# Topological bound on structure factor

Yugo Onishi  Department of Physics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  Liang Fu  Department of Physics, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA

###### Abstract

We show that the static structure factor of general many-body systems with
$U(1)$ symmetry has a lower bound determined only by the ground state Chern
number. Our bound relies only on causality and non-negative energy
dissipation, and holds for a wide range of systems. We apply our theory to
(fractional) Chern insulators, (fractional) quantum spin Hall insulators,
topological superconductors, and chiral spin liquids. Our results uncover a
universal feature of topological phases beyond the quantized response.

It is well known that topological invariants of ground states often govern
quantized physical responses of quantum many-body systems: for example, the
many-body Chern number determines the quantized (anomalous) Hall conductivity
[1, 2, 3, 4, 5, 6, 7]. Recently, it has been shown [8] that the ground-state
topology not only dictates the quantization of certain observables, but also
sets universal bounds on other physical quantities. Specifically, we found
that the topology of (fractional) Chern insulators sets an upper bound on the
energy gap, which is determined only by the electron density, mass and the
many-body Chern number. This topological bound on the energy gap reveals new
information about Chern insulators in general. Interestingly, this bound is
found to be saturated for Landau level systems and fairly tight for the zero-
field Chern insulator in twisted semiconductor bilayers near the magic angle
[9].

In this work, we uncover a universal bound on the structure factor that
applies to general chiral topological phases with a quantized Hall response.
We show that the static structure factor — the equal-time density-density
correlation function — at long wavelengths has a lower bound solely determined
by the many-body Chern number, which governs the Hall response. We discuss
applications of our theory to various chiral topological phases, including
integer and fractional Chern insulators, topological superconductors, and
quantum spin liquids. Our topological bound on structure factor is shown to be
useful for Chern insulators in twisted semiconductor bilayers. It also
provides valuable guidance for identifying topological superconductors and
quantum spin liquids from the spin structure factor.

Our finding is derived by considering the response of the system to an
external $U(1)$ gauge field, and relies only on the fundamental principle that
the energy absorption is always non-negative; this leads to a nontrivial
relation between the quantized Hall response and the longitudinal responses.
Further relating the longitudinal response to the structure factor by the
dissipation-fluctuation theorem, we find a topological bound on the structure
factor.

General theory — Consider a two-dimensional (2D) system with a global $U(1)$
symmetry, with corresponding $U(1)$ charge and current densities denoted as
$\rho,{\bf\it j}$. For example, $\rho$ and ${\bf\it j}$ can represent the
electric charge and current, or spin density and spin current in systems with
spin $U(1)$ symmetry. The static structure factor, i.e., the equal-time
density-density correlation function, is defined as:

$\displaystyle S_{{\bf\it q}}$
$\displaystyle=\frac{1}{V}\expectationvalue{\rho_{{\bf\it q}}{\rho}_{-{\bf\it
q}}}$ (1)

with $\rho_{{\bf\it q}}=\int\differential{{\bf\it r}}e^{-i{\bf\it
q}\dotproduct{{\bf\it r}}}\rho({\bf\it r})$ the Fourier transform of the
charge density and $V$ the volume. In our convention, $S_{{\bf\it
q}=0}=Vn^{2}$ with total charge density $n$. By the dissipation-fluctuation
theorem, the static structure factor $S_{{\bf\it q}}$ is related to the
density response function $\Pi({\bf\it q},\omega)$, which describes the
density response to a scalar potential $\phi$, $\delta\rho({\bf\it
q},\omega)=\Pi({\bf\it q},\omega)\phi({\bf\it q},\omega)$:

$\displaystyle-\int_{0}^{\infty}\differential\omega\imaginary\Pi({\bf\it
q},\omega)=\frac{\pi}{\hbar}S_{{\bf\it q}}.$ (2)

By the continuity equation $\partial\rho/\partial t+\nabla\cdot j=0$, the
density response function $\Pi$ can be further related to the current response
to the (longitudinal) “electric” field associated with the potential $\phi$,
${\bf\it\mathcal{E}}=-\nabla\phi$. Denoting the conductivity as $\sigma$,
$\Pi$ can be written as

$\displaystyle\Pi({\bf\it
q},\omega)=-i\frac{q_{\alpha}q_{\beta}\sigma_{\alpha\beta}({\bf\it
q},\omega)}{\omega}.$ (3)

Thus combining Eq. (2) and (3), we find a relation:

$\displaystyle
q_{\alpha}q_{\beta}\int_{0}^{\infty}\differential{\omega}\frac{\real\sigma_{\alpha\beta}({\bf\it
q},\omega)}{\omega}=\frac{\pi}{\hbar}S_{{\bf\it q}}.$ (4)

Eq. (4) is a sum rule that relates the negative-first moment of optical
conductivity and static structure factor. Note that the left hand side of Eq.
(4) only involves the real and symmetric part of $\sigma_{\alpha\beta}$, which
describes the absorption of linearly polarized light. In this work, we
consider 2D systems with an energy gap, so that optical absorption vanishes at
low frequency. This allows us to take ${\bf\it q}\rightarrow 0$ limit for
$\sigma({\bf\it q},\omega)$ and obtain the small-$q$ expansion for $S_{{\bf\it
q}}$:

$\displaystyle S_{{\bf\it q}}$ $\displaystyle=$
$\displaystyle\frac{K_{\alpha\beta}}{2\pi}q_{\alpha}q_{\beta}+...,$ (5)
$\displaystyle\text{with}\quad K_{\alpha\beta}$ $\displaystyle=$
$\displaystyle
2\hbar\int_{0}^{\infty}\differential{\omega}\frac{\real\sigma_{\alpha\beta}(\omega)}{\omega}.$
(6)

Eq. (6) shows that the static structure factor of gapped phases vanishes
quadratically at small ${\bf\it q}$. The quadratic coefficient $K$ (recently
termed quantum weight) characterizes long-wavelength density fluctuation in
the ground state. Eq. (6) directly relates the quantum weight of a 2D gapped
system to the negative-first moment of its optical conductivity [10].

