# Engineering geometrically flat Chern bands with Fubini-Study Kähler
structure

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  Tomoki Ozawa  Advanced Institute for Materials Research (WPI-AIMR),
Tohoku University, Sendai 980-8577, Japan

###### Abstract

We describe a systematic method to construct models of Chern insulators whose
Berry curvature and the quantum volume form coincide and are flat over the
Brillouin zone; such models are known to be suitable for hosting fractional
Chern insulators. The bands of Chern insulator models where the Berry
curvature and the quantum volume form coincide, and are nowhere vanishing, are
known to induce the structure of a Kähler manifold in momentum space, and thus
we are naturally led to define Kähler bands to be Chern bands satisfying such
properties. We show how to construct a geometrically flat Kähler band, with
Chern number equal to minus the total number of bands in the system, using the
idea of Kähler quantization and properties of Bergman kernel asymptotics. We
show that, with our construction, the geometrical properties become flatter as
the total number of bands in the system is increased; we also show the no-go
theorem that it is not possible to construct geometrically perfectly flat
Kähler bands with a finite number of bands. We give an explicit realization of
this construction in terms of theta functions and numerically confirm how the
constructed Kähler bands become geometrically flat as we increase the number
of bands. We also show the effect of truncating hoppings at a finite length,
which will generally result in deviation from a perfect Kähler band but does
not seem to seriously affect the flatness of the geometrical properties.

##  I Introduction

Chern insulators are prototypical lattice models of topological insulators [1,
2]. Without interparticle interactions, fermionic Chern insulators exhibit the
integer quantum Hall effect, where the Hall conductivity is proportional to
the first Chern number of the occupied bands. Including appropriate
interactions, Chern insulators with certain filling factors are expected to
become fractional Chern insulators, which exhibit the fractional quantum Hall
effect [3, 4, 5, 6, 7, 8, 9]. To theoretically study fractional Chern
insulators on lattices, a natural guiding principle is to explore possible
analogies from the existing studies of fractional quantum Hall states in the
continuum. A challenge here is to find analogies between the Landau level
physics in the continuum and the eigenstates of Chern insulators on lattices.
In previous works, it has been noticed that when the geometrical properties of
noninteracting Chern insulators fulfill certain conditions, we can draw good
analogies between Chern insulators on lattices and the fractional quantum Hall
states in the continuum [10, 11]. In particular, it is desirable to have bands
whose Berry curvature $F_{12}(\mathbf{k})$ and the quantum metric
$g_{ij}(\mathbf{k})$ as a function of quasimomentum $\mathbf{k}$ in the
Brillouin zone satisfy the equality
$\sqrt{\det(g(\mathbf{k}))}=|F_{12}(\mathbf{k})/2|$. When this condition is
met, the Bloch states of a Chern band become analogous to the lowest Landau
levels in the continuum, from which we can expect stable construction of
fractional Chern insulators. This condition is related to the possibility to
choose Bloch states as holomorphic functions on momentum space. Earlier works
have proposed to construct Chern insulators with such properties using
variational states involving elliptic functions [10, 11].

The condition of $\sqrt{\det(g(\mathbf{k}))}=|F_{12}(\mathbf{k})/2|$ together
with $\det(g(\mathbf{k}))>0$ is equivalent to momentum space being a Kähler
manifold with the quantum metric as the Kähler metric and the Berry curvature
(up to a constant) as the Kähler form [10, 11, 12, 13]; in this paper we
define Kähler bands to be energy bands fulfilling these conditions. Here we
propose a systematic method to construct a sequence of Kähler bands, labeled
by the total number of bands in the system, with asymptotically flat geometry,
that is, $\sqrt{\det(g(\mathbf{k}))}=|F_{12}(\mathbf{k})/2|$ is asymptotically
constant over the Brillouin zone. We note that Kähler bands with flat energy
dispersion have been discussed under the name of (near-)ideal flatbands in the
recent works of Refs. [14, 15], in which the flatness of geometry is not
imposed.

In our method, once we fix an auxiliary Hermitian holomorphic line bundle on
the Brillouin zone, $L\to\mathrm{BZ}^{2}$, with its curvature satisfying
certain conditions, we can construct Kähler bands from an orthogonal basis of
the vector space of (global) holomorphic sections of the tensor-product line
bundle $L^{\otimes p}\to\mathrm{BZ}^{2}$. The total number of bands of the
constructed model equals the opposite of the Chern number of the constructed
Kähler band which, in turn, is equal to $-p\mathcal{C}$, where $\mathcal{C}$,
with $\mathcal{C}>0$, is the Chern number of $L\to\textnormal{\text{BZ}}^{2}$.
The Kähler band approaches flat geometry in the limit of large $p$, which is
guaranteed from a known mathematical theorem on the Bergman kernel asymptotics
[16]. Notably, in order to obtain flat Kähler bands, we do not need to
determine any parameter in a variational manner. Our method, being exact in
the infinite total number of bands limit, yields asymptotically geometrically
flat bands with a high Chern number. This should be contrasted to typical
studies of fractional Chern insulators which focus on bands with low Chern
number, in particular, with Chern number equal to $1$ due to the analogy with
lowest Landau level physics. More recent studies [17] are focusing on
fractional Chern insulators in bands with higher Chern numbers as they have
been recently realized experimentally [18] and they can exhibit richer
physical phenomena. We confirm the validity of our method by explicitly
constructing a model using theta functions. Although our model generally
contains long-range hoppings in real space to an arbitrary distance, we also
numerically show that truncating the hopping at a reasonable length would not
seriously affect the resulting geometry of the model.

Below, in Sec. II, we first introduce the basic terminology of band geometry
and describe how to introduce a complex structure in momentum space. In Sec.
III, we then review arguments on how Chern bands with certain geometrical
properties are preferred to obtain fractional Chern insulators. In Sec. IV, we
discuss the general expressions of Kähler (and quasi-Kähler) bands as
holomorphic maps from the Brillouin zone to complex projective spaces. Section
V discusses a systematic construction of flat Kähler bands, which forms the
central result of our paper. In Sec. VI, we provide an explicit construction
of Kähler bands based on our method, including numerical calculations.
Finally, we present the conclusions in Sec. VII. Some detailed derivations are
given in Appendix, including a proof that it is not possible to construct a
perfectly flat Kähler band with a finite number of bands.

##  II Band geometry and complex structure

We first summarize basic terminology of band geometry and how it can give rise
to a complex structure on the Brillouin zone. The geometry of the Bloch states
of a single isolated band is characterized by a map
$P:\textnormal{\text{BZ}}^{2}\to\mathbb{C}P^{n-1}$, where
$\textnormal{\text{BZ}}^{2}$ is the two-dimensional Brillouin zone and
$\mathbb{C}P^{n-1}$ is the $(n-1)$-dimensional complex projective space;
physically $n$ corresponds to the total number of bands. The map $P$ induces a
metric on the Brillouin zone
$ds^{2}=\sum_{i,j}g_{ij}(\textnormal{{k}})dk_{i}dk_{j}$, where
$\mathbf{k}\in\textnormal{\text{BZ}}^{2}$, called the _quantum metric_ ,
through the pullback of the standard Fubini-Study metric on
$\mathbb{C}P^{n-1}$ [19, 20]. Similarly, from the pullback of the Fubini-Study
symplectic form on $\mathbb{C}P^{n-1}$, we obtain a symplectic two-form
$\omega=-iF/2$ on momentum space, whose only nonzero component is
$\omega_{12}=-\omega_{21}$. The two-form $F$ is called the Berry curvature,
whose integral gives the first Chern number of the band
$\int_{\mathrm{BZ^{2}}}iF/2\pi\in\mathbb{Z}$. Generally, an inequality between
the determinant of the quantum metric, $\det(g)=g_{11}g_{22}-g_{12}^{2}$, and
the Berry curvature holds:
$\sqrt{\det(g(\textnormal{{k}}))}\geq|F_{12}(\textnormal{{k}})|/2$ as first
noticed by Roy [21]. Since $\sqrt{\det(g)}dk_{1}\wedge dk_{2}$, if
$\det(g)\neq 0$ everywhere, provides the natural volume form based on the
quantum metric and the standard orientation of the Brillouin zone, we call
this quantity the quantum volume form in this paper. We note, however, that
although the quantum volume form is always nonnegative, it can generally be
zero in certain points (regions) of the Brillouin zone, so it is not a volume
form in the strict mathematical sense. From the quantum metric, it is
sometimes possible to introduce a complex structure on the Brillouin zone
which we proceed to describe.

First, we assume that the map $P$ is an immersion, in which case, as we show,
a complex structure can always be defined on the entire Brillouin zone. Since
$P$ is an immersion, the metric is everywhere nondegenerate on the Brillouin
zone, namely $\det(g(\textnormal{{k}}))\neq 0$ for any
$\textnormal{{k}}\in\textnormal{\text{BZ}}^{2}$. In this case the quantum
metric is a Riemannian metric defined everywhere on the Brillouin zone. For a
given point on the Brillouin zone,
$\textnormal{{k}}\in\textnormal{\text{BZ}}^{2}$, it is known that there exists
a system of coordinates $(u,v)$ in a neighborhood of $\mathbf{k}$ which
satisfies $ds^{2}=\sum_{ij}g_{ij}dk_{i}dk_{j}=\rho(du^{2}+dv^{2})$, where
$\rho$ is a positive function. Such coordinates $(u,v)$ are called _isothermal
coordinates_ , and by defining $z=u+iv$, where $(u,v)$ are taken consistent
with the standard orientation on the Brillouin zone, we can introduce the
complex coordinate $z$ on a neighborhood of $\mathbf{k}$. Such complex
coordinates defined around different points on the Brillouin zone can be
patched together to give a complex atlas of the Brillouin zone, giving the
Brillouin zone the structure of a complex manifold of complex dimension one,
namely a _Riemann surface_. We note that for a given metric on the Brillouin
zone and a choice of orientation, the choice of the local complex coordinate
is unique up to local orientation preserving conformal transformations, i.e.,
up to local bi-holomorphisms—this ensures that the local coordinates patch
together nicely in a holomorphic manner, and also it tells us that the complex
structure on the Brillouin zone is uniquely determined by the quantum metric.
Since the complex projective space $\mathbb{C}P^{n-1}$ is also a complex
manifold, the map $P$ can now be considered as a map between complex
manifolds. As shown in Refs. [12, 13], when $P$ is a holomorphic immersion,
momentum space is a Kähler manifold with the quantum metric as the Kähler
metric and the symplectic two form $\omega=-iF/2$ as the Kähler form. We thus
define Kähler bands to be bands with $P$ being a holomorphic immersion, with
respect to the complex structure on the Brillouin zone defined above. For
Kähler bands, the equality
$\sqrt{\det(g(\textnormal{{k}}))}=|F_{12}(\textnormal{{k}})|/2$ holds [11, 12,
13]. A primary objective of this paper is to find flat Kähler bands, which are
Kähler bands whose geometrical properties, i.e., the quantum metric, the
symplectic form (which equals the quantum volume form) and the complex
structure, are flat, i.e., their components in the (periodic) coordinates
$(k_{1},k_{2})$ are constant.

We stress that the word _flat_ in the present work refers to the situation
where the geometric structures of interest are constant with respect to the
$(k_{1},k_{2})$ coordinates 111In a coordinate free description, the periodic
$(k_{1},k_{2})$ coordinates give rise to two independent globally defined
vector fields over the Brillouin zone $X_{1}=\partial/\partial k_{1}$ and
$X_{2}=\partial/\partial k_{2}$. Then, what we mean by _flatness_ of a Kähler
structure over the Brillouin zone is that the Lie derivatives, with respect to
these vector fields, of the symplectic form $\omega$, the complex structure
$j$ and the metric $g$, denoted by, respectively, $\mathcal{L}_{X_{i}}\omega$,
$\mathcal{L}_{X_{i}}j$ and $\mathcal{L}_{X_{i}}g$, $i=1,2$, vanish
identically., and does not refer to flatness of the energy dispersion unless
explicitly stated. We also want to stress that geometric flatness does not
mean vanishing Berry curvature, i.e., it does not mean that the Berry
connection is a flat connection; rather the Berry curvature takes a constant
nonzero value over the Brillouin zone.

Next, we consider the situation where the map $P$ is not an immersion, in
which case existence of the complex structure on the entire Brillouin zone is
not always guaranteed. When $P$ is not an immersion, there are certain points
on the Brillouin zone where $\det(g(\textnormal{{k}}))=0$. Let us denote the
(closed) set of points on the Brillouin zone where
$\det(g(\textnormal{{k}}))=0$ by
$S=\\{\textnormal{{k}}\in\textnormal{\text{BZ}}^{2}|\det(g(\textnormal{{k}}))=0\\}$.
If $\textnormal{\text{BZ}}^{2}\backslash S$, namely the set of points where
$\det(g(\textnormal{{k}}))\neq 0$, has more than one connected component, with
each component necessarily separated by a loop, it is not possible to define a
complex structure throughout the Brillouin zone. We note that since the Berry
curvature can change sign from one connected component to the other, the
choice of local orientations in each connected component, based on the sign of
the Berry curvature (more precisely using the pullback under $P$ of the
Fubini-Study symplectic form as a volume form), cannot in general be made
globally consistent over the whole Brillouin zone—something that would
necessarily happen if the map was holomorphic or anti-holomorphic (a standard
argument for this is presented Sec. IV). Over each connected component of
$\textnormal{\text{BZ}}^{2}\backslash S$, we can find local complex coordinate
systems consistent with the quantum metric by finding a local complex
coordinate in sufficiently small open neighbourhoods of each point on
$\textnormal{\text{BZ}}^{2}\backslash S$ as in the first case and patching
them together. Whether or not the complex coordinates defined in this way on
$\textnormal{\text{BZ}}^{2}\backslash S$ belong to some complex atlas on the
entire Brillouin zone and, hence, come from a globally defined complex
structure, depends on specific situations. If the map
$P:\textnormal{\text{BZ}}^{2}\to\mathbb{C}P^{n-1}$ is known to be holomorphic
_a priori_ with respect to some complex structure $j$, then we know that we
can extend the complex structure on $\textnormal{\text{BZ}}^{2}\backslash S$
as determined from the quantum metric to be defined on the entire
$\textnormal{\text{BZ}}^{2}$ since the former will just be the restriction of
$j$ to $\textnormal{\text{BZ}}^{2}\backslash S$. We define quasi-Kähler bands
to be bands where $P$ is not an immersion but we can define the complex
structure on the entire Brillouin zone and thus $P$ is represented by a
holomorphic function. Also for the quasi-Kähler bands, the equality
$\sqrt{\det(g(\textnormal{{k}}))}=|F_{12}(\textnormal{{k}})|/2$ holds
throughout the Brillouin zone. Note that since zeros of holomorphic functions
are isolated, if $P$ is holomorphic (and non-constant) then $S$, which is
determined by the simultaneous vanishing of the first derivatives of the local
holomorphic functions determining $P$ locally—hence determined by zeros of
(non-constant) holomorphic functions—, will consist of a collection of
isolated points in the Brillouin zone and, due to compactness, this collection
will necessarily be finite.

