and a quadratic band crossing model in class C.

###  IV.1 2D class A

The minimal model of 2D class A is expanded by all three components of Pauli
matrices $H=\sum_{i=1}^{3}d_{i}\sigma_{i}$, and the model regularized on the
whole BZ

$\displaystyle d_{1}=A\sin k_{x},\;\;\;d_{2}=A\sin k_{y},$ $\displaystyle
d_{3}=M+4B-2B\cos k_{x}-2B\cos k_{y},$ (69)

gives the Chern insulator, whose lattice model has been given previouslyChen
(2020); Molignini  _et al._ (2022). Since all Pauli matrices are used, we have
$W=I$, and the normalization factor is $N_{D}=2\pi i$. The projection to
lattice sites $|{\bf r}\rangle$ is equivalent to the original Chern marker
that have been intensively studiedBianco and Resta (2011), and the off-
diagonal elements has been called nonlocal Chern markerMolignini  _et al._
(2022). Nevertheless, for sake of completeness of the presentation, we perform
simulations using the parameters $t=A/2=1$, $t^{\prime}=B=1$,
$\left\\{M_{1},M_{2},M_{3},M_{4}\right\\}=\left\\{-2,-0.8,0.8,2\right\\}$.

###  IV.2 2D class D

A concrete system that realizes 2D class D is the spinless chiral $p$-wave
SCSchnyder  _et al._ (2008), described by the lattice Hamiltonian

$\displaystyle
H=\sum_{i\delta}t\left(c_{i}^{{\dagger}}c_{i+\delta}+c_{i+\delta}^{{\dagger}}c_{i}\right)-\mu\sum_{i}c_{i}^{{\dagger}}c_{i}$
$\displaystyle+\sum_{i}\Delta\left(-ic_{i}c_{i+x}+ic_{i+x}^{{\dagger}}c_{i}^{{\dagger}}+c_{i}c_{i+y}+c_{i+y}^{{\dagger}}c_{i}^{{\dagger}}\right),$
(70)

where $\delta=\left\\{x,y\right\\}$, and $c_{i}$ is the spinless fermion
operator at site $i$. All three components of Pauli matrices are used, so we
also use $W=I$ and $N_{D}=2\pi i$. The parameters examined are $t=-1$,
$\Delta=0.5$,
$\left\\{\mu_{1},\mu_{2},\mu_{3},\mu_{4}\right\\}=\left\\{-3,-3.7,-4.3,-5\right\\}$.

###  IV.3 2D class C

The minimal model of 2D class C is a $2\times 2$ Dirac modelChen and Schnyder
(2019), where the off-diagonal pairing term around the HSP can be expanded by
$d_{1}-id_{2}=k_{+}^{n_{+}}k_{-}^{n_{-}}$, with $k_{\pm}=k_{x}\pm ik_{y}$. We
consider a spinless model of particle-hole basis $\eta_{\bf
k}^{{\dagger}}=(c_{\bf k}^{{\dagger}},c_{\bf-k})$, where the PH operator
$C=\sigma_{y}K$ requires all $d_{i}({\bf k})$ to be even in momentum, i.e.,
the model has even-order band crossing at TPTs. For concreteness, we choose to
examine the model with the power $n_{+}=0$, $n_{-}=2$, which may be
regularized on a lattice to give the following pairing term

$\displaystyle\Delta_{\bf k}=\Delta(2\cos k_{x}-2\cos k_{y}-2i\sin k_{x}\sin
k_{y}).$ (71)

This leads us to consider the lattice model that contains both nearest- and
next-nearest-neighbor pairings of the same amplitude but a phase difference

$\displaystyle
H=\sum_{i\delta}t\left(c_{i}^{{\dagger}}c_{i+\delta}+c_{i+\delta}^{{\dagger}}c_{i}\right)-\mu\sum_{i}c_{i}^{{\dagger}}c_{i}$
$\displaystyle+\Delta\sum_{i,\sigma=\pm}\left(c_{i}c_{i+\sigma x}+c_{i+\sigma
x}^{{\dagger}}c_{i}^{{\dagger}}-c_{i}c_{i+\sigma y}-c_{i+\sigma
y}^{{\dagger}}c_{i}^{{\dagger}}\right)$
$\displaystyle+\frac{\Delta}{2}\sum_{i}\left\\{-ic_{i}c_{i+x+y}+ic_{i}c_{i+x-y}+ic_{i}c_{i-x+y}-ic_{i}c_{i-x-y}\right.$
$\displaystyle\left.+ic_{i+x+y}^{{\dagger}}c_{i}^{{\dagger}}-ic_{i+x-y}^{{\dagger}}c_{i}^{{\dagger}}-ic_{i-x+y}^{{\dagger}}c_{i}^{{\dagger}}+ic_{i-x-y}^{{\dagger}}c_{i}^{{\dagger}}\right\\}.$
(72)