We now show that the quantum weight has a universal lower bound determined by
the quantized dc Hall conductivity, or the many-body Chern number. To derive
this bound, consider the absorbed power of a 2D system from a monochromatic
external field
${\bf\it\mathcal{E}}(t)={\bf\it\mathcal{E}}_{0}(\omega)e^{-i\omega
t}+\mathrm{c.c.}$, which is given by

$\displaystyle P$ $\displaystyle=\frac{1}{T}\int\differential{t}{\bf\it
j}\cdot{\bf\it\mathcal{E}}={\bf\it\mathcal{E}}_{0}(\omega)^{*}\dotproduct{\bf\it
j}(\omega)+\mathrm{c.c.}$
$\displaystyle={\bf\it\mathcal{E}}_{0}^{\dagger}(\sigma(\omega)+\sigma^{\dagger}(\omega)){\bf\it\mathcal{E}}_{0},$
(7)

with $T=2\pi/\omega$ the period of $\mathcal{E}(t)$. Importantly, the absorbed
power $P$ is non-negative for arbitrary $\omega$ and
${\bf\it\mathcal{E}}_{0}$. In particular, for circularly polarized “light”
with $x$ and $y$ components equal in magnitude and $90^{\circ}$ out of phase,
${\bf\it\mathcal{E}}_{0}=(1,\pm i)$, the non-negativity of absorbed power
$P\geq 0$ leads to the following inequality [8]:

$\displaystyle\real\sigma_{xx}(\omega)\geq\absolutevalue{\imaginary\sigma_{xy}(\omega)},$
(8)

which holds at an arbitrary frequency $\omega$.

From Eq. (8) we immediately obtain inequalities between all moments of
$\real\sigma_{xx}(\omega)$ and $\imaginary\sigma_{xy}(\omega)$:

$\displaystyle\int_{0}^{\infty}\differential{\omega}\frac{\real\sigma_{xx}(\omega)}{\omega^{n}}\geq\absolutevalue{\int_{0}^{\infty}\differential{\omega}\frac{\imaginary\sigma_{xy}(\omega)}{\omega^{n}}}.$
(9)

For gapped systems, since there is no absorption at frequency below the gap,
the frequency integral in Eq. (9) does not have infrared divergence for any
$n$. Consider the negative-first moments ($n=1$) of $\real\sigma_{xx}$ and
$\imaginary\sigma_{xy}$: the former is directly related to the quantum weight
by Eq. (6), while the latter is related to the dc Hall conductivity by the
Kramers-Kronig relation due to the causality of the response:

$\displaystyle\int_{0}^{\infty}\differential{\omega}\frac{\imaginary\sigma_{xy}(\omega)}{\omega}=\frac{\pi}{2}\sigma_{xy}(0)=\frac{1}{4\hbar}C.$
(10)

Here, we have used the fact that the dc Hall conductivity of a 2D gapped
system is a topological invariant—the many-body Chern number $C$, which may
take integer or fractionally quantized values [2, 3]: $\sigma_{xy}=C/h$.

Combining Eq. (9) with Eqs. (10) and (6), we obtain a topological bound
$K\equiv K_{xx}+K_{yy}\geq\absolutevalue{C}$, or equivalently,

$\displaystyle S_{{\bf\it q}}\geq\frac{\hbar
q^{2}}{2}\absolutevalue{\sigma_{xy}}=\frac{q^{2}}{4\pi}\absolutevalue{C}\quad({\bf\it
q}\to 0).$ (11)

Eq. (11) applies to general gapped 2D systems with $U(1)$ symmetry, including
strongly interacting systems such as fractional Chern insulators. The same
result holds for lattice models where $\rho_{{\bf\it q}}=\sum_{i}e^{i{\bf\it
q}\dotproduct{\bf\it r}_{i}}\rho_{i}$ is defined with the density operator
$\rho_{i}$ on site $i$. Below, we will apply Eq. (11) to various systems:
Chern insulators, topological superconductors and chiral spin liquids.

Chern insulators —  Let us first consider Chern insulators, where the $U(1)$
symmetry corresponds to electric charge conservation. In this case, $\rho$ and
${\bf\it j}$ correspond to the electron number density and current
respectively. The electric conductivity $\sigma^{e}$ is related to the
conductivity $\sigma$ in our general theory as $\sigma^{e}=e^{2}\sigma$, and
the many-body Chern number $C$ governs the quantized electrical Hall
conductivity $\sigma^{e}_{xy}=Ce^{2}/h$. Eq. (11) imposes a lower bound on the
charge structure factor for Chern insulators: $S_{{\bf\it q}}\geq
q^{2}\absolutevalue{C}/(4\pi)$ at small $q$.

For noninteracting insulators, the inequality (11) can be explicitly verified.
The structure factor is given by

$\displaystyle S_{{\bf\it q}}$ $\displaystyle=\int_{\rm
BZ}\frac{\differential^{d}{{\bf\it k}}}{(2\pi)^{d}}\Tr[P({\bf\it k})(P({\bf\it
k})-P({\bf\it k}+{\bf\it q}))]$ (12)
$\displaystyle=\frac{q_{\alpha}q_{\beta}}{2\pi}\int_{\rm
BZ}\frac{\differential^{2}{{\bf\it k}}}{2\pi}g_{\alpha\beta}({\bf\it
k})\quad({\bf\it q}\to 0),$ (13)

where $P({\bf\it k})=\sum_{n}^{\rm occ}\ket{u_{n{\bf\it k}}}\bra{u_{n{\bf\it
k}}}$ is the projection operator onto the occupied bands at wavevector
${\bf\it k}$ with $\ket{u_{n{\bf\it k}}}$ the cell-periodic Bloch wavefunction
for $n$-th band. $g_{\mu\nu}({\bf\it k})=\real Q_{\mu\nu}({\bf\it k})$ is the
quantum metric with $Q_{\mu\nu}({\bf\it k})\equiv\sum_{n}^{\rm
occ}\matrixelement{\partial_{\mu}u_{n{\bf\it k}}}{(1-P({\bf\it
k}))}{\partial_{\nu}u_{n{\bf\it k}}}$ the quantum geometric tensor of the
occupied Bloch bands. Eq. (13) shows that the quadratic coefficient of
structure factor of band insulators, $K=K_{xx}+K_{yy}$, reduces to the
integral of the quantum metric over the Brillouin zone. Because
$g_{xx}+g_{yy}\geq\absolutevalue{\Omega_{xy}}$ with the Berry curvature
$\Omega_{\mu\nu}=-2\imaginary Q_{\mu\nu}$ [11, 12], $K\geq\absolutevalue{C}$
follows, consistent with our general result (11). This inequality is related
to the fact that the Wannier functions in Chern bands cannot be exponentially
localized [13] and must have a minimum spread [14, 12].