The quantum metric $g(\mathbf{k})$ is physically related to localization of
states [23, 24, 25], and it is experimentally observable. Recent experiments
have reported measurements of the quantum metric in various setups of
synthetic quantum matter [26, 27]. The complex structure, which as we saw is
determined from the quantum metric, in turn, is related to the anisotropy in
localization of the insulating state where the entire band is occupied by
fermions [28].

##  III Background on fractional Chern insulators and band geometry

Following Ref. [11], we now recall some notions on fractional Chern insulators
and the role of band geometry on their stability.

Consider the Hamiltonian for an isotropic free electron gas in two-dimensions
in the presence of an external uniform magnetic field:

$\displaystyle
H=\frac{1}{2m}\sum_{i}\left(\textnormal{{p}}_{i}-e\textnormal{{A}}_{i}\right)^{2}+\sum_{i<j}V(\textnormal{{r}}_{i}-\textnormal{{r}}_{j}),$

where $m$ is the effective mass, $e\textnormal{{A}}=(1/2)(-y,x)$ is the
electromagnetic gauge field in the symmetric gauge and we have set the
magnetic length $\ell_{B}^{2}=1/eB=1$. This is the canonical model for the
fractional quantum Hall effect. Assume filling fraction $\nu<1$ and assume
that the cyclotron frequency, i.e., the Landau level gap, is much larger than
the scale of interactions, and, hence, an effective description based on
projection onto the lowest Landau level (LLL) subspace is valid. The single-
body electron problem can be written as

$\displaystyle H_{0}=\omega_{c}(b^{\dagger}b+1/2),$

where $\omega_{c}=eB/m$ is the cyclotron frequency,
$b=\left(-i\partial_{\overline{z}}-iz/2\right)/\sqrt{2}=(-i/\sqrt{2})\nabla_{\partial/\partial\overline{z}}$
is the Cauchy-Riemann operator determined by the covariant derivative
$\nabla=d-i(xdy-ydx)/2=d+(zd\overline{z}-\overline{z}dz)/4$, where $d$ is the
exterior derivative, and the isotropic flat complex structure determined by
the complex coordinate $z=x+iy$ in the plane. The LLL wave functions are then
square-integrable holomorphic functions, with respect to an appropriate inner
product [29], which can also be understood as holomorphic sections of the
electromagnetic gauge bundle
$L=\mathbb{R}^{2}\times\mathbb{C}\to\mathbb{R}^{2}$ which is equipped with the
connection $\nabla=d+A$, $A=-i\textnormal{{A}}^{\flat}=-i(xdy-ydx)/2$, where
$\flat$ is the musical isomorphism sending vector fields to $1-$forms. There
is another oscillator algebra associated with the single-particle problem,
namely,

$\displaystyle a=-i\left(\partial_{z}+z/2\right)/\sqrt{2},$

which commutes with the one determined by $b$. Because of this, if
$|\psi\rangle$ is in the LLL so is $a^{\dagger}|\psi\rangle$. One can then
build all the states in the LLL from the vacuum of the $a$’s within the LLL,
which is the state that satisfies $a|\Psi_{0}\rangle=b|\Psi_{0}\rangle=0$.
This state is most simply the Gaussian $\Psi_{0}(z)\sim e^{-|z|^{2}/4}$.
Laughlin’s many-body trial wave functions may now be elegantly expressed in
terms of the $a$’s (one for each particle):

$\displaystyle\Psi_{\text{Laughlin}}=\prod_{i<j}\left(a_{i}^{\dagger}-a_{j}^{\dagger}\right)^{1/\nu}\Psi_{0}^{\otimes
N},$

where $N$ is the number of particles and $n=1/\nu$ is odd (to satisfy Fermi
statistics). The choice of the guiding-center basis
$\left(\left(a^{\dagger}\right)^{m}/\sqrt{m!}\right)\Psi_{0}\sim
z^{m}e^{-|z|^{2}/4}$, eigenstates of $L_{z}$, makes perfect sense in the
isotropic case. However, when the effective mass is anisotropic this choice is
not appropriate. The anisotropy is related to a choice of a _complex
structure_ in the plane, independent of the kinetic energy and the interacting
potential (which is assumed to be isotropic), which enters as a variational
degree of freedom determining the geometry of the ground state wave function.

For Chern insulators, one is forced to abandon isotropy in order to be able to
have a fractional quantum Hall fluid on the lattice. While Bloch bands,
labeled by $\textnormal{{k}}\in\textnormal{\text{BZ}}^{2}$, seem to have a
different structure from the wave functions in the LLL, one can still define
an analogous guiding-center basis on the lattice. The way to do this is to
define the FQHE states in terms of a basis of eigenstates of a small
anisotropic confining potential which is projected to the flat Chern band
[10]. Namely, one makes the observation that while the angular momentum does
not directly translate to the lattice, there is an equivalent description of
the associated eigenstates,
$\Psi_{m}=\left(\left(a^{\dagger}\right)^{m}/\sqrt{m!}\right)\Psi_{0}\sim
z^{m}e^{-|z|^{2}/4}$, as eigenstates of a parabolic confinement potential
$V(\textnormal{{r}})=(\lambda/2)\textnormal{{r}}^{2}=(\lambda/2)\bar{z}z$
projected to the $LLL$,

$\displaystyle\left(P_{LLL}V(\textnormal{{r}})P_{LLL}\right)\Psi_{n}=\frac{\lambda}{2}(n+1)\Psi_{n},\
n\in\mathbb{N},$

where $\lambda$ is a small real constant – the latter having a well-defined
lattice version. Indeed, we can swiftly adapt this construction to fractional
Chern insulators by taking an anisotropic confinement potential on the
lattice:

$\displaystyle V(\textnormal{{r}})=\frac{1}{2}\lambda\;\eta_{ij}x^{i}x^{j},$

where $\eta=(\eta_{ij})$ is a unimodular Galilean metric which is to be
interpreted as a variational degree of freedom and where we assume the
Einstein summation convention for repeated indices. Since under the Zak
transform [30] the position operators get mapped to $x^{j}=i\partial/\partial
k_{j}$, $j=1,2$, it follows that the projection onto a Chern band, described
by a rank $1$ smooth projector $P(\textnormal{{k}})$, yields

$\displaystyle\overline{V}(\textnormal{{k}})$ $\displaystyle\equiv
P(\mathbf{k})\circ V(\textnormal{{r}})\circ P(\textnormal{{k}})$
$\displaystyle=-\frac{1}{2}\lambda\;\eta_{ij}P(\textnormal{{k}})\circ\frac{\partial}{\partial
k_{i}}\circ\frac{\partial}{\partial k_{j}}\circ P(\textnormal{{k}}),$

where $\circ$ means operator composition. Expanding the derivatives, one finds

$\displaystyle\overline{V}(\textnormal{{k}})=-\frac{\lambda}{2}\eta_{ij}\nabla_{\frac{\partial}{\partial
k_{i}}}\nabla_{\frac{\partial}{\partial
k_{j}}}+\frac{\lambda}{2}\eta_{ij}g_{ij}(\textnormal{{k}}),$ (1)

where $\nabla=P\circ d\circ P$ and $g_{ij}(\textnormal{{k}})$ are,
respectively, the Berry connection and the components of the quantum metric of
the band under consideration. The metric $\eta$ identifies a flat complex
structure on $\textnormal{\text{BZ}}^{2}$, described by a (multi-valued)
complex coordinate $z=k_{1}+\tau k_{2}$, through the formula

$\displaystyle\eta^{ij}dk_{i}dk_{j}=\eta^{11}|dk_{1}+\tau
dk_{2}|^{2}=\eta^{11}|dz|^{2},\ \tau\in\mathcal{H},$ (2)

where $\eta^{ij}$ are the matrix elements of the inverse matrix $\eta^{-1}$.
Here $\mathcal{H}$ is the upper half of the complex plane, and thus
$\mathrm{Im}(\tau)>0$. We can then write

$\displaystyle\eta_{ij}\nabla_{i}\nabla_{j}$
$\displaystyle=2\eta_{11}\left(\nabla_{z}\nabla_{\overline{z}}+\nabla_{\overline{z}}\nabla_{z}\right)$
$\displaystyle=2\eta_{11}\left(2\nabla_{z}\nabla_{\overline{z}}-[\nabla_{z},\nabla_{\overline{z}}]\right)$
$\displaystyle=4\eta_{11}\nabla_{z}\nabla_{\overline{z}}-2\eta_{11}F_{z\overline{z}},$

and, equivalently,

$\displaystyle\eta_{ij}\nabla_{i}\nabla_{j}$
$\displaystyle=2\eta_{11}\left(\nabla_{z}\nabla_{\overline{z}}+\nabla_{\overline{z}}\nabla_{z}\right)$
$\displaystyle=2\eta_{11}\left(2\nabla_{\overline{z}}\nabla_{z}-[\nabla_{\overline{z}},\nabla_{z}]\right)$
$\displaystyle=4\eta_{11}\nabla_{\overline{z}}\nabla_{z}+2\eta_{11}F_{z\overline{z}},$

where $F=F_{z\overline{z}}dz\wedge
d\bar{z}=(\overline{\tau}-\tau)F_{z\overline{z}}dk_{1}\wedge
dk_{2}=F_{12}dk_{1}\wedge dk_{2}$ is the Berry curvature, so that we have the
following two equivalent expressions for $\overline{V}(\textnormal{{k}})$

$\displaystyle\overline{V}(\textnormal{{k}})$
$\displaystyle=-2\lambda\eta_{11}\nabla_{z}\nabla_{\overline{z}}+\lambda\left(\frac{i\eta_{11}}{2\tau_{2}}F_{12}+\frac{1}{2}\eta_{ij}g_{ij}\right)$
(3)

and

$\displaystyle\overline{V}(\textnormal{{k}})$
$\displaystyle=-2\lambda\eta_{11}\nabla_{\overline{z}}\nabla_{z}+\lambda\left(-\frac{i\eta_{11}}{2\tau_{2}}F_{12}+\frac{1}{2}\eta_{ij}g_{ij}\right).$
(4)

We remark that the first term in Eq. (3) is minimized for holomorphic sections
of the complex line bundle $L\to\textnormal{\text{BZ}}^{2}$ whose fiber at k
is the image of $P(\textnormal{{k}})$, i.e., Bloch wave functions
$|u_{\textnormal{{k}}}\rangle$ living in the Chern band
[$P(\textbf{k})|u_{\textnormal{{k}}}\rangle=|u_{\textnormal{{k}}}\rangle$] and
satisfying

$\displaystyle\nabla_{\overline{z}}|u_{\textnormal{{k}}}\rangle=0.$

The vector space of (global) solutions of the above equation is denoted
$H^{0}(\textnormal{\text{BZ}}^{2},L)$. If the Chern number $\mathcal{C}$ of
$L$ is positive then the above equation has $\mathcal{C}$ linearly independent
solutions by the Riemann-Roch theorem. If the Chern number of $L$ is negative,
then, the above equation has no (global) solutions and we have to turn to Eq.
(4), in which the first term is minimized by antiholomorphic sections of $L$,
i.e., those sections satisfying

$\displaystyle\nabla_{z}|u_{\textnormal{{k}}}\rangle=0,$

which will have $|\mathcal{C}|$ linearly independent solutions as it
corresponds to Bloch wave functions on the Chern band, i.e., sections of $L$,
which satisfy the constraint that they are holomorphic with respect to the
opposite complex structure on the Brillouin zone.

The second term in Eqs. (3) and (4) vanishes when

$\displaystyle\mp
iF_{12}=\eta^{11}\tau_{2}\eta_{ij}g_{ij}=\sqrt{\det(\eta^{-1})}\eta_{ij}g_{ij}=\eta_{ij}g_{ij},$

where the last equality follows from unimodularity of $\eta$. This identity
holds when the triple of structures $(g,-iF/2=\omega,\pm j_{\tau})$ satisfies
$\omega(\cdot,\pm j_{\tau}\cdot)=g$, so that

$\displaystyle g_{ij}(\textnormal{{k}})=f(\textnormal{{k}})\eta^{ij},$

for some non-negative smooth function $f(\textnormal{{k}})$, and

$\displaystyle\eta_{ij}g_{ij}=2f(\textnormal{{k}}),$

and also

$\displaystyle\mp\frac{iF_{12}}{2}=\sqrt{\det(g)}=f(\textnormal{{k}}).$

We note that, for any Chern insulator, an inequality
$\sqrt{\det(g)}\geq|F_{12}/2|$ holds [21], which follows from the Cauchy-
Schwarz inequality; the saturation of the inequality is related to the Bloch
states being holomorphic or anti-holomorphic as described in Refs. [12, 13].
We thus see that vanishing of the expectation value of
$\overline{V}(\textnormal{{k}})$ is equivalent to the condition that we can
choose local Bloch states that are holomorphic or anti-holomorphic functions
(depending on whether the sign of the first Chern number is negative or
positive, respectively).

The guiding-center orbitals of the Chern band are then defined to be a basis
of the Hilbert space of Bloch wave functions on the Chern band, i.e., sections
of $L$ [a Hilbert subspace of the total single-particle Hilbert space of the
system defined by those Bloch states $|u_{\textnormal{{k}}}\rangle$ which
satisfy
$P(\textnormal{{k}})|u_{\textnormal{{k}}}\rangle=|u_{\textnormal{{k}}}\rangle$
for all $\textnormal{{k}}\in\textnormal{\text{BZ}}^{2}$] composed of
eigenvectors of the confining potential $\overline{V}(\textnormal{{k}})$, with
$\eta$ variationally chosen to optimize the dispersive second term of Eqs. (3)
and (4) (see Refs. [10, 11]). Observe that if the band is Kähler with respect
to some flat Kähler structure with complex structure $j_{\tau}$ ($-j_{\tau}$),
then the second term in Eq. (3) (Eq. (3)) is naturally minimized and the
conformal factor $f(\textbf{k})$ appearing above is simply a constant.

