# Measurement of Topological Order based on Metric-Curvature Correspondence

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

###### Abstract

A unified expression for topological invariants has been proposed recently to
describe the topological order in Dirac models belonging to any dimension and
symmetry class. We uncover a correspondence between the curvature function
that integrates to this unified topological invariant and the quantum metric
that measures the distance between properly defined many-body Bloch states in
momentum space. Based on this metric-curvature correspondence, a time-resolved
and angle-resolved photoemission spectroscopy experiment is proposed to
measure the violation of spectral sum rule caused by a pulse electric field to
detect the quantum metric, from which the topological properties of the system
may be extracted.

##  I Introduction

Topological order in materials differs from the usual Landau order parameters
in that it does not require breaking a continuous symmetry, but rather
represents a certain geometric property of the Bloch state in momentum space.
In addition, topological order manifests as different physical phenomena
according to the dimension and symmetry of the systemSchnyder  _et al._
(2008); Ryu  _et al._ (2010); Kitaev (2009); Chiu  _et al._ (2016), such as
quantized Hall conductanceThouless  _et al._ (1982); Niu  _et al._ (1985) and
Majorana fermionsKitaev (2001); Oreg  _et al._ (2010); Lutchyn  _et al._
(2010), many of which rely on the metallic edge state that only exists at the
boundary and in topologically nontrivial phases. Nevertheless, drawing analogy
with Landau order parameters, an intriguing question is whether a bulk
spectroscopy method exists to measure the geometric property of the Bloch
state, through which the topological order can be detected. Should such a bulk
spectroscopy exist ubiquitously for any topological insulator (TI) and
topological superconductor (TSC), the aforementioned features seem to suggest
that the detection principle cannot rely on the existence of local order
parameter or edge state, which rules out many existing methods.

Two recent progresses shed a light on this issue. The first is the recognition
that for topological materials described by two-band Dirac models, the modulus
of the Berry connection or Berry curvature that integrates to the topological
invariant is equal to the quantum metricProvost and Vallee (1980); Berry
(1989) that measures the distance between single-particle Bloch state in
momentum spaceMa, Yu-Quan  _et al._ (2013); Kolodrubetz  _et al._ (2013); Ma
(2014); Yang  _et al._ (2015); Piéchon  _et al._ (2016); Kolodrubetz  _et al._
(2017); Ozawa and Goldman (2018); Palumbo and Goldman (2018); Palumbo (2018);
Lapa and Hughes (2019); Yu  _et al._ (2019); Chen  _et al._ (2020); Ma (2020);
Salerno  _et al._ (2020); Lin and Hsiao (2021). As a result, direct
measurement to the quantum metric may yield information about the topological
order. The second is that all the topological invariants for Dirac models in
any dimension and symmetry classSchnyder  _et al._ (2008); Ryu  _et al._
(2010); Kitaev (2009); Chiu  _et al._ (2016) can be unified into a single
formula called wrapping number, which counts how many times the Brillouin zone
(BZ) wraps around a target sphere induced by the Dirac Hamiltonianvon
Gersdorff  _et al._ (2021). In particular, the wrapping number is calculated
from integrating the cyclic derivative of the components of the Dirac
Hamiltonian, which we refer to as the curvature function. In two-band systems,
the curvature function is simply the Berry connection or Berry curvature.

The goal of this Letter is to demonstrate that this equivalence between the
modulus of the Berry connection or Berry curvature and the quantum metric in
fact holds for Dirac models in any dimension and symmetry class. We elaborate
that the modulus of the curvature function that integrates to the wrapping
number corresponds to the quantum metric of a properly defined many-body Bloch
state, a relation that we call metric-curvature correspondence. Motivated by
this correspondence, we generalized a previously proposed measurement protocol
for the single-particle quantum metricOzawa and Goldman (2018) to degenerate
bands, and propose a time-resolved and angle-resolved photoemission
spectroscopy (trARPES) measurementHajlaoui  _et al._ (2013); Sobota  _et al._
(2014a); Lv  _et al._ (2019); Sobota  _et al._ (2014b); Wang  _et al._ (2012)
that detects the violation of spectral sum rule caused by a pulse electric
field as a universal spectroscopy to probe the topological property of
materials, using graphene as a concrete example.

##  II Metric-curvature correspondence

###  II.1 Theoretical formalism

We consider the TIs and TSCs described by the Hamiltonian and Bloch
eigenstates $H({\bf k})|\psi_{n}({\bf k})\rangle=\epsilon_{n}({\bf
k})|\psi_{n}({\bf k})\rangle$, and focus on the filled valence band or
fermionic quasiparticle states with $\epsilon_{n}({\bf k})<0$. The $D$
dimensional BZ $T^{D}$ is parametrized by Cartesian coordinates $k^{\mu}$ with
$\mu=1,2...D$, where the Einstein notation is used throughout the article. For
a generic single or many particle Bloch state $\ket{\psi({\bf k})}$ the
overlap of this eigenstate at ${\bf k}$ with itself at a slightly different
momentum $|\langle\psi({\bf k})|\psi({\bf k}+\delta{\bf
k})\rangle|=1-\frac{1}{2}g_{\mu\nu}\delta k^{\mu}\delta k^{\nu}$ defines the
quantum metric tensor of that state Provost and Vallee (1980)

$\displaystyle g^{\psi}_{\mu\nu}({\bf k})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\langle\partial_{\mu}\psi|\partial_{\nu}\psi\rangle+\frac{1}{2}\langle\partial_{\nu}\psi|\partial_{\mu}\psi\rangle$
(1)
$\displaystyle-\langle\partial_{\mu}\psi|\psi\rangle\langle\psi|\partial_{\nu}\psi\rangle.$

which is invariant under a local $U(1)$ gauge rotation $\ket{\psi({\bf k})}\to
e^{i\phi({\bf k})}\ket{\psi({\bf k})}$. We remark that this metric tensor on
the BZ is the one inherited from the well known Fubini-Study metric on state
space 111 For instance, in the case of one-particle Bloch states, this state
space is $\mathbb{CP}^{N-1}$. Technically, $g_{\mu\nu}^{\psi}$ is the pullback
of the Fubini-Study metric over the map $\ket{\psi({\bf k})}$. .

