# Relations between topology and the quantum metric for Chern insulators

Tomoki Ozawa  Advanced Institute for Materials Research, Tohoku University,
Sendai 980-8577, Japan  Bruno Mera  Instituto de Telecomunicações, 1049-001
Lisboa, Portugal  Departmento de Física, Instituto Superior Técnico,
Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Departmento de Matemática, Instituto Superior Técnico, Universidade de Lisboa,
Av. Rovisco Pais, 1049-001 Lisboa, Portugal

###### Abstract

We investigate relations between topology and the quantum metric of two-
dimensional Chern insulators. The quantum metric is the Riemannian metric
defined on a parameter space induced from quantum states. Similar to the Berry
curvature, the quantum metric provides a geometrical structure associated to
quantum states. We consider the volume of the parameter space measured with
the quantum metric, which we call the quantum volume of the parameter space.
We establish an inequality between the quantum volume of the Brillouin zone
and that of the twist-angle space. Exploiting this inequality and the
inequality between the Chern number and the quantum volume, we investigate how
the quantum volume can be used as a good measure to infer the Chern number.
The inequalities are found to be saturated for fermions filling Landau levels.
Through various concrete models, we elucidate conditions when the quantum
volume gives a good estimate of the topology of the system.

##  I Introduction

Topological band structure and associated geometrical properties of quantum
and classical states have become an important subject of study in a wide
variety of disciplines in physics, ranging from solid-state electrons [1, 2,
3], where the concept of topological insulator was first invented [4, 5, 6, 7,
8, 9], to atomic, molecular, and optical (AMO) physics [10, 11], acoustics
[12, 13, 14], active matter [15, 16], and even in geophysics [17].

Topological properties, such as Chern numbers and winding numbers,
characterize the global nature of the states defined in a suitable parameter
space, such as momentum space. On the other hand, geometry characterizes local
properties at each point in the parameter space. Topology and geometry are
closely related to each other; for example, the integral of the Berry
curvature, a geometrical quantity, over a two-dimensional compact surface
gives rise to the topological Chern number. Recently, there is an increasing
interest in the study of a geometrical property called the quantum metric
[18]. The quantum metric provides a positive semi-definite metric on a
parameter space, and is known to be relevant in various phenomena such as the
width of the maximally localized Wannier function [19, 20, 21], responses in
dissipative systems [22, 23, 24, 25], response upon periodical modulations
[26, 27, 28], non-adiabatic responses [29], orbital magnetism [30, 31, 32, 33,
34], superfluid density [35, 36, 37, 38], Lamb-shift-like energy shift in
excitons [39], Landau-level spreading in flat band systems [40], and in the
construction of fractional Chern insulators [41, 42]. Very recently,
experimental measurements of the quantum metric have been reported in various
systems [43, 44, 45, 46, 47, 48].

The aim of this paper is to study relations between the topological properties
of the system and the quantum metric, focusing on two-dimensional Chern
insulators. Chern insulators are characterized by topology, in particular, by
the Chern numbers of families of quantum states defined on the first Brillouin
zone, which is the compact momentum space obtained in the presence of
translation symmetry with respect to a lattice in real space. When considering
many-body systems, another relevant parameter space is the twist-angle space;
one considers a periodic boundary condition with a phase twist, and the many-
body wave function is parametrized by the phase [49]. When considering non-
interacting fermions filling a band, the geometrical properties of the twist-
angle space are given by an average of the corresponding quantities in
momentum space, and hence the topological Chern numbers calculated in both
spaces are identical. Besides the Chern number, we can construct a non-
topological global quantity characterizing the geometrical structure of the
states from the quantum metric. Since the quantum metric provides a Riemannian
structure in the parameter space, we can consider the volume of the parameter
space measured using this metric; we call it quantum volume of the parameter
space. Inspired by the known inequality between the Chern number and the
quantum volume, we investigate how and when the quantum volume can act as a
useful measure to infer the topology of the system. Furthermore, we establish
an inequality between the quantum volume of momentum space and the quantum
volume of the twist-angle space. We show that in the limit of bands
approaching Landau levels, we can obtain a very good estimate of the topology
(Chern number) just by studying the quantum volume. The relations between
topology and the quantum volume have intricate mathematical structures as
well, which are analyzed in detail in the accompanying paper from the view
point of Kähler geometry [50].

The structure of the paper is as follows. In Sec. II, we introduce the quantum
metric, the Berry curvature, and the Chern number for a general parameter
space. In Sec. III, we introduce the momentum space and the twist-angle space
and discuss general relations between the topology and the quantum volumes. In
Sec. IV, we then discuss specific models to see how the general relations can
be applied to understand the topological features of various cases. We first
analyze Landau levels, which are the energy levels of a two-dimensional system
in the presence of a uniform magnetic field. We then discuss two-band models,
taking the Haldane model [51] and the two-band model introduced in Yang et al.
[52] as explicit examples. As we will see, two-band models have characteristic
features which are not present in models with more than two bands. We finally
discuss multi-band models, taking the Harper-Hofstadter model [53, 54] and the
three-band models introduced in Ref. 55 and Ref. 52 as examples. We finally
give conclusions in Sec. V. Detailed derivations of some formulas and
relations are given in the Appendix.

##  II Definitions and Properties

###  II.1 Definitions

We first define the quantum metric and the Berry curvature for a set of
mutually orthogonal quantum states $|\psi_{m}(\bm{\lambda})\rangle$, labeled
by an integer $m=1,2,\cdots$, which depend on a set of parameters
$\bm{\lambda}=(\lambda_{1},\lambda_{2},\cdots)$. In what follows,
$\bm{\lambda}$ will either be a quasimomentum $\mathbf{k}$ or the twist angle
$(\theta_{x},\theta_{y},\cdots)$.

The geometrical structure of a set of quantum states
$|\psi_{m}(\bm{\lambda})\rangle$ with $m=1,2,\cdots,r$, where $r$ is the
number of occupied bands for band insulators, is characterized by the quantum
geometric tensor, whose real part gives the quantum metric and the imaginary
part gives the Berry curvature [18]. The quantum geometric tensor for a set of
states $|\psi_{m}(\bm{\lambda})\rangle$ with $1\leq m\leq r$ is defined by
[24]

$\displaystyle\chi_{ij}(\bm{\lambda})$
$\displaystyle\equiv\sum_{m=1}^{r}\langle\partial_{\lambda_{i}}\psi_{m}(\bm{\lambda})|(1-P(\bm{\lambda}))|\partial_{\lambda_{j}}\psi_{m}(\bm{\lambda})\rangle,$
(1)

where
$P(\bm{\lambda})\equiv\sum_{m=1}^{r}|\psi_{m}(\bm{\lambda})\rangle\langle\psi_{m}(\bm{\lambda})|$
is the projector onto the set of states with $m=1,2,\cdots,r$. The quantum
metric $g_{ij}$ and the Berry curvature $\Omega_{ij}$ are defined through the
real and imaginary parts of the quantum geometric tensor as

$\displaystyle g_{ij}(\bm{\lambda})$
$\displaystyle\equiv\mathrm{Re}[\chi_{ij}(\bm{\lambda})].$ (2)
$\displaystyle\Omega_{ij}(\bm{\lambda})$
$\displaystyle\equiv-2\,\mathrm{Im}[\chi_{ij}(\bm{\lambda})].$ (3)

It is straightforward to show $g_{ij}=g_{ji}$ and $\Omega_{ij}=-\Omega_{ji}$
from the definition of the quantum geometric tensor.

In this paper, we focus on two-dimensional systems:
$\bm{\lambda}=(\lambda_{1},\lambda_{2})$. Then, in matrix notation,

$\displaystyle\chi(\bm{\lambda})=\begin{pmatrix}g_{11}(\bm{\lambda})&g_{12}(\bm{\lambda})-i\Omega_{12}(\bm{\lambda})/2\\\
g_{12}(\bm{\lambda})+i\Omega_{12}(\bm{\lambda})/2&g_{22}(\bm{\lambda})\end{pmatrix}.$
(4)

###  II.2 Basic properties

The quantum metric $g_{ij}(\bm{\lambda})$ provides a positive semidefinite
metric in the parameter space described by
$\bm{\lambda}=(\lambda_{1},\lambda_{2})$ satisfying
$\det(g)=g_{11}g_{22}-g_{12}^{2}\geq 0$. We note that the quantum metric is
not a positive definite metric; there can be points in the parameter space
where $\det(g)=0$ holds. The Berry curvature
$\Omega_{ij}(\bm{\lambda})d\lambda_{i}\wedge d\lambda_{j}$ provides the
curvature $2$-form of a connection on a vector bundle whose fiber at
$\bm{\lambda}$ is spanned by the set of states
$|\psi_{m}(\bm{\lambda})\rangle$ with $m=1,2,\cdots,r$.

Even though the quantum metric is a geometrical property of the tangent bundle
of the parameter space and the Berry curvature is a property of the fiber
bundle made of the quantum states, there are relations between the two; after
all, they are both defined in terms of the same family of quantum states. An
important relation that we want to explore in detail in this paper is the
inequality that holds between the volume form and the Berry curvature:

$\displaystyle\sqrt{\det(g)}\geq\frac{|\Omega_{12}|}{2}.$ (5)

This inequality is well known in the literature and is due to the fact that
the quantum geometric tensor $\chi_{ij}$ is a positive semi-definite matrix;
the inequality was first shown for the single-band case ($r=1$) in Roy [56],
and then later generalized to include multi-band cases by Peotta and Törmä
[35]. For completeness, we give a proof of this inequality in Appendix A using
Cauchy-Schwarz inequality, clarifying the condition under which the equality
holds. By integrating this inequality, we obtain an important relation between
the topological Chern number and the quantum volume as we describe in the
following. The Chern number can be calculated by integrating the Berry
curvature over the parameter space (when the parameter space is compact, which
will be the case below):

$\displaystyle\mathcal{C}=\frac{1}{2\pi}\int
d\lambda_{1}d\lambda_{2}\;\Omega_{12}(\bm{\lambda}).$ (6)

The Chern number is guaranteed to be an integer and reflects a topological
property of the states. The quantum volume $\mathrm{vol}_{g}$ is the volume of
the parameter space calculated using the quantum metric as the metric, and it
is defined by

$\displaystyle\mathrm{vol}_{g}$ $\displaystyle\equiv\int
d\lambda_{1}d\lambda_{2}\sqrt{\det(g)}.$ (7)

Using the inequality Eq. (5), the following inequality between the quantum
volume and the Chern number holds [56, 35]:

$\displaystyle\mathrm{vol}_{g}\geq\int
d\lambda_{1}d\lambda_{2}\frac{|\Omega_{12}|}{2}\geq\left|\int
d\lambda_{1}d\lambda_{2}\frac{\Omega_{12}}{2}\right|=\pi|\mathcal{C}|.$ (8)

Thus, the Chern number multiplied by $\pi$ is upper bounded by the quantum
volume of the parameter space. This relation has, for example, been applied to
topological superconductors to find a relation between the superfluid weight
and the topological properties of the system [35]. In this paper, we will
investigate consequences of this inequality applied to the quasimomentum and
twist-angle spaces as relevant parameter spaces. Before studying concrete
parameter spaces, we briefly discuss how the quantum geometric tensor is
related to the Hamiltonian of the system.