The challenge of obtaining a microscopic description of fractional quantum
Hall states associated with fractionally filled Chern bands reduces, according
to [10], to determining the deformed guiding-center orbitals, above defined as
eigenstates of $\overline{V}(\textbf{k})$, upon placing a fractional quantum
Hall liquid on the lattice. Given an appropriate $\eta$, any fractional Chern
insulator can in principle be captured by Laughlin-like many-body trial ground
states, constructed from the single-body eigenstates of Eq. (1).

The discussion above refers to the choice of “guiding-center” bases in the
conventional fractional quantum Hall effect and in fractional Chern
insulators. In the latter case, to ensure the stability with respect to
interactions one further needs that the Berry curvature is uniform [21, 31] to
ensure that the resulting projected density algebra is isomorphic to the
$W_{\infty}$-algebra found in the ordinary FQHE. For $\tau=i$ it is not just
isomorphic but exactly the same algebra.

The construction in the present manuscript, see Sec. V, provides
asymptotically flat Kähler bands and, hence, it provides a way to obtain
optimal Chern bands for hosting fractional Chern insulators, for arbitrary
anisotropy as described by the modular parameter $\tau\in\mathcal{H}$. The
fact that our formalism allows for arbitrary anisotropy $\tau$, allows to
account for scenarios in which, due to symmetry reasons, it will be more
favourable to form a fractional Hall fluid on the lattice if the geometry of
the Chern band is anisotropic. Intuitively, this should be the case for
lattices that are not conformal to the square lattice.

##  IV General expressions for (quasi-)Kähler bands from holomorphic maps from
$\textnormal{\text{BZ}}^{2}$ to $\mathbb{C}P^{n-1}$

We now explain how Kähler and quasi-Kähler bands can be generally expressed as
holomorphic maps from $\textnormal{\text{BZ}}^{2}$ to $\mathbb{C}P^{n-1}$,
using meromorphic functions on complex tori. We first consider the case where
$n=2$, which are two-band models and thus one can only construct quasi-Kähler
bands. We then extend the argument to $n\geq 2$ and present a general
expression for a map $\textnormal{\text{BZ}}^{2}\to\mathbb{C}P^{n-1}$, which
any Kähler and quasi-Kähler band should obey.

For two-band models a quasi-Kähler band is determined by a holomorphic map
$P:\textnormal{\text{BZ}}^{2}\to\mathbb{C}P^{1}$, or equivalently, because we
can identify $\mathbb{C}P^{1}\cong\mathbb{C}\cup\\{\infty\\}$ (Riemann
sphere), by a meromorphic function over a torus equipped with the structure of
a complex manifold. Holomorphic maps, unlike smooth maps, are very “rigid”,
something that is ultimately related to the fact that holomorphic functions
are Taylor series expandable everywhere. In the following, for illustration
purposes, we describe meromorphic functions over a complex torus and we will
see how they are parametrized by their zeros and poles. For simplicity, we
equip the Brillouin zone $\textnormal{\text{BZ}}^{2}$ with the
$2\pi\left(\mathbb{Z}+i\mathbb{Z}\right)-$periodic complex coordinate
$z=k_{1}+ik_{2}$, making it a complex torus. We note that generalization to a
general case of $z=k_{1}+\tau k_{2}$ with $\tau\in\mathcal{H}$ is
straightforward, as we will see below. Observe that we are using coordinates
$(k_{1},k_{2})$ such that the action of the reciprocal lattice by translations
is given by shifts by integer multiples of $2\pi$ on each coordinate. Let

$\displaystyle\theta(z,\tau=i):=\theta(z)=\sum_{n\in\mathbb{Z}}e^{-\pi
n^{2}+2\pi inz},$

be the associated _theta function_.

Observe that

$\displaystyle\theta(z+m)=\theta(z),\text{ for }m\in\mathbb{Z},$

and

$\displaystyle\theta(z+mi)$ $\displaystyle=\sum_{n\in\mathbb{Z}}e^{-\pi
n^{2}+2\pi inz-2\pi mn}$
$\displaystyle=\sum_{n\in\mathbb{Z}}e^{-\pi(n+m)^{2}+2\pi i(n+m)z+\pi
m^{2}-2\pi imz}$ $\displaystyle=e^{\pi m^{2}-2\pi imz}\theta(z),\text{ for all
}m\in\mathbb{Z}.$

The $\theta$ function has a unique simple zero at
$1/2+i/2\mod\mathbb{Z}+i\mathbb{Z}$ [32]. We define the translated theta
functions by

$\displaystyle\theta^{(x)}(z)=\theta(z-(1/2+i/2)-x),\ x\in\mathbb{C},$

which has zeros at $x+\mathbb{Z}+i\mathbb{Z}$. A holomorphic map
$R:\textnormal{\text{BZ}}^{2}\to\mathbb{C}P^{1}$, i.e., a meromorphic function
on $\textnormal{\text{BZ}}^{2}$, can be constructed by taking products and
ratios of the translated theta functions

$\displaystyle
R(z)=\frac{\prod_{i}\theta^{(x_{i})}\left(\frac{z}{2\pi}\right)}{\prod_{j}\theta^{(y_{j})}\left(\frac{z}{2\pi}\right)}$
(5)

for a set of complex numbers $x_{1},x_{2},\cdots,x_{N}$ and
$y_{1},y_{2},\cdots,y_{N}$ satisfying the condition

$\displaystyle\sum_{i=1}^{N}x_{i}-\sum_{j=1}^{N}y_{j}\in\mathbb{Z}.$ (6)

One can explicitly confirm that $R(z+2\pi i)=R(z)$ holds if and only if the
condition Eq. (6) is satisfied. Note that in Eq. (5) we performed a rescaling
on the $z$ variable to ensure the correct periodicity properties under
translations by reciprocal lattice vectors.

Actually, every meromorphic function on the complex torus
$\mathbb{C}/\mathbb{Z}^{2}$ is, _up to a multiplicative constant_ , of this
form (see, for instance, Proposition 2.7 and Lemma 3.14 of [32]).

For a given holomorphic map $R:\textnormal{\text{BZ}}^{2}\to\mathbb{C}P^{1}$,
we can construct a two-band momentum-space Hamiltonian
$H(\mathbf{k})=-\varepsilon n(\mathbf{k})\cdot\sigma$, where $\varepsilon$ is
a positive constant, $\sigma=(\sigma_{1},\sigma_{2},\sigma_{3})$ are the Pauli
matrices and $n$ is a map $n:\textnormal{\text{BZ}}^{2}\to
S^{2}\subset\mathbb{R}^{3}$ with

$\displaystyle
n(z)=\frac{1}{1+|R(z)|^{2}}\left(2\mbox{Re}\left(R(z)\right),2\mbox{Im}\left(R(z)\right),1-|R(z)|^{2}\right).$

The eigenvalues of the Hamiltonian are $\pm\varepsilon$, and thus flat over
the Brillouin zone, and the eigenvectors are, up to normalization,
$(-\overline{R(z)},1)$ and $(1,R(z))$. Therefore, the lower band of this
Hamiltonian is described by the map
$f:\textnormal{\text{BZ}}^{2}\to\mathbb{C}P^{1}$ given by

$\displaystyle f:\textnormal{\text{BZ}}^{2}\to\mathbb{C}P^{1};\ z\mapsto
f(z)=\left[\prod_{j}\theta^{(y_{j})}\\!\left(\frac{z}{2\pi}\right)\\!:\\!\prod_{i}\theta^{(x_{i})}\\!\left(\frac{z}{2\pi}\right)\\!\right],$

in homogeneous coordinates. The lowest band of this Hamiltonian is, by
construction, quasi-Kähler with respect to the flat complex structure
determined by $\tau=i$. We note that, since this is a two-band Hamiltonian,
there must be points in the Brillouin zone where $\det(g)=0$ due to Theorem 3
of Ref. [13]. That means that the Berry curvature will be always non-negative.
The reason is that, given that
$R:\textnormal{\text{BZ}}^{2}\to\mathbb{C}P^{1}$ is holomorphic, it will
necessarily be orientation preserving at every point. This follows from the
fact that, locally, $R$ is described by a holomorphic map $z\mapsto w(z)$,
where $z=k_{1}+ik_{2}$ is the holomorphic coordinate on the Brillouin zone and
$w$ is a holomorphic coordinate on the sphere. The orientations on both
manifolds are locally determined by the $2-$forms $dk_{1}\wedge
dk_{2}=\frac{i}{2}dz\wedge d\bar{z}$ and $\frac{i}{2}dw\wedge d\bar{w}$,
respectively. The pullback of $\frac{i}{2}dw\wedge d\bar{w}$ then reads
$\frac{i}{2}|\frac{\partial w}{\partial z}|^{2}dz\wedge d\bar{z}$. Since
$|\frac{\partial w}{\partial z}|^{2}\geq 0$, it follows that the map will be
orientation preserving on each point where the derivative does not vanish,
i.e., away from the points where the map is not an immersion or, equivalently,
$\det(g)=0$.

We can generalize the above construction of a holomorphic map to the case of
$\mathbb{C}P^{n-1}$ by:

$\displaystyle f:\textnormal{\text{BZ}}^{2}\to\mathbb{C}P^{n-1};\
z\\!\mapsto\left[c_{1}\prod_{i_{1}}\theta^{(x^{1}_{i_{1}})}\left(\frac{z}{2\pi}\right):c_{2}\prod_{i_{2}}\theta^{(x^{2}_{i_{2}})}\left(\frac{z}{2\pi}\right)\\!:\\!\cdots\\!:\\!c_{n}\prod_{i_{n}}\theta^{(x^{n}_{i_{n}})}\left(\frac{z}{2\pi}\right)\right],$

where $c_{1},\dots,c_{n}\in\mathbb{C}$, with at least one of the $c_{i}$’s
different from zero, $1\leq i_{j}\leq N$, for all $j=1,\dots,n$, and

$\displaystyle\sum_{j=1}^{N}x^{i}_{j}-x^{k}_{j}\in\mathbb{Z},\text{ for all
}i<k.$ (7)

To see that this is completely general, note that to have a general well-
defined holomorphic map
$f:\textnormal{\text{BZ}}^{2}\to\mathbb{C}P^{n-1};z\mapsto[f_{1}(z):\dots:f_{n}(z)]$
we need a collection of functions $f_{i}:\mathbb{C}\to\mathbb{C}$ depending
holomorphically in $z$ and which satisfy

$\displaystyle f_{i}(z+G)=e_{G}(z)f_{i}(z),\text{ for all
}i\in\\{1,\dots,n\\},$ (8)

where $G=G_{1}+iG_{2}$ represents a reciprocal lattice vector and $e_{G}(z)$
is a holomorphic multiplier which is the same for all functions. This is the
case because the projective space does not care about the overall scale of the
vector. The equation above tells us that the $f_{i}(z)$’s define $n$ sections
of a holomorphic line bundle over the complex torus. Note, however that we can
write

$\displaystyle[f_{1}(z):\dots:f_{n}(z)]$
$\displaystyle=\left[1:\frac{f_{2}(z)}{f_{1}(z)}:\dots:\frac{f_{n}(z)}{f_{1}(z)}\right]$
$\displaystyle=[1:R_{1}(z):\dots:R_{n}(z)],$ (9)

where the $R_{i}(z)$’s are now meromorphic functions over the complex torus.
Since any meromorphic function over the complex torus can be written, up to a
multiplicative constant, as a ratio of theta functions we understand that the
prescription is general.

Thus, any Kähler and quasi-Kähler band, having a flat complex structure
defined by $\tau=i$, can be written in this form, and can be expressed by
appropriately choosing the values of $c_{1},\cdots,c_{n}$ and $x_{j}^{i}$ to
satisfy Eq. (7). The discussion above is easily generalized to a complex torus
$\mathbb{C}/\left(\mathbb{Z}\oplus\tau\mathbb{Z}\right)$, for some
$\tau\in\mathcal{H}$, by replacing $\theta(z)=\theta(z,\tau=i)$ with

$\displaystyle\theta(z,\tau)=\sum_{n\in\mathbb{Z}}e^{i\pi\tau n^{2}+2\pi
inz},$ (10)

where now $z=k_{1}+\tau k_{2}$. In fact, since any torus with the structure of
a complex manifold, i.e., a Riemann surface of genus $1$, is biholomorphic to
a complex torus $\mathbb{C}/\left(\mathbb{Z}\oplus\tau\mathbb{Z}\right)$, for
some $\tau\in\mathcal{H}$, see Proposition 5.2 of [32], it follows that any
Kähler and quasi-Kähler band, after composing the map with a suitable map
giving the biholomorphism, can be described in this way 222To be precise, it
means that if we equip the Brillouin zone with the structure of a complex
manifold, denoted $(\textnormal{\text{BZ}}^{2},j)$, with $j$ being the complex
structure, there exists a biholomorphism (i.e., an isomorphism of complex
manifolds)
$\varphi:(\textnormal{\text{BZ}}^{2},j)\to\mathbb{C}/\left(\mathbb{Z}\oplus\tau\mathbb{Z}\right)$
such that $d\varphi\circ j=j_{\tau}\circ d\varphi,$ with $j_{\tau}$ the
uniform flat complex structure associated with $\tau\in\mathcal{H}$. Note
that, since $\varphi$ is in particular a diffeomorphism, we also have $j\circ
d\varphi^{-1}=d\varphi^{-1}\circ j_{\tau}.$ This means that if
$P:(\textnormal{\text{BZ}}^{2},j)\to\mathbb{C}P^{n-1}$ is a holomorphic map,
meaning $dP\circ j=J_{FS}\circ dP,$ where $J_{FS}$ is the Fubini-Study complex
structure, then,
$\tilde{P}=P\circ\varphi^{-1}:\mathbb{C}/\left(\mathbb{Z}\oplus\tau\mathbb{Z}\right)\to\mathbb{C}P^{n-1}$
is holomorphic with respect to the flat complex structure $j_{\tau}$ defined
by $\tau\in\mathcal{H}$, i.e.,  $\displaystyle d\tilde{P}\circ j_{\tau}$
$\displaystyle=dP\circ d\varphi^{-1}\circ j_{\tau}=dP\circ j\circ
d\varphi^{-1}$ $\displaystyle=J_{FS}\circ dP\circ d\varphi^{-1}=J_{FS}\circ
d\tilde{P},$ where we used the chain rule for differentials
$d\tilde{P}=d(P\circ\varphi^{-1})=dP\circ d\varphi^{-1}$. Now the projector
$\tilde{P}$ will have a description in terms of meromorphic functions on a
complex torus parametrized by a complex coordinate $z=k_{1}+\tau k_{2}$ as
described in the main text. .