For reasons that will become transparent in a moment, we will consider the
quantum metric constructed from the following $N_{-}$-particle Bloch state

$\ket{\psi^{\rm val}({\bf
k})}\equiv\frac{1}{\sqrt{N_{-}!}}\epsilon^{a_{1}\cdots
a_{N_{-}}}\ket{u^{-}_{a_{1}}}\ket{u^{-}_{a_{2}}}\cdots\ket{u^{-}_{a_{N_{-}}}}$
(2)

where the $\ket{u^{-}_{a}({\bf k})}$ form a basis of the $N_{-}$ filled bands
(with negative energy). The metric for this state has also been considered
previously in Ref.Matsuura and Ryu, 2010. We can interpret this state as the
Fermi sea for fixed ${\bf k}$. Notice that when two or more of these negative
energy states are degenerate (as will be the case in the Dirac models to be
considered below), the basis is only defined modulo a ${\bf k}$ dependent
”gauge rotation” $U({\bf k})\in U(N_{-})$. Under this non-Abelian gauge
transformation the state $\ket{\psi^{\rm val}}$ transforms with $\det U({\bf
k})$ which is a pure phase, thus rendering the metric tensor completely
invariant under such a basis redefinition.

Physically, the quantum metric $g^{\rm val}_{\mu\nu}$ of the state
$\ket{\psi^{\rm val}}$ defined by Eq. (1) measures how much the unit vector
$\ket{\psi^{\rm val}}$ has rotated in the $N_{-}$–particle Hilbert space as
one moves from ${\bf k}$ to ${\bf k}+\delta{\bf k}$. It is possible to express
$g^{\rm val}_{\mu\nu}$ explicitly in terms of one-particle states using
standard techniques from second quantization. The result is

$g^{\rm val}_{\mu\nu}({\bf
k})=\frac{1}{2}\sum_{a}\biggl{(}\langle\partial_{\mu}u_{a}^{-}|Q_{+}|\partial_{\nu}u_{a}^{-}\rangle+\langle\partial_{\nu}u_{a}^{-}|Q_{+}|\partial_{\mu}u_{a}^{-}\rangle\biggr{)}$
(3)

where we have defined the projectors onto the positive and negative
eigenstates:

$\displaystyle
Q_{\pm}\equiv\sum_{a=1}^{N_{\pm}}\ket{u^{\pm}_{a}}\bra{u^{\pm}_{a}}\,,$ (4)

which satisfy $Q_{+}+Q_{-}=1$ and $Q_{\pm}^{2}=Q_{\pm}$. We futher define the
spectrally flattened Hamiltonian $Q({\bf k})\equiv Q_{+}({\bf k})-Q_{-}({\bf
k})$, which has the same eigenstates as $H({\bf k})$ but with eigenvalues $\pm
1$. The Hamiltonians $H({\bf k})$ and $Q({\bf k})$ can be continuously
deformed into each other without closing the band gap, and thus have identical
topological properties. We remark that $Q({\bf k})$ takes values in the
complex Grassmannian $\frac{U(N)}{U(N_{+})\times U(N_{-})}$, since it can be
specified by a diagonalizing unitary matrix $\in U(N)$ with two such matrices
yielding the same $Q$ if they differ by a gauge transformation $\in
U(N_{+})\times U(N_{-})$Ryu  _et al._ (2010). This manifold has a canonical
Riemannian metric, and $g^{\rm val}({\bf k})$ is precisely the pullback of
this metric to the BZ along the map $Q({\bf k})$. By differentiating
$Q_{-}\ket{u^{-}_{a}}=\ket{u^{-}_{a}}$ w.r.t. $k^{\mu}$, one obtains
$Q_{+}\ket{\partial_{\mu}u^{-}_{a}}=\partial_{\mu}Q_{-}\ket{u^{-}_{a}}$ and
hence after some straightforward projector algebra

$g^{\rm val}_{\mu\nu}({\bf
k})=\frac{1}{8}\operatorname{tr}\,\partial_{\mu}Q\,\partial_{\nu}Q\,.$ (5)

Indeed, $Q$ is manifestly invariant under the aforementioned basis
redefinitions, and so is $g^{\rm val}$, hence $g^{\rm val}$ is in principle
measurable. Detailed derivation of the above formalism is given in Appendix A.

We will focus on the Dirac Hamiltonians that realize TIs and TSCs according to
their symmetry classes, which take the form

$H({\bf k})=\sum_{i=0}^{D}d^{i}({\bf k})\Gamma_{i}$ (6)

where ${\bf d}=(d^{0},d^{1}...d^{D})$ is the vector that parametrizes the
Dirac Hamiltonian, and $\Gamma_{i}$ are $N=2^{n}$ dimensional Dirac matrices
satisfying the Clifford algebra $\\{\Gamma_{i},\Gamma_{j}\\}=2\delta_{ij}$.
The Hamiltonian in Eq. (6) has $N/2=2^{n-1}$ degenerate eigenvalues $|{\bf
d}|$ and $2^{n-1}$ eigenvalues $-|{\bf d}|$ (in our context corresponding to
empty and filled bands respectively). The spectrally flattened Hamiltonian can
be expressed as

$Q({\bf k})=\sum_{i=0}^{D}n^{i}({\bf k})\Gamma_{i}$ (7)

where throughout the article we define the unit vector ${\bf n}({\bf
k})\equiv{\bf d}({\bf k})/|{\bf d}({\bf k})|$.