###  II.3 Relation to the Hamiltonian

So far we have not introduced any Hamiltonian. Typically, quantum states
$|\psi_{m}(\bm{\lambda})\rangle$ are defined as eigenstates of a certain
Hamiltonian $H(\bm{\lambda})$, which depends smoothly on the same parameter
$\bm{\lambda}$. We then define that $|\psi_{m}(\bm{\lambda})\rangle$ is an
$m$th eigenstate of $H(\bm{\lambda})$, where the corresponding eigenvalue
$\epsilon_{m}(\bm{\lambda})$ is the $m$th smallest of all the eigenvalues of
$H(\bm{\lambda})$. The eigenvalue equation is
$H(\bm{\lambda})|\psi_{m}(\bm{\lambda})\rangle=\epsilon_{m}(\bm{\lambda})|\psi_{m}(\bm{\lambda})\rangle$.
Using the Hamiltonian, we can find an expression for the quantum geometric
tensor, which does not involve the derivative of quantum states. From the
definition of the quantum geometric tensor, we see

$\displaystyle\chi_{ij}(\bm{\lambda})$
$\displaystyle=\sum_{m=1}^{r}\sum_{m^{\prime}>r}\langle\partial_{\lambda_{1}}\psi_{m}(\bm{\lambda})|\psi_{m^{\prime}}(\bm{\lambda})\rangle\langle\psi_{m^{\prime}}(\bm{\lambda})|\partial_{\lambda_{j}}\psi_{m}(\bm{\lambda})\rangle.$
(9)

Using the relation $\langle\psi_{m^{\prime}}(\bm{\lambda})|\frac{\partial
H}{\partial\lambda_{i}}|\psi_{m}(\bm{\lambda})\rangle=(\epsilon_{m}(\bm{\lambda})-\epsilon_{m^{\prime}}(\bm{\lambda}))\langle\psi_{m^{\prime}}(\bm{\lambda})|\partial_{\lambda_{i}}\psi_{m}(\bm{\lambda})\rangle$,
we can rewrite the quantum geometric tensor as

$\displaystyle\chi_{ij}(\bm{\lambda})$
$\displaystyle=\sum_{m=1}^{r}\sum_{m^{\prime}>r}\frac{\langle\psi_{m}(\bm{\lambda})|\frac{\partial
H}{\partial\lambda_{i}}|\psi_{m^{\prime}}(\bm{\lambda})\rangle\langle\psi_{m^{\prime}}(\bm{\lambda})|\frac{\partial
H}{\partial\lambda_{j}}|\psi_{m}(\bm{\lambda})\rangle}{(\epsilon_{m^{\prime}}(\bm{\lambda})-\epsilon_{m}(\bm{\lambda}))^{2}}.$
(10)

This is an expression which does not involve any derivatives of the quantum
states themselves, and is useful when numerically calculating the quantum
geometric tensor. We note that there is also a technique of calculating the
quantum geometric tensor without computing the eigenstates at all as discussed
in Refs. [57, 58].

##  III Momentum space and twist-angle space

Now we explicitly consider momentum space and twist-angle space as parameter
spaces which are relevant in the study of two-dimensional Chern insulators. An
$n$-band model is characterized by an $n$-by-$n$ Hamiltonian $H(k_{x},k_{x})$,
which depends on quasi momenta $(k_{x},k_{y})$ defined in the first Brillouin
zone, which is periodic and is, topologically, a two-torus. The quantum state
$|\psi_{m}(k_{x},k_{y})\rangle$, often called the Bloch state, is the
eigenvector of $H(k_{x},k_{y})$ with the $m$-th smallest eigenvalue. From the
Bloch states with $r$ smallest eigenvalues, we construct the quantum metric
and the Berry curvature, which we write as $g_{ij}(k_{x},k_{y})$ and
$\Omega_{ij}(k_{x},k_{y})$. Topology and geometry of the Bloch states are
directly relevant in the study of topological insulators; Chern insulators are
two-dimensional systems where the Chern number of the Bloch states, associated
with the filled bands, over momentum space is nonzero. An important physical
consequence of nonzero Chern number is the bulk-edge correspondence; the
number of edge modes of a given energy gap is equal to the sum of the Chern
numbers of energy bands below the gap [59, 60].

Another natural parameter space to analyze Chern insulators is the twist-angle
space [49], which is more abstract than momentum space but can be applied also
to disordered and interacting many-body systems. The geometrical structure in
the twist-angle space is defined by introducing a periodic boundary condition
with a twist phase $\theta_{x}$ and $\theta_{y}$ along $x$ and $y$ directions,
respectively, and defining quantum states in the parameter space of
$(\theta_{x},\theta_{y})$. Twisted boundary conditions can be gauged away by a
suitable gauge transformation, but at a cost of changing the Hamiltonian from
$H(k_{x},k_{y})$ to $H(k_{x}+\theta_{x}/L_{x},k_{y}+\theta_{y}/L_{y})$, where
$L_{x}$ and $L_{y}$ are the lengths of the system in $x$ and $y$ directions,
respectively. It then follows that $\frac{\partial H}{\partial
k_{i}}=L_{i}\frac{\partial H}{\partial\theta_{i}}$. Then, from the expression
of the quantum geometric tensor in terms of the derivative of the Hamiltonian,
Eq. (10), we can see that the quantum geometric tensor in twist-angle space is
$1/L_{i}L_{j}$ times the quantum geometric tensor defined in momentum space.
We are most interested in the situation where $r$ bands labeled by
$m=1,2,\cdots,r$ are completely filled. We let
$\tilde{\chi}_{ij}(\theta_{x},\theta_{y})$ denote the quantum geometric tensor
in twist-angle space when $r$ bands are filled. Assuming non-interacting
fermions, the quantum geometric tensor for the entire band is a sum of the
quantum geometric tensor of each quasimomentum, namely

$\displaystyle\tilde{\chi}_{ij}(\theta_{x},\theta_{y})$
$\displaystyle=\frac{1}{L_{i}L_{j}}\sum_{\mathbf{k}}\chi_{ij}(k_{x},k_{y})$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int d^{2}k\chi_{ij}(k_{x},k_{y}),$ (11)

which shows that the quantum geometric tensor for a set of bands in the twist-
angle space is nothing but the average of the quantum geometric tensor defined
in momentum space. One can show that this integral is $\theta_{x}$ and
$\theta_{y}$ independent in the thermodynamic limit [61, 62]. Explicitly
writing out each component of the quantum geometric tensor, we obtain

$\displaystyle\tilde{g}_{ij}(\theta_{x},\theta_{y})$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int d^{2}k\;g_{ij}(k_{x},k_{y}),$ (12)
$\displaystyle\tilde{\Omega}_{xy}(\theta_{x},\theta_{y})$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int
d^{2}k\;\Omega_{xy}(k_{x},k_{y})=\frac{1}{2\pi}\mathcal{C},$ (13)

where $\tilde{g}_{ij}$ and $\tilde{\Omega}_{ij}$ are the quantum metric and
the Berry curvature for the filled $r$ bands defined in the twist-angle space.
We note again that these quantities do not depend on the twist angles in the
thermodynamic limit, or in other words, these quantities are flat in the
twist-angle space. As it is clear from the relation Eq. (13), the Chern number
calculated from $\Omega_{xy}(k_{x},k_{y})$ and
$\tilde{\Omega}_{xy}(\theta_{x},\theta_{y})$ are the same:

$\displaystyle\mathcal{C}=\frac{1}{2\pi}\int
d^{2}k\;\Omega_{xy}(k_{x},k_{y})=\frac{1}{2\pi}\int
d^{2}\theta\;\tilde{\Omega}_{xy}(\theta_{x},\theta_{y}).$ (14)

Therefore, for non-interacting fermions completely filling a set of bands, the
Chern number in twist-angle space gives the same information as the Chern
number in momentum space. On the other hand, the quantum volumes of momentum
space and the twist-angle space are generally different. We note that the
quantum metric in twist-angle space is proportional to the localization tensor
of the material [63], which is directly related to the longitudinal response
of the system upon periodic modulation when a band is filled [43, 27]. The
quantum volume of twist-angle space has the interpretation of the geometric
mean of the eigenvalues of the localization tensor, giving an average notion
of the spread of the many-particle ground state in real space [64].

Since the Chern numbers for non-interacting fermions are the same for momentum
space and the twist-angle space, the inequality Eq.(8) holds for both spaces
with the same Chern number:

$\displaystyle\pi|\mathcal{C}|$ $\displaystyle\leq\mathrm{vol}_{g}\equiv\int
dk_{x}dk_{y}\sqrt{\mathrm{det}[g(k_{x},k_{y})]},$
$\displaystyle\pi|\mathcal{C}|$
$\displaystyle\leq\mathrm{vol}_{\tilde{g}}\equiv\int
d\theta_{x}d\theta_{y}\sqrt{\mathrm{det}[]\tilde{g}(\theta_{x},\theta_{y})]}.$
(15)

The quantum volumes $\mathrm{vol}_{g}$ and $\mathrm{vol}_{\tilde{g}}$ are
generally different, and in fact the inequality
$\mathrm{vol}_{g}\leq\mathrm{vol}_{\tilde{g}}$ holds; the quantum volume of
momentum space is always smaller or equal to the quantum volume of the twist-
angle space. The proof of $\mathrm{vol}_{g}\leq\mathrm{vol}_{\tilde{g}}$ is
given in Appendix B. Combining all the inequalities, we finally obtain

$\displaystyle\pi|\mathcal{C}|\leq\mathrm{vol}_{g}\leq\mathrm{vol}_{\tilde{g}},$
(16)

which is the central relation between the topology and the quantum metric
whose consequence we will explore in the paper. In the accompanying paper
[50], this inequality is analyzed from the mathematical viewpoint of Kähler
geometry. In particular, it is discussed that the first inequality is
saturated, provided the quantum metric on momentum-space is non-degenerate,
when the parameter space equipped with the quantum metric and the Berry
curvature, playing the role of a symplectic form, forms a Kähler manifold.

One immediate consequence of this inequality is that if $\mathrm{vol}_{g}<\pi$
or $\mathrm{vol}_{\tilde{g}}<\pi$, then $|\mathcal{C}|=0$, i.e., a set of
bands is topologically trivial. For systems with $\mathrm{vol}_{g}\geq\pi$, we
cannot make a definite statement about their topology. However, as we confirm
below with several models, $\mathrm{vol}_{g}/\pi$ often gives a very good
estimate of the topology, and we can also see that in a proper limit
approaching the situation similar to the filled Landau levels,
$\mathrm{vol}_{g}/\pi$ approaches the Chern number from above.