Because of the number of free parameters at hand, tuning these parameters to
obtain the desired (quasi-)Kähler band is generally a nontrivial task. As
discussed in Sec. III, in the context of the fractional Chern insulators, one
is interested in finding bands with the quantum metric and the Berry curvature
which are as uniform as possible over the Brillouin zone. One approach taken
by Lee _et al._ [11] was to set an ansatz wave function in terms of the
Weierstrass zeta functions, instead of the theta functions, and to flatten the
geometrical properties by tuning a collection of variational parameters. In
the next section, we present an alternative approach, for which one needs to
determine only an auxiliary Hermitian holomorphic line bundle, for a fixed
value of the modular parameter $\tau$, and total number of bands. After the
explicit construction of the holomorphic line bundle for a given $\tau$, which
is made possible using the theory of theta functions, the only parameter in
our approach becomes the number of total bands. Despite the simplicity of our
approach, we show that, in an appropriate asymptotic limit of large total
number of bands, our method gives rise to the desired flat quantum metric and
Berry curvature, satisfying $|F_{12}|/2=\sqrt{\det(g)}$. Our approach is
inspired by the one by Kovrizhin _et al._ [34], in which skyrmion lattices
with desired properties were constructed using theta functions.

##  V The Bergman kernel prescription

We want to build a holomorphic map
$f:\textnormal{\text{BZ}}^{2}\to\mathbb{C}P^{n-1}$, for some $n>1$, which
induces a flat geometrical structure over the Brillouin zone
$\textnormal{\text{BZ}}^{2}$. We first show that Kähler bands which are
geometrically flat are solely determined by the modular parameter $\tau$ and
the first Chern number $-\mathcal{C}$, with $\mathcal{C}>0$ (the minus sign
coming from the relation between our convention of the Berry curvature and the
pullback of the Fubini-Study form, as we will see more explicitly below). Flat
Kähler bands are characterized by a compatible triple, $(\omega,j_{\tau},g)$,
where each of the structures is uniform over the Brillouin zone. Let us denote
the modular parameter of the system by $\tau$, which means that the complex
variable $z=k_{1}+\tau k_{2}$ gives a complex coordinate on the Brillouin zone
satisfying $ds^{2}=\sum_{i,j}g_{ij}dk_{i}dk_{j}=g_{11}|dz|^{2}$. From this
relation, we can deduce $\tau=(g_{12}+i\sqrt{\det g})/g_{11}$. The almost
complex structure $j_{\tau}$ satisfies $j_{\tau}(\partial_{z})=i\partial_{z}$
and $j_{\tau}(\partial_{\bar{z}})=-i\partial_{\bar{z}}$, from which we can
deduce

$\displaystyle
j_{\tau}=\frac{1}{\mathrm{Im}(\tau)}\begin{pmatrix}-\mathrm{Re}(\tau)&-|\tau|^{2}\\\
1&\mathrm{Re}(\tau)\end{pmatrix}=\frac{1}{\sqrt{\det(g)}}\begin{pmatrix}-g_{12}&-g_{22}\\\
g_{11}&g_{12}\end{pmatrix}$ (11)

in the basis of $\partial/\partial k_{1}$ and $\partial/\partial k_{2}$. Since
$g_{ij}=\sum_{k}\omega_{ik}(j_{\tau})_{j}^{k}$ from the compatibility of the
Kähler structure, we can then show $\omega=\sqrt{\det(g)}dk_{1}\wedge dk_{2}$.
On the other hand, the negative of the Chern number of the Kähler band is
$\mathcal{C}=\int_{\textnormal{\text{BZ}}^{2}}\omega/\pi=4\pi\sqrt{\det(g)}$.
Therefore, for a given value of the modular parameter $\tau$ and of the Chern
number $-\mathcal{C}$, the flat Kähler structure we want to obtain is

$\displaystyle\omega$ $\displaystyle=\sqrt{\det(g)}dk_{1}\wedge
dk_{2}=\frac{\mathcal{C}}{4\pi}dk_{1}\wedge
dk_{2}=\frac{\mathcal{C}}{4\pi}\frac{1}{\bar{\tau}-\tau}dz\wedge
d\overline{z},$ $\displaystyle j_{\tau}$
$\displaystyle=\frac{1}{\mathrm{Im}(\tau)}\begin{pmatrix}-\mathrm{Re}(\tau)&-|\tau|^{2}\\\
1&\mathrm{Re}(\tau)\end{pmatrix},\
g=\frac{\mathcal{C}}{4\pi\mathrm{Im}(\tau)}|dz|^{2},$ (12)

which is the Kähler structure that we want the map $f$ to induce on
$\textnormal{\text{BZ}}^{2}$ (up to a constant scaling on $\omega$ and $g$).
Hence, the triple $(\omega,j_{\tau},g)$ associated with a Kähler band is
completely determined by $\tau$ and $-\mathcal{C}$, as claimed. We note the
symplectic form $\omega$ can be written in terms of a positive function
$h_{*}$ as $\omega=-(i/2)\partial\bar{\partial}\log h_{*}$, where

$\displaystyle
h_{*}=\exp\left(\frac{\mathcal{C}}{4\pi}\frac{i}{\tau-\overline{\tau}}(z-\overline{z})^{2}\right)=\exp\left(-\frac{\mathcal{C}}{2\pi}\mathrm{Im}(\tau)k_{2}^{2}\right),$
(13)

and $\partial=dz\frac{\partial}{\partial z}\wedge\cdot$,
$\overline{\partial}=d\overline{z}\frac{\partial}{\partial\overline{z}}\wedge\cdot$
are the Dolbeault operators. With this function $h_{*}$, the Kähler potential
$K$, satisfying $\omega=i\partial\bar{\partial}K$, can be written as

$\displaystyle K=-\frac{1}{2}\log
h_{*}=\frac{\mathcal{C}}{4\pi}\mathrm{Im}(\tau)k_{2}^{2}\text{ or
}h_{*}=e^{-2K}.$ (14)

In order to build the map $f$, we are inspired by the idea of the _Kähler
quantization_. The theory of geometric quantization [35, 36] gives us a
prescription to quantize a Kähler manifold
$(\textnormal{\text{BZ}}^{2},\omega,J,g)$ provided the symplectic form
$\omega$ satisfies the quantization condition, namely that $\omega/2\pi$
represents an integer cohomology class – $[\omega/2\pi]\in
H^{2}(\textnormal{\text{BZ}}^{2};\mathbb{Z})$ or, equivalently, that
$\int_{\textnormal{\text{BZ}}^{2}}\omega$ is $2\pi$ times an integer. If this
is the case, then $-i\omega$ represents the 1st Chern class of a Hermitian
holomorphic line bundle $L\to\textnormal{\text{BZ}}^{2}$ equipped with the
Chern connection [i.e., the unique connection whose $(0,1)$ part coincides
with the Cauchy-Riemann operator of $L$]. Once we are given the data
$L\to\textnormal{\text{BZ}}^{2}$, known as the _pre-quantum line bundle_ ,
then, the Kähler quantization of $\textnormal{\text{BZ}}^{2}$ is defined as
the space
$H^{0}(\textnormal{\text{BZ}}^{2},L)\subset\Gamma(\textnormal{\text{BZ}}^{2},L)$
(equipped with the $L^{2}$-norm and completed with respect to it) consisting
of (square-integrable) global holomorphic sections of $L$. From the physics
point of view, this prescription appears naturally in the physics of fermions,
in two spatial dimensions, in the presence of an external uniform magnetic
field, i.e., in the quantum Hall effect, where the lowest Landau level is
precisely described by $H^{0}(M,L)$ where $M$ is the surface corresponding to
the physical sample [37]. For example, if $M=\mathbb{R}^{2}$, and
$L=\mathbb{R}^{2}\times\mathbb{C}$ with magnetic field represented by the
Faraday $2-$form $F=-idx\wedge dy$ we recover the space of (square-integrable)
holomorphic functions on the plane with measure $e^{-|z|^{2}/4}d^{2}x$. The
difference here, is that $M$ is not the real space physical sample, but rather
the quasi-momentum space. This geometric quantization procedure appears also
naturally in the discussion of Sec. III, where the confining potential
$\overline{V}(\textnormal{{k}})$ isolates
$H^{0}(\textnormal{\text{BZ}}^{2},L)$, where $L\to\textnormal{\text{BZ}}^{2}$
is the line bundle associated to the Chern band. There, the complex structure
over $\textnormal{\text{BZ}}^{2}$ was singled out by the variational flat
metric $\eta$, much like here we will fix it _a priori_ below. Finally, for
the construction below, it will be convenient and, in fact, crucial to take
not $H^{0}(\textnormal{\text{BZ}}^{2},L)$ but rather
$H^{0}(\textnormal{\text{BZ}}^{2},L^{\otimes p})$, for some integer $p>0$,
which, as we will soon see, determines the total number of bands in the
system.

We are now in condition to build the map
$f:\textnormal{\text{BZ}}^{2}\to\mathbb{C}P^{n-1}$. For that, let us consider
an auxiliary Hermitian holomorphic line bundle
$L\to\textnormal{\text{BZ}}^{2}$, which, at this point, is arbitrary besides
having nontrivial Chern number $\mathcal{C}>0$, which is also known as the
degree of $L$ (note that this auxiliary holomorphic line bundle $L$ will have
positive Chern number $\mathcal{C}>0$, unlike the resulting Kähler band which
will have negative Chern number $-p\mathcal{C}$). Note that the fact that $L$
is a holomorphic line bundle means that its transition functions are
holomorphic with respect to the complex structure specified by $\tau$ in the
Brillouin zone. More specifically, they will depend holomorphically on the
complex variable $z=k_{1}+\tau k_{2}$. The fact that $L$ is Hermitian means
that it comes equipped with a Hermitian metric on the fibers, which on a given
(holomorphic) gauge is represented by positive function $h>0$. The Chern
connection on $L$ has the property that, in the local holomorphic gauges, the
local gauge field and the associated curvature are given by, respectively,

$\displaystyle A=\partial\log h\text{ and }F=-\partial\overline{\partial}\log
h,$ (15)

where $h$ is the representative of the Hermitian metric in this gauge. Observe
how the $(0,1)$ part of $A$ vanishes, and hence a local holomorphic gauge is
holomorphic with respect to the connection $\nabla$, i.e., it is annhihilated
by the covariant derivative $\nabla_{\frac{\partial}{\partial\bar{z}}}$. This
is the local form of the defining property of the Chern connection, namely,
that it is the unique connection whose $(0,1)$ part coincides with the Cauchy-
Riemann operator of $L$ (see Sec. 6 of Ref. [38] and Sec. 4 of Ref. [39], in
particular, Proposition (4.9)).

Borrowing the ideas from the Kähler quantization, we choose the line bundle
$L$ so that the Hermitian metric $h$ coincides with the positive function
$h_{*}$ defined in Eq.(13), which is related to the Kähler potential of the
flat Kähler band we want to achieve. Taking $h=h_{*}$, the curvature of the
line bundle is

$\displaystyle F$ $\displaystyle=-\partial\bar{\partial}\log
h=-2i\omega=-i\frac{\mathcal{C}}{2\pi}dk_{1}\wedge dk_{2}.$ (16)

A line bundle $L$ with such curvature $F$ will have first Chern number

$\displaystyle\int_{\textnormal{\text{BZ}}^{2}}\frac{iF}{2\pi}=\mathcal{C}.$
(17)

Now suppose for a moment that such Hermitian holomorphic $L$ exists and we
have built it. An explicit construction, where, without loss of generality,
$\mathcal{C}=1$, will be given below in Sec. VI. We now take $p$ copies of the
line bundle $L$, where the number $n$ in $\mathbb{C}P^{n-1}$ is related to the
Chern number through $n=p\mathcal{C}$. The fact that $\dim
H^{0}(\textnormal{\text{BZ}}^{2},L^{\otimes p})=\deg(L^{\otimes
p})=p\mathcal{C}=n$ follows from the celebrated Riemann-Roch theorem [32, 40].
We collect a basis of $H^{0}(\textnormal{\text{BZ}}^{2},L^{\otimes p})$, and
call them $\\{s_{j}\\}_{j=1}^{p\mathcal{C}}$. We take the map
$f:\textnormal{\text{BZ}}^{2}\to\mathbb{C}P^{n-1}$ given by

$\displaystyle
f:\textnormal{\text{BZ}}^{2}\to\mathbb{C}^{n-1};\;z\mapsto[a_{1}(z):\dots:a_{p\mathcal{C}}(z)],$

with $a_{j}(z)$ being the holomorphic components of $s_{j}$,
$j=1,\dots,p\mathcal{C}$, in a holomorphic gauge $s$ [in which gauge the
$(0,1)$ part of the holomorphic gauge field representing the Chern connection,
which is compatible with the Hermitian metric defined by the $p-$th power of
$h$, $h^{p}$, vanishes]. Note that the $a_{j}$’s will satisfy appropriate
boundary conditions so as to define sections of $L^{\otimes p}$, more
precisely, the section $s_{j}$ as determined by $s_{j}=sa_{j}$ will be
periodic, but $s$ and $a_{j}$ will satisfy

$\displaystyle
s(\textbf{k}+\textbf{G})=s(\textbf{k})\left(e_{\textbf{G}}(z)\right)^{-1}\text{
and }a_{j}(z+G_{1}+\tau G_{2})=e_{\textbf{G}}(z)a_{j}(z),\text{ for all
}j=1,\dots,p\mathcal{C},$ (18)

for a given _system of holomorphic multipliers_ $\\{e_{\textbf{G}}(z)\\}$,
where $\textbf{G}=(G_{1},G_{2})$ is an arbitrary element of the reciprocal
lattice; i.e., a collection of functions $\\{e_{\textbf{G}}(z)\\}$, labeled by
the reciprocal lattice, depending holomorphically in $z$ and that satisfy

$\displaystyle
e_{\textbf{G}+\textbf{G}^{\prime}}(z)=e_{\textbf{G}^{\prime}}(z+G_{1}+\tau
G_{2})e_{\textbf{G}}(z),$ (19)

where $\textbf{G}=(G_{1},G_{2})$ and
$\textbf{G}^{\prime}=(G^{\prime}_{1},G^{\prime}_{2})$ are arbitrary reciprocal
lattice vectors written in the $(k_{1},k_{2})-$coordinates. The holomorphic
multipliers are enough to reconstruct $L^{\otimes p}$ completely [41]. The
transformation rules of Eq. (18) ensure that the map to the projective space
is well-defined (provided $s_{1},\dots,s_{p\mathcal{C}}$ do not vanish
simultaneously) because, for all $z$,