The image of the map ${\bf n}({\bf k})$ lies in the $D$ dimensional sphere
$S^{D}\subset\mathbb{R}^{D+1}$, that we will refer to as the Dirac sphere (DS)
in what follows. In Ref. von Gersdorff  _et al._ , 2021 it was shown that all
nontrivial topological invariants can be expressed in terms of the degree
$\deg[{\bf n}]$ which was also referred to as the wrapping number. The latter
counts how many times the BZ torus wraps around the DS under the map ${\bf
n}({\bf k})$. In particular, all known integer topological invariants are
given by $\deg[{\bf n}]$ or $2\deg[{\bf n}]$, while all the binary ones are
given by $(-1)^{\deg[{\bf n}]}$. One explicit representation of the wrapping
number is given by von Gersdorff  _et al._ (2021)

$\deg[{\bf n}]=\frac{1}{V_{D}}\int_{\rm BZ}J_{{\bf n}}(k)\,d^{D}k\,,$ (8)

where $V_{D}=2\pi^{\frac{D+1}{2}}/\Gamma(\frac{D+1}{2})$ is the volume of
$S^{D}$, and the ”curvature function”Chen (2016); Chen  _et al._ (2016, 2017);
Chen and Schnyder (2019) $J_{\bf n}$ is defined as

$J_{{\bf n}}({\bf k})\equiv\det\left({\bf n},\frac{\partial{\bf n}}{\partial
k^{1}},\dots,\frac{\partial{\bf n}}{\partial k^{D}}\right)\equiv\det E_{{\bf
n}},$ (9)

which is sometimes referred to as the cyclic derivative of the ${\bf
n}$-vector.

We proceed to relate the curvature function $J_{{\bf n}}$ to the quantum
metric $g_{\mu\nu}^{\rm val}$. First we apply our general equation for the
quantum metric of the valence band, Eq. (5) to Dirac models to find

$g_{\mu\nu}^{\rm val}=\frac{N}{8}\,\partial_{\mu}{\bf
n}\cdot\partial_{\nu}{\bf n}$ (10)

where we used Eq. (7) and applied the Clifford algebra. This equation is
remarkably simple. In fact, it is nothing but the pullback of the canonical
metric of the $D$-sphere of radius $\sqrt{N/8}$ to the BZ along the map ${\bf
n}({\bf k})$. Next we write

$J_{{\bf n}}^{2}=\det E^{T}_{{\bf n}}E_{{\bf n}}=\det\begin{pmatrix}{\bf
n}\cdot{\bf n}&{\bf n}\cdot\partial_{\nu}{\bf n}\\\ \partial_{\mu}{\bf
n}\cdot{\bf n}&\partial_{\mu}{\bf n}\cdot\partial_{\nu}{\bf n}\end{pmatrix}\\\
=\det\partial_{\mu}{\bf n}\cdot\partial_{\nu}{\bf n}\,.$ (11)

where it was used that ${\bf n}^{2}=1$ and ${\bf n}\cdot\partial_{\mu}{\bf
n}=0$. Combining Eq. (11) and Eq. (10) we arrive at one of our main results

$|J_{\bf n}|=\left(\frac{8}{N}\right)^{\frac{D}{2}}\sqrt{\det g^{\rm val}}$
(12)

a relation that we refer to as the metric-curvature correspondence. A similar
result that relates the integrand of Chern number and winding number,
individually, to the canonical metric on the Dirac sphere given by Eq. (10)
has also been proposed recentlyMera and Goldman . Nevertheless, besides giving
a more unified formalism, we emphasize that it is the link to the many-body
quantum metric $g_{\mu\nu}^{\rm val}$ we uncovered that makes this
correspondence experimentally relevant, as demonstrated below.

###  II.2 Experimental implications

The metric-curvature correspondence, Eq. (12), prompts us to seek for a
measurement for the momentum-profile of the quantum metric $g_{\mu\nu}^{\rm
val}({\bf k})$, since it may give direct information about the topological
invariant in any dimension and symmetry class. Out of several existing
proposals Neupert  _et al._ (2013); Kolodrubetz  _et al._ (2013); Tan  _et
al._ (2019); Gianfrate  _et al._ (2020), we focus on the one based on time-
dependent perturbation theory Ozawa and Goldman (2018), which has been
verified experimentally in a single atom Rabi oscillation in NV centers in
diamonds Chen  _et al._ ; Yu  _et al._ (2019). Our aim is to generalize this
theory to our Dirac Hamiltonian that has $N/2$-fold degeneracy in both the
filled and empty bands, and consider the application of a pulse electric field
of magnitude ${\bf E}^{0}$ and pulse profile $g(t)$,

${\bf E}(t)={\bf E}^{0}g(t)$ (13)

to the Bloch Hamiltonian $H({\bf k})$, and the pulse profile is assumed to
satisfy $g(\pm\infty)=0$ but is otherwise arbitrary. Electromagnetic gauge
invariance

$\phi\to\phi-\partial_{t}\Lambda\qquad{\bf A}\to{\bf A}+\nabla\Lambda$ (14)

allows for various equivalent implementations of the electric field, Eq. (13).
We find it convenient to work in the gauge

${\bf A}=0\qquad\phi=-g(t){\bf E}_{0}\cdot{\bf x}\,.$ (15)

We can then straightforwardly apply first-order time dependent perturbation
theory for large times, yielding the transition amplitude from the initial to
the final state