##  IV Models

Now we explore how the topology and geometry are related through various
examples and models. We first consider the paradigmatic case of Landau levels
associated to the problem of two-dimensional electrons in the presence of a
uniform magnetic field. We then consider two-band models, and then finally
discuss multi-band models.

###  IV.1 Landau levels

A charged particle in a uniform magnetic field is the simplest example of a
system with nontrivial Chern number. The Hamiltonian of a charged particle
with mass $M$ in a uniform magnetic fields takes the following form:

$\displaystyle H=\frac{\hat{p}_{x}^{2}+(\hat{p}_{y}-B\hat{x})^{2}}{2M},$ (17)

where we assumed that the charge is unity, and we are taking the Landau gauge
for the magnetic vector potential $\vec{A}(\mathbf{r})=(0,Bx)$. The
Hamiltonian does not have a translational symmetry, but there is a magnetic
translational symmetry, which is a symmetry of a discrete translation followed
by a phase twist. To understand that there is a magnetic translational
symmetry, we consider a fictitious magnetic unit cell, which is a rectangle
with lengths $a_{x}$ and $a_{y}$ in $x$ and $y$ directions, respectively,
which satisfies $|B|a_{x}a_{y}=2\pi$. Note that the choice of $a_{x}$ and
$a_{y}$ is not unique. Using the magnetic translational symmetry, we can apply
Bloch’s theorem and introduce quasimomentum, which takes a value in the
Brillouin zone: $-\pi/a_{x}\leq k_{x}\leq\pi/a_{x}$ and $-\pi/a_{y}\leq
k_{y}\leq\pi/a_{y}$.

The quantum metric and the Berry curvature of each Landau level have been
evaluated in Ref. [35] by taking an appropriate limit of the Harper-Hofstadter
model. We give an alternative derivation of the quantum geometric tensor of
the Landau levels in Appendix C directly using the wave functions of the
Landau levels, which allow us to derive formulas for the quantum geometric
tensor of multiple Landau levels. The quantum metric and the Berry curvature
for the collection of lowest $r$ Landau levels is

$\displaystyle g(k_{x},k_{y})$
$\displaystyle=\frac{r}{2|B|}\begin{pmatrix}1&0\\\ 0&1\end{pmatrix},$
$\displaystyle\Omega(k_{x},k_{y})$ $\displaystyle=-\frac{r}{B}.$ (18)

Thus, the quantum geometric tensor is completely flat in the Brillouin zone.
This flatness implies that the quantum geometric tensor $\chi_{ij}$ of the
Brillouin zone and $\tilde{\chi}_{ij}$ of the twist-angle space coincide. The
Chern number and the quantum volumes are $\mathcal{C}=-r\ \mathrm{sign}(B)$,
$\mathrm{vol}_{g}=\mathrm{vol}_{\tilde{g}}=r\pi$ and thus, the equality

$\displaystyle\pi|\mathcal{C}|=r\pi=\mathrm{vol}_{g}=\mathrm{vol}_{\tilde{g}}$
(19)

holds for the collection of $r$ lowest Landau levels. As we see below for the
case of the multi-band models, we can heuristically say that, as the system
approaches the Landau levels, the equality is approached.

###  IV.2 Two-band models

There are two special properties that only hold for two-band models, where the
quantum states are just $|\psi_{1}(\bm{\lambda})\rangle$ and
$|\psi_{2}(\bm{\lambda})\rangle$, and we consider geometrical properties of
one of the two states. The first property is:

> When the parameter space is a two-torus, there must be some points in the
> parameter space where $\det(g)=|\Omega_{12}|=0$.

This property and its proof are presented in the accompanying paper [50] as
Theorem 3, exploiting the difference in the fundamental groups of the two-
torus and a complex projective space, which, in this case, is just a two-
sphere. A consequence of this property is that, for two-band Chern insulators
with a nontrivial Chern number, one cannot achieve a situation where either
the quantum metric or the Berry curvature becomes flat in the parameter space,
which has also been noted in Ref. [42]. This implies that it is not possible
to achieve the limit of lowest Landau levels with two-band models. On the
other hand, there is a second property which holds only for two-band models,
which is:

> The equality $\displaystyle\sqrt{\det(g)}=\frac{|\Omega_{12}|}{2}$ always
> holds.

This relation has been noticed by several earlier works [65, 37, 66]. One can
see this property by confirming that the condition for the saturation of the
inequality Eq. (16) always holds for two-band models as noticed in Ref. [66]
and also detailed in Appendix A, or also by calculating directly the volume
form and the Berry curvature for two-band models, as in Refs. [65, 37]. A
simple consequence of the second property is that, if $\Omega_{12}$ is
everywhere non-negative in the parameter space,
$\mathrm{vol}_{g}=\pi\mathcal{C}$, and if $\Omega_{12}$ is everywhere
nonpositive in the parameter space, $\mathrm{vol}_{g}=-\pi\mathcal{C}$. This
implies that the quantum volume directly tells us about the topology of the
quantum states if there is no change of sign of the Berry curvature over the
parameter space. We will see shortly that it is often the case that the Berry
curvature does not change its sign in the Brillouin zone, and thus the quantum
volume coincides with $\pi|\mathcal{C}|$.

####  IV.2.1 Haldane model

As an explicit example of a two-band model, we first analyze the Haldane model
[51]. The Haldane model is a tight-binding model on a honeycomb lattice with
complex next-nearest-neighbor hopping amplitudes. The Haldane model has
various topological phases as one varies $\phi$ and $M/(3\sqrt{3}t_{2})$,
where $\phi$ is the phase of next-nearest-neighbor hopping, and $t_{2}$ is its
magnitude. $M$ is the on-site energy difference between the two sublattices
within a unit cell. The Hamiltonian is

$\displaystyle H(\mathbf{k})=$
$\displaystyle\begin{pmatrix}2t_{2}\sum_{j}\cos(\phi-\mathbf{k}\cdot\mathbf{R}_{j}^{\prime})+M&t_{1}\sum_{j}e^{-i\mathbf{k}\cdot\mathbf{R}_{j}}\\\
t_{1}\sum_{j}e^{i\mathbf{k}\cdot\mathbf{R}_{j}}&2t_{2}\sum_{j}\cos(\phi+\mathbf{k}\cdot\mathbf{R}_{j}^{\prime})-M\end{pmatrix},$
(20)

where $t_{1}$ is the nearest-neighbor hopping, $\mathbf{R}_{j}$ and
$\mathbf{R}_{j}^{\prime}$ with $j=$ 1, 2, and 3 are the vectors connecting
nearest- and next-nearest-neighbor sites in the honeycomb lattice. In Fig. 1,
we plot $|\mathcal{C}|$, $\mathrm{vol}_{g}/\pi$, and
$\mathrm{vol}_{\tilde{g}}/\pi$ as a function of $M/(3\sqrt{3}t_{2})$ for a
fixed value of $\phi=\pi/2$. There is a topological phase transition at
$M/(3\sqrt{3}t_{2})=1$.

The figure shows that, in the topologically nontrivial regime,
$|\mathcal{C}|=\mathrm{vol}_{g}/\pi=1$ holds, which is a consequence of a fact
that $\sqrt{\det(g)}=|\Omega_{12}|/2$ strictly holds in the case of two-band
models, together with the fact that the Berry curvature does not change sign
within the Brillouin zone for the Haldane model. In the topologically trivial
regime, we see $\mathrm{vol}_{g}/\pi<1$, which implies the correct relation
$|\mathcal{C}|=0$. Therefore, the quantum volume in momentum space precisely
captures the topology of the model.

Figure 1: The Haldane model: The Chern number and the quantum volumes as a
function of a parameter $M$ in the model. The solid line corresponds to the
Chern number, and the circles and triangles are $\mathrm{vol}_{g}/\pi$ and
$\mathrm{vol}_{\tilde{g}}/\pi$, respectively. There is a topological phase
transition at $M/(3\sqrt{3}t_{2})=1$.

We also note that the quantum volume of the twist-angle space,
$\mathrm{vol}_{\tilde{g}}/\pi$, is not equal to the Chern number in the
topological regime. This is because the relation
$\sqrt{\det(g)}=|\Omega_{12}|/2$ holds if the Hilbert space of quantum states
for a particular value of a parameter is two-dimensional, but in the twist-
angle space, the relevant Hilbert space is a many-body Fock space; the Hilbert
space is much larger than two even for two-band models.

####  IV.2.2 Square lattice model with higher Chern numbers

Next, as an example of a topological two-band model with the Chern number
greater than one, we consider the square lattice model introduced in Yang et
al. [52], which has two orbital degrees of freedom per site. The Hamiltonian
is

$\displaystyle
H(\mathbf{k})=\begin{pmatrix}2t_{3}(\cos(2k_{x})+\cos(2k_{y}))-4t_{2}\sin(k_{x})\sin(k_{y})+M&2t_{1}(e^{i\pi/4}\cos(k_{x})+e^{-i\pi/4}\cos(k_{y}))\\\
2t_{1}(e^{-i\pi/4}\cos(k_{x})+e^{i\pi/4}\cos(k_{y}))&2t_{3}(\cos(2k_{x})+\cos(2k_{y}))+4t_{2}\sin(k_{x})\sin(k_{y})-M\end{pmatrix}$
(21)

The orbital-dependent energy $M$ is not in the original paper of Yang et al.
[52]; we add this factor to induce a topological phase transition. For other
parameters, we take $t_{1}=1$, $t_{2}=1/(2+\sqrt{2})$, and
$t_{3}=1/(2+2\sqrt{2})$ as in Ref. [52]. When $M=0$, the lower (upper) band
has the Chern number of $-2$($+2$). As one increases $M$, the bands go through
a topological phase transition and become topologically trivial. As a function
of $M$, we plot the Chern number and the quantum volumes in Fig. 2. We see
that the equality $|\mathcal{C}|=\mathrm{vol}_{g}/\pi=2$ is again achieved in
the topological region.

Although achieving $|\mathcal{C}|=\mathrm{vol}_{g}/\pi$ requires that the
Berry curvature does not change its sign over the Brillouin zone, we have
confirmed through these examples that this equality is often realized in
topological two-band models.

Figure 2: The square-lattice two-band model with higher Chern numbers from
Yang et al. [52]: The Chern number and the quantum volumes as a function of a
parameter $M$ in the model. The solid line corresponds to the Chern number,
and the circles and triangles are $\mathrm{vol}_{g}/\pi$ and
$\mathrm{vol}_{\tilde{g}}/\pi$, respectively. There is a topological phase
transition around $M\approx 1.3$.