$\displaystyle[a_{1}(z+G_{1}+\tau G_{2}):\dots:a_{p\mathcal{C}}(z+G_{1}+\tau
G_{2})]=[e_{\textbf{G}}(z)a_{1}(z):\dots:e_{\textbf{G}}(z)a_{p\mathcal{C}}(z)]=[a_{1}(z):\dots:a_{p\mathcal{C}}(z)],$
(20)

since the equivalence class does not care about an overall scale. Furthermore
the map $f$ is holomorphic in $z$, by construction. The pullback of the
Fubini-Study symplectic form under $f$ is

$\displaystyle
f^{*}\omega_{FS}=\frac{i}{2}\partial\overline{\partial}\log\left(\sum_{j}|a_{j}(z)|^{2}\right).$

Furthermore

$\displaystyle p\omega-f^{*}\omega_{FS}$
$\displaystyle=-\frac{i}{2}\partial\overline{\partial}\log
h^{p}-\frac{i}{2}\partial\overline{\partial}\log\left(\sum_{j}|a_{j}|^{2}\right)$
$\displaystyle=-\frac{i}{2}\partial\overline{\partial}\log B,$

with $B$ being the (diagonal) of the so-called _Bergman kernel_ [16]:

$\displaystyle B=\sum_{j}h^{p}|a_{j}|^{2}=\sum_{j}h^{p}(s_{j},s_{j}),$ (21)

provided $\\{s_{j}\\}_{j=1}^{p\mathcal{C}}$ form an orthonormal basis with
respect to the $L^{2}$-inner product induced by $h$ and $\omega$ on
$\Gamma(\textnormal{\text{BZ}}^{2},L^{\otimes p})$, i.e.,

$\displaystyle\langle s_{i},s_{j}\rangle_{L^{2}}$
$\displaystyle=\int_{\textnormal{\text{BZ}}^{2}}h^{p}(s_{i},s_{j})\;\omega$
$\displaystyle=\frac{\mathcal{C}}{4\pi}\int
d^{2}k\;e^{-\frac{p\mathcal{C}\tau_{2}}{2\pi}k_{2}^{2}}\overline{a_{i}(z)}a_{j}(z)=\delta_{ij}.$
(22)

In Eqs. (21) and (22), $h^{p}(s_{i},s_{j})=h^{p}\overline{a}_{i}a_{j}$ denotes
the evaluation of the Hermitian metric $h^{p}$ on the sections $s_{i}$ and
$s_{j}$ determined by the functions $a_{i}$ and $a_{j}$ – it is, therefore, a
periodic function. For large $p$, it is known that there is an asymptotic
expansion assuming the form [16]

$\displaystyle B=p+A_{1}p^{0}+A_{2}p^{-1}+\dots+A_{k}p^{1-k}+\dots,$

where the $A_{i}$’s are smooth functions. In particular, as $p$ becomes large,
we see that $f^{*}\omega_{FS}$ goes to $p\omega$, meaning that the symplectic
structure, and consequently, by compatibility, the Kähler structure, becomes
flat.

From a general consideration, one can show that it is not possible to
construct a Kähler band which is geometrically completely flat, that is,
$f^{*}\omega_{FS}$ is a constant over the entire Brillouin zone, with a model
with a finite number of bands. We prove this no-go theorem in Appendix C.
Allowing an infinite number of bands, it is possible to construct bands which
are geometrically completely flat; Landau levels are such examples.

We note that the diagonal of the Bergman kernel also appears, in real space
rather than in momentum space, in the context of the lowest Landau level over
a Riemann surface as the particle density for the associated many-particle
integer Hall effect state [37] and it has been previously approached through
path integrals in Ref. [42].

##  VI Explicit construction

We now provide an explicit construction of geometrically nearly flat Kähler
bands along the prescription described in the previous section. For this
purpose, we first need to specify the auxiliary Hermitian holomorphic line
bundle $L\to\mathrm{BZ}^{2}$, whose curvature satisfies Eq. (16). A line
bundle over $\textnormal{\text{BZ}}^{2}$ is uniquely characterized by
specifying a system of holomorphic multipliers, Eq. (18). We define the line
bundle $L$ to have the holomorphic multipliers, denoted
$\\{e^{L}_{\textnormal{{G}}}(z)\\}$, by

$\displaystyle e^{L}_{\textnormal{{G}}=(2\pi m,2\pi n)}(z)=e^{-i\pi\tau
n^{2}-2\pi in\left(\frac{z}{2\pi}\right)},$

where G is in the reciprocal lattice, so that $m,n\in\mathbb{Z}$. Observe that
for a gauge field $A$ to be consistent with the above system of holomorphic
multipliers and to define a connection on $L$, it must satisfy

$\displaystyle
A(\textnormal{{k}}+\textnormal{{G}})-A(\textnormal{{k}})=-\left(e^{L}_{\textnormal{{G}}}\right)^{-1}de^{L}_{\textnormal{{G}}}=-d\log
e^{L}_{\textnormal{{G}}},$ (23)

for all G in the reciprocal lattice, because the holomorphic multipliers
define the holomorphic transition functions of the line bundle $L$. We note
that the $(1,0)$ form

$\displaystyle A=2\pi ik_{2}dz=\partial\log
h=\partial\log\exp\left(\frac{1}{4\pi}\frac{i}{\tau-\overline{\tau}}(z-\overline{z})^{2}\right)$
(24)

does satisfy this property and is, indeed, a sensible gauge field for the
bundle $L$. It is also clear that the prescribed gauges are holomorphic since
$A$ is a $(1,0)-$form. Furthermore, from Eq. (24) , we can read off a
Hermitian metric [for which $A$ is representing the Chern connection] as
described by the positive function
$h=e^{\frac{1}{4\pi}\frac{i}{\tau-\overline{\tau}}(z-\overline{z})^{2}}=e^{-\frac{\mathrm{Im}(\tau)k_{2}^{2}}{2\pi}}$.
The associated curvature is

$\displaystyle F=-\partial\overline{\partial}\log h=2\pi idk_{2}\wedge
dz=-2\pi idk_{1}\wedge dk_{2},$ (25)

thus, Eq. (16), with
$\deg(L)=\mathcal{C}=\int_{\textnormal{\text{BZ}}^{2}}idA/2\pi=1$, is
satisfied by $L$ equipped with this connection.

The holomorphic multipliers for $L^{\otimes p}$ are simply the $p-$th power of
those of $L$, namely,

$\displaystyle e^{L^{\otimes p}}_{2\pi(m,n)}(z)$
$\displaystyle=\left(e^{L}_{2\pi(m,n)}(z)\right)^{p}=e^{-\pi i\tau
pn^{2}-inpz},\text{ for }m,n\in\mathbb{Z}.$ (26)

The holomorphic sections of such line bundle $L^{\otimes p}$ can be explicitly
given by theta functions as we proceed to describe below.

For $\tau\in\mathcal{H}$, we define the theta function with characteristics
$a$ and $b$ by

$\displaystyle\vartheta\left[\begin{array}[]{c}a\\\
b\end{array}\right](z,\tau)=\sum_{n\in\mathbb{Z}}e^{i\tau\pi(n+a)^{2}+2\pi
i(n+a)(z+b)},\text{ for }a,b\in\mathbb{R}.$

This function satisfies

$\displaystyle\vartheta\left[\begin{array}[]{c}a\\\
b\end{array}\right](z\\!+\\!m\\!+\\!ni,\tau)\\!=\\!e^{-i\pi\tau
n^{2}\\!-\\!2\pi inz\\!+\\!2\pi
i(am\\!-\\!bn)}\vartheta\left[\begin{array}[]{c}a\\\
b\end{array}\right](z,\tau),$

for $m,n\in\mathbb{Z}$. The relation to the standard theta function

$\displaystyle\theta(z,\tau)=\sum_{n\in\mathbb{Z}}e^{\pi in^{2}\tau+2\pi
inz},$

is

$\displaystyle\vartheta\left[\begin{array}[]{c}a\\\
b\end{array}\right](z,\tau)=e^{\pi ia^{2}\tau+2\pi
ia(z+b)}\theta(z+a\tau+b,\tau).$

The functions

$\displaystyle a_{j}(z)$
$\displaystyle=\vartheta\left[\begin{array}[]{c}\frac{j}{p}\\\
0\end{array}\right]\left(p\frac{z}{2\pi},p\tau\right)$ (29)
$\displaystyle=e^{\pi
i\tau\frac{j^{2}}{p}+ijz}\theta\left(p\frac{z}{2\pi}+j\tau,p\tau\right),\
j=0,\dots,p-1,$ (30)

define a basis of holomorphic sections $\\{s_{j}\\}_{j=0}^{p-1}$ of a line
bundle $L^{\otimes p}\to\textnormal{\text{BZ}}^{2}$. The proof that these
functions do satisfy the periodicity required by the holomorphic multipliers
is given in Appendix A. We note that, in real space, the same theta functions
with characteristics arise when looking at the lowest Landau level over the
torus, see Refs. [43, 37].

A natural gauge field consistent with this system of multipliers of
$L^{\otimes p}$ is simply $pA$, with $A$ as in Eq. (24). It immediately
follows that the associated curvature is $pF=-ip(dk_{1}\wedge dk_{2})/(2\pi)$
and that the first Chern number of $L^{\otimes p}$ is $p$.

Furthermore we can show that (see Ref. [37] for an analogous formula)

$\displaystyle\langle s_{i},s_{j}\rangle_{L^{2}}$
$\displaystyle=\frac{1}{(2\pi)^{2}}\int_{\textnormal{\text{BZ}}^{2}}e^{-p\frac{k_{2}^{2}}{2\pi}}\overline{a_{i}}a_{j}dk_{1}\wedge
dk_{2}$ $\displaystyle=\sqrt{\frac{i}{p(\tau-\overline{\tau})}}\delta_{ij},$

hence, the map we want is induced by the vector

$\displaystyle z\mapsto(a_{0}(z),\dots,a_{p-1}(z)),$

for large $p$, namely,

$\displaystyle
f_{p}:\textnormal{\text{BZ}}^{2}\to\mathbb{C}P^{p-1};\;z\mapsto[a_{0}(z):\dots:a_{p-1}(z)],$

or, in terms of orthogonal rank $1$ projectors, a smooth map $z\mapsto
P_{p}(z)$ with

$\displaystyle\langle
i|P_{p}(z)|j\rangle=\frac{a_{i}(z)\overline{a_{j}(z)}}{\sum_{k}|a_{k}(z)|^{2}},$

with $|i\rangle$, $i=0,\dots,p-1$, the canonical basis of $\mathbb{C}^{p}$.
Observe that, since $\theta(z,\tau)$ has zeros in
$\frac{1}{2}(1+\tau)+\mathbb{Z}+\tau\mathbb{Z}$, it follows that $a_{j}$, or
equivalently, $s_{j}$ has zeros at positions

$\displaystyle\frac{2\pi}{p}\left(\frac{1}{2}+r\right)+2\pi\left(\frac{1}{2}-\frac{j}{p}\right)\tau+2\pi\mathbb{Z}+2\pi\tau\mathbb{Z},\
r=0,\dots,p-1,$ (31)

for $j=0,\dots,p-1$. Thus, the sections do not vanish simultaneously and the
map $f_{p}$ is well-defined. For large enough $p$ (actually for $p>2$), the
band is Kähler. In the large $p$ limit the band is Kähler and actually the
Kähler structure is asymptotically flat
$(f_{p}^{*}\omega_{FS},j_{\tau},f_{p}^{*}g_{FS})\sim(p\,\omega,j_{\tau},p\,g)$.
Having built an approximately flat Kähler band (for finite $p$) one can always
build a tight-binding model with a flat dispersion by declaring the momentum-
space Hamiltonian to be $H(\textbf{k})=I_{p}-2P_{p}(\textbf{k})$,
$\textnormal{{k}}\in\textnormal{\text{BZ}}^{2}$ and then inverse Fourier
transforming it to obtain the model in the lattice, namely,
$H(\textnormal{{r}}_{i},\textnormal{{r}}_{j})=\int_{\textnormal{\text{BZ}}^{2}}\frac{d^{2}k}{(2\pi)^{2}}e^{i\textnormal{{k}}\cdot\left(\textnormal{{r}}_{i}-\textnormal{{r}}_{j}\right)}H(\textnormal{{k}})$
with $p$ orbitals/internal degrees of freedom per site. We can always build a
tight-binding model with a flat dispersion with this prescription, but it will
always have long range hoppings provided its Chern number is nontrivial [44].
We can make the tight-binding model strictly local by truncating the hoppings.
This truncation procedure will, however, always violate (even if weakly) the
flat structures. In particular, we will see that the resulting Chern bands
will not, in general, be Kähler bands.

###  VI.1 Numerical results

We now perform numerical simulation of the constructed model. We consider the
isotropic case $\tau=i$, and also anisotropic cases $\tau=e^{i\pi/3}$ and
$\tau=2e^{i\pi/7}$, and show how the Berry curvature and the quantum volume
form flatten for large $p$. By construction, our model is Kähler and thus
$\sqrt{\det(g(\textnormal{{k}}))}=|F_{12}(\textnormal{{k}})|/2=\omega_{12}(\textnormal{{k}})$
always holds, where $\omega_{12}dk_{1}\wedge dk_{2}=(-i/2)F_{12}dk_{1}\wedge
dk_{2}$ is the pullback of the Fubini-Study symplectic form, and we have also
numerically confirmed this equality. In Figs. 1(a), 1(b), and 1(c), we plot
$4\pi\omega_{xy}(\textnormal{{k}})$, for $p=2,4,6$, and for $\tau=i$,
$e^{i\pi/3}$, and $2e^{i\pi/7}$, respectively. We see that already for $p=6$,
$4\pi\omega_{xy}(\textnormal{{k}})$ reaches the flat value of
$4\pi\omega=4\pi\sqrt{\det(g)}dk_{1}\wedge dk_{2}=p\,dk_{1}\wedge dk_{2}$ very
well.

(a) $\tau=i$ (b) $\tau=e^{i\pi/3}$

(c) $\tau=2e^{i\pi/7}$

Figure 1: Plot of $4\pi\omega_{12}$, where $\omega_{12}$ the only independent
component of the pullback $f_{p}^{*}\omega_{FS}$, as a function of the
quasimomentum $\textnormal{{k}}\in\textnormal{\text{BZ}}^{2}$, for $p=2$
(orange), $p=4$ (blue), $p=6$ (green), for various anisotropies $\tau$. We
have numerically confirmed that $4\pi\sqrt{\det(g)}$, representing the quantum
volume form, takes the same value as $4\pi\omega_{12}$ for all the cases shown
here.