Since the model already uses all the Dirac matrices, one has $W=I$ and
$N_{D}=2\pi i$. We use the parameters $t=1$, $\Delta=0.5$, and
$\left\\{\mu_{1},\mu_{2},\mu_{3},\mu_{4}\right\\}=\left\\{2,3,4.3,5\right\\}$.
Note that deep inside the bulk and in the topologically nontrivial phase, we
obtain ${\cal C}({\bf r})\approx 2$, consistent with that expected from the
quadratic band crossingChen and Schnyder (2019).

###  IV.4 2D class AII

For 2D class AII that has TR symmetry, we consider the prototype BHZ model
with spinful $s$ and $p$ orbitals
$\psi=\left(s\uparrow,p\uparrow,s\downarrow,p\downarrow\right)^{T}$, which
uses the Dirac matrices of this model already given in Eq. (38).
Interestingly, the unused Dirac matrices combined to give the spin operator
$W=\Gamma_{3}\Gamma_{4}=iI\otimes\sigma^{z}$, and the normalization factor is
found to be $N_{D}=\pi$. The diagonal and off-diagonal elements of this
topological operator has been called local and nonlocal spin Chern markers
previously, whose validity applied to the BHZ model has been elaborated
explicitly. The numerical result is exactly the same as the Chern insulator in
Sec. IV.1 if the same parameters are used, as expected since BHZ model is
equivalently two copies of Chern insulators, one for each spin species.

###  IV.5 2D class DIII

The lattice model of 2D class DIII can be obtained from that of the 3D class
DIII presented in Sec. III.2 via a dimensional reduction, which turns off all
the $\sin k_{z}$ and $\cos k_{z}$ terms in the momentum space Hamiltonian in
Eq. (39), and equivalently all the terms that contain $c_{i+z\sigma}$ or
$c_{i+z\sigma}^{{\dagger}}$ in the lattice Hamiltonian in Eq. (40). Since we
arrange our spinor by $\eta_{\bf k}^{{\dagger}}=(c_{\bf
k\uparrow}^{{\dagger}},c_{\bf-k\uparrow},c_{\bf
k\downarrow}^{{\dagger}},c_{\bf-k\downarrow})$, this results in a Hamiltonian
that is block-diagonal, where each block corresponds to a chiral $p$-wave SC
addressed in Sec. IV.2 for one spin species, and therefore describes a helical
$p$-wave SC.

The 3D class DIII model in Sec. III.2 already omits the $\Gamma^{5}$
component, and the dimensional reduction to 2D turns off the $\Gamma^{4}$
matrix, so the unused $\Gamma$-matrices multiplied together
$W=\Gamma^{3}\Gamma^{5}=iI\otimes\sigma^{z}$ gives the spin polarization
operator, indicating that the topological operator is precisely the spin Chern
operator discussed in Sec. IV.4 with a normalization factor $N_{D}=\pi$.
Physically, this means that the topological invariant is given by the
difference between the Chern number of the spin up chiral $p$-wave SC and that
of the spin down component. As a result, the spin Chern marker of the helical
$p$-wave SC is identical to the Chern marker of the chiral $p$-wave SC given
in Sec. IV.2 at the same parameters.

##  V Topological markers in one dimension

The topological operator in 1D takes the form

$\displaystyle{\hat{\cal C}}_{1D}=N_{D}W\left[Q{\hat{x}}P+P{\hat{x}}Q\right].$
(73)

Interestingly, for all the symmetry classes in 1D that preserve chiral
symmetry (AIII, BDI, CII, DIII), the product of unused Dirac matrices is
always proportional to the chiral operator $W\propto S$, whereas the class D
that does not preserve chiral symmetry has a different interpretation of $W$.
The numerical results for these 5 classes are given in Fig. 4 and are
described in detail below.

Figure 4: Local (left column) and nonlocal (right column) topological markers
for the lattice models in the five topologically nontrivial symmetry classes
in 1D, including (a) the SSH model in class BDI, (b) a regularized lattice
model in class AIII, (c) a class DIII model obtained from dimensional
reduction, (d) the Kitaev $p$-wave SC chain in class D, and (e) a regularized
lattice model in class CII.