To illustrate our topological bound on structure factor, we consider a two-
band $k\cdot p$ model described by the following Hamiltonian:

$\displaystyle H=\sum_{{\bf\it k}}\begin{pmatrix}{c}^{\dagger}_{{\bf\it
k}\uparrow}&{c}^{\dagger}_{{\bf\it k}\downarrow}\end{pmatrix}H({\bf\it
k})\begin{pmatrix}{c}_{{\bf\it k}\uparrow}\\\ {c}_{{\bf\it
k}\downarrow}\end{pmatrix},$ (14) $\displaystyle H({\bf\it
k})=\begin{pmatrix}\xi({\bf\it k})&\Delta({\bf\it k})\\\ \Delta({\bf\it
k})^{*}&-\xi({\bf\it k})\end{pmatrix},$ (15)

where ${c}_{{\bf\it k}\uparrow(\downarrow)}$ is the annihilation operator of
an electron with wavevector ${\bf\it k}$ and spin up (down) and $\xi({\bf\it
k})=k^{2}/(2m)-\mu$. The hybridization term is either $s$-wave,
$\Delta({\bf\it k})=\Delta_{s}$, or $p$-wave, $\Delta({\bf\it
k})=\Delta_{p}(k_{x}-ik_{y})$. For the $p$-wave case, the system becomes a
Chern insulator with $\absolutevalue{C}=1$ when $\mu>0$ and is trivial for
$\mu<0$. At $\mu=0$, the system becomes gapless and undergoes the topological
phase transition.

Figure 1: (a, b) The structure factor $S(q_{x},0)$ for the Hamiltonian (14)
when $\Delta({\bf\it k})$ is (a) $s$-wave and (b) $p$-wave. The dashed lines
are the quadratic curves determined by the quantum weight. (c, d) The quantum
weight $K$ as a function of $\mu$ for (c) $s$-wave and (d) $p$-wave. Here,
$k_{s}=(2m\Delta_{s})^{1/2}$ and $k_{p}=2m\Delta_{p}$. The results are
identical to the spin structure factor for the superconductors described by
(18).

The structure factor $S_{{\bf\it q}}$ and the quantum weight $K$ for both
$s$-wave and $p$-wave $\Delta({\bf\it k})$ for several $\mu$ are shown in Fig.
1. In all the cases except at the gapless point $\mu=0$ in the $p$-wave case,
the structure factor is quadratic in $q$ near $q=0$, with the coefficient
given by the quantum weight $K$ (dashed line in the insets). In $s$-wave
cases, $K$ continuously decreases as $\mu$ is reduced, and vanishes as
$\mu\rightarrow-\infty$.

In contrast, for $p$-wave case, the quantum weight $K$ in the topological
phase ($\mu>0$) is always larger than or equal to $\absolutevalue{C}=1$, being
consistent with our general result (11). Interestingly, when
$\mu=m\Delta_{p}^{2}/2$, the energy dispersion is parabolic as
$\varepsilon_{\pm}({\bf\it k})=\pm\quantity(k^{2}/(2m)+\mu)$ and the quantum
weight $K$ is equal to $1$ [15, 16], saturating its topological bound. At the
topological phase transition $\mu=0$, $K$ has a logarithmic divergence
$\sim\log\absolutevalue{\mu}$ as previously shown [8]. See Supplemental
Materials for more details.

The topological bound on charge structure factor is also saturated for a
particular type of Chern insulators, quantum Hall states in Landau level
systems. For 2D electron systems with parabolic dispersion $p^{2}/(2m)$ under
a magnetic field $B$, the static structure factor at long wavelength is given
by

$\displaystyle S^{\rm LL}_{{\bf\it q}}=\nu q^{2}/(4\pi)+\dots.$ (16)

where $\nu=n/(2\pi l^{2})$ is the Landau level filling factor with $n$ the
carrier density and $l=\sqrt{\hbar/(eB)}$ is the magnetic length. This follows
from the fact that the optical spectral weight is exhausted by Kohn’s mode at
cyclotron frequency $\omega_{c}=eB/m$ [17, 18, 19] and the relation between
structure factor and optical conductivity (6). The Galilean invariance also
dictates the quantized Hall conductivity $|C|=\nu$. Importantly, Eq. (16) only
relies on Kohn’s theorem and therefore holds for both integer and fractional
quantum Hall states. Therefore, quantum Hall states in Landau level systems
saturate our general bound on the structure factor. Note that the structure
factor $S^{\rm LL}_{\bm{q}}$ is different from the guiding center structure
factor $\bar{S}_{\bm{q}}$ or the Landau-level-projected density-density
correlation function; the latter is proportional to $q^{4}$ [18].

Figure 2: (a) Quantum weight for $t$MoTe2 and $t$WSe2 at filling factor
$\nu=-1$. (b) The static structure factor $S_{q}$ for $t$MoTe2 at
$\theta=$$,$$$ and the lowest Landau level. $S_{q}$ is scaled with the unit
cell area $A_{UC}=\sqrt{3}a^{2}/2$. For $t$MoTe2, $a$ is moiré lattice
constant $a_{M}$. For Landau level, $a$ is defined so that
$A_{UC}=\sqrt{3}a^{2}/2$. See Supplemental Materials for details.