###  IV.3 Multi-band models

For multi-band systems, we do not have an equality
$\sqrt{\det(g)}=|\Omega_{12}|/2$ in general. On the other hand, we do not have
the constraint that $\sqrt{\det(g)}$ and $|\Omega_{12}|$ vanish somewhere in
the Brillouin zone. The latter property implies that we can consider the limit
where the quantum metric and the Berry curvature become flat (constant) over
the Brillouin zone, which is similar to the situation of the Landau levels.
For examples of multi-band models, we consider the Harper-Hofstadter model
[53, 54], the three-band model with a unit Chern number introduced in Sun et
al. [55], and the three-band model with a higher Chern number introduced in
Yang et al. [52].

####  IV.3.1 Harper-Hofstadter model

The Harper-Hofstadter model is a square-lattice tight-binding model with a
magnetic flux $\phi$. When $\phi=2\pi p/q$ with mutually prime integers $p$
and $q$, the model has $q$ bands. We consider the lowest one and two bands of
the Harper-Hofstadter model for $p=1$ to show that, in the large $q$ limit,
the model approaches the limit of the Landau levels and the inequality Eq.
(16) is saturated.

First, we consider the lowest band of $\phi=2\pi/q$. It is known in this case
that the lowest band has the Chern number of $-1$. We calculate the Chern
number and the quantum volume of the lowest band for various values of $q$.
The results are plotted in Fig. 3. We see that the inequality Eq. (16) becomes
saturated as one increases $q$. In the Harper-Hofstadter model, the large $q$
limit corresponds to the Landau level limit [67], and thus the lowest band
should approach the lowest Landau level. The saturation of the inequality is
consistent with the property of the lowest Landau level that the equality is
achieved.

Figure 3: The Harper-Hofstadter model: The Chern number and quantum volumes as
a function of $q$, where $2\pi/q$ is the magnitude of the magnetic flux
through a plaquette. The left figure is for the lowest band, and the right
figure is for the lowest two bands. The solid line corresponds to the Chern
number, and the circles and triangles are $\mathrm{vol}_{g}/\pi$ and
$\mathrm{vol}_{\tilde{g}}/\pi$, respectively.

Next we consider the lowest two bands of $\phi=2\pi/q$. When $q\geq 5$, lowest
two bands both have the Chern number of $-1$, and thus the Chern number of the
two bands combined is $-2$. In Fig. 3 we also plot the Chern number and the
quantum volume of the lowest two bands. The inequality Eq. (16) is saturated
again in the large $q$ limit, confirming that the limit of Landau levels is
achieved also when more than one band are considered.

####  IV.3.2 Three-band model with unit Chern number

Next, we consider the three-band model introduced in Sun et al. Ref. [55]. In
this model, one can make a band very flat by properly choosing the parameters.
We investigate the relation between the band flatness and the saturation of
the inequality Eq. (16). The momentum-space Hamiltonian of the model is

$\displaystyle H(\mathbf{k})=\begin{pmatrix}-2t_{dd}(\cos k_{x}+\cos
k_{y})+\delta&2it_{pd}\sin k_{x}&2it_{pd}\sin k_{y}\\\ -2it_{pd}\sin
k_{x}&2t_{pp}\cos k_{x}-2t_{pp}^{\prime}\cos k_{y}&i\Delta\\\ -2it_{pd}\sin
k_{y}&-i\Delta&2t_{pp}\cos k_{y}-2t^{\prime}_{pp}\cos k_{x}\end{pmatrix}.$
(22)

We focus on the lowest band. A topological phase transition can be induced by
changing $\Delta$. We use a set of parameters discussed in Ref. [55], which
are $t_{dd}=t_{pd}=t_{pp}=1$ and
$\delta=-4t_{dd}+2t_{pp}+\Delta-2t_{pp}\Delta/(4t_{pp}+\Delta)$ and
$t_{pp}^{\prime}=t_{pp}\Delta/(4t_{pp}+\Delta)$. The Chern number and the
quantum volumes are plotted in Fig. 4. There is a topological phase transition
at $\Delta=4$. We also plot the band flatness in the inset of Fig. 4. The band
flatness is evaluated by the ratio between the band width and the band gap;
the smaller value of this ratio implies that the band is flatter. In the
topologically trivial regime, the quantum volume in momentum space predicts
the Chern number being zero correctly apart from region in the vicinity of the
topological phase transition. In the topologically nontrivial regime, the
agreement between the Chern number and the quantum volume becomes better as
the band becomes flatter. We expect this tendency that flattening of a band
makes the inequality Eq. (16) saturated to hold in general from an analogy
with the Landau levels, which are flat and topological.

Figure 4: Three band model from Sun et al. [55]: The Chern number and quantum
volumes as a function of a parameter $\Delta$. The solid line corresponds to
the Chern number, and the circles and triangles are $\mathrm{vol}_{g}/\pi$ and
$\mathrm{vol}_{\tilde{g}}/\pi$, respectively. The inset shows the band
flatness, which is defined by the ratio between the bandwidth and the band
gap.

####  IV.3.3 Three-band model with higher Chern numbers

The tendency that flatter bands lead to $|\mathcal{C}|\approx\mathrm{vol}_{g}$
also holds when we have a band with the Chern number greater than one. To see
this property, we consider another three band model proposed in Yang et al.
[52]. The lowest band of the model has the Chern number of three, and
depending on parameters, the band can become very flat. The Hamiltonian is

$\displaystyle
H(\mathbf{k})=\begin{pmatrix}\mathcal{F}_{\mathbf{k}}(\phi)/2&\mathcal{G}_{2,\mathbf{k}}(5\phi)&\mathcal{G}_{1,\mathbf{k}}(6\phi)\\\
\mathcal{G}_{1,\mathbf{k}}(2\phi)&\mathcal{F}_{\mathbf{k}}(3\phi)/2&\mathcal{G}_{2,\mathbf{k}}(7\phi)\\\
\mathcal{G}_{2,\mathbf{k}}(3\phi)&\mathcal{G}_{1,\mathbf{k}}(4\phi)&\mathcal{F}_{\mathbf{k}}(5\phi)/2\end{pmatrix}+\mathrm{H.c.},$
(23)

where

$\displaystyle\mathcal{F}_{\mathbf{k}}(\Phi)$ $\displaystyle\equiv
2t_{2}\cos(k_{x}+k_{y}-\Phi)$ (24)
$\displaystyle\mathcal{G}_{1,\mathbf{k}}(\Phi)$ $\displaystyle\equiv
t_{1}e^{ik_{x}}+t_{1}e^{-ik_{y}+i\Phi}$ (25)
$\displaystyle\mathcal{G}_{2,\mathbf{k}}(\Phi)$ $\displaystyle\equiv
t_{2}e^{ik_{x}-ik_{y}+i\Phi}.$ (26)

As in Ref. [52], we take $t_{1}=1$ and $\phi=\pi/3$. We take
$t_{2}=-\Delta/\sqrt{3}$, and vary $\Delta$ to tune the band flatness of the
lowest band. In Fig. 5, we plot the Chern number and quantum volumes as a
function of the tuning parameter $\Delta$, as well as the band flatness. The
band becomes most flat when $\Delta=1$, and indeed the equality
$|\mathcal{C}|\approx\mathrm{vol}_{g}/\pi\approx\mathrm{vol}_{\tilde{g}}/\pi$
almost holds around the region where the band is flat. As the band flatness is
reduced, the discrepancy among $|\mathcal{C}|$, $\mathrm{vol}_{g}/\pi$, and
$\mathrm{vol}_{\tilde{g}}/\pi$ becomes larger. With these examples, we can
heuristically say that the inequality Eq. (16) approaches equality when the
band becomes flatter.

Figure 5: Three-band model from Yang et al. [52]: The Chern number and quantum
volumes as a function of a parameter $\Delta$. The solid line corresponds to
the Chern number, and the circles and triangles are $\mathrm{vol}_{g}/\pi$ and
$\mathrm{vol}_{\tilde{g}}/\pi$, respectively. The inset shows the band
flatness, which is defined by the ratio between the bandwidth and the bandgap.

##  V Conclusion

Although Chern number and the quantum metric are both derived from the same
map from the parameter space to the space of quantum states, their physical
manifestations are quite different. As a bulk response, the former is related
to an anti-symmetric responses such as the Hall conductivity and anomalous
velocity, whereas the latter is more related to the localization properties of
the system. Despite the difference, we have seen that because of the
inequality

$\displaystyle\pi|\mathcal{C}|\leq\mathrm{vol}_{g}\leq\mathrm{vol}_{\tilde{g}}$
(27)

we can infer topological properties just by looking at the quantum metric and
vice versa. Because of the inequality, the quantum volume of momentum space,
$\mathrm{vol}_{g}$, is always closer to $\pi|\mathcal{C}|$ than the quantum
volume of the twist-angle space, $\mathrm{vol}_{\tilde{g}}$. On the other
hand, we note that when one has a fermionic band insulator, what is
experimentally relevant is often the quantum metric in the twist-angle space,
$\mathrm{vol}_{\tilde{g}}$, as manifested in the experiment Ref. [43]. Since
each term of the inequality Eq. (27) is related to observables in different
experimental setups, the inequality provides us the possibility of studying
topology and geometry from multiple perspectives.

If similar inequalities hold for higher-dimensional Chern insulators and/or
symmetry-protected topological phases is left for future studies. Furthermore,
as discussed in more detail in the accompanying paper [50], the saturation of
the inequality is related to holomorphic structure of the map from the
parameter space to the space of the quantum states. Studying consequences of
the inequality both from physical and mathematical points of view will open a
new avenue towards a unified understanding of the two apparently different
concepts of topology and localization.

###### Acknowledgements.

We thank Hosho Katsura for suggesting a simple proof for Appendix A. T.O.
acknowledges support from JSPS KAKENHI Grant No. JP20H01845, JST PRESTO Grant
No. JPMJPR19L2, JST CREST Grant No. JPMJCR19T1, and RIKEN iTHEMS. B.M.
acknowledges the support from SQIG – Security and Quantum Information Group,
the Instituto de Telecomunicações (IT) Research Unit, Ref. UIDB/50008/2020,
funded by Fundação para a Ciência e a Tecnologia (FCT), European funds,
namely, H2020 project SPARTA, as well as projects QuantMining
POCI-01-0145-FEDER-031826 and PREDICT PTDC/CCI-CIF/29877/2017.