###  V.1 1D class BDI

For 1D class BDI, we use the prototype spinless Su-Schrieffer-Heeger (SSH)
model as an example, which is described by the lattice Hamiltonian Su  _et
al._ (1979)

$\displaystyle{\cal H}_{0}$ $\displaystyle=$ $\displaystyle\sum_{i}(t+\delta
t)c_{Ai}^{{\dagger}}c_{Bi}+(t-\delta t)c_{Ai+1}^{{\dagger}}c_{Bi}+h.c.$ (74)

where $c_{Ai}$ and $c_{Bi}$ are the fermion annihilation operators on
sublattice $A$ and $B$ in the unit cell $i$, respectively, and $t\pm\delta t$
are the alternating hopping amplitudes. The $2\times 2$ Hamiltonian expressed
in momentum space with the basis $(c_{Ak},c_{Bk})$ is expanded by $\sigma_{x}$
and $\sigma_{y}$, so the only Pauli matrix that has not been used is the
chiral operator $W=S=\sigma_{z}$, and the normalization factor is unity
$N_{D}=1$. Note that as discussed after Eq. (8), to realize the position
operator ${\hat{x}}$ as a diagonal matrix, the $A$ and $B$ sublattices within
the same unit cell located at $i$ are assigned with the same position $x_{i}$,
even though they are frequently drawn as a certain distance apart. We choose
parameters $t=1$, $\left\\{\delta t_{1},\delta t_{2},\delta t_{3},\delta
t_{4}\right\\}=\left\\{-0.5,-0.2,0.2,0.5\right\\}$ and 20 lattice sites in the
numerical simulation.

###  V.2 1D class AIII

The low energy linear Dirac modelChen and Schnyder (2019) for 1D class AIII
can be expanded by $H({\bf k})=Ak_{x}\sigma_{x}+M\sigma_{z}$, with the unused
Pauli matrix being the chiral symmetry operator $W=S=\sigma_{y}$ and the
normalization factor $N_{D}=1$. The regularization in Eqs. (35) and (36) leads
to a lattice model

$\displaystyle
H=\sum_{i}(M-2t^{\prime})\left(c_{i1}^{{\dagger}}c_{i1}-c_{i2}^{{\dagger}}c_{i2}\right)$
$\displaystyle+\sum_{i}t^{\prime}\left(c_{i1}^{{\dagger}}c_{i+x1}+c_{i+x1}^{{\dagger}}c_{i1}+c_{i2}^{{\dagger}}c_{i+x2}+c_{i+x2}^{{\dagger}}c_{i2}\right)$
$\displaystyle+\sum_{i}t\left(-ic_{i1}^{{\dagger}}c_{i+x2}+ic_{i+x2}^{{\dagger}}c_{i1}-ic_{i2}^{{\dagger}}c_{i+x1}+ic_{i+x1}^{{\dagger}}c_{i2}\right).$

We have used $t=1$, $t^{\prime}=0.4$,
$\left\\{M_{1},M_{2},M_{3},M_{4}\right\\}=\left\\{0.6,0.2,-0.2,-0.6\right\\}$
and 60 lattice sites in the numerical simulation.

###  V.3 1D class DIII

We construct a lattice model of 1D class DIII by performing dimensional
reduction twice on the 3D class DIII model in Sec. III.2, which is done by
turning off all the $\left\\{\sin k_{y},\sin k_{z},\cos k_{y},\cos
k_{z}\right\\}$ terms in Eq. (39), and analogously turning off all the
$\left\\{c_{i+y\sigma},c_{i+y\sigma}^{{\dagger}},c_{i+z\sigma},c_{i+z\sigma}^{{\dagger}}\right\\}$
terms in Eq. (40). The resulting Hamiltonian omits
$\left\\{\Gamma^{2},\Gamma^{4},\Gamma^{5}\right\\}$ matrices defined in Eq.
(38), so $W=\Gamma^{2}\Gamma^{4}\Gamma^{5}=S$ is given by the chiral operator,
and the normalization factor is $N_{D}=i/2$. We use the parameters $t=1$,
$\Delta=0.5$,
$\left\\{\mu_{1},\mu_{2},\mu_{3},\mu_{4}\right\\}=\left\\{1,1.8,2.2,3\right\\}$
and 20 lattice sites in the numerical calculation.