Our model of the Kähler band is constructed in momentum space. When
interpreted as a real-space lattice model, it contains long-range hoppings at
any length. We now consider the effect of truncating the hopping to obtain
short-range models. We consider how our momentum space model can be translated
into a square lattice model in real space, and analyze the consequence of
truncation of hopping on the geometrical structure in momentum space.

The real-space tight-binding model associated to
$H(\textnormal{{k}})=I_{p}-2P_{p}(\textnormal{{k}})$ is

$\displaystyle
H(\textnormal{{r}}_{i},\textnormal{{r}}_{j})=\int_{\textnormal{\text{BZ}}^{2}}\frac{d^{2}k}{(2\pi)^{2}}H(\textnormal{{k}})e^{i\textnormal{{k}}\cdot\left(\textnormal{{r}}_{i}-\textnormal{{r}}_{j}\right)},$
(32)

where $\textnormal{{r}}_{i},\textnormal{{r}}_{j}\in\mathbb{Z}^{2}$ label the
positions on the lattice. The truncation of the hoppings can be done by taking
the function $f_{R}(\textnormal{{r}})=f_{R}(x_{1},x_{2})$ defined by

$\displaystyle f_{R}(x_{1},x_{2})=\begin{cases}1,\text{ if }-\lfloor
R\rfloor\leq x_{i}\leq\lfloor R\rfloor,\ i=1,2,\\\ 0,\text{
otherwise.}\end{cases},$ (33)

for some $R$ and considering the truncated real space Hamiltonian
$H_{R}(\textnormal{{r}}_{i},\textnormal{{r}}_{j})=H(\textnormal{{r}}_{i},\textnormal{{r}}_{j})f(\textnormal{{r}}_{i}-\textnormal{{r}}_{j})$.
It is convenient, to simplify notation, to write $B_{R}$ for the set of
translations of the lattice r with $f_{R}(\textnormal{{r}})=1$. Note that
$f_{R}(\textnormal{{r}})$ is just the indicator function of the set $B_{R}$.
Since under Fourier transformation the product becomes a convolution and the
Fourier coefficients of $f_{R}(\textnormal{{r}})$ are, evidently,

$\displaystyle\widetilde{f}_{R}(\textnormal{{k}})=\sum_{\textnormal{{r}}\in
B_{R}}e^{-i\textnormal{{k}}\cdot\textnormal{{r}}},$ (34)

we finally arrive at the truncated Hamiltonian

$\displaystyle H_{R}(\textnormal{{k}})$
$\displaystyle=\int_{\textnormal{\text{BZ}}^{2}}\frac{dk^{\prime}}{(2\pi)^{2}}H(\textnormal{{k}}-\textnormal{{k}}^{\prime})\widetilde{f}_{R}(\textnormal{{k}}^{\prime})$
$\displaystyle=\int_{\textnormal{\text{BZ}}^{2}}\frac{dk^{\prime}}{(2\pi)^{2}}\sum_{\textnormal{{r}}\in
B_{R}}H(\textnormal{{k}}-\textnormal{{k}}^{\prime})e^{-i\textnormal{{k}}^{\prime}\cdot\textnormal{{r}}}.$
(35)

Note that, as $R\to\infty$, we approach the original Hamiltonian we started
with, because $\widetilde{f}_{R}(\textnormal{{k}})$ approaches the Dirac delta
distribution on the Brillouin zone $(2\pi)^{2}\delta^{2}(\textnormal{{k}})$.
Now observe that a Riemann sum for the above expression is provided by

$\displaystyle\frac{1}{N_{1}N_{2}}\sum_{\textnormal{{n}}\in\mathbb{Z}_{N_{1}}\times\mathbb{Z}_{N_{2}}}\sum_{\textnormal{{r}}\in
B_{R}}H(\textnormal{{k}}-\textnormal{{k}}^{\prime}_{\textnormal{{n}}})e^{-i\textnormal{{k}}^{\prime}_{\textnormal{{n}}}\cdot\textnormal{{r}}},$
(36)

with $\textnormal{{k}}^{\prime}_{\textnormal{{n}}}=(2\pi n_{1}/N_{1},2\pi
n_{2}/N_{2})$ with
$\textnormal{{n}}=(n_{1},n_{2})\in\mathbb{Z}_{N_{1}}\times\mathbb{Z}_{N_{2}}$,
where $\mathbb{Z}_{N}$ denote the integers modulo $N$. Provided
$R<\min(N_{1},N_{2})$, the above expression corresponds to the tight-binding
model one would get by truncating the one in the finite system with $N_{1}$
and $N_{2}$ sites in the $x_{1}$ and $x_{2}$ directions, respectively, with
periodic boundary conditions. In the limit when $N_{1}$, $N_{2}$ are large,
the above expression is a good approximation for the Bloch Hamiltonian
obtained by truncating the tight-binding model in $\mathbb{Z}^{2}$.

In our simulation, we take a square lattice with $N_{1}=N_{2}=50$ sites and
periodic boundary conditions on the $x_{1}$ and $x_{2}$ directions, we fix
$\tau=i$ and we take $p=6$, and consider how the quantum geometry of the
lowest energy band of the Hamiltonian is affected as we truncate the allowed
hoppings by shrinking $R$. In Fig. 2, we present the numerically calculated
$\sqrt{\det(g)}$ and $\omega_{12}$ for $R=1$, $R=2$, and $R=3$, corresponding
to allowing up to first, second, and third nearest neighbor hoppings,
respectively, and compared them to the $R=\infty$ long-range case, where there
is no truncation. Additionally, in Fig. 3, we present the relative
fluctuations of $\sqrt{\det(g)}$ and $\omega_{12}$ with respect to the flat
value $p/4\pi$ (with $p=6$), for $R=1$, $R=2$, and $R=3$, and $R=\infty$,
along the submanifold defined by $k_{2}=0$. We can see that when truncating
the hopping at $R=3$, namely, including up to third-nearest-neighbor hoppings,
the geometrical structure is already almost identical to the long-range case
of $R=\infty$. When $R=1$ and $R=2$, we can see the effect of truncation more
clearly. The effect of truncation is two-fold. The first effect is that
$\sqrt{\det(g)}$ and $\omega_{12}$ are no longer equal, implying the breaking
of holomorphicity of the Chern bands. However, the difference between
$\sqrt{\det(g)}$ and $\omega_{12}$ is not so large even for the case of
nearest-neighbor model of $R=1$; in our simulation the difference is around
one percent as one can read off from Fig. 2. The second effect is that the
flatness of the geometrical quantities, $\sqrt{\det(g)}$ and $\omega_{12}$,
will change. What we have numerically observed is that the geometrical
quantities will not become more dispersive, and sometimes they can become even
flatter when truncating the hopping. We note that, even when $R=1$ and $R=2$,
the Chern number, $-\int_{\textnormal{\text{BZ}}^{2}}\omega/\pi=-6$, is the
same as $R=\infty$, and thus the bands are adiabatically connected to the
ideal Kähler band. We have also numerically checked the cases with smaller
$p$, such as $p=3$ or $p=4$, and the overall behavior remains the same. We
note that $p=2$ is special; the band cannot be Kähler because of the
constraints that $\det(g(\mathbf{k}))=0$ should hold somewhere in the
Brillouin zone, and thus bands cannot be made geometrically flat.

(a) $R=\infty$ vs. $R=1$

(b) $R=\infty$ vs. $R=2$

(c) $R=\infty$ vs. $R=3$

Figure 2: The geometrical structure, $4\pi\sqrt{\det(g)}$ and
$4\pi\omega_{12}$, of truncated models as a function of
$\textnormal{{k}}\in\textnormal{\text{BZ}}^{2}$, for $p=6$. In the top panel,
$R=1$ is compared to the long range case $R=\infty$; in the middle panel $R=2$
is compared to the $R=\infty$ case, and in the bottom panel $R=3$ is compared
to $R=\infty$. On the left column we present the variation of the geometric
quantities along the whole of $\textnormal{\text{BZ}}^{2}$ and on the right
column, for clarity, we present the variation along the cut at $k_{2}=0$. On
the right column, the markers represent the data points and the lines are
there for a clear visualization of the data.

(a) Relative fluctuations of $\omega_{12}$ (b) Relative fluctuations of
$\sqrt{\det(g)}$

Figure 3: The relative fluctuations, for $p=6$, of $\omega_{12}$ and
$\sqrt{\det(g)}$ as measured from the flat value $p/4\pi$, as a function of
$k_{1}$ along the cut $k_{2}=0$. The markers represent the data points and the
lines are there for a clear visualization of the data.

##  VII Conclusions

In this manuscript, we described a systematic method to obtain Kähler bands,
whose metric and curvature satisfies the equality
$\sqrt{\det(g(\mathbf{k}))}=|F_{12}(\mathbf{k})|/2$, become flat in momentum
space as the total number of bands is increased. We also showed that it is not
possible to obtain a perfectly flat Kähler band with a finite number of total
bands. We provided an explicit construction of our method using theta
functions, and we have numerically observed that we can achieve fairly flat
Kähler bands with a reasonably small number of total bands. We have also shown
that truncating the lattice model to include only local hoppings will not
affect the flatness of the bands, but does make the band deviate from the
perfect Kähler bands, leading to
$\sqrt{\det(g(\mathbf{k}))}\geq|F_{12}(\mathbf{k})|/2$.

The Bergman kernel prescription can be applied to the more general case in
which one replaces the flat Kähler structure $(\omega,g,j_{\tau})$ in Eq. (12)
by some arbitrary Kähler structure $(\omega^{\prime},g^{\prime},j^{\prime})$.
In that case, the resulting map $f_{p}$, built from an orthonormal basis of
holomorphic sections of the $p$-th tensor power of an auxiliary Hermitian
holomorphic (with respect to the new complex structure $j^{\prime}$) line
bundle $L^{\prime}$ satisfying the quantization condition
$-\partial\overline{\partial}\log h^{\prime}=-2i\omega^{\prime}$, will also
yield, asymptotically in $p$, the compatible triple
$(f_{p}^{*}\omega_{FS},j^{\prime},f_{p}^{*}g_{FS})\sim(p\omega^{\prime},j^{\prime},pg^{\prime})$.
This prescription would then allow us to engineer Kähler bands with a
prescribed profile of Berry curvature and quantum metric which may be
potentially interesting for engineering of Chern bands with desired
properties. Such generalizations of our method will be discussed in more
detail in future works.

Upon submission of this work, we became aware of a related no-go theorem which
was very recently presented in Ref. [45]. They showed that it is impossible to
have a flat Kähler band coming from a lattice system with a finite number of
lattice sites per unit cell (hence with a finite total number of bands), which
agrees with our no-go theorem.

###### Acknowledgements.

B.M. acknowledges very stimulating discussions with J. P. Nunes and J. M.
Mourão. T.O. acknowledges support from JSPS KAKENHI Grant Number JP20H01845,
JST PRESTO Grant Number JPMJPR19L2, JST CREST Grant Number 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 Additional properties of theta functions

We had

$\displaystyle a_{j}(z)$
$\displaystyle=\vartheta\left[\begin{array}[]{c}\frac{j}{p}\\\
0\end{array}\right]\left(p\frac{z}{2\pi},p\tau\right)$ $\displaystyle=e^{\pi
i\tau\frac{j^{2}}{p}+ijz}\theta(p\frac{z}{2\pi}+j\tau,p\tau),\ j=0,\dots,p-1,$

We can then write

$\displaystyle a_{j}(z)=e^{\pi
i\tau\frac{j^{2}}{p}+ijz}\sum_{n\in\mathbb{Z}}e^{\pi i\tau pn^{2}+2\pi
in\left(\frac{pz}{2\pi}+j\tau\right)}$

We have, for $m,n\in\mathbb{Z}$,

$\displaystyle a_{j}(z+2\pi m+2\pi n\tau)$ $\displaystyle=e^{\pi
i\tau\frac{j^{2}}{p}+ij(z+2\pi m+2\pi n\tau)}\theta(p\frac{z}{2\pi}+pm+\tau
pn+j\tau,p\tau)$ $\displaystyle=e^{\pi i\tau\frac{j^{2}}{p}\\!+\\!ij(z+2\pi
m+2\pi n\tau)\\!-\\!\pi i\tau pn^{2}\\!-\\!2\pi
in\left(p\frac{z}{2\pi}+j\tau\right)}\theta(p\frac{z}{2\pi}\\!+\\!j\tau,pi)$
$\displaystyle=e^{-\pi i\tau pn^{2}-inpz-2\pi inj\tau+2\pi ijn\tau+2\pi
ijm}a_{j}(z)$ $\displaystyle=e^{-\pi i\tau pn^{2}-inpz}a_{j}(z),$

which agrees with the system of holomorphic multipliers,
$\\{e_{\textbf{G}}(z)\\}$, for the holomorphic line bundle $L^{\otimes p}$,

$\displaystyle e_{2\pi(m,n)}(z)=e^{-\pi i\tau pn^{2}-inpz},\text{ for
}m,n\in\mathbb{Z}.$ (37)

The associated gauge field is given by

$\displaystyle A_{p}$
$\displaystyle=pA=ipk_{2}\frac{dz}{2\pi}=\partial\log\exp\left(-p\frac{\tau_{2}k_{2}^{2}}{2\pi}\right)$
$\displaystyle=\frac{-i}{2}\frac{\partial}{\partial
k_{2}}\left[\log\exp\left(-p\frac{\tau_{2}k_{2}^{2}}{2\pi}\right)\right]dz,$

where $p=\deg(L^{\otimes p})$ is the first Chern number of $L^{\otimes p}$.

##  Appendix B Formulae for performing Numerics

Let $|\Psi(\textnormal{{k}})\rangle$ be the unit vector in $\mathbb{C}^{p}$
defined by

$\displaystyle|\Psi(\textnormal{{k}})\rangle=\frac{(a_{0}(z),\dots,a_{p-1}(z))}{\sqrt{\sum_{j}|a_{j}(z)|^{2}}},$
(38)

with the $a_{j}$’s defined by Eq. (30), $z=k_{1}+\tau k_{2}$, and
$\textnormal{{k}}\in\textnormal{\text{BZ}}^{2}$. The discussion below allows
us to numerically evaluate the quantum metric and the Berry curvature in terms
of $|\Psi(\textnormal{{k}})\rangle$. Additionally, we will also use the
formulas below to compute the quantum metric and the Berry curvature for the
case when $|\Psi(\textnormal{{k}})\rangle$ is not defined by the above formula
but rather defined as the lowest energy eigenstate of the short-range tight-
binding model specified by the truncation of the Bloch Hamiltonian
$H(\textnormal{{k}})=I_{p}-2P_{p}(\textnormal{{k}})$, denoted
$H_{R}(\textnormal{{k}})$ and defined in Eq. (35).