##  Appendix A Proof of $\sqrt{\det(g)}\geq|\Omega_{12}|/2$

In this appendix, we give a proof of the inequality Eq.(5), i.e.,
$\sqrt{\det(g(\bm{\lambda}))}\geq|\Omega_{12}(\bm{\lambda})|/2$ identifying
the condition under which the inequality is saturated. Let us consider two
sets of vectors

$\displaystyle|\alpha\rangle$
$\displaystyle=\begin{pmatrix}|\alpha_{1}\rangle\\\ |\alpha_{2}\rangle\\\
\vdots\\\ |\alpha_{r}\rangle\end{pmatrix},$ $\displaystyle|\beta\rangle$
$\displaystyle=\begin{pmatrix}|\beta_{1}\rangle\\\ |\beta_{2}\rangle\\\
\vdots\\\ |\beta_{r}\rangle\end{pmatrix},$ (28)

where each $|\alpha_{m}\rangle$ and $|\beta_{m}\rangle$ is an element of the
Hilbert space spanned by the quantum states for a given value of
$\bm{\lambda}$. As in the main text, $r$ denotes the number of occupied bands.
Let us define a complex inner product $\langle\alpha|\beta\rangle$ between two
sets of vectors by

$\displaystyle\langle\alpha|\beta\rangle\equiv\sum_{m=1}^{r}\langle\alpha_{m}|\beta_{m}\rangle.$
(29)

It is straightforward to show that this definition fulfills the condition of
inner product. Then, the following Cauchy-Schwarz inequality holds [68]:

$\displaystyle\langle\alpha|\alpha\rangle\langle\beta|\beta\rangle\geq|\langle\alpha|\beta\rangle|^{2},$
(30)

and the equality holds if either $|\alpha\rangle$ or $|\beta\rangle$ is zero,
or if there exists a complex number $c$, which satisfies
$|\alpha\rangle=c|\beta\rangle$. We now take

$\displaystyle|\alpha_{m}\rangle$
$\displaystyle=(1-P)|\partial_{i}\psi_{m}(\bm{\lambda})\rangle,$ (31)
$\displaystyle|\beta_{m}\rangle$
$\displaystyle=(1-P)|\partial_{j}\psi_{m}(\bm{\lambda})\rangle,$ (32)

where $P$ is, as in the main text, the projector
$P=\sum_{m=1}^{r}|\psi_{m}(\bm{\lambda})\rangle\langle\psi_{m}(\bm{\lambda})|$.
Using the property $(1-P)^{2}=(1-P)$, one can see that

$\displaystyle\langle\alpha|\alpha\rangle$
$\displaystyle=\chi_{ii}(\bm{\lambda}),$
$\displaystyle\langle\beta|\beta\rangle$
$\displaystyle=\chi_{jj}(\bm{\lambda}),$
$\displaystyle\langle\alpha|\beta\rangle$
$\displaystyle=\chi_{ij}(\bm{\lambda}).$ (33)

The Cauchy-Schwarz inequality Eq. (30) then gives us

$\displaystyle\chi_{ii}(\bm{\lambda})\chi_{jj}(\bm{\lambda})\geq|\chi_{ij}(\bm{\lambda})|^{2}.$
(34)

Note that, for two-dimensional parameter spaces, this relation is nothing but
the positive semi-definiteness of the quantum geometric tensor $\det(\chi)\geq
0$ [35]. Since $\chi_{ii}=g_{ii}$ and
$|\chi_{ij}(\bm{\lambda})|^{2}=(\mathrm{Re}[\chi_{ij}])^{2}+(\mathrm{Im}[\chi_{ij}])^{2}=g_{ij}^{2}+\Omega_{ij}^{2}/4$,
we obtain

$\displaystyle g_{ii}g_{jj}-g_{ij}^{2}\geq\Omega_{ij}^{2}/4$ (35)

Now, considering a two-dimensional parameter space and taking $i=1$ and $j=2$,
we obtain

$\displaystyle\det(g)\geq|\Omega_{12}|^{2}/4.$ (36)

Taking the square root of both sides, we obtain the desired inequality. The
condition for the inequality to be saturated is either
$\chi_{11}=\chi_{12}=\chi_{22}=0$ or $\exists\,c\in\mathbb{C}$ such that
$\forall m,\
(1-P)|\partial_{1}\psi_{m}(\bm{\lambda})\rangle=c(1-P)|\partial_{2}\psi_{m}(\bm{\lambda})\rangle$.
One can see that, for two-band models, the inequality is always saturated
because $1-P$ projects to a one-dimensional subspace, and thus the condition
$\exists\,c\in\mathbb{C}$ such that
$(1-P)|\partial_{1}\psi_{1}(\bm{\lambda})\rangle=c(1-P)|\partial_{2}\psi_{1}(\bm{\lambda})\rangle$
always holds regardless of the details of $|\psi_{1}(\bm{\lambda})\rangle$.

We also note that, since the two eigenvalues of the matrix $g$ are
nonnegative, $\mathrm{tr}(g)\geq 2\sqrt{\det(g)}$ holds in general. This
implies $\mathrm{tr}(g)\geq 2\sqrt{\det(g)}\geq|\Omega_{12}|$, which was also
first noted by Roy [56].

##  Appendix B Proof of $\mathrm{vol}_{g}\leq\mathrm{vol}_{\tilde{g}}$

We now prove $\mathrm{vol}_{g}\leq\mathrm{vol}_{\tilde{g}}$, which states that
the quantum volume of momentum space is smaller or equal to the quantum volume
of the twist-angle space. Since the quantum metric in the twist-angle space,
$\tilde{g}_{ij}(\theta_{x},\theta_{y})$, is flat, we have
$\mathrm{vol}_{\tilde{g}}=(2\pi)^{2}\det(\tilde{g}_{ij})$. Then, explicitly
writing out the expressions for the determinant, the statement we want to
prove can be rewritten in the following form:

$\displaystyle\mathrm{vol}_{g}=\int d^{2}k\sqrt{g_{xx}g_{yy}-g_{xy}^{2}}$
$\displaystyle\leq\sqrt{\int d^{2}kg_{xx}\int d^{2}kg_{yy}-\left(\int
d^{2}kg_{xy}\right)^{2}}=\mathrm{vol}_{\tilde{g}}.$ (37)

To prove this relation, we use the Cauchy-Schwarz inequality again, which, in
a formulation relevant to the current case, states that for any square-
integrable real nonnegative functions $\phi(\mathbf{k})$ and
$\psi(\mathbf{k})$,

$\displaystyle\int d^{2}k\;\phi(\mathbf{k})\psi(\mathbf{k})\leq\sqrt{\int
d^{2}k[\phi(\mathbf{k})]^{2}\cdot\int d^{2}k[\psi(\mathbf{k})]^{2}}.$ (38)

To use the Cauchy-Schwarz inequality, we write

$\displaystyle\mathrm{vol}_{g}=\int
d^{2}k\sqrt{\sqrt{g_{xx}g_{yy}}-g_{xy}}\cdot\sqrt{\sqrt{g_{xx}g_{yy}}+g_{xy}}.$
(39)

Using the Cauchy-Schwarz inequality, we obtain

$\displaystyle\mathrm{vol}_{g}$ $\displaystyle\leq\sqrt{\int
d^{2}k\left(\sqrt{g_{xx}g_{yy}}-g_{xy}\right)\cdot\int
d^{2}k\left(\sqrt{g_{xx}g_{yy}}+g_{xy}\right)}$
$\displaystyle=\sqrt{\left(\int
d^{2}k\sqrt{g_{xx}g_{yy}}\right)^{2}-\left(\int d^{2}kg_{xy}\right)^{2}}.$
(40)

Using again the Cauchy-Schwarz inequality to the first term inside the square
root, we obtain the desired relation:

$\displaystyle\mathrm{vol}_{g}$ $\displaystyle\leq\sqrt{\int d^{2}kg_{xx}\int
d^{2}kg_{yy}-\left(\int d^{2}kg_{xy}\right)^{2}}=\mathrm{vol}_{\tilde{g}}.$
(41)

The condition for the Cauchy-Schwarz inequality to hold as an equality, when
both functions $\phi$ and $\psi$ are not identically zero, is that the two
functions are linearly dependent, namely, one is a scalar multiple of the
other. Upon deriving $\mathrm{vol}_{g}\leq\mathrm{vol}_{\tilde{g}}$, we used
the Cauchy-Schwarz inequality twice. For the equality to hold in both steps,
we need that $g_{xy}$ and $g_{yy}$ are both scalar multiples of $g_{xx}$. This
is equivalent to saying that the quantum metric in momentum space takes the
following form:

$\displaystyle g(k_{x},k_{y})$
$\displaystyle=f(k_{x},k_{y})\begin{pmatrix}c_{xx}&c_{xy}\\\
c_{xy}&c_{yy}\end{pmatrix},$ (42)

where $c_{xx}$, $c_{xy}$, and $c_{yy}$ are $\mathbf{k}$-independent real
numbers and $f(k_{x},k_{y})$ is a nonnegative function.

##  Appendix C Geometry and topology of Landau levels

Here we briefly summarize the derivation of the geometrical properties and the
first Chern number of Landau levels. For single Landau levels, the similar
quantities have also been evaluated in Ref. [35] by taking appropriate limits
of the Harper-Hofstadter model. In the alternative derivation given here, we
work directly on the Landau level wave functions, and provide the quantum
geometric tensor of a collection of $r$ Landau levels. The Hamiltonian of a
charged particle in a uniform magnetic field $B$ in a two-dimensional space is
given by

$\displaystyle
H=\frac{(\hat{p}_{x}-\hat{A}_{x})^{2}+(\hat{p}_{y}-\hat{A}_{y})^{2}}{2M},$
(43)

with a magnetic vector potential $\mathbf{A}=(\hat{A}_{x},\hat{A}_{y})$
satisfying $B=\partial_{x}\hat{A}_{y}-\partial_{y}\hat{A}_{x}$. Later we take
the Landau gauge $\mathbf{A}=(0,B\hat{x})$. We take the charge to be unity.
This Hamiltonian has a translational symmetry only in $y$ direction, and the
momentum along $x$ direction is not a good quantum number. We can, however,
consider a magnetic translational symmetry, which is a translation followed by
a phase shift. Using the magnetic translational symmetry, we can label
eigenstates using quasimomenta along both $x$ and $y$ directions, and the
geometrical and topological properties can be defined as in Sec. II.

To introduce the magnetic translational symmetry, we consider a fictitious
magnetic unit cell, which is a rectangle of length $a_{x}$ in $x$ direction
and $a_{y}$ in $y$ direction satisfying

$\displaystyle|B|a_{x}a_{y}=2\pi.$ (44)

Note that the choice of $a_{x}$ and $a_{y}$ satisfying the relation Eq. (44)
is not unique, and the result below does not depend on the choice.