$a_{i\to f}=\frac{ie}{\hbar}\int_{-\infty}^{\infty}e^{i\omega
t}g(t)\,\braket{\phi_{f}}{{\bf E}_{0}\cdot{\bf x}}{\phi_{i}}$ (16)

where $\hbar\omega\equiv\epsilon_{f}-\epsilon_{i}$. Suppose that an electron,
initially in a filled-band Bloch state $\ket{u^{-}_{a}({\bf k})}$, under the
influence of the electric field makes a transition to the empty-band state
$\ket{u^{+}_{b}({\bf k}^{\prime})}$. 222Notice that we do not need to consider
transitions within the fully occupied degenerate valence bands since they are
forbidden by the Pauli principle at zero temperature. At finite temperature
and for very small band gaps, a non-negligible number of valence electrons
might be thermally excited and one would have to also consider intra-band
transitions, which should be treated elsewhere. The dipole operator matrix
elements between two general Bloch states is given byKarplus and Luttinger
(1954)

$\displaystyle\braket{u^{\sigma^{\prime}}_{b}({\bf k}^{\prime})}{{\bf
x}}{u^{\sigma}_{a}({\bf k})}$ (17)
$\displaystyle=-i\delta_{ab}\delta_{\sigma\sigma^{\prime}}\nabla_{{\bf
k}}\delta_{{\bf k},{\bf k}^{\prime}}+i\braket{u^{\sigma^{\prime}}_{b}({\bf
k})}{\nabla_{{\bf k}}}{u^{\sigma}_{a}({\bf k})}\delta_{{\bf k},{\bf
k}^{\prime}}$

and hence for $\sigma^{\prime}=+$ and $\sigma=-$ we have simply 333More
rigorously, we should consider the transition from the $n_{\rm val}$-particle
valence band state $\ket{\phi_{i}}=\ket{\rm val}$, where $n_{\rm val}$ is the
total number of valence electrons, to the state $\ket{\phi_{f}}=\ket{\rm
val^{\prime}}=c_{{\bf k}^{\prime},b}^{+\,\dagger}c_{{\bf k},a}^{-}\ket{\rm
val}$. However, it is easily seen that $\braket{\rm val^{\prime}}{{\bf x}}{\rm
val}$ is precisely equal to the left hand side of Eq.(18).

$\braket{u^{+}_{b}({\bf k}^{\prime})}{{\bf x}}{u^{-}_{a}({\bf
k})}=i\braket{u^{+}_{b}({\bf k})}{\nabla_{{\bf k}}}{u^{-}_{a}({\bf
k})}\delta_{{\bf k},{\bf k}^{\prime}}$ (18)

which is momentum conserving.

Assuming that the pulse is only on for a finite amount of time, the first
order perturbation theory yields the probability for such a transition:

$p_{b}^{(a)}({\bf
k})=\left(\frac{e}{\hbar}\right)^{2}\,\left|\tilde{g}(\omega({\bf
k}))\right|^{2}\,\left|\braket{u_{b}^{+}({\bf k})}{{\bf
E}^{0}\cdot\nabla_{{\bf k}}}{u_{a}^{-}({\bf k})}\right|^{2}$ (19)

where $\hbar\omega\equiv\epsilon_{b}-\epsilon_{a}$ and
$\tilde{g}(\omega)\equiv\int_{-\infty}^{\infty}e^{i\omega t}g(t)dt$ denotes
the Fourier transform of the pulse. We would like to briefly remark on the
shape of the pulse $g(t)$: If it has a dominant frequency such that
$\tilde{g}(\omega)$ is strongly peaked at some $\Omega$, only part of the
filled band will be excited (those with $\omega\sim\Omega$) and one might have
to scan over different values of $\Omega$ to eventually cover the whole BZ. On
the other hand, one could also envisage an extremely short pulse which has a
rather flat profile $\tilde{g}(\omega)$ and thus the whole filled band will be
democratically excited.

The transition probability into any of the conduction band states is then
simply $p^{(a)}({\bf k})=\sum_{b\in c}p_{b}^{(a)}({\bf k})$. Suppose we direct
the electric field to be along $\mu$ direction ${\bf
E}^{0}=E^{0}{\hat{\bm{\mu}}}$, one finds

$\displaystyle p^{(a)}({\bf
k})=\left(\frac{eE^{0}}{\hbar}\right)^{2}\,\left|\tilde{g}(\omega({\bf
k}))\right|^{2}\,\sum_{b}\left|\braket{u_{b}^{+}({\bf
k})}{\partial_{\mu}u_{a}^{-}({\bf k})}\right|^{2}$ (20)

For the nondegenerate case of a single valence band, this is directly
proportional to the diagonal element of the quantum metric $g_{\mu\mu}^{\rm
val}({\bf k})=g^{\psi}_{\mu\mu}({\bf k})$ of the single-particle Bloch state
$|\psi\rangle=|u_{1}^{-}\rangle$Ozawa and Goldman (2018). However, for the
degenerate case, the right hand side is no longer proportional to the quantum
metric of any particular state, and moreover the above formalism is not gauge
invariant. Instead, the truly gauge-invariant and measurable object is the
probability summed over the degenerate occupied valence bands