Chern insulators in twisted transition metal dichalcogenides provide an
excellent platform to test our theory. Following theoretical proposals [9, 20,
21], integer and fractional quantum anomalous Hall effect have been
experimentally observed in twisted MoTe2 ($t$MoTe2) and WSe2 ($t$WSe2) [5, 22,
6, 23, 24]. Our general bound (11) applies to both integer and fractional
Chern insulators in these materials. Here, we explicitly calculate quantum
weight and structure factor of the integer Chern insulator at $\nu=1$ where
charge carriers completely fill the lowest Chern band of one spin [25, 9, 26]
(assuming that band mixing is negligible). The results show the inequality
(11) always holds and is fairly tight: $K=1.07$ at $\theta=$$$ for $t$MoTe2,
and $K=1.04$ at $\theta=$$$ for $t$WSe2, being close to the bound
$\absolutevalue{C}=1$. Interestingly, the structure factor of $t$MoTe2 is
nearly isotropic and close to that of the lowest Landau level, especially at
$\theta=$$$, indicating that the system is close to the lowest Landau level.

Quantum spin Hall insulators — The topological bound for Chern insulators can
be straightforwardly extended to quantum spin Hall insulators. In time-
reversal invariant systems with spin $s^{z}$ conservation, the Chern number of
all the occupied bands must be zero, but occupied spin-$\uparrow$ and
spin-$\downarrow$ bands can have equal and opposite Chern numbers:
$C_{\uparrow}=-C_{\downarrow}\equiv C_{s}$. Correspondingly, both the static
structure factor for spin-$\uparrow$ and spin-$\downarrow$ have a lower bound
of $\absolutevalue{C_{s}}$, and thus the total static structure factor is
bounded by $2\absolutevalue{C_{s}}$: $S_{{\bf\it q}}\geq
q^{2}\absolutevalue{C_{s}}/(2\pi)$.

Spin $U(1)$ symmetry —We can also apply our general theory to systems with
spin $U(1)$ symmetry. When the total spin along $z$ direction $s^{z}$ is
conserved, we can relate $\rho$ and ${\bf\it j}$ to the spin density
$s^{z}({\bf\it r})$ and the spin current ${\bf\it j}^{s}({\bf\it r})$ as
$s^{z}({\bf\it r})\equiv s_{0}\rho({\bf\it r})$ and ${\bf\it j}^{s}({\bf\it
r})\equiv s_{0}{\bf\it j}({\bf\it r})$. Here, $s_{0}$ has a dimension of spin
angular momentum. The potential $\phi$ in this case corresponds to the Zeeman
field, $-s_{0}B_{z}({\bf\it r})$, which couples to the spin density as
$\rho\phi=-B_{z}s^{z}$. The corresponding “electric” field is the gradient of
the magnetic field, ${\bf\it\mathcal{E}}=s_{0}\nabla B_{z}$. By defining the
spin conductivity as ${\bf\it j}^{s}=\sigma^{s}\nabla B_{z}$, the conductivity
in our general theory $\sigma$ is related to $\sigma^{s}$ as
$\sigma^{s}=s_{0}^{2}\sigma$, while the static structure factor is related to
the spin static structure factor $S^{s}_{{\bf\it
q}}\equiv\expectationvalue{\hat{s}^{z}_{{\bf\it q}}\hat{s}^{z}_{-{\bf\it
q}}}/V$ as $S^{s}_{{\bf\it q}}=s_{0}^{2}S_{{\bf\it q}}$.

For systems with a spin gap, we can apply our general theory to obtain a bound
on the spin static structure factor in terms of the Hall conductivity for
spin:

$\displaystyle S^{s}_{{\bf\it q}}$ $\displaystyle\geq\frac{\hbar
q^{2}}{2}\absolutevalue{\sigma^{s}_{xy}}=\frac{q^{2}s_{0}^{2}}{4\pi}\absolutevalue{C}\quad({\bf\it
q}\to 0).$ (17)

In the following, we examine two examples with the spin $U(1)$ symmetry :
chiral topological superconductors and chiral spin liquids.

Chiral topological superconductors — For superconductors with a full
superconducting gap and spin $U(1)$ symmetry, the spin conductivity vanishes
at frequencies below the gap. This allows us to apply our theory to the spin
structure factor.

As an example, we consider a superconductor with $s_{z}$ conservation
described by the following mean-field Hamiltonian:

$\displaystyle H$ $\displaystyle=\sum_{{\bf\it
k}}\begin{pmatrix}{c}^{\dagger}_{{\bf\it k}\uparrow}&{c}_{-{\bf\it
k}\downarrow}\end{pmatrix}H({\bf\it k})\begin{pmatrix}{c}_{{\bf\it
k}\uparrow}\\\ {c}^{\dagger}_{-{\bf\it
k}\downarrow}\end{pmatrix}\equiv\sum_{{\bf\it k}}{\Psi}^{\dagger}_{{\bf\it
k}}H({\bf\it k}){\Psi}_{{\bf\it k}}.$ (18)

where $H({\bf\it k})$ takes the same form as the two-band model (14) for
insulators. As a mean-field Hamiltonian for superconductors, $\Delta({\bf\it
k})$ represents the pairing function; $\Delta({\bf\it k})=\Delta_{s}$
describes a $s$-wave pairing while $\Delta({\bf\it
k})=\Delta_{p}(k_{x}-ik_{y})$ describes a $p$-wave pairing [27]. The $p$-wave
pairing considered here is a spin triplet with zero spin angular momentum in
$z$-direction. The parameter $\mu$ describes the evolution of the
superconducting state from BCS ($\mu>0$) to BEC ($\mu<0$) regime.

Since the Hamiltonian conserves $U(1)$ charge ${\Psi}^{\dagger}{\Psi}$, we can
apply our theory to the structure factor for the density
${\Psi}^{\dagger}{\Psi}$. Importantly, ${\Psi}^{\dagger}{\Psi}$ corresponds to
the spin density $s^{z}$, the structure factor corresponds to the spin
structure factor, and Eq. (17) holds with $s_{0}=\hbar/2$. Therefore, the spin
structure factor of $s_{z}$-conserving superconductors is (up to a factor of
$s_{0}^{2}$) identical to the charge structure factor for insulators, which is
shown in Fig. 1. For the chiral $p$-wave topological superconductor, the spin
structure factor and the corresponding quantum weight have a lower bound by
the Chern number. As another example, we discuss $d+id$-wave spin-singlet
superconductors on a lattice in Supplemental Materials.