A lattice vector $\mathbf{R}$ is any linear combination of
$\mathbf{a}_{x}\equiv(a_{x},0)$ and $\mathbf{a}_{y}\equiv(0,a_{y})$ with
integer coefficients. Now, we define a usual translation operator by

$\displaystyle\mathcal{T}(\mathbf{R})\equiv
e^{i\mathbf{R}\cdot\hat{\mathbf{p}}}=e^{\mathbf{R}\cdot\nabla},$ (45)

which satisfies

$\displaystyle\mathcal{T}(\mathbf{R})f(\mathbf{r})=f(\mathbf{r}+\mathbf{R}),$
(46)

when acted to a smooth function $f(\mathbf{r})$. The translational operator
$\mathcal{T}(\mathbf{R})$ does not commute with the Hamiltonian. However, by
introducing a function $\xi_{\mathbf{R}}(\mathbf{r})$, which satisfies

$\displaystyle\mathbf{A}(\mathbf{r}+\mathbf{R})=\mathbf{A}(\mathbf{r})+\nabla\xi_{\mathbf{R}}(\mathbf{r}),$
(47)

we can define a magnetic translation operator:

$\displaystyle\mathcal{M}(\mathbf{R})\equiv
e^{-i\xi_{\mathbf{R}}(\mathbf{r})}\mathcal{T}(\mathbf{R}).$ (48)

The magnetic translation operators $\mathcal{M}(\mathbf{R})$ commute with
themselves and also with the Hamiltonian. In addition, one can show
$\mathcal{M}(\mathbf{R}_{1})\mathcal{M}(\mathbf{R}_{2})=\mathcal{M}(\mathbf{R}_{1}+\mathbf{R}_{2})$.
With these properties, we can prove the Bloch theorem for the magnetic
translation operators.

The Bloch theorem now states that the Hamiltonian and the magnetic translation
operators can be simultaneously diagonalized. An eigenstate
$\psi_{m,\mathbf{k}}(\mathbf{r})$ is characterized by a band index $m$ and a
quasimomentum $\mathbf{k}$ and satisfies

$\displaystyle\mathcal{M}(\mathbf{R})\psi_{m,\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{R}}\psi_{m,\mathbf{k}}(\mathbf{r}).$
(49)

On the other hand, using (48), one obtains

$\displaystyle\mathcal{M}(\mathbf{R})\psi_{m,\mathbf{k}}(\mathbf{r})$
$\displaystyle=e^{-i\xi_{\mathbf{R}}(\mathbf{r})}\mathcal{T}(\mathbf{R})\psi_{m,\mathbf{k}}(\mathbf{r})$
$\displaystyle=e^{-i\xi_{\mathbf{R}}(\mathbf{r})}\psi_{m,\mathbf{k}}(\mathbf{r}+\mathbf{R}).$
(50)

This means

$\displaystyle\psi_{m,\mathbf{k}}(\mathbf{r}+\mathbf{R})=e^{i\mathbf{k}\cdot\mathbf{R}+i\xi_{\mathbf{R}}(\mathbf{r})}\psi_{m,\mathbf{k}}(\mathbf{r}).$
(51)

The Bloch wave function $u_{m,\mathbf{k}}(\mathbf{r})$ is defined by

$\displaystyle\psi_{m,\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{m,\mathbf{k}}(\mathbf{r}).$
(52)

From (51) and (52), one obtains the twisted boundary condition for the Bloch
wave functions upon translation by a lattice vector:

$\displaystyle
u_{m,\mathbf{k}}(\mathbf{r}+\mathbf{R})=e^{i\xi_{\mathbf{R}}(\mathbf{r})}u_{m,\mathbf{k}}(\mathbf{r}).$
(53)

In the absence of the magnetic field, the Bloch wave function satisfies
$u_{m,\mathbf{k}}(\mathbf{r}+\mathbf{R})=u_{m,\mathbf{k}}(\mathbf{r})$. So,
the additional factor of $e^{i\xi_{\mathbf{R}}(\mathbf{r})}$ is the
peculiarity of having nonzero magnetic field. We note that the condition Eq.
(53) defines the sections of a non-trivial line bundle over the unit magnetic
cell in real space, which topologically is a torus. The quantities
$\\{e^{i\xi_{\bf{R}}}\\}$, where $\bf{R}$ ranges over the lattice, form what
is known in the mathematics literature as a system of multipliers for the line
bundle.

Now, let us explicitly take the Landau gauge $\mathbf{A}(\mathbf{r})=(0,Bx)$.
Then we can take $\xi_{\mathbf{a}_{x}}(\mathbf{r})=Ba_{x}y$ and
$\xi_{\mathbf{a}_{y}}(\mathbf{r})=0$. The boundary condition for the Bloch
wave function for the Landau gauge is then

$\displaystyle u_{m,\mathbf{k}}(x+a_{x},y)$
$\displaystyle=e^{iBa_{x}y}u_{m,\mathbf{k}}(x,y),$ $\displaystyle
u_{m,\mathbf{k}}(x,y+a_{y})$ $\displaystyle=u_{m,\mathbf{k}}(x,y).$ (54)

We now want to find eigenstates of $H$, which satisfy

$\displaystyle
He^{i\mathbf{k}\cdot\mathbf{r}}u_{m,\mathbf{k}}(\mathbf{r})=E_{m,\mathbf{k}}e^{i\mathbf{k}\cdot\mathbf{r}}u_{m,\mathbf{k}}(\mathbf{r})$
(55)

and the boundary conditions (54). Once we find the Bloch wave functions
$u_{m,\mathbf{k}}(\mathbf{r})$, it is straightforward to calculate the Berry
curvature and the Chern number from their definitions.

Under the Landau gauge, the momentum along $y$ direction is a good quantum
number. Writing an eigenstate as $e^{ik_{y}y}\psi(x,t)$, the Schrödinger
equation becomes

$\displaystyle i\frac{\partial}{\partial
t}e^{ik_{y}y}\psi(x,t)=\frac{\hat{p}_{x}^{2}+(k_{y}-B\hat{x})^{2}}{2M}e^{ik_{y}y}\psi(x,t),$
(56)

This equation can be rewritten to become

$\displaystyle i\frac{\partial}{\partial
t}\psi(x,t)=\left[\frac{\hat{p}_{x}^{2}}{2M}+\frac{B^{2}}{2M}\left(\hat{x}-\frac{k_{y}}{B}\right)^{2}\right]\psi(x,t).$
(57)

This is the Schrödinger equation of a particle in a one-dimensional harmonic
oscillator with the frequency $\omega=|B|/M$ and the origin shifted at
$x_{0}=k_{y}/B$. This means that, for a given value of $k_{y}$, the energy
level is $\omega(m+1/2)$ with an integer $m\geq 0$. We note that the lowest
Landau level is given by $m=0$. The (unnormalized) eigenstate is

$\displaystyle
h_{m}(x-k_{y}/B)=e^{-(x-k_{y}/B)^{2}/2l_{B}^{2}}\mathrm{H}_{m}((x-k_{y}/B)/l_{B}),$
(58)

where $l_{B}\equiv 1/\sqrt{|B|}$ is the magnetic length and $H_{m}$ is the
Hermite polynomial with degree $m$. This eigenstate does not obey the
periodicity of the Bloch wave function Eq. (54). To obtain the Bloch wave
function, we consider the following linear combination of states where values
of $k_{y}$ are separated by $2\pi/a_{y}$ times an integer:

$\displaystyle\psi_{m,\mathbf{k}}(\mathbf{r})$
$\displaystyle\equiv\sum_{l=-\infty}^{\infty}e^{ik_{x}a_{x}l}e^{i(k_{y}+2\pi
l/a_{y})y}h_{m}(x-(k_{y}+2\pi l/a_{y})/B)$
$\displaystyle=\sum_{l=-\infty}^{\infty}e^{ik_{x}a_{x}l}e^{i(k_{y}+2\pi
l/a_{y})y}h_{m}(x-a_{x}l-k_{y}/B)$ (59)

Each term in the sum is an eingenstate of the original Hamiltonian with the
eigenvalue $\omega(m+1/2)$, so the linear combiation
$\psi_{m,\mathbf{k}}(\mathbf{r})$ is still an eigenstate of the original
Hamiltonian with the same eigenvalue:

$\displaystyle
H\psi_{m,\mathbf{k}}(\mathbf{r})=\omega\left(m+\frac{1}{2}\right)\psi_{m,\mathbf{k}}(\mathbf{r}).$
(60)

We now define

$\displaystyle u_{m,\mathbf{k}}(\mathbf{r})$ $\displaystyle\equiv
e^{-i\mathbf{k}\cdot\mathbf{r}}\psi_{m,\mathbf{k}}(\mathbf{r})$ (61)
$\displaystyle=\sum_{l=-\infty}^{\infty}e^{-ik_{x}(x-a_{x}l)+i2\pi
ly/a_{y}}h_{m}(x-a_{x}l-k_{y}/B).$

One can confirm that this function $u_{m,\mathbf{k}}(\mathbf{r})$ is the
desired Bloch wave function, which obeys the boundary condition Eq. (54).

Using the Bloch wave function (61), we can now calculate the quantum geometric
tensor and the Chern number. We note that the Bloch wave function (61) is not
normalized. We introduce an unnormalized Bloch wave function in a “ket” form
$|\tilde{u}_{m,\mathbf{k}}\rangle$ which satisfies, in position
representation,
$\langle\mathbf{r}|\tilde{u}_{m,\mathbf{k}}\rangle=u_{m,\mathbf{k}(\mathbf{r})}$.
Its inner product with itself satisfies

$\displaystyle\langle\tilde{u}_{m,\mathbf{k}}|\tilde{u}_{m,\mathbf{k}}\rangle=\int_{0}^{a_{x}}dx\int_{0}^{a_{y}}dy|u_{m,\mathbf{k}}(x,y)|^{2}.$
(62)

In terms of the unnormalized Bloch wave function, one can show that the
quantum geometric tensor when the lowest $r$ bands are occupied takes the
following form:

$\displaystyle\chi_{ij}(\mathbf{k})=\sum_{m=0}^{r-1}\frac{\langle\partial_{k_{i}}\tilde{u}_{m,\mathbf{k}}|(1-P(\mathbf{k}))|\partial_{k_{j}}\tilde{u}_{m,\mathbf{k}}\rangle}{\langle\tilde{u}_{m,\mathbf{k}}|\tilde{u}_{m,\mathbf{k}}\rangle}.$
(63)

where

$\displaystyle
P(\mathbf{k})=\sum_{m=0}^{r-1}\frac{|\tilde{u}_{m,\mathbf{k}}\rangle\langle\tilde{u}_{m,\mathbf{k}}|}{\langle\tilde{u}_{m,\mathbf{k}}|\tilde{u}_{m,\mathbf{k}}\rangle}.$
(64)

The denominators take care of the normalization of the wave function.