###  V.4 1D class D

For 1D class D, we examine the spinless Kitaev $p$-wave SC chain described
byKitaev (2001)

$\displaystyle
H=\sum_{i}t\left(c_{i}^{{\dagger}}c_{i+1}+c_{i+1}^{{\dagger}}c_{i}\right)-\mu\sum_{i}c_{i}^{{\dagger}}c_{i}$
$\displaystyle+\sum_{i}\Delta\left(c_{i}c_{i+1}+c_{i+1}^{{\dagger}}c_{i}^{{\dagger}}\right),$
(76)

where $c_{i}$ is the spinless fermion annihilation operator at site $i$. The
Hamiltonian in momentum space in the basis of $(c_{k},c_{-k}^{{\dagger}})^{T}$
is spanned by $\sigma_{z}$ and $\sigma_{y}$, so the only Pauli matrix that has
not been used is $W=\sigma_{x}$, and $N_{D}=1$. A 30-site lattice with the
parameters $t=1$, $\Delta=0.5$,
$\left\\{\mu_{1},\mu_{2},\mu_{3},\mu_{4}\right\\}=\left\\{1,1.8,2.2,3\right\\}$
is used in the numerical simulation.

###  V.5 1D class CII

For 1D class CII, we adopt the $\Gamma$-maticesZhao and Wang (2014)

$\displaystyle\Gamma^{a}=\left\\{\sigma_{x}\otimes\tau_{z},\sigma_{y}\otimes\tau_{z},I\otimes\sigma_{x},I\otimes\tau_{y},\sigma_{z}\otimes\tau_{z}\right\\}\;,$
(77)

and the TR and PH operators are interpreted by $T=\sigma_{y}\otimes IK$ and
$C=I\otimes\tau_{y}K$. The minimal model in momentum space is
$H=d_{2}\Gamma^{2}+d_{3}\Gamma^{3}=Ak\Gamma^{2}+M\Gamma^{3}$, and we denote
the spinor by
$\psi_{k}^{{\dagger}}=(c_{1k}^{{\dagger}},c_{2k}^{{\dagger}},c_{3k}^{{\dagger}},c_{4k}^{{\dagger}})$
where the four degrees of freedom are enumerated by $1\sim 4$. The
regularization on a lattice gives

$\displaystyle
H=\sum_{i}t\left\\{-c_{i1}^{{\dagger}}c_{i+x2}+c_{i2}^{{\dagger}}c_{i+x1}+c_{i3}^{{\dagger}}c_{i+x4}-c_{i4}^{{\dagger}}c_{i+x3}\right\\}$
$\displaystyle+\sum_{i}(-t^{\prime})\left\\{c_{i1}^{{\dagger}}c_{i+x3}+c_{i3}^{{\dagger}}c_{i+x1}+c_{i2}^{{\dagger}}c_{i+x4}+c_{i4}^{{\dagger}}c_{i+x2}\right\\}.$
$\displaystyle+\sum_{i}(M+2t^{\prime})\left\\{c_{i1}^{{\dagger}}c_{i3}+c_{i2}^{{\dagger}}c_{i4}\right\\}+h.c.$
(78)

The unused Dirac matrices multiplied together is proportional to the chiral
operator $W=\Gamma^{1}\Gamma^{4}\Gamma^{5}=-iS$, and the normalization factor
is $N_{D}=i/2$. We use the parameters $t=1$, $t^{\prime}=0.5$,
$\left\\{M_{1},M_{2},M_{3},M_{4}\right\\}=\left\\{-1,-1.8,-2.2,-3\right\\}$
and a 30-site lattice in the simulation.

##  VI Conclusions

In summary, we show that topological marker can be constructed in a unified
manner for TIs and TSCs in any dimension and symmetry class. The central
object in our formalism is the topological operator in Eq. (15) derived from
the universal topological invariant in momentum space, which takes the form of
alternating projectors to the lattice eigenstates and the position operators,
multiplied by the Dirac matrices that are omitted in the Hamiltonian. The
${\bf r}$-th diagonal element of the topological operator gives the local
topological marker that recovers the topological invariant for lattice sites
deep inside the bulk. In addition, the $({\bf r+R,r})$-th off-diagonal element
yields a nonlocal topological marker that decays with ${\bf R}$, whose decay
length diverges at TPT and may be interpreted as a Wannier state correlation
function, thereby serving as a faithful correlator to identify TPTs in real
space. The topological operator is constructed explicitly for each of the 15
topologically nontrivial symmetry classes in 1D to 3D. For 13 out of these 15
cases, we perform numerical calculation on concrete lattice models to
demonstrate the validity of our topological marker, which cover a great number
of prototype TIs and TSCs including the SSH model, Majorana chain, Chern
insulator, BHZ model, chiral and helical $p$-wave SCs, 3D TR invariant TIs,
lattice model of 3He B-phase, among many others, suggesting the ubiquity of
our formalism.