Chiral spin liquids — Our theory also applies to chiral spin liquids with spin
$U(1)$ symmetry [28]. For example, consider a spin-$1/2$ Heisenberg model on a
lattice with total spin $s^{z}$ conserved. The $U(1)$ symmetry becomes more
clear when the model is mapped to a tight-binding model of hard-core bosons: a
spin-up state at site $i$ is represented by one boson at site $i$, with an
infinite onsite repulsion preventing more than one bosons on the same site.
Formally, this mapping is achieved by rewriting the spin operator ${\bf\it
s}_{i}=(\hbar/2){\bf\it\sigma}$ with
${a}_{i}\equiv\quantity(s^{x}_{i}-is^{y}_{i})/\hbar$. The total number of
bosons is conserved due to the spin $U(1)$ symmetry, i.e., total $s^{z}$
conservation.

The $U(1)$ gauge field couples to the density
$\rho_{i}={a}^{\dagger}_{i}{a}_{i}$ and the corresponding current ${\bf\it
j}$, determined by the derivative of the Hamiltonian with respect to the gauge
field. Since one boson carries spin of $\hbar$, the natural choice for the
coupling constant is $s_{0}=\hbar$, and the spin density and the spin current
are related to the density and the current of the bosons as
$s^{z}_{i}=\hbar(\rho_{i}-1/2),{\bf\it j}^{s}=\hbar{\bf\it j}$.

With these, we can apply the topological bound (17). The quantum weight $K$
and the Chern number $C$ are related to the spin structure factor and the spin
conductivity as $S^{s}_{{\bf\it q}}=\hbar^{2}q^{2}K/(4\pi)$,
$\sigma^{s}_{xy}=\hbar C/(2\pi)$, and the topological bound (17) reduces to
the following relation between $S^{s}_{{\bf\it q}}$ and the Chern number $C$:

$\displaystyle S^{s}_{{\bf\it q}}$
$\displaystyle\geq\frac{\hbar^{2}}{4\pi}q^{2}\absolutevalue{C}\quad({\bf\it
q}\to 0)$ (19)

This analysis is readily generalized to spin-$S$ systems.

Discussion — While our work focuses on the topological bound on the structure
factor which is a ground state property, it is also closely related to the
topological bound on the energy gap shown recently [8]. In fact, the upper
bound on the energy gap $\Delta$ can be obtained from our lower bound on the
structure factor by using the Feynmann-Bijl formula [29]:

$\displaystyle\Delta\leq\frac{\expectationvalue{\commutator{\rho_{{\bf\it
q}}}{\commutator{H}{\rho_{-{\bf\it q}}}}}}{2VS_{{\bf\it q}}},$ (20)

where $\expectationvalue{\dots}$ denotes the expectation value in the ground
state. The derivation of Eq. (20) for general $U(1)$ symmetric systems is
given in Supplemental Materials. Using the continuity equation
$\dot{\rho}_{{\bf\it q}}+i{\bf\it q}\dotproduct{\bf\it j}_{{\bf\it q}}=0$ and
the relation ${\bf\it j}_{{\bf\it q}}=-\partialderivative*{H}{{\bf\it
A}_{-{\bf\it q}}}$, we can rewrite Eq. (20) as

$\displaystyle\Delta\leq\frac{\hbar^{2}q_{\mu}q_{\nu}}{2VS_{{\bf\it
q}}}\expectationvalue{\partialderivative{H}{A^{\mu}_{{\bf\it
q}}}{A^{\nu}_{-{\bf\it q}}}},$ (21)

By taking ${\bf\it q}\to 0$ limit and applying the topological bound on the
structure factor (11), we find

$\displaystyle\Delta$
$\displaystyle\leq\frac{\pi\hbar^{2}}{K}\expectationvalue{\frac{1}{V}\partialderivative[2]{H}{{\bf\it
A}}}\leq\frac{\pi\hbar^{2}}{\absolutevalue{C}}\expectationvalue{\frac{1}{V}\partialderivative[2]{H}{{\bf\it
A}}},$ (22)

where ${\bf\it A}$ is the uniform $U(1)$ vector potential. Eq. (22) is the
topological gap bound that applies to general topological phases with a finite
Chern number. In particular, when the Hamiltonian is quadratic in the momentum
${\bf\it p}$ of the particles, $H=\sum_{i}({\bf\it p}_{i}-{\bf\it
A})^{2}/(2m)+\dots$, Eq. (22) reduces to the previously derived topological
gap bound for Chern insulators: $\Delta\leq
2\pi\hbar^{2}n/(m\absolutevalue{C})$ with the charge density $n$. It is worth
noting that Eq. (22) also applies to systems with spin-$U(1)$ symmetry as we
discussed in this work. For $s_{z}$-conserving superconductors, the gap
$\Delta$ is the spin gap, i.e., the energy of spin excitation at long
wavelength ${\bf\it q}\rightarrow 0$. The upper bound on the spin gap of
topological superconductors also gives an upper bound on the superconducting
gap.

Since the structure factor represents density fluctuations, the existence of a
lower bound shows that nontrivial topology necessarily requires quantum
fluctuation in the $U(1)$ charge density. The static structure factor is a
physical observable for all systems with $U(1)$ symmetry and can be measured
by X-ray or neutron scattering, or determined by the optical conductivity
measurement through the sum rule (4). Since our bound applies to interacting
systems, it will be interesting to investigate fractional Chern insulators in
twisted semiconductor bilayers and multilayer graphene/h-BN numerically and
experimentally.

Our topological bound on the structure factor may be useful in identifying
topological superconductors and quantum spin liquids in experiments. If the
spin structure factor is smaller than the lower bound (19) determined by a
Chern number $C$, we can rule out the possibility of a chiral spin liquid with
the Chern number $C$. A large spin structure factor at small ${\bf\it q}$ is a
necessary criterion for a system to be considered as a quantum spin liquid
candidate.

While this work focuses on the bound in topological phases characterized by
the Chern number, an interesting research direction is to find the topological
bound in other topological phases of matter.