The matrix elements can be calculated using standard properties of Hermite
polynomials. For example,

$\displaystyle\langle\tilde{u}_{m,\mathbf{k}}|\tilde{u}_{m,\mathbf{k}}\rangle$
$\displaystyle=\int_{0}^{a_{x}}dx\int_{0}^{a_{y}}dy|u_{m,\mathbf{k}}(x,y)|^{2}$
$\displaystyle=a_{y}\int_{0}^{a_{x}}dx\sum_{l=-\infty}^{\infty}h_{m}(x-a_{x}l-k_{y}/B)^{2}$
$\displaystyle=a_{y}\int_{-\infty}^{\infty}h_{m}(x-k_{y}/B)^{2}=a_{y}\int_{-\infty}^{\infty}h_{m}(x)^{2}$
$\displaystyle=a_{y}l_{B}\int_{-\infty}^{\infty}dXd^{-X^{2}}H_{n}(X)^{2}$
$\displaystyle=a_{y}l_{B}\sqrt{\pi}2^{n}n!.$ (65)

Also,

$\displaystyle\langle\partial_{k_{x}}\tilde{u}_{m,\mathbf{k}}|\partial_{k_{x}}\tilde{u}_{m,\mathbf{k}}\rangle$
$\displaystyle=\int_{0}^{a_{x}}dx\int_{0}^{a_{y}}dy\partial_{k_{x}}u^{*}_{m,\mathbf{k}}(x,y)\partial_{k_{x}}u_{m,\mathbf{k}}(x,y)$
$\displaystyle=a_{y}\int_{0}^{a_{x}}dx\sum_{l=-\infty}^{\infty}(x-a_{x}l)^{2}h_{m}(x-a_{x}l-k_{y}/B)^{2}$
$\displaystyle=a_{y}\int_{-\infty}^{\infty}dx(x+k_{y}/B)^{2}h_{m}(x)^{2}$
$\displaystyle=a_{y}\int_{-\infty}^{\infty}dx(x^{2}+k_{y}^{2}/B^{2})h_{m}(x)^{2},$
(66)

where in the final step we used that $h_{m}(x)^{2}$ is an even function of
$x$. Using the recurrence relations of the Hermite polynomial, one can show

$\displaystyle x^{2}h_{m}(x)$ (67)
$\displaystyle=l_{B}^{2}\left[\frac{1}{4}h_{m+2}(x)+\left(m+\frac{1}{2}\right)h_{m}(x)+m(m-1)h_{m-2}(x)\right].$

Then, using the orthogonality of the Hermite polynomials, we obtain

$\displaystyle\frac{\langle\partial_{k_{x}}\tilde{u}_{m,\mathbf{k}}|\partial_{k_{x}}\tilde{u}_{m,\mathbf{k}}\rangle}{\langle\tilde{u}_{m,\mathbf{k}}|\tilde{u}_{m,\mathbf{k}}\rangle}=\frac{1}{|B|}\left(m+\frac{1}{2}\right)+\frac{k_{y}^{2}}{B^{2}}.$
(68)

Similarly, one can show

$\displaystyle\frac{\langle\partial_{k_{y}}\tilde{u}_{m,\mathbf{k}}|\partial_{k_{y}}\tilde{u}_{m,\mathbf{k}}\rangle}{\langle\tilde{u}_{m,\mathbf{k}}|\tilde{u}_{m,\mathbf{k}}\rangle}$
$\displaystyle=\frac{1}{|B|}\left(m+\frac{1}{2}\right)$ (69)
$\displaystyle\frac{\langle\partial_{k_{x}}\tilde{u}_{m,\mathbf{k}}|\partial_{k_{y}}\tilde{u}_{m,\mathbf{k}}\rangle}{\langle\tilde{u}_{m,\mathbf{k}}|\tilde{u}_{m,\mathbf{k}}\rangle}$
$\displaystyle=\frac{i}{2B}.$ (70)

Furthermore, the inter-band matrix elements are

$\displaystyle\frac{\langle\tilde{u}_{m^{\prime},\mathbf{k}}|\partial_{k_{x}}\tilde{u}_{m,\mathbf{k}}\rangle}{\langle\tilde{u}_{m,\mathbf{k}}|\tilde{u}_{m,\mathbf{k}}\rangle}$
$\displaystyle=-i\left(l_{B}(m+1)\delta_{m+1,m^{\prime}}+(l_{B}/2)\delta_{m-1,m^{\prime}}+(k_{y}/B)\delta_{m,m^{\prime}}\right)$
(71)
$\displaystyle\frac{\langle\tilde{u}_{m^{\prime},\mathbf{k}}|\partial_{k_{y}}\tilde{u}_{m,\mathbf{k}}\rangle}{\langle\tilde{u}_{m,\mathbf{k}}|\tilde{u}_{m,\mathbf{k}}\rangle}$
$\displaystyle=-\frac{1}{l_{B}B}\left(-(m+1)\delta_{m+1,m^{\prime}}+(1/2)\delta_{m-1,m^{\prime}}\right).$
(72)

From these relations, we can compute the full quantum geometric tensor. Below
we first consider the quantum geometric tensor or each band, and then consider
the quantum geometric tensor of multiple bands combined.

###  C.1 Single band

We first derive the quantum geometric tensor when only single $m$-th band is
in consideration. The quantum metric of $m$-th band is

$\displaystyle g_{m,xx}(\mathbf{k})$
$\displaystyle=g_{m,yy}(\mathbf{k})=\frac{1}{|B|}\left(m+\frac{1}{2}\right),$
(73) $\displaystyle g_{m,xy}(\mathbf{k})$ $\displaystyle=0,$ (74)

which has been obtained in Ref. [35]. The quantum volume for each band is then

$\displaystyle\mathrm{vol}_{g}$
$\displaystyle=\int_{-\pi/a_{x}}^{\pi/a_{x}}dk_{x}\int_{-\pi/a_{y}}^{\pi/a_{y}}dk_{y}\sqrt{\det(g)}$
$\displaystyle=\int_{-\pi/a_{x}}^{\pi/a_{x}}dk_{x}\int_{-\pi/a_{y}}^{\pi/a_{y}}dk_{y}\frac{1}{|B|}\left(m+\frac{1}{2}\right)$
$\displaystyle=\frac{2\pi}{a_{x}}\frac{2\pi}{a_{y}}\frac{1}{|B|}\left(m+\frac{1}{2}\right)=2\pi\left(m+\frac{1}{2}\right).$
(75)

Note that since the quantum metric is flat in momentum space, the quantum
volume of the twist-angle space takes the same value:
$\mathrm{vol}_{g}=\mathrm{vol}_{\tilde{g}}$.

For the Berry curvature,

$\displaystyle\Omega_{xy}(k_{x},k_{y})=-\frac{1}{B}.$ (76)

The Chern number is then

$\displaystyle\mathcal{C}$
$\displaystyle=\frac{1}{2\pi}\int_{-\pi/a_{x}}^{\pi/a_{x}}dk_{x}\int_{-\pi/a_{y}}^{\pi/a_{y}}dk_{y}\Omega_{z}(k_{x},k_{y})$
$\displaystyle=\frac{1}{2\pi}\frac{2\pi}{a_{x}}\frac{2\pi}{a_{y}}\frac{-1}{B}=-\mathrm{sign}(B).$
(77)

Therefore, the Berry curvature of Landau levels is a constant $-1/B$
irrespective of the band index $m$, and the absolute value of the Chern number
is always one.

###  C.2 Multiple bands

We finally consider the situation where the lowest $r$ bands are all occupied,
and thus the quantum geometric tensor is given by the expression Eq. (63).
From a straightforward calculation, one obtains

$\displaystyle g_{xx}(\mathbf{k})$
$\displaystyle=g_{yy}(\mathbf{k})=\frac{r}{2|B|},$ $\displaystyle
g_{xy}(\mathbf{k})$ $\displaystyle=0,$ $\displaystyle\Omega_{xy}$
$\displaystyle=-\frac{r}{B}.$ (78)

Therefore, the quantum volume and the Chern number are

$\displaystyle\mathrm{vol}_{g}$ $\displaystyle=\mathrm{vol}_{\tilde{g}}=r\pi,$
$\displaystyle\mathcal{C}$ $\displaystyle=-r\ \mathrm{sign}(B).$ (79)

We see that the equality
$\mathrm{vol}_{g}=\mathrm{vol}_{\tilde{g}}=\pi|\mathcal{C}|$ holds for the
collection of the lowest $r$ Landau levels. The fact that the equality
$\mathrm{vol}_{g}=\mathrm{vol}_{\tilde{g}}=\pi|\mathcal{C}|$ holds even though
each single Landau level does not satisfy the equality apart form the lowest
Landau level reflects the property that the quantum metric and quantum volume
of combined bands is not just a sum of each band [35].