To compute the _quantum metric_ , we used the fidelity between neighboring
states

$\displaystyle\left|\langle\Psi(\textnormal{{k}})|\Psi(\textnormal{{k}}+\varepsilon
v)\rangle\right|=1-\frac{1}{2}g(v,v)\varepsilon^{2}+\cdots,$

where $v=(v^{1},v^{2})$ defines a tangent vector
$\sum_{i}v_{i}\frac{\partial}{\partial k_{i}}\in
T_{\textnormal{{k}}}\textnormal{\text{BZ}}^{2}$ and $\varepsilon$ is a small
number.

To compute the _Berry curvature_ , we used the fact that for a given path
$\textnormal{{k}}(t)$, $(0\leq t\leq 1)$, we have, for large $N$,

$\displaystyle\prod_{i=0}^{N-1}\langle\Psi(\textnormal{{k}}(\frac{i+1}{N}))|\Psi(\textnormal{{k}}(\frac{i}{N}))\rangle\approx\exp\left(-\int_{0}^{1}A\left(\frac{d\textnormal{{k}}}{dt}\right)dt\right),$

In particular, if we take a loop that is a boundary of a surface
$\Sigma\subset\textnormal{\text{BZ}}^{2}$, we have

$\displaystyle\prod_{i=0}^{N-1}\langle\Psi(\textnormal{{k}}(\frac{i+1}{N}))|\Psi(\textnormal{{k}}(\frac{i}{N}))\rangle\approx\exp(-\int_{\Sigma}F).$

We can take $\Sigma$ to be an infinitesimal rectangle with vertices
$\textnormal{{k}}_{1}\equiv\textnormal{{k}}_{5}=\textnormal{{k}}$,
$\textnormal{{k}}_{2}=\textnormal{{k}}+\varepsilon_{1}\hat{\textnormal{{x}}}_{1}$,
$\textnormal{{k}}_{3}=\textnormal{{k}}+\varepsilon_{1}\hat{\textnormal{{x}}}_{1}+\varepsilon_{2}\hat{\textnormal{{x}}}_{2}$
and
$\textnormal{{k}}_{4}=\textnormal{{k}}+\varepsilon_{2}\hat{\textnormal{{x}}}_{2}$
in an orientation consistent with that of the standard one in
$\textnormal{\text{BZ}}^{2}$, for small $\varepsilon_{1}$ and
$\varepsilon_{2}$. Here $\hat{\textnormal{{x}}}_{1}=(1,0)$ and
$\hat{\textnormal{{x}}}_{2}=(0,1)$. This then gives

$\displaystyle-i\text{Im}\log\left[\prod_{i=1}^{4}\langle\Psi(\textnormal{{k}}_{i+1})|\Psi(\textnormal{{k}}_{i})\rangle\right]\approx
F_{12}(\textnormal{{k}})\varepsilon_{1}\varepsilon_{2}.$

Using this notation we also see that

$\displaystyle-\log\left[|\langle\Psi(\textnormal{{k}}_{2})|\Psi(\textnormal{{k}}_{1})\rangle|^{2}\right]\approx$
$\displaystyle g_{11}(\textnormal{{k}})\varepsilon_{1}^{2},$
$\displaystyle-\log\left[|\langle\Psi(\textnormal{{k}}_{4})|\Psi(\textnormal{{k}}_{1})\rangle|^{2}\right]\approx$
$\displaystyle g_{22}(\textnormal{{k}})\varepsilon_{2}^{2},$
$\displaystyle-\log\left[|\langle\Psi(\textnormal{{k}}_{3})|\Psi(\textnormal{{k}}_{1})\rangle|^{2}\right]\approx$
$\displaystyle
g_{11}(\textnormal{{k}})\varepsilon_{1}^{2}+2g_{12}(\textnormal{{k}})\varepsilon_{1}\varepsilon_{2}$
$\displaystyle+g_{22}(\textnormal{{k}})\varepsilon_{2}^{2}.$

This justifies why we have to take the imaginary part in the previous
equation:

$\displaystyle-\text{Re}\log\left[\prod_{i=1}^{4}\langle\Psi(\textnormal{{k}}_{i+1})|\Psi(\textnormal{{k}}_{i})\rangle\right]$
$\displaystyle\approx\frac{1}{2}g_{11}(\textnormal{{k}})\varepsilon_{1}^{2}+\frac{1}{2}g_{22}(\textnormal{{k}}+\varepsilon_{1}\hat{\textnormal{{x}}}_{1})\varepsilon_{2}^{2}+\frac{1}{2}g_{11}(\textnormal{{k}}+\varepsilon_{2}\hat{\textnormal{{x}}}_{2})\varepsilon_{1}^{2}$
$\displaystyle+\frac{1}{2}g_{22}(\textnormal{{k}})\varepsilon_{2}^{2}$
$\displaystyle\approx
g_{11}(\textnormal{{k}})\varepsilon_{1}^{2}+g_{22}(\textnormal{{k}})\varepsilon_{2}^{2}.$

Hence

$\displaystyle\prod_{i=1}^{4}\langle\Psi(\textnormal{{k}}_{i+1})|\Psi(\textnormal{{k}}_{i})\rangle$
$\displaystyle\approx\exp\left[-\left(g_{11}(\textnormal{{k}})\varepsilon_{1}^{2}+g_{22}(\textnormal{{k}})\varepsilon_{2}^{2}\right)-F_{12}(\textnormal{{k}})\varepsilon_{1}\varepsilon_{2}\right].$

##  Appendix C No-go theorem: impossibility of having nontrivial flat Kähler
bands for finite total number of bands

Suppose we are given a Kähler band described by a holomorphic map
$f:\textnormal{\text{BZ}}^{2}\to\mathbb{C}P^{n-1};z\mapsto[Z_{1}(z):\dots:Z_{n}(z)]$,
with $n$ the number of bands, where the map is holomorphic with respect to a
flat complex structure determined by $\tau=\tau_{1}+i\tau_{2}\in\mathcal{H}$
and complex coordinate $z=k_{1}+\tau k_{2}$. Note that the orthogonal
projector $P(\textnormal{{k}})$ describing the band associated to $f$ is
simply

$\displaystyle
P(\textnormal{{k}})=\sum_{i,j=1}^{n}\frac{Z_{i}(z)\overline{Z_{j}(z)}}{\sum_{k=1}^{n}|Z_{k}(z)|^{2}}|i\rangle\langle
j|,$

where $|i\rangle$, $i=1,\dots,n$, is the canonical basis of $\mathbb{C}^{n}$
describing the internal degrees of freedom.

We want to show that $f^{*}\omega_{FS}$ cannot be uniform in the Brillouin
zone. We want to compare the compatible tripe
$(f^{*}\omega_{FS},j_{\tau},f^{*}g_{FS})$ and the flat anisotropic one given
by

$\displaystyle\omega$ $\displaystyle=\sqrt{\det(g)}dk_{1}\wedge
dk_{2}=\frac{\mathcal{C}}{4\pi}dk_{1}\wedge
dk_{2}=\frac{\mathcal{C}}{4\pi}\frac{1}{\bar{\tau}-\tau}dz\wedge
d\overline{z},$ $\displaystyle J$
$\displaystyle=\left[\begin{array}[]{cc}-\frac{\tau_{1}}{\tau_{2}}&-\frac{|\tau|^{2}}{\tau_{2}}\\\
\frac{1}{\tau_{2}}&\frac{\tau_{1}}{\tau_{2}}\end{array}\right],$
$\displaystyle g$
$\displaystyle=\frac{\mathcal{C}}{4\pi}\left(\frac{-2i}{\overline{\tau}-\tau}\right)|dz|^{2},$

where

$\displaystyle\mathcal{C}=\int_{\textnormal{\text{BZ}}^{2}}\frac{f^{*}\omega_{FS}}{\pi}>0.$

We have $\mathcal{C}>0$ because the map $f$ is Kähler by assumption. Observe
that $\omega=(-i/2)\partial\overline{\partial}\log h$, for
$h=e^{-\frac{\mathcal{C}}{2\pi}\tau_{2}k_{2}^{2}}$. For the comparison we take
the difference,

$\displaystyle\omega-f^{*}\omega_{FS}=-\frac{i}{2}\partial\overline{\partial}\log
F,$ (39)

where

$\displaystyle
F=|C(z)|^{2}\sum_{j=1}^{n}e^{-\frac{\mathcal{C}}{2\pi}\tau_{2}k_{2}^{2}}|Z_{j}(z)|^{2},$
(40)

for some $C(z)$ holomorphic (not periodic function), defines a smooth function
on $\textnormal{\text{BZ}}^{2}$. From the point of view of the formula of Eq.
(39), the $C(z)$ may seem arbitrary, however, by the
$\partial\overline{\partial}-$lemma (Corollary 3.2.10 of Ref. [40]), it can be
chosen so that $F$ is indeed a globally defined smooth function, because
$\omega$ and $f^{*}\omega_{FS}$ lie in the same de Rham cohomology class
(because they integrate to the same value $\pi\mathcal{C}$). As pointed out in
Ref. [11], the uniformity of Berry curvature amounts to finding a map such
that the Kähler potential for $f^{*}\omega_{FS}$ is Laplacian free. They also
point out in the Appendix that, in their prescription, they would need an
infinite number of parameters. Here we will go a bit further, and show that
for finite total number of bands $n$ it is impossible to have a flat Kähler
band. Since if the symplectic form is flat so will be the quantum metric by
compatibility, it is enough to consider the flatness of the former. Note that
the functions $b_{j}(z)=|C(z)|^{2}|Z_{j}(z)|^{2}$, $j=1,\dots,n$, must
transform, under lattice translations, in a complementary way to
$e^{-\frac{\mathcal{C}}{2\pi}\tau_{2}k_{2}^{2}}$. They must also transform in
a way which is independent of $j$, because for the map $f$ to be well defined,
$Z_{j}(z+2\pi m+2\pi n\tau)/Z_{j}(z)$ must be independent of $j$, for all
$m,n\in\mathbb{Z}$. Put differently, the quantities $C(z)Z_{j}(z)$,
$j=1,\dots,n$, must transform as holomorphic sections of
$L^{\otimes\mathcal{C}}$, where $L$ is the basic line bundle as defined in
Sec. VI. In particular, it means that $C(z)Z_{j}(z)$, $j=1,\dots,n$, can be
written as linear combinations of a basis of
$H^{0}(\textnormal{\text{BZ}}^{2},L^{\otimes\mathcal{C}})$ whose dimension is,
by Riemann-Roch, $\dim
H^{0}(\textnormal{\text{BZ}}^{2},L^{\otimes\mathcal{C}})=\mathcal{C}$. Making
the replacements $p\leftrightarrow\mathcal{C}$ in the formulas of Sec. VI, we
can then write, using the basis of theta functions,

$\displaystyle C(z)Z_{j}(z)$
$\displaystyle=\sum_{l=0}^{\mathcal{C}-1}A_{j}^{l}\vartheta\left[\begin{array}[]{c}\frac{l}{\mathcal{C}}\\\
0\end{array}\right]\left(\mathcal{C}\frac{z}{2\pi},\mathcal{C}\tau\right)$
$\displaystyle=\sum_{l=0}^{\mathcal{C}-1}A_{j}^{l}a_{l}(z),$

where $A_{j}^{l}$ are complex numbers. Thus,

$\displaystyle
F=\sum_{j=1}^{n}\sum_{l,l^{\prime}=0}^{\mathcal{C}-1}e^{-\frac{\mathcal{C}}{2\pi}\tau_{2}k_{2}^{2}}A_{j}^{l}\overline{A_{j}^{l^{\prime}}}a_{l}(z)\overline{a_{l^{\prime}}(z)}.$
(41)

We remark that the fact that there are $\mathcal{C}$ independent holomorphic
sections ($\dim
H^{0}(\textnormal{\text{BZ}}^{2},L^{\otimes\mathcal{C}})=\mathcal{C}$) shows
that for $n>\mathcal{C}$ the matrix $A=[A_{j}^{l}]_{1\leq j\leq n,0\leq
l\leq\mathcal{C}}$ is singular. This means that, at most, there are only
$\mathcal{C}\times\mathcal{C}$ linearly independent coefficients. Thus,
without loss of generality, we may assume that the sum on $j$ goes only up
until $\mathcal{C}$. The requirement for uniformity of $f^{*}\omega_{FS}$ is
now seen to be equivalent to $F$ being a constant. We can even determine what
that constant must be, because

$\displaystyle\frac{1}{(2\pi)^{2}}\int_{BZ^{2}}F(z)dk_{1}\wedge dk_{2}$
$\displaystyle=\sum_{j=1}^{n}\sum_{l,l^{\prime}=0}^{\mathcal{C}-1}A_{j}^{l}\overline{A_{j}^{l^{\prime}}}\langle
s_{l},s_{l^{\prime}}\rangle_{L^{2}}$
$\displaystyle=\sum_{j=1}^{n}\sum_{l,l^{\prime}=0}^{\mathcal{C}-1}A_{j}^{l}\overline{A_{j}^{l^{\prime}}}\delta_{ll^{\prime}}\sqrt{\frac{i}{\mathcal{C}(\tau-\bar{\tau})}}$
$\displaystyle=\sum_{j=1}^{n}\sum_{l=0}^{\mathcal{C}-1}|A_{j}^{l}|^{2}\sqrt{\frac{i}{\mathcal{C}(\tau-\bar{\tau})}}.$

The quantity $a_{l}(z)\overline{a_{l^{\prime}}(z)}$ can be written as

$\displaystyle a_{l}(z)\overline{a_{l^{\prime}}(z)}$
$\displaystyle=\left(\sum_{m,n\in\mathbb{Z}}e^{i\frac{\pi\tau}{\mathcal{C}}\left(l+m\mathcal{C}\right)^{2}-\frac{i\pi\overline{\tau}}{\mathcal{C}}\left(l^{\prime}+n\mathcal{C}\right)^{2}+izm-i\bar{z}n)\mathcal{C}}\right)e^{izl-
izl^{\prime}}$