Our results point to many open questions that remain to be clarified. Firstly,
it is known that the deviation of the Chern marker at the boundary sites of
the 2D lattice, which occur even if periodic boundary condition is imposed,
comes from the fact that the position operators $\hat{i}$ in Eq. (15) do not
respect the translational invariance, and may be cured by exponentiating the
position operatorsProdan  _et al._ (2010); Prodan (2010, 2011). Whether such
an exponentiating trick can be generally applied to topological markers in any
dimension and symmetry class, and whether it has some non-commutative
interpretation on the topological order in general, remain to be investigated.
Secondly, a major category of topological materials that we did not address
are the topological semimetals in $D$-dimension, such as graphene in 2D, in
which the momentum space topological invariant is the wrapping number in Eq.
(1) but integrated over a $(D-1)$-dimensional surface enclosing a nodal point.
Because the momentum integration is one dimension lower, it is unclear to us
at present whether the projector algebra in Sec. II.1 still applies, or how it
may be modified to construct a topological marker for semimetals. Thirdly,
concerning the experimental measurement of the topological marker, it has been
pointed out that the Chern marker in 2D TR-breaking systems can be measured by
circular dichroismMolignini  _et al._ (2022), and the spin-Chern marker in 2D
TR-symmetric systems can be detected by spin-resolved circular dichroismChen
(2022), both are due to the linear response of valence electrons to polarized
electric field. However, because TIs and TSCs in other dimensions do not
respond linearly to the electric fieldBernevig and Hughes (2013), it remains
to be investigated whether higher order responses can help to extract the
topological marker in other dimensions, or if one has to resort to some other
kinds of experimental protocol. Finally, an obvious question is whether our
universal topological marker can still describe systems that are beyond the
paradigm of Dirac models, such as 2D class AII systems with spin-orbit
couplingKane and Mele (2005). All these open questions, together with the
applications of the universal topological marker on issues such as real space
inhomogeneity and topological quantum criticality, are intriguing subjects
that await to be explored.