In conclusion, we have proved a topological bound on static structure factor
in general systems with $U(1)$ symmetry. Our derivation relies only on basic
physical principles and holds for a broad range of systems, such as integer or
fractional Chern insulators, topological superconductors, and quantum spin
liquid, revealing universal information about correlation function obtained
from topology.

###### Acknowledgements.

This work was supported by the U.S. Army Research Laboratory and the U.S. Army
Research Office through the Institute for Soldier Nanotechnologies, under
Collaborative Agreement Number W911NF-18-2-0048. YO is grateful for the
support provided by the Funai Overseas Scholarship. LF was partly supported by
a Simons Investigator Award from the Simons Foundation.

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##  Appendix A Supplemental Materials

##  Appendix B Calculation details for model of Chern insulators and
topological superconductors

Here we present details of the calculation for the model discussed in the main
text:

$\displaystyle H$ $\displaystyle=\sum_{{\bf\it
k}}\begin{pmatrix}{c}^{\dagger}_{{\bf\it k}\uparrow}&{c}^{\dagger}_{{\bf\it
k}\downarrow}\end{pmatrix}H({\bf\it k})\begin{pmatrix}{c}_{{\bf\it
k}\uparrow}\\\ {c}_{{\bf\it k}\downarrow}\end{pmatrix},$ (23) $\displaystyle
H({\bf\it k})$ $\displaystyle=\begin{pmatrix}\xi({\bf\it k})&\Delta({\bf\it
k})\\\ \Delta({\bf\it k})^{*}&-\xi({\bf\it k})\end{pmatrix},$ (24)

where $\xi({\bf\it k})=\hbar^{2}k^{2}/(2m)-\mu$. If the annihilation and
creation operators are appropriately replaced, this Hamiltonian will be a
model for superconductors as well. When $\Delta({\bf\it k})=\Delta_{s}$
(constant), the system is topologically trivial, while the system can be
topological with Chern number $\absolutevalue{C}=1$ for $\Delta({\bf\it
k})=\Delta_{p}(k_{x}-ik_{y})$. We shall refer to the former case as $s$-wave
and the latter as $p$-wave.

The quantum weight of this system can be calculated with the quantum geometric
tensor. Here we first present the general expression for the quantum geometric
tensor for the two-band systems, and then we apply it to the specific case of
the $s$-wave and $p$-wave.

###  B.1 General expression of the quantum geometric tensor for two-band
systems

Consider general Hamiltonian given by

$\displaystyle H({\bf\it k})={\bf\it d}({\bf\it k})\dotproduct{\bf\it\sigma},$
(25)

with ${\bf\it\sigma}$ is the vector of Pauli matrices. The energy dispersion
is given by

$\displaystyle\varepsilon_{\pm}({\bf\it k})$
$\displaystyle=\pm\absolutevalue{{\bf\it d}({\bf\it k})}$ (26)

The Bloch wavefunction $\ket{u_{{\bf\it k},\pm}}$ is determined by a unit
vector ${\bf\it n}({\bf\it k})={\bf\it d}({\bf\it k})/\absolutevalue{{\bf\it
d}}({\bf\it k})=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ with
$\theta$ and $\phi$ functions of ${\bf\it k}$, and is given by

$\displaystyle\ket{u_{{\bf\it k},+}}=\begin{pmatrix}\cos(\theta/2)\\\
e^{i\phi}\sin(\theta/2)\end{pmatrix}$ (27) $\displaystyle\ket{u_{{\bf\it
k},-}}=\begin{pmatrix}\sin(\theta/2)\\\ -e^{i\phi}\cos(\theta/2)\end{pmatrix}$
(28)

The quantum geometric tensor
$Q_{\mu\nu}^{\pm}=g_{\mu\nu}\mp(i/2)\Omega_{\mu\nu}$ for each band $(\pm)$ is
then given by

$\displaystyle g_{\mu\nu}$
$\displaystyle=\partialderivative{\theta}{k_{\mu}}\partialderivative{\theta}{k_{\nu}}g_{\theta\theta}+\partialderivative{\phi}{k_{\mu}}\partialderivative{\phi}{k_{\nu}}g_{\phi\phi}$
$\displaystyle=\frac{1}{4}\partialderivative{\theta}{k_{\mu}}\partialderivative{\theta}{k_{\nu}}+\frac{\sin^{2}\theta}{4}\partialderivative{\phi}{k_{\mu}}\partialderivative{\phi}{k_{\nu}}$
(29) $\displaystyle\Omega_{\mu\nu}$
$\displaystyle=\quantity(\partialderivative{\theta}{k_{\mu}}\partialderivative{\phi}{k_{\nu}}-\partialderivative{\phi}{k_{\mu}}\partialderivative{\theta}{k_{\nu}})\Omega_{\theta\phi}$
$\displaystyle=\frac{\sin\theta}{2}\quantity(\partialderivative{\theta}{k_{\mu}}\partialderivative{\phi}{k_{\nu}}-\partialderivative{\phi}{k_{\mu}}\partialderivative{\theta}{k_{\nu}})$
(30)

Now let us apply these formulas to the $s$-wave case and $p$-wave case of the
Hamiltonian (23).

###  B.2 $s$-wave case

The Hamiltonian for $s$-wave can be written as $H({\bf\it
k})=\xi_{k}\sigma_{z}+\Delta_{s}\sigma_{x}$ with $\sigma_{\mu}$ is the Pauli
matrices. Thus the energy dispersion is given by

$\displaystyle\varepsilon_{\pm}({\bf\it k})$
$\displaystyle=\pm\sqrt{\xi_{k}^{2}+\Delta_{s}^{2}}\equiv\pm\varepsilon({\bf\it
k}).$ (31)

The Bloch wavefunction $\ket{u_{{\bf\it k},\pm}}$ is given by

$\displaystyle\ket{u_{{\bf\it k},+}}=\begin{pmatrix}\cos(\theta_{k}/2)\\\
\sin(\theta_{k}/2)\end{pmatrix}$ (32) $\displaystyle\ket{u_{{\bf\it
k},-}}=\begin{pmatrix}\sin(\theta_{k}/2)\\\ -\cos(\theta_{k}/2)\end{pmatrix}$
(33)

where $\theta_{k}$ is determined so that it satisfies

$\displaystyle\cos\theta_{k}=\xi_{k}/\varepsilon({\bf\it k}),$ (34)
$\displaystyle\sin\theta_{k}=\Delta_{s}/\varepsilon({\bf\it k}).$ (35)