$\displaystyle\nu({\bf k})$ $\displaystyle=$
$\displaystyle\left(\frac{eE^{0}}{\hbar}\right)^{2}\,\left|\tilde{g}(\omega({\bf
k}))\right|^{2}\sum_{b\in c,a\in v}\left|\braket{u_{b}^{+}({\bf
k})}{\partial_{\mu}u_{a}^{-}({\bf k})}\right|^{2}$ (21) $\displaystyle=$
$\displaystyle\left(\frac{eE^{0}}{\hbar}\right)^{2}\,\left|\tilde{g}(\omega({\bf
k}))\right|^{2}\,g_{\mu\mu}^{\rm val}({\bf k}),$

where we have used Eq. (3). Similarly, we can consider two different driving
protocols $E_{\mu}=E^{0}g(t)$ and $E_{\nu}=\pm E^{0}g(t)$ in different spatial
directions. By subtracting the results of the two driving protocols
$\nu^{+}({\bf k})-\nu^{-}({\bf
k})=\left(\frac{eE_{0}}{\hbar}\right)^{2}\,\left|\tilde{g}(\omega({\bf
k}))\right|^{2}\,4g_{\mu\nu}^{\rm val}({\bf k})$, one can obtain the off-
diagonal elements of the metricOzawa and Goldman (2018).

Figure 1:  Schematics of the proposed trARPES experiment. All the particles
are in the degenerate filled bands $\varepsilon_{v}$ before the pulse electric
field ${\bf E}(t)$ is applied. Immediately after the pulse, some particles are
in the degenerate empty bands $\varepsilon_{c}$, causing a violation of
spectral sum rule in the filled bands at momentum ${\bf k}$, which can be
measured by detecting the spectral function times distribution function
$z({\bf k})A({\bf k},\omega)f^{\ast}(\omega)$ using trARPES, where the matrix
element effect is taken care of by $z({\bf k})$. The quantum metric
$g_{\mu\nu}({\bf k})$ can be extracted from the measurement according to Eqs.
(21) and (24).

Encouraged by the result in Eq. (21), we further propose the following
protocol of trARPES experiment to measure the quantity $\nu({\bf k})$. Our
proposal is similar to measuring the depletion rate of the ultracold atom
version of Chern insulator under periodic perturbation Tran  _et al._ (2017);
Asteria  _et al._ (2019). Because in the summation of Eq. (20), only the final
states not equal to the initial state $b\neq a$ contribute, this implies one
should measure how many electrons are leaving the degenerate filled band
states at momentum ${\bf k}$. In practice, how many electrons are leaving the
filled bands can be measured by the ARPES spectral function times the Fermi
distribution, which now has been modified due to the pump pulse. To illustrate
this, consider a static Dirac model at zero temperature, which gives the
imaginary time Green’s function $G({\bf k},i\omega)^{-1}=i\omega-H({\bf k})$.
After an analytical continuation $i\omega\rightarrow\omega+i\eta$ and taking
the scattering rate $\eta\rightarrow 0$, one obtains the retarded Green’s
function $G^{ret}({\bf k},\omega)$ and subsequently the ARPES spectral
function

$\displaystyle A({\bf k},\omega)=-\frac{1}{\pi}{\rm Im}\left\\{{\rm
tr}\left[G^{ret}({\bf k},\omega)\right]\right\\}$
$\displaystyle=\frac{N}{2}\delta(\omega-d({\bf
k}))+\frac{N}{2}\delta(\omega+d({\bf k})),$ (22)

signifying the $N/2$-fold degeneracy. Ideally, the spectral function satisfies
the spectral sum rule $\int_{-\infty}^{\infty}d\omega\,A({\bf
k},\omega)f(\omega)=N/2$, where $f(\omega)=\theta(-\omega)$ is the Fermi
distribution at zero temperature. However, the ARPES experiment in reality
contains the matrix element effect that affects the absolute scale of $A({\bf
k},\omega)$ and renders this normalization ${\bf k}$-dependent. For our
purpose of extracting quantum metric without ambiguity, one may add a ${\bf
k}$-dependent factor $z({\bf k})$ to fit the spectral sum rule until it is
rigorously satisfied at equilibrium at any momentum $z({\bf
k})\int_{-\infty}^{\infty}d\omega\,A({\bf k},\omega)f(\omega)=N/2$, such that
the matrix element effect does not obscure our formalism below.

When the time dependent electric field ${\bf E}^{0}g(t)$ is applied, the
spectral sum rule becomes time-dependent. The average number of electrons in
the valence band, shortly after the pulse has ceased (such that relaxation has
not yet set in) is

$\displaystyle z({\bf k})\int_{-\infty}^{0}d\omega\,A({\bf
k},\omega)f^{*}(\omega)=\frac{N}{2}-\sum_{a\in v}\sum_{b\in c}p^{(a)}_{b}({\bf
k}),$ (23)

where we assume that the spectral function $z({\bf k})A({\bf k},\omega)$
remains a $\delta$-function, but the distribution function is no longer the
equilibrium Fermi function $f^{*}(\omega)\neq\theta(-\omega)$ because the
electron has a finite probability to enter other final states $b$. Then

$\displaystyle\nu({\bf k})=\frac{N}{2}-z({\bf
k})\int_{-\infty}^{0}d\omega\,A({\bf k},\omega)f^{*}(\omega).$ (24)

Through comparing Eqs. (21) and (24), we see that the quantum metric can be
extracted from trARPES by measuring the particle density loss in the
degenerate filled bands at ${\bf k}$ immediately after the pulse, as shown
schematically in Fig. 1. In Sec.II.3, we use the measurement of topological
charge of graphene to elaborate the feasibility of our proposal. We also
emphasize that our proposal based on Eqs. (21) and (24) is a universal
protocol to measure the many-body quantum metric in any gapped degenerate
fermionic systems, not only limited to topological materials. Moreover, ${\bf
k}$ is not limited to momentum but can be any system parameters, which may
also help to measure the fidelity susceptibility associated with quantum phase
transitions in generalYou  _et al._ (2007); Zanardi  _et al._ (2007); Gu  _et
al._ (2008); Yang  _et al._ (2008); Albuquerque  _et al._ (2010); Gu (2010);
Carollo  _et al._ (2020), provided the driving field ${\bf E}(t)$ couples to
the system parameter ${\bf k}$ in the same way as that described in Eqs. (13)
to (19). It is only through the metric-curvature correspondence in Eq. (12)
that the measurement performed on TIs and TSCs would directly reveal the
topological order.