## References

  * Hasan and Kane [2010] M. Z. Hasan and C. L. Kane, Colloquium: Topological insulators, Rev. Mod. Phys. 82, 3045 (2010).
  * Qi and Zhang [2011] X.-L. Qi and S.-C. Zhang, Topological insulators and superconductors, Rev. Mod. Phys. 83, 1057 (2011).
  * Chiu  _et al._ [2016] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu, Classification of topological quantum matter with symmetries, Rev. Mod. Phys. 88, 035005 (2016).
  * Klitzing  _et al._ [1980] K. v. Klitzing, G. Dorda, and M. Pepper, New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance, Phys. Rev. Lett. 45, 494 (1980).
  * Thouless  _et al._ [1982] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs, Quantized Hall conductance in a two-dimensional periodic potential, Phys. Rev. Lett. 49, 405 (1982).
  * Kohmoto [1985] M. Kohmoto, Topological invariant and the quantization of the Hall conductance, Annals of Physics 160, 343 (1985).
  * Kane and Mele [2005a] C. L. Kane and E. J. Mele, ${Z}_{2}$ topological order and the quantum spin Hall effect, Phys. Rev. Lett. 95, 146802 (2005a).
  * Kane and Mele [2005b] C. L. Kane and E. J. Mele, Quantum spin Hall effect in graphene, Phys. Rev. Lett. 95, 226801 (2005b).
  * Bernevig and Zhang [2006] B. A. Bernevig and S.-C. Zhang, Quantum spin Hall effect, Phys. Rev. Lett. 96, 106802 (2006).
  * Cooper  _et al._ [2019] N. R. Cooper, J. Dalibard, and I. B. Spielman, Topological bands for ultracold atoms, Rev. Mod. Phys. 91, 015005 (2019).
  * Ozawa  _et al._ [2019] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, Topological photonics, Rev. Mod. Phys. 91, 015006 (2019).
  * Yang  _et al._ [2015] Z. Yang, F. Gao, X. Shi, X. Lin, Z. Gao, Y. Chong, and B. Zhang, Topological acoustics, Phys. Rev. Lett. 114, 114301 (2015).
  * Huber [2016] S. D. Huber, Topological mechanics, Nature Physics 12, 621 (2016).
  * Ma  _et al._ [2019] G. Ma, M. Xiao, and C. T. Chan, Topological phases in acoustic and mechanical systems, Nature Reviews Physics 1, 281 (2019).
  * Yamauchi  _et al._ [2020] L. Yamauchi, T. Hayata, M. Uwamichi, T. Ozawa, and K. Kawaguchi, Chirality-driven edge flow and non-Hermitian topology in active nematic cells, arXiv:2008.10852 (2020).
  * Shankar  _et al._ [2020] S. Shankar, A. Souslov, M. J. Bowick, M. C. Marchetti, and V. Vitelli, Topological active matter, arXiv:2010.00364 (2020).
  * Delplace  _et al._ [2017] P. Delplace, J. B. Marston, and A. Venaille, Topological origin of equatorial waves, Science 358, 1075 (2017).
  * Provost and Vallee [1980] J. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Communications in Mathematical Physics 76, 289 (1980).
  * Marzari and Vanderbilt [1997] N. Marzari and D. Vanderbilt, Maximally localized generalized Wannier functions for composite energy bands, Phys. Rev. B 56, 12847 (1997).
  * Marzari  _et al._ [2012] N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Maximally localized Wannier functions: Theory and applications, Rev. Mod. Phys. 84, 1419 (2012).
  * Matsuura and Ryu [2010] S. Matsuura and S. Ryu, Momentum space metric, nonlocal operator, and topological insulators, Physical Review B 82, 245113 (2010).
  * Neupert  _et al._ [2013] T. Neupert, C. Chamon, and C. Mudry, Measuring the quantum geometry of Bloch bands with current noise, Phys. Rev. B 87, 245103 (2013).
  * Albert  _et al._ [2016] V. V. Albert, B. Bradlyn, M. Fraas, and L. Jiang, Geometry and response of Lindbladians, Phys. Rev. X 6, 041031 (2016).
  * Kolodrubetz  _et al._ [2017] M. Kolodrubetz, D. Sels, P. Mehta, and A. Polkovnikov, Geometry and non-adiabatic response in quantum and classical systems, Physics Reports 697, 1 (2017).
  * Ozawa [2018] T. Ozawa, Steady-state Hall response and quantum geometry of driven-dissipative lattices, Phys. Rev. B 97, 041108 (2018).
  * Ozawa and Goldman [2018] T. Ozawa and N. Goldman, Extracting the quantum metric tensor through periodic driving, Phys. Rev. B 97, 201117 (2018).
  * Ozawa and Goldman [2019] T. Ozawa and N. Goldman, Probing localization and quantum geometry by spectroscopy, Phys. Rev. Research 1, 032019 (2019).
  * Klees  _et al._ [2021] R. L. Klees, J. C. Cuevas, W. Belzig, and G. Rastelli, Ground-state quantum geometry in superconductor–quantum dot chains, Phys. Rev. B 103, 014516 (2021).
  * Bleu  _et al._ [2018] O. Bleu, G. Malpuech, Y. Gao, and D. D. Solnyshkov, Effective theory of nonadiabatic quantum evolution based on the quantum geometric tensor, Phys. Rev. Lett. 121, 020401 (2018).
  * Raoux  _et al._ [2015] A. Raoux, F. Piéchon, J.-N. Fuchs, and G. Montambaux, Orbital magnetism in coupled-bands models, Phys. Rev. B 91, 085120 (2015).
  * Gao  _et al._ [2015] Y. Gao, S. A. Yang, and Q. Niu, Geometrical effects in orbital magnetic susceptibility, Phys. Rev. B 91, 214405 (2015).
  * Piéchon  _et al._ [2016] F. Piéchon, A. Raoux, J.-N. Fuchs, and G. Montambaux, Geometric orbital susceptibility: Quantum metric without Berry curvature, Phys. Rev. B 94, 134423 (2016).
  * Combes  _et al._ [2016] F. Combes, M. Trescher, F. Piéchon, and J.-N. Fuchs, Statistical mechanics approach to the electric polarization and dielectric constant of band insulators, Phys. Rev. B 94, 155109 (2016).
  * Freimuth  _et al._ [2017] F. Freimuth, S. Blügel, and Y. Mokrousov, Geometrical contributions to the exchange constants: Free electrons with spin-orbit interaction, Phys. Rev. B 95, 184428 (2017).
  * Peotta and Törmä [2015] S. Peotta and P. Törmä, Superfluidity in topologically nontrivial flat bands, Nature communications 6, 8944 (2015).
  * Julku  _et al._ [2016] A. Julku, S. Peotta, T. I. Vanhala, D.-H. Kim, and P. Törmä, Geometric origin of superfluidity in the Lieb-lattice flat band, Phys. Rev. Lett. 117, 045303 (2016).
  * Liang  _et al._ [2017] L. Liang, T. I. Vanhala, S. Peotta, T. Siro, A. Harju, and P. Törmä, Band geometry, Berry curvature, and superfluid weight, Phys. Rev. B 95, 024515 (2017).
  * Iskin [2018] M. Iskin, Quantum-metric contribution to the pair mass in spin-orbit-coupled Fermi superfluids, Phys. Rev. A 97, 033625 (2018).
  * Srivastava and Imamoğlu [2015] A. Srivastava and A. Imamoğlu, Signatures of Bloch-band geometry on excitons: Nonhydrogenic spectra in transition-metal dichalcogenides, Phys. Rev. Lett. 115, 166802 (2015).
  * Rhim  _et al._ [2020] J.-W. Rhim, K. Kim, and B.-J. Yang, Quantum distance and anomalous Landau levels of flat bands, Nature 584, 59 (2020).
  * Claassen  _et al._ [2015] M. Claassen, C. H. Lee, R. Thomale, X.-L. Qi, and T. P. Devereaux, Position-momentum duality and fractional quantum hall effect in chern insulators, Phys. Rev. Lett. 114, 236802 (2015).
  * Lee  _et al._ [2017] C. H. Lee, M. Claassen, and R. Thomale, Band structure engineering of ideal fractional chern insulators, Phys. Rev. B 96, 165150 (2017).
  * Asteria  _et al._ [2019] L. Asteria, D. T. Tran, T. Ozawa, M. Tarnowski, B. S. Rem, N. Fläschner, K. Sengstock, N. Goldman, and C. Weitenberg, Measuring quantized circular dichroism in ultracold topological matter, Nature physics 15, 449 (2019).
  * Tan  _et al._ [2019] X. Tan, D.-W. Zhang, Z. Yang, J. Chu, Y.-Q. Zhu, D. Li, X. Yang, S. Song, Z. Han, Z. Li, Y. Dong, H.-F. Yu, H. Yan, S.-L. Zhu, and Y. Yu, Experimental measurement of the quantum metric tensor and related topological phase transition with a superconducting qubit, Phys. Rev. Lett. 122, 210401 (2019).
  * Yu  _et al._ [2020] M. Yu, P. Yang, M. Gong, Q. Cao, Q. Lu, H. Liu, S. Zhang, M. B. Plenio, F. Jelezko, T. Ozawa, _et al._ , Experimental measurement of the quantum geometric tensor using coupled qubits in diamond, National Science Review 7, 254 (2020).
  * Klees  _et al._ [2020] R. L. Klees, G. Rastelli, J. C. Cuevas, and W. Belzig, Microwave spectroscopy reveals the quantum geometric tensor of topological Josephson matter, Phys. Rev. Lett. 124, 197002 (2020).
  * Gianfrate  _et al._ [2020] A. Gianfrate, O. Bleu, L. Dominici, V. Ardizzone, M. De Giorgi, D. Ballarini, G. Lerario, K. West, L. Pfeiffer, D. Solnyshkov, _et al._ , Measurement of the quantum geometric tensor and of the anomalous Hall drift, Nature 578, 381 (2020).
  * Liu  _et al._ [2020] Y. Liu, M. Yu, P. Yang, M. Gong, Q. Cao, S. Zhang, H. Liu, M. Heyl, T. Ozawa, N. Goldman, _et al._ , Saturating the quantum Cramér-rao bound and measuring the related quantum Fisher information in a nitrogen-vacancy center in diamond, arXiv:2003.08373 (2020).
  * Niu  _et al._ [1985] Q. Niu, D. J. Thouless, and Y.-S. Wu, Quantized Hall conductance as a topological invariant, Phys. Rev. B 31, 3372 (1985).
  * Mera and Ozawa [2021] B. Mera and T. Ozawa, Kähler geometry and Chern insulators: Relations between topology and the quantum metric, Phys. Rev. B 104, 045104 (2021).
  * Haldane [1988] F. D. M. Haldane, Model for a quantum Hall effect without landau levels: Condensed-matter realization of the “parity anomaly”, Phys. Rev. Lett. 61, 2015 (1988).
  * Yang  _et al._ [2012] S. Yang, Z.-C. Gu, K. Sun, and S. Das Sarma, Topological flat band models with arbitrary Chern numbers, Phys. Rev. B 86, 241112 (2012).
  * Harper [1955] P. G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proceedings of the Physical Society. Section A 68, 874 (1955).
  * Hofstadter [1976] D. R. Hofstadter, Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14, 2239 (1976).
  * Sun  _et al._ [2011] K. Sun, Z. Gu, H. Katsura, and S. Das Sarma, Nearly flatbands with nontrivial topology, Phys. Rev. Lett. 106, 236803 (2011).
  * Roy [2014] R. Roy, Band geometry of fractional topological insulators, Phys. Rev. B 90, 165139 (2014).
  * Pozo and de Juan [2020] O. Pozo and F. de Juan, Computing observables without eigenstates: Applications to Bloch Hamiltonians, Phys. Rev. B 102, 115138 (2020).
  * Graf and Piéchon [2021] A. Graf and F. Piéchon, Eigenprojectors, bloch vectors and quantum geometry of $n$-band systems, arXiv:2102.09899 (2021).
  * Hatsugai [1993a] Y. Hatsugai, Chern number and edge states in the integer quantum Hall effect, Phys. Rev. Lett. 71, 3697 (1993a).
  * Hatsugai [1993b] Y. Hatsugai, Edge states in the integer quantum Hall effect and the riemann surface of the Bloch function, Phys. Rev. B 48, 11851 (1993b).
  * Watanabe [2018] H. Watanabe, Insensitivity of bulk properties to the twisted boundary condition, Phys. Rev. B 98, 155137 (2018).
  * Kudo  _et al._ [2019] K. Kudo, H. Watanabe, T. Kariyado, and Y. Hatsugai, Many-body Chern number without integration, Phys. Rev. Lett. 122, 146601 (2019).
  * Souza  _et al._ [2000] I. Souza, T. Wilkens, and R. M. Martin, Polarization and localization in insulators: Generating function approach, Physical Review B 62, 1666 (2000).
  * Mera [2020] B. Mera, Localization anisotropy and complex geometry in two-dimensional insulators, Phys. Rev. B 101, 115128 (2020).
  * Ma  _et al._ [2013] Y.-Q. Ma, S.-J. Gu, S. Chen, H. Fan, and W.-M. Liu, The Euler number of Bloch states manifold and the quantum phases in gapped fermionic systems, EPL (Europhysics Letters) 103, 10008 (2013).
  * Ma [2020] Y.-Q. Ma, Euler characteristic number of the energy band and the reason for its non-integer values, arXiv:2001.05946 (2020).
  * Harper  _et al._ [2014] F. Harper, S. H. Simon, and R. Roy, Perturbative approach to flat chern bands in the Hofstadter model, Phys. Rev. B 90, 075104 (2014).
  * Gowers [2008] T. Gowers, ed.,  _The Princeton Companion to Mathematics_ (Princeton University Press, 2008).