The quantum weight $K=K_{xx}+K_{yy}$ can be calculated with the expression
with the quantum metric (Eq. (13) in the main text):

$\displaystyle K=\int\frac{\differential^{2}k}{2\pi}\quantity(g_{xx}+g_{yy}).$
(36)

Here, the integrand $g_{xx}+g_{yy}$ is given by

$\displaystyle g_{xx}+g_{yy}$
$\displaystyle=\frac{\Delta_{s}^{2}\hbar^{4}}{4m^{2}}\frac{k^{2}}{\varepsilon_{k}^{4}}$
(37)

and thus the quantum weight is

$\displaystyle K$
$\displaystyle=\frac{1}{2}\int_{0}^{\infty}\differential{x}\frac{x}{\quantity((x-(\mu/\Delta_{s}))^{2}+1)^{2}}$
$\displaystyle=\frac{1}{8}\quantity(2+\frac{\mu}{\Delta_{s}}(\pi+\arctan(\mu/\Delta_{s})))$
(38)

###  B.3 $p$-wave case

The ${\bf\it d}({\bf\it k})$ in this case is given by ${\bf\it d}({\bf\it
k})=(\xi_{{\bf\it k}},\Delta_{p}k_{x},\Delta_{p}k_{y})$. The energy dispersion
is given by

$\displaystyle\varepsilon_{\pm}({\bf\it k})=\pm\sqrt{\xi_{{\bf\it
k}}^{2}+\Delta_{p}^{2}k^{2}}.$ (39)

The quantum weight $K$ is given by

$\displaystyle
K=\frac{1}{4}\int_{0}^{\infty}x\differential{x}\frac{2y^{2}+2x^{4}+x^{2}}{\quantity{\quantity(y-x^{2})^{2}+x^{2}}^{2}}$
(40)

where $y=\hbar^{2}\mu/(2m\Delta_{p}^{2})$.

Interestingly, when $y=1/4$, i.e., $\mu=m\Delta_{p}^{2}/(2\hbar^{2})$, the
energy dispersion is exactly parabolic:

$\displaystyle\varepsilon_{\pm}({\bf\it
k})=\pm\quantity(\frac{\hbar^{2}k^{2}}{2m}+\mu),$ (41)

and $K$ can be analytically calculated and is equal to 1:

$\displaystyle K(\mu=m\Delta_{p}^{2}/(2\hbar^{2}))$ $\displaystyle=1.$ (42)

Namely, the quantum weight saturates the bound determined by the Chern number
$\absolutevalue{C}=1$.

##  Appendix C Quantum geometry in $U(1)$ symmetric systems

We consider ground states with degeneracy $r$ in the $U(1)$ symmetric system.
Under twisted boundary condition, the ground states satisfies

$\displaystyle\Psi_{a{\bf\it\kappa}}({\bf\it r}_{1},\dots,{\bf\it
r}_{i}+{\bf\it L}_{\mu},\dots,{\bf\it r}_{N})$
$\displaystyle=e^{i{\bf\it\kappa}\dotproduct{\bf\it
L}_{\mu}}\Psi_{a{\bf\it\kappa}}({\bf\it r}_{1},\dots,{\bf\it
r}_{i},\dots,{\bf\it r}_{N}),$ (43)

where $\Psi_{a{\bf\it\kappa}}$ is the $a$-th ground state wavefunction in the
first quantization representation, and ${\bf\it r}_{i}$ specifies the position
of $i$-th particle corresponding to the conserved $U(1)$ charge. ${\bf\it
L}_{\mu}=(0,\dots,L_{\mu},\dots,0)$ specifies the system size in
$\mu$-direction $L_{\mu}$, and the vector ${\bf\it\kappa}$ specifies the
twisted boundary condition. Note that the number of particle is a good quantum
number due to the global $U(1)$ symmetry.

Introducing the twisted boundary condition ${\bf\it\kappa}$ is equivalent to
coupling the system to a uniform $U(1)$ gauge field, ${\bf\it
A}={\bf\it\kappa}$. Therefore, the current response to the gauge field can be
described by the quantum geometric quantity associated with the twisted
boundary condition ${\bf\it\kappa}$, and the corresponding conductivity for
the $U(1)$ charge with the twisted boundary condition ${\bf\it\kappa}$
$\sigma(\omega;\kappa)$ is given by

$\displaystyle\sigma_{\mu\nu}(\omega;\kappa)=\frac{1}{\hbar}\sum_{n,m}\frac{-iE_{nm}A^{\mu}_{nm}A^{\nu}_{mn}f_{nm}}{\hbar\omega+E_{nm}+i\delta},$
(44)

where $E_{nm}=E_{n}-E_{m}$ is the energy difference between $n$-th and $m$-th
many body eigenstate, and $\partial_{\mu}$ is the derivative with respect to
$\kappa_{\mu}$. $f_{nm}=f_{n}-f_{m}$ with the probability $f_{n}$ that $n$-th
eigenstate is realized. At zero temperature, the canonical distribution gives
$f_{n}=1/r$ when the state $n$ is one of the $r$-fold degenerated ground
states and otherwise $f_{n}=0$.
$A^{\mu}_{nm}=\matrixelement{n,{\bf\it\kappa}}{i\partial_{\mu}}{m,{\bf\it\kappa}}$
is the interband Berry connection for the $n$-th and $m$-th eigenstates the
interacting system under the boundary condition ${\bf\it\kappa}$.