We now comment on several issues one may encounter in realistic ARPES
measurements. Firstly, our proposal only allows to measure the modulus of the
integrand $J_{\bf n}$ of the wrapping number ${\rm deg}[{\bf n}]$ via Eq.
(12), but not the sign of $J_{\bf n}$. In reality, the BZ consists of domains
of different signs of $J_{\bf n}$, and to fix these signs to unambiguously
determine the wrapping number requires some other input, such as band
structure calculations. Nevertheless, the modulus of $J_{\bf n}$ itself
already yields various valuable information about the topological order, such
as the correlation length, scaling lawsChen  _et al._ (2017); Chen and Sigrist
(2019), and fidelity susceptibilityPanahiyan  _et al._ (2020); Molignini  _et
al._ (2021). Secondly, for systems beyond the Dirac model, it remains to be
clarified how the metric-curvature correspondence will be modified, which may
need to be dealt with case by case. Thirdly, many-body effects can broaden the
spectral function and invalidate the sharp $\delta$-function in Eq. (22).
Effects of this kind require a fully nonequilibrium many-body version of our
formalism, which awaits further investigations. Finally, since ARPES is a
surface probe, our proposal is presumably more suitable to detect the topology
of 2D systems, provided the ARPES laser spot is smaller than the systems size
such that the edge states do not interrupt. For 3D systems, most likely the
method will be detecting the topology of the surface states if the system is
in the topologically nontrivial phase.

###  II.3 ARPES protocol applied to graphene

As a concrete example, we discuss the aforementioned ARPES technique applied
to measuring the topological charge of graphene, which is a topological
semimetal that contains two Dirac points ${\bf K}$ and ${\bf K}^{\prime}$ that
have opposite topological charges. For a $D$-dimensional topological semimetal
in general, our wrapping number formalism still applies, but the integration
in Eq. (8) is over a compact $(D-1)$-dimensional surface enclosing each nodal
point, and the metric-curvature correspondence is defined on this
$(D-1)$-dimensional surface, as we shall see below for graphene with $D=2$. We
choose graphene because its spectral function is extremely sharp due to the
long mean free time Bostwick  _et al._ (2007); Sprinkle  _et al._ (2009), and
moreover the spin degeneracy is well preserved since spin-orbit coupling is
negligibleHuertas-Hernando  _et al._ (2006); Min  _et al._ (2006); Yao  _et
al._ (2007). Consider only one spin species and expand the Hamiltonian around
the two Dirac points

$\displaystyle{\bf
K}=\left(\frac{2\pi}{3},\frac{2\pi}{3\sqrt{3}}\right)\;,\;\;\;{\bf
K}^{\prime}=\left(\frac{2\pi}{3},-\frac{2\pi}{3\sqrt{3}}\right)$ (25)

yields the linear Dirac Hamiltonian Bernevig and Hughes (2013)

$\displaystyle H_{0}({\bf K}+{\bf
k})=\frac{3}{2}t\left(k_{y}\sigma_{x}-k_{x}\sigma_{y}\right),$ $\displaystyle
H_{0}({\bf K}^{\prime}+{\bf
k})=\frac{3}{2}t\left(-k_{y}\sigma_{x}-k_{x}\sigma_{y}\right)\;,$ (26)

where $t$ is the nearest-neighbor hopping on the honeycomb lattice. This
linear dispersion is well satisfied up to energy $\sim 1$eV away from the
Dirac point, which covers a large enough momentum space to perform the
proposed pump-probe experimentGierz  _et al._ (2013). The eigenenergies and
eigenstates for one spin species, say spin up, are

$\displaystyle\varepsilon_{\pm}^{\bf K}({\bf
k})=\pm\frac{3}{2}t_{1}k,\;\;\;|u_{\bf k\pm}^{\bf
K}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\ \mp
ie^{i\phi}\end{array}\right),$ (29) $\displaystyle\varepsilon_{\pm}^{\bf
K^{\prime}}({\bf k})=\pm\frac{3}{2}t_{1}k,\;\;\;|u_{\bf k\pm}^{\bf
K^{\prime}}\rangle=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}1\\\ \mp
ie^{-i\phi}\end{array}\right),$ (32)

where $\phi$ is the polar angle of the momentum ${\bf k}=(k,\phi)$.
Integrating the valence band Berry connection $\langle u_{\bf k-}^{\bf
K}|i\partial_{\varphi}|u_{\bf k-}^{\bf K}\rangle$ along a loop of radius $k$
circulating the Dirac points yields the topological charges

$\displaystyle\frac{1}{2\pi}\oint d\phi\langle u_{\bf k-}^{\bf
K}|i\partial_{\phi}|u_{\bf k-}^{\bf K}\rangle=-\frac{1}{2\pi}\oint
d\phi\langle u_{\bf k-}^{\bf K^{\prime}}|i\partial_{\phi}|u_{\bf k-}^{\bf
K^{\prime}}\rangle$ $\displaystyle=-1/2,$ (33)

as shown schematically in Fig. 2 (a). The two spin degrees of freedom are
completely decoupled and the Hamiltonian is block-diagonal. In our choice of
basis this means that the valence band metric reduces to the sum of the
metrics of the spin-up and spin-down states $g_{\phi\phi}^{\rm
val}=g_{\phi\phi}^{\uparrow}({\bf k})+g_{\phi\phi}^{\downarrow}({\bf k})$.
Focusing on the ${\bf K}$ point $|u\rangle\equiv|u_{\bf k-}^{\bf K}\rangle$,
the Dirac Hamiltonian $H({\bf K+k})=d_{1}\sigma_{1}+d_{2}\sigma_{2}$ gives a
quantum metric