The dependence $e^{izl-izl^{\prime}}$, which gives a factor
$e^{ik_{1}(l-l^{\prime})}$, is the only $k_{1}-$dependence on the whole of
$e^{-\frac{\mathcal{C}}{2\pi}\tau_{2}k_{2}^{2}}a_{l}(z)\overline{a_{l^{\prime}}(z)}$
and, together with the fact that $a_{l}$ is $\mathcal{C}-$periodic, i.e.,
$a_{l}=a_{l+\mathcal{C}}$, and $\int
e^{ik_{1}(l-l^{\prime})}dk_{2}=2\pi\delta_{ll^{\prime}}$, justifies why
$\langle s_{l},s_{l^{\prime}}\rangle_{L^{2}}\propto\delta_{ll^{\prime}}$.
Furthermore, this same dependence implies that if $F$ were to be constant, the
sum over $0\leq l,l^{\prime}\leq\mathcal{C}-1$ must be restricted to the
diagonal $l=l^{\prime}$, otherwise there will be independent oscillatory terms
in $k_{1}$ which will not allow for a constant $F$. Hence, we have reduced $F$
to be of the form

$\displaystyle F$
$\displaystyle=\sum_{j=1}^{n}\sum_{l=0}^{\mathcal{C}-1}e^{-\frac{\mathcal{C}}{2\pi}\tau_{2}k_{2}^{2}}|A_{j}^{l}|^{2}|a_{l}(z)|^{2}$
$\displaystyle=\sum_{l=0}^{\mathcal{C}-1}e^{-\frac{\mathcal{C}}{2\pi}\tau_{2}k_{2}^{2}}|A_{l}|^{2}|a_{l}(z)|^{2},$
(42)

where we defined $|A_{l}|^{2}=\sum_{j=1}^{n}|A_{j}^{l}|^{2}$ for the sake of
simplicity. We will now show that $F$ cannot be constant, by showing that the
integral over $k_{1}$ defines a nonconstant function. Note

$\displaystyle\int_{0}^{2\pi}\frac{dk_{1}}{2\pi}F(k_{1},k_{2})=\sum_{l=0}^{\mathcal{C}-1}|A_{l}|^{2}\int_{0}^{2\pi}\frac{dk_{1}}{2\pi}e^{-\frac{\mathcal{C}}{2\pi}\tau_{2}k_{2}^{2}}|a_{l}(z)|^{2}.$

We then first compute

$\displaystyle\int_{0}^{2\pi}\frac{dk_{1}}{2\pi}e^{-\frac{\mathcal{C}}{2\pi}\tau_{2}k_{2}^{2}}|a_{l}(z)|^{2}$
$\displaystyle=\int_{0}^{2\pi}\frac{dk_{1}}{2\pi}e^{-\frac{\mathcal{C}}{2\pi}\tau_{2}k_{2}^{2}}$
$\displaystyle\times\sum_{m,n\in\mathbb{Z}}e^{i\frac{\pi\tau}{\mathcal{C}}\left(l+m\mathcal{C}\right)^{2}-\frac{i\pi\overline{\tau}}{\mathcal{C}}\left(l+n\mathcal{C}\right)^{2}+iz(l+m\mathcal{C})-i\bar{z}(l+n\mathcal{C})}$
$\displaystyle=e^{-\frac{\mathcal{C}}{2\pi}\tau_{2}k_{2}^{2}}\sum_{m\in\mathbb{Z}}e^{i\frac{\pi\tau}{\mathcal{C}}\left(l+m\mathcal{C}\right)^{2}-\frac{i\pi\overline{\tau}}{\mathcal{C}}\left(l+m\mathcal{C}\right)^{2}+i(z-\bar{z})(l+m\mathcal{C})}$
$\displaystyle=e^{-\frac{\mathcal{C}}{2\pi}\tau_{2}k_{2}^{2}}\sum_{m\in\mathbb{Z}}e^{-\frac{2\pi\tau_{2}}{\mathcal{C}}\left(l+m\mathcal{C}\right)^{2}-2\tau_{2}k_{2}(l+m\mathcal{C})}$
$\displaystyle=\sum_{m\in\mathbb{Z}}e^{-\frac{\mathcal{C}\tau_{2}}{2\pi}\left(k_{2}+\frac{2\pi}{\mathcal{C}}\left(l+m\mathcal{C}\right)\right)^{2}}.$

Thus, we conclude that

$\displaystyle\int_{0}^{2\pi}\frac{dk_{1}}{2\pi}F(k_{1},k_{2})=\sum_{l=0}^{\mathcal{C}-1}|A_{l}|^{2}\sum_{m\in\mathbb{Z}}e^{-\frac{\mathcal{C}\tau_{2}}{2\pi}\left(k_{2}+\frac{2\pi}{\mathcal{C}}\left(l+m\mathcal{C}\right)\right)^{2}},$
(43)

which is a sum of Gaussians centered at the points
$\frac{2\pi}{\mathcal{C}}l+2\pi m$, with $l=0,\dots,\mathcal{C}-1$ and
$m\in\mathbb{Z}$ (ensuring periodicity in $k_{2}$). Provided the $A_{l}$’s are
nontrivial, which needs to happen for the map $f$ to be well-defined (there
are further constraints on the $A_{l}$’s as for the map to be well-defined the
$Z_{j}$’s cannot vanish or have poles simultaneously, but this is not relevant
for the proof), this sum cannot be made constant in the case $\mathcal{C}$ is
finite and the result is, thus, proved: there are no flat Kähler bands for
finite total number of bands.

## 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).
  * Tang  _et al._ [2011] E. Tang, J.-W. Mei, and X.-G. Wen, High-temperature fractional quantum Hall states, Phys. Rev. Lett. 106, 236802 (2011).
  * 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).
  * Neupert  _et al._ [2011] T. Neupert, L. Santos, C. Chamon, and C. Mudry, Fractional quantum Hall states at zero magnetic field, Phys. Rev. Lett. 106, 236804 (2011).
  * Regnault and Bernevig [2011] N. Regnault and B. A. Bernevig, Fractional Chern insulator, Phys. Rev. X 1, 021014 (2011).
  * Sheng  _et al._ [2011] D. Sheng, Z.-C. Gu, K. Sun, and L. Sheng, Fractional quantum Hall effect in the absence of Landau levels, Nature Communications 2, 1 (2011).
  * Parameswaran  _et al._ [2013] S. A. Parameswaran, R. Roy, and S. L. Sondhi, Fractional quantum Hall physics in topological flat bands, Comptes Rendus Physique 14, 816 (2013), topological insulators / Isolants topologiques.
  * Bergholtz and Liu [2013] E. J. Bergholtz and Z. Liu, Topological flat band models and fractional Chern insulators, International Journal of Modern Physics B 27, 1330017 (2013).
  * 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).
  * Ozawa and Mera [2021] T. Ozawa and B. Mera, Relations between topology and the quantum metric for Chern insulators, Phys. Rev. B 104, 045103 (2021).
  * 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).
  * Ledwith  _et al._ [2020] P. J. Ledwith, G. Tarnopolsky, E. Khalaf, and A. Vishwanath, Fractional Chern insulator states in twisted bilayer graphene: An analytical approach, Phys. Rev. Research 2, 023237 (2020).
  * Wang  _et al._ [2021] J. Wang, J. Cano, A. J. Millis, Z. Liu, and B. Yang, Exact Landau level description of geometry and interaction in a flatband, arXiv:2105.07491 (2021).
  * Zelditch [1998] S. Zelditch, Szegö kernels and a theorem of Tian, International Mathematics Research Notices 1998, 317 (1998), https://academic.oup.com/imrn/article-pdf/1998/6/317/2123138/1998-6-317.pdf .
  * Andrews  _et al._ [2021] B. Andrews, T. Neupert, and G. Möller, Stability, phase transitions, and numerical breakdown of fractional Chern insulators in higher Chern bands of the Hofstadter model, Phys. Rev. B 104, 125107 (2021).
  * Spanton  _et al._ [2018] E. M. Spanton, A. A. Zibrov, H. Zhou, T. Taniguchi, K. Watanabe, M. P. Zaletel, and A. F. Young, Observation of fractional Chern insulators in a van der Waals heterostructure, Science 360, 62 (2018), https://www.science.org/doi/pdf/10.1126/science.aan8458 .
  * Provost and Vallee [1980] J. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Communications in Mathematical Physics 76, 289 (1980).
  * Page [1987] D. N. Page, Geometrical description of Berry’s phase, Phys. Rev. A 36, 3479 (1987).
  * Roy [2014] R. Roy, Band geometry of fractional topological insulators, Phys. Rev. B 90, 165139 (2014).
  * Note [1] In a coordinate free description, the periodic $(k_{1},k_{2})$ coordinates give rise to two independent globally defined vector fields over the Brillouin zone $X_{1}=\partial/\partial k_{1}$ and $X_{2}=\partial/\partial k_{2}$. Then, what we mean by _flatness_ of a Kähler structure over the Brillouin zone is that the Lie derivatives, with respect to these vector fields, of the symplectic form $\omega$, the complex structure $j$ and the metric $g$, denoted by, respectively, $\mathcal{L}_{X_{i}}\omega$, $\mathcal{L}_{X_{i}}j$ and $\mathcal{L}_{X_{i}}g$, $i=1,2$, vanish identically.
  * Marzari and Vanderbilt [1997] N. Marzari and D. Vanderbilt, Maximally localized generalized wannier functions for composite energy bands, Phys. Rev. B 56, 12847 (1997).
  * Souza  _et al._ [2000] I. Souza, T. Wilkens, and R. M. Martin, Polarization and localization in insulators: Generating function approach, Phys. Rev. B 62, 1666 (2000).
  * Ozawa and Goldman [2019] T. Ozawa and N. Goldman, Probing localization and quantum geometry by spectroscopy, Phys. Rev. Research 1, 032019 (2019).
  * 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).
  * 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, D. Sanvitto, and G. Malpuech, Measurement of the quantum geometric tensor and of the anomalous hall drift, Nature 578, 381 (2020).
  * Mera [2020] B. Mera, Localization anisotropy and complex geometry in two-dimensional insulators, Phys. Rev. B 101, 115128 (2020).
  * Girvin and Jach [1984] S. M. Girvin and T. Jach, Formalism for the quantum Hall effect: Hilbert space of analytic functions, Phys. Rev. B 29, 5617 (1984).
  * Zak [1968] J. Zak, Dynamics of electrons in solids in external fields, Physical Review 168, 686 (1968).
  * Jackson  _et al._ [2015] T. S. Jackson, G. Möller, and R. Roy, Geometric stability of topological lattice phases, Nature Communications 6, 10.1038/ncomms9629 (2015).
  * Miranda [1995] R. Miranda,  _Algebraic curves and Riemann surfaces_ , Vol. 5 (American Mathematical Soc., 1995).
  * Note [2] To be precise, it means that if we equip the Brillouin zone with the structure of a complex manifold, denoted $(\textnormal{\text{BZ}}^{2},j)$, with $j$ being the complex structure, there exists a biholomorphism (i.e., an isomorphism of complex manifolds) $\varphi:(\textnormal{\text{BZ}}^{2},j)\to\mathbb{C}/\left(\mathbb{Z}\oplus\tau\mathbb{Z}\right)$ such that $d\varphi\circ j=j_{\tau}\circ d\varphi,$ with $j_{\tau}$ the uniform flat complex structure associated with $\tau\in\mathcal{H}$. Note that, since $\varphi$ is in particular a diffeomorphism, we also have $j\circ d\varphi^{-1}=d\varphi^{-1}\circ j_{\tau}.$ This means that if $P:(\textnormal{\text{BZ}}^{2},j)\to\mathbb{C}P^{n-1}$ is a holomorphic map, meaning $dP\circ j=J_{FS}\circ dP,$ where $J_{FS}$ is the Fubini-Study complex structure, then, $\tilde{P}=P\circ\varphi^{-1}:\mathbb{C}/\left(\mathbb{Z}\oplus\tau\mathbb{Z}\right)\to\mathbb{C}P^{n-1}$ is holomorphic with respect to the flat complex structure $j_{\tau}$ defined by $\tau\in\mathcal{H}$, i.e.,  $\displaystyle d\tilde{P}\circ j_{\tau}$ $\displaystyle=dP\circ d\varphi^{-1}\circ j_{\tau}=dP\circ j\circ d\varphi^{-1}$ $\displaystyle=J_{FS}\circ dP\circ d\varphi^{-1}=J_{FS}\circ d\tilde{P},$ where we used the chain rule for differentials $d\tilde{P}=d(P\circ\varphi^{-1})=dP\circ d\varphi^{-1}$. Now the projector $\tilde{P}$ will have a description in terms of meromorphic functions on a complex torus parametrized by a complex coordinate $z=k_{1}+\tau k_{2}$ as described in the main text.
  * Kovrizhin  _et al._ [2013] D. L. Kovrizhin, B. Douçot, and R. Moessner, Multicomponent skyrmion lattices and their excitations, Phys. Rev. Lett. 110, 186802 (2013).
  * Woodhouse [1997] N. M. J. Woodhouse,  _Geometric quantization_ (Oxford university press, 1997).
  * Hall [2013] B. C. Hall,  _Quantum theory for mathematicians_ , Vol. 267 (Springer, 2013).
  * Klevtsov [2016] S. Klevtsov, Geometry and large N limits in Laughlin states (2016), arXiv:1608.02928 [cond-mat.str-el] .
  * Chern [1967] S.-S. Chern,  _Complex manifolds without potential theory_ , 15 (Springer, 1967).
  * Kobayashi [2014] S. Kobayashi,  _Differential geometry of complex vector bundles_ (Princeton University Press, 2014).
  * Huybrechts [2005] D. Huybrechts,  _Complex geometry: an introduction_ (Springer Science & Business Media, 2005).
  * Mumford  _et al._ [1974] D. Mumford, C. P. Ramanujam, and I. I. Manin,  _Abelian varieties_ , Vol. 2 (Oxford university press Oxford, 1974).
  * Douglas and Klevtsov [2010] M. R. Douglas and S. Klevtsov, Bergman kernel from path integral, Communications in Mathematical Physics 293, 205 (2010).
  * Onofri [2001] E. Onofri, Landau levels on a torus, International Journal of Theoretical Physics 40, 537 (2001).
  * Chen  _et al._ [2014] L. Chen, T. Mazaheri, A. Seidel, and X. Tang, The impossibility of exactly flat non-trivial Chern bands in strictly local periodic tight binding models, Journal of Physics A: Mathematical and Theoretical 47, 152001 (2014).
  * Varjas  _et al._ [2021] D. Varjas, A. Abouelkomsan, K. Yang, and E. J. Bergholtz, Topological lattice models with constant Berry curvature, arXiv:2107.06902 (2021).