## References

  * Hasan and Kane (2010) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010).
  * Qi and Zhang (2011) X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
  * Schnyder  _et al._ (2008) A. P. Schnyder, S. Ryu, A. Furusaki, and A. W. W. Ludwig, Phys. Rev. B 78, 195125 (2008).
  * Ryu  _et al._ (2010) S. Ryu, A. P. Schnyder, A. Furusaki, and A. W. W. Ludwig, New J. Phys. 12, 065010 (2010).
  * Kitaev (2009) A. Kitaev, AIP Conf. Proc. 1134, 22 (2009).
  * Chiu  _et al._ (2016) C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Rev. Mod. Phys. 88, 035005 (2016).
  * Bianco and Resta (2011) R. Bianco and R. Resta, Phys. Rev. B 84, 241106 (2011).
  * Prodan  _et al._ (2010) E. Prodan, T. L. Hughes, and B. A. Bernevig, Phys. Rev. Lett. 105, 115501 (2010).
  * Prodan (2010) E. Prodan, New J. Phys. 12, 065003 (2010).
  * Prodan (2011) E. Prodan, J. Phys. A: Math. Theor. 44, 113001 (2011).
  * Loring and Hastings (2010) T. A. Loring and M. B. Hastings, EPL 92, 67004 (2010).
  * Bianco and Resta (2013) R. Bianco and R. Resta, Phys. Rev. Lett. 110, 087202 (2013).
  * Mondragon-Shem  _et al._ (2014) I. Mondragon-Shem, T. L. Hughes, J. Song, and E. Prodan, Phys. Rev. Lett. 113, 046802 (2014).
  * Marrazzo and Resta (2017) A. Marrazzo and R. Resta, Phys. Rev. B 95, 121114 (2017).
  * Cardano  _et al._ (2017) F. Cardano, A. D’Errico, A. Dauphin, M. Maffei, B. Piccirillo, C. de Lisio, G. De Filippis, V. Cataudella, E. Santamato, L. Marrucci, M. Lewenstein, and P. Massignan, Nat. Commun. 8, 15516 (2017).
  * Meier  _et al._ (2018) E. J. Meier, F. A. An, A. Dauphin, M. Maffei, P. Massignan, T. L. Hughes, and B. Gadway, Science 362, 929 (2018).
  * Huang and Liu (2018a) H. Huang and F. Liu, Phys. Rev. B 98, 125130 (2018a).
  * Huang and Liu (2018b) H. Huang and F. Liu, Phys. Rev. Lett. 121, 126401 (2018b).
  * Focassio  _et al._ (2021) B. Focassio, G. R. Schleder, F. Crasto de Lima, C. Lewenkopf, and A. Fazzio, Phys. Rev. B 104, 214206 (2021).
  * Sykes and Barnett (2021) J. Sykes and R. Barnett, Phys. Rev. B 103, 155134 (2021).
  * Jezequel  _et al._ (2022) L. Jezequel, C. Tauber, and P. Delplace, arXiv:2203.17099 (2022).
  * Wang  _et al._ (2022) C. Wang, T. Cheng, Z. Liu, F. Liu, and H. Huang, Phys. Rev. Lett. 128, 056401 (2022).
  * Hannukainen  _et al._ (2022) J. D. Hannukainen, M. F. Martinez, J. H. Bardarson, and T. K. Kvorning, arXiv:2207.01646 (2022).
  * Molignini  _et al._ (2022) P. Molignini, B. Lapierre, R. Chitra, and W. Chen, arXiv:2207.00016 (2022).
  * Chen (2022) W. Chen, arXiv:2207.02973 (2022).
  * von Gersdorff  _et al._ (2021) G. von Gersdorff, S. Panahiyan, and W. Chen, Phys. Rev. B 103, 245146 (2021).
  * Chen (2016) W. Chen, J. Phys. Condens. Matter 28, 055601 (2016).
  * Chen  _et al._ (2016) W. Chen, M. Sigrist, and A. P. Schnyder, J. Phys. Condens. Matter 28, 365501 (2016).
  * Chen  _et al._ (2017) W. Chen, M. Legner, A. Rüegg, and M. Sigrist, Phys. Rev. B 95, 075116 (2017).
  * Chen and Sigrist (2019) W. Chen and M. Sigrist,  _Advanced Topological Insulators, Ch. 7_ (Wiley-Scrivener, 2019).
  * Chen and Schnyder (2019) W. Chen and A. P. Schnyder, New J. Phys. 21, 073003 (2019).
  * Provost and Vallee (1980) J. P. Provost and G. Vallee, Comm. Math. Phys. 76, 289 (1980).
  * von Gersdorff and Chen (2021) G. von Gersdorff and W. Chen, Phys. Rev. B 104, 195133 (2021).
  * Panahiyan  _et al._ (2020) S. Panahiyan, W. Chen, and S. Fritzsche, Phys. Rev. B 102, 134111 (2020).
  * Balian and Werthamer (1963) R. Balian and N. R. Werthamer, Phys. Rev. 131, 1553 (1963).
  * Volovik (2009) G. E. Volovik,  _The Universe in a Helium Droplet_ (Oxford University Press, 2009).
  * Bernevig  _et al._ (2006) B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Science 314, 1757 (2006).
  * König  _et al._ (2007) M. König, S. Wiedmann, C. Brüne, A. Roth, H. Buhmann, L. W. Molenkamp, X.-L. Qi, and S.-C. Zhang, Science 318, 766 (2007).
  * Zhang  _et al._ (2009) H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Nat. Phys. 5, 438 (2009).
  * Liu  _et al._ (2010) C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and S.-C. Zhang, Phys. Rev. B 82, 045122 (2010).
  * Chen (2020) W. Chen, Phys. Rev. B 101, 195120 (2020).
  * Su  _et al._ (1979) W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett. 42, 1698 (1979).
  * Kitaev (2001) A. Y. Kitaev, Phys. Usp. 44, 131 (2001).
  * Zhao and Wang (2014) Y. X. Zhao and Z. D. Wang, Phys. Rev. B 90, 115158 (2014).
  * Bernevig and Hughes (2013) B. A. Bernevig and T. L. Hughes,  _Topological Insulators and Topological Superconductors_ (Princeton University Press, 2013).
  * Kane and Mele (2005) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005).