$\displaystyle
g^{u}_{\phi\phi}=\langle\partial_{\phi}u|\partial_{\phi}u\rangle-\langle\partial_{\phi}u|u\rangle\langle
u|\partial_{\phi}u\rangle=|\langle u|i\partial_{\phi}|u\rangle|^{2}$
$\displaystyle=\left|\frac{1}{2}\epsilon^{ab}{\bf n}_{a}\partial_{\phi}{\bf
n}_{b}\right|^{2}=\frac{1}{4}.$ (34)

We see that indeed the metric-curvature correspondence is satisfied with
$\sqrt{g^{\rm val}}=\sqrt{g^{\rm val}_{\phi\phi}}=1/\sqrt{2}$. Therefore, if
the proposed ARPES experiment yields a constant quantum metric in the angular
direction $g^{\rm val}_{\phi\phi}=1/2$ at any momentum ${\bf k}$, then the
topological charge is verified.

Figure 2: (a) The topological charge ${\cal C}=\pm 1/2$ obtained by
integrating the Berry connection along a circle of arbitrary radius
circulating the Dirac points ${\bf K}$ and ${\bf K}^{\prime}$. The orange line
denotes the BZ. The corresponding quantum metric in the angular direction
should be a constant $g_{\phi\phi}^{\rm val}=1/2$. (b) Schematics of using a
pulse electric field to excite the electrons from valence to conduction band.

Although it may be difficult to directly measure $g_{\phi\phi}$, one may apply
an oscillating electric field in planar directions to extract
$\left\\{g_{xx},g_{xy},g_{yy}\right\\}$, and then use the conversion between
the derivatives $\partial_{k}=\cos\phi\partial_{x}+\sin\phi\partial_{y}$ and
$\partial_{\phi}=-k\sin\phi\partial_{x}+k\cos\phi\partial_{y}$ to extract
$g_{\phi\phi}$ from $\left\\{g_{xx},g_{xy},g_{yy}\right\\}$ by

$\displaystyle g_{\phi\phi}=k^{2}\sin^{2}\phi\,g_{xx}-k^{2}\sin
2\phi\,g_{xy}+k^{2}\cos^{2}\phi\,g_{yy}.\;\;\;\;\;$ (35)

To give an order of magnitude estimation, we use the numbers in the pump-probe
experiment performed on grapheneGierz  _et al._ (2013). To measure the quantum
metric, we suppose that the pump pulse excites the valence electron at ${\bf
k}$ that has a band gap of the order of an electron volt
$\hbar\omega\sim\varepsilon_{+}-\varepsilon_{-}\sim\,$eV, which corresponds to
a frequency $\sim 10^{15}\,$Hz, as shown schematically in Fig. 2 (b). For the
matrix element we can estimate

$\braket{u_{+}}{\partial_{\mu}u_{-}}=\Braket{u_{+}}{\frac{\partial_{\mu}H}{\epsilon_{+}-\epsilon_{-}}}{u_{-}}\sim\frac{v_{F}}{\omega}$
(36)

where $v_{F}\sim 10^{6}$m/s is the Fermi velocity of graphene. If the pulse is
on for a short time $T\sim 0.1\,$ps, then $\tilde{g}(\omega)\sim T$ and

$\nu({\bf k})\sim\left(\frac{eE^{0}v_{F}T}{\hbar\omega}\right)^{2}$ (37)

Suppose we aim to excite $\nu({\bf k})\sim 10\%$ electrons after the pulse.
This requires an electric field square of the order of $(E^{0})^{2}\sim
10^{13}$V${}^{2}/$m2. The fluence after applying the pulse is

$\displaystyle F\sim\frac{c\,\varepsilon_{0}}{2}|E^{0}|^{2}T\sim
10^{-4}\frac{\rm mJ}{\rm cm^{2}},$ (38)

where $c$ is the speed of light and $\varepsilon_{0}$ is the vacuum
permittivity. This fluence is much smaller than that delivered in the graphene
pump-probe experiment $\sim{\rm mJ}/{\rm cm^{2}}$Gierz  _et al._ (2013), hence
should be easily achievable and heating can be ignored. In fact, the pump-
probe trARPES experiment has already revealed a significant amount of
excitation from the valence to the conduction band, although the pump pulse
${\bf E}(t)$ is usually not polarized. Finally, we remark that the light-
matter interactionsSchüler  _et al._ (2021) and light-induced anomalous Hall
effectSato  _et al._ (2019) in graphene, as well as the nonequilibrium
response of graphene in the pump-probe setup in the presence of many-body
interactions has been addressed previouslySchüler  _et al._ (2020). Precisely
how these realistic factors, in particular the many-body interactions, can
affect the measurement of quantum metric is a fundamental issue that requires
further investigations.

##  III Conclusions

In summary, we elaborate the metric-curvature correspondence between the
modulus of the integrand of the wrapping number and the quantum metric of an
appropriate many-body Bloch state described by Eq. (12). Based on the validity
of this correspondence in any dimension and symmetry class, we propose an
ARPES measurement protocol to ubiquitously detect the momentum profile of the
quantum metric, from which information about the topological properties of the
system can be extracted. By generalizing a recently proposed time-dependent
perturbation theory to degenerate bands, our proposal suggests to measure the
violation of spectral sum rule caused by a pulse electric field to extract the
quantum metric. Various complications in reality, such as systems beyond Dirac
models or containing electronic correlations, requires further generalization
of our formalism that awaits to be explored.

###### Acknowledgements.

The authors acknowledge stimulating discussions with M. Manske, S. Panahiyan,
L. Kemper, B. Mera, and the experimental input from S. Kaiser. W. C. is
financially supported by the productivity in research fellowship from CNPq.

##  Appendix A Detailed derivation of the quantum metric formalism

We now give some detailed calculation for the quantum metric. We would like to
compute the metric for the $N_{-}$ particle state $|\psi^{\rm val}({\bf
k})\rangle$ defined in Eq. (2), i.e.

$\displaystyle g^{{\rm val}}_{\mu\nu}({\bf k})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\langle\partial_{\mu}\psi^{\rm
val}|\partial_{\nu}\psi^{\rm
val}\rangle+\frac{1}{2}\langle\partial_{\nu}\psi^{\rm
val}|\partial_{\mu}\psi^{\rm val}\rangle$ (39)
$\displaystyle-\langle\partial_{\mu}\psi^{\rm val}|\psi^{\rm
val}\rangle\langle\psi^{\rm val}|\partial_{\nu}\psi^{\rm val}\rangle.$

The simplest approach is to use second-quantization formalism with fermionic
annihilation operators $c_{a,\sigma}$ where $\sigma=\pm$ denotes positive and
negative energy eigenstates respectively. Then by the standard formula for
multi-particle operators we have

$\displaystyle\frac{\partial}{\partial
k^{\mu}}=\sum_{\begin{subarray}{c}\sigma=\pm\\\
\sigma^{\prime}=\pm\end{subarray}}\sum_{a=1}^{N_{\sigma}}\sum_{a^{\prime}=1}^{N_{\sigma^{\prime}}}\langle
u_{a^{\prime}}^{\sigma^{\prime}}|\partial_{\mu}|u_{a}^{\sigma}\rangle
c_{a^{\prime}\sigma^{\prime}}^{\dagger}c_{a\sigma}.$ (40)

Acting with this on $|\psi^{\rm
val}\rangle=\prod_{a=1}^{N_{-}}c^{\dagger}_{a-}|0\rangle$, one gets

$\displaystyle|\partial_{\mu}\psi^{\rm val}\rangle$ $\displaystyle=$
$\displaystyle\sum_{a=1}^{N_{-}}\biggl{(}\langle
u_{a}^{-}|\partial_{\mu}u_{a}^{-}\rangle+\sum_{a^{\prime}=1}^{N_{+}}\langle
u_{a^{\prime}}^{+}|\partial_{\mu}u_{a}^{-}\rangle
c_{a^{\prime}+}^{\dagger}c_{a-}\biggr{)}|\psi^{\rm val}\rangle.$

Then it is straightforward to compute

$\displaystyle\langle\psi^{\rm val}|\partial_{\mu}\psi^{\rm val}\rangle$
$\displaystyle=$ $\displaystyle\sum_{a=1}^{N_{-}}\langle
u_{a}^{-}|\partial_{\mu}u_{a}^{-}\rangle$
$\displaystyle\langle\partial_{\mu}\psi^{\rm val}|\partial_{\nu}\psi^{\rm
val}\rangle$ $\displaystyle=$
$\displaystyle\left(\sum_{a=1}^{N_{-}}\langle\partial_{\mu}u_{a}^{-}|u_{a}^{-}\rangle\right)\left(\sum_{a=1}^{N_{-}}\langle
u_{a}^{-}|\partial_{\nu}u_{a}^{-}\rangle\right)$ (42)
$\displaystyle+\sum_{a=1}^{N_{-}}\langle\partial_{\mu}u_{a}^{-}|Q_{+}|\partial_{\nu}u_{a}^{-}\rangle,\;\;\;\;\;\;$

where $Q_{+}$ was defined in Eq. (4). Applying Eqs. (LABEL:eq:A1) and (42) to
Eq. (39), we obtain Eq. (3).

To further express $g_{\mu\nu}^{\rm val}$ in terms of the spectrally flattened
Hamiltonian $Q({\bf k})$, we use $Q_{+}^{2}=Q_{+}$, $Q_{\pm}=(1\pm Q)/2$, and
$\left(\partial_{\mu}Q_{-}\right)|u_{a}^{-}\rangle=Q_{+}|\partial_{\mu}u_{a}^{-}\rangle$
to write

$\displaystyle g_{\mu\nu}^{\rm val}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\sum_{a=1}^{N_{-}}\left(\langle
u_{a}^{-}|\partial_{\mu}Q_{-}\partial_{\nu}Q_{-}|u_{a}^{-}\rangle+\langle
u_{a}^{-}|\partial_{\nu}Q_{-}\partial_{\mu}Q_{-}|u_{a}^{-}\rangle\right)$ (43)
$\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
tr}\left(Q_{-}\partial_{\mu}Q_{-}\partial_{\nu}Q_{-}+Q_{-}\partial_{\nu}Q_{-}\partial_{\mu}Q_{-}\right)$
$\displaystyle=$ $\displaystyle\frac{1}{2}{\rm
tr}\,\partial_{\mu}Q_{-}\partial_{\nu}Q_{-}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\rm
tr}\,\partial_{\mu}\left[\frac{1}{2}(1-Q)\right]\partial_{\nu}\left[\frac{1}{2}(1-Q)\right]$
$\displaystyle=$ $\displaystyle\frac{1}{8}{\rm
tr}\,\partial_{\mu}Q\,\partial_{\nu}Q,$

which gives Eq. (5).Matsuura and Ryu (2010)

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