Further we introduce the quantum geometric tensor $Q^{\mu\nu}$ for the many-
body ground states as

$\displaystyle Q^{\mu\nu}_{ab}$
$\displaystyle=\matrixelement{\partial_{\mu}\Psi_{a{\bf\it\kappa}}}{(1-P_{{\bf\it\kappa}})}{\partial_{\nu}\Psi_{b{\bf\it\kappa}}}.$
(45)

where $\partial_{\mu}$ refers to the derivative with respect to
$\kappa_{\mu}$, and $P_{{\bf\it\kappa}}$ is the projection operator onto the
ground state subspace for the boundary condition ${\bf\it\kappa}$.
$Q^{\mu\nu}$ is in general an $r\times r$ matrix and non-Abelian quantum
geometric tensor. The symmetric and antisymmetric components of $Q^{\mu\nu}$
define the quantum metric and Berry curvature for the many-body states
$\ket{\Psi_{{\bf\it\kappa}}}$ respectively:
$Q^{\mu\nu}=G^{\mu\nu}-iF^{\mu\nu}/2$. The trace of them gives the Abelian
quantum geometry for the entire ground state subspace: $g^{\mu\nu}_{\rm
tot}\equiv\Tr G^{\mu\nu},\Omega^{\mu\nu}_{\rm tot}\equiv\Tr F^{\mu\nu}$. If
the ground state is unique, the quantum geometric tensor reduces to the
Abelian quantum geometric tensor.

Since the relation between the conductivity $\sigma(\omega)$ and the Berry
connection $A^{\mu}$ is identical to the electric conductivity [8], we can
readily show the sum rule for the negative-first moment:

$\displaystyle\int_{0}^{\infty}\differential{\omega}\frac{\real\sigma_{\mu\nu}(\omega)}{\omega}=\frac{1}{2\hbar}g_{\mu\nu}$
(46)
$\displaystyle\int_{0}^{\infty}\differential{\omega}\frac{\imaginary\sigma_{\mu\nu}(\omega)}{\omega}=\frac{1}{4\hbar}\Omega_{\mu\nu}$
(47)

where $g_{\mu\nu}$, $\Omega_{\mu\nu}$ is the quantum metric and the Chern
number for the ground state(s) defined as

$\displaystyle g_{\mu\nu}$ $\displaystyle=\frac{1}{r}\Tr G^{\mu\nu},$ (48)
$\displaystyle\Omega_{\mu\nu}$ $\displaystyle=\frac{1}{r}\Tr F^{\mu\nu}.$ (49)

If the ground state is unique ($r=1$), $g_{\mu\nu}$ and $\Omega_{\mu\nu}$
reduces to the Abelian quantum geometric tensor.

If we further assume that the bulk conductivity $\sigma(\omega)$ does not
depend on the boundary condition, we can replace $g_{\mu\nu}$ and
$\Omega^{\mu\nu}$ with the averaged ones over ${\bf\it\kappa}$, and in
particular $\Omega_{\mu\nu}$ is replaced by $C_{\rm tot}/r$ with the Chern
number of the ground state subspace $C_{\rm tot}$.

With the sum rules (46), (47), we can repeat the same discussion in Ref. [10]
to show the relation between the structure factor or the quantum weight $K$
and the quantum metric for 2D systems as

$\displaystyle
K=\int\frac{\differential^{2}{{\bf\it\kappa}}}{2\pi}(g_{xx}+g_{yy}),$ (50)

and this is the generalization of Eq. (13) in the main text to many-body
systems with $U(1)$ symmetry. From an inequality
$g_{xx}+g_{yy}\geq\absolutevalue{\Omega_{xy}}$, we can show the bound on the
quantum weight $K\geq\absolutevalue{C}$ and the topological bound (11) in the
main text.

##  Appendix D Topological bound in $d+id$ topological superconductors

As an example of a topological superconductor, consider a
$d_{x^{2}-y^{2}}+id_{xy}$ superconductor in two dimensions [30]. A model for
such a state on a lattice is described by the following mean-field Hamiltonian
[31]:

$\displaystyle H$ $\displaystyle=\sum_{{\bf\it
k}}\begin{pmatrix}{c}^{\dagger}_{{\bf\it k}\uparrow}&{c}_{-{\bf\it
k}\downarrow}\end{pmatrix}\begin{pmatrix}\varepsilon({\bf\it
k})&\Delta({\bf\it k})\\\ \Delta(-{\bf\it k})^{*}&-\varepsilon({\bf\it
k})\end{pmatrix}\begin{pmatrix}{c}_{{\bf\it k}\uparrow}\\\
{c}^{\dagger}_{-{\bf\it k}\downarrow}\end{pmatrix}$
$\displaystyle\equiv\sum_{{\bf\it k}}{\Psi}^{\dagger}_{{\bf\it
k}}\quantity({\bf\it h}({\bf\it k})\dotproduct{\bf\it\sigma}){\Psi}_{{\bf\it
k}}.$ (51)

$\varepsilon({\bf\it k})$ is the single particle dispersion and here we take
$\varepsilon({\bf\it k})=\varepsilon_{0}(\cos k_{x}+\cos k_{y})$, and
$\Delta({\bf\it k})=\Delta_{0}(\cos k_{x}-\cos k_{y})-i\Delta_{xy}\sin
k_{x}\sin k_{y}$ is the pairing potential. As in Eq. (18), we have a
topological bound for the $U(1)$ charge ${\Psi}^{\dagger}{\Psi}$ and hence for
the spin structure factor with $s_{0}=\hbar/2$.

This system is in a topological phase with the quantized spin Hall conductance
determined by Chern number when $\Delta_{xy}\neq 0$ [31]. This can be
understood with the Dirac dispersion at four points, ${\bf\it
k}=(\pm\pi/2,\pm\pi/2)$. When $\Delta_{xy}=0$, the quasiparticle excitation at
these points has gapless Dirac dispersion, but once $\Delta_{xy}$ becomes
finite, these Dirac points are gapped. The Chern number is the sum of the
contributions from each Dirac point and is $C=2~{}\mathrm{sgn}(\Delta_{xy})$.
The Chern number can be also directly calculated by
$C=(1/4\pi)\int\differential^{2}{{\bf\it k}}~{}{\bf\it
n}\dotproduct\quantity(\partialderivative*{{\bf\it
n}}{k_{x}}\crossproduct\partialderivative*{{\bf\it n}}{k_{y}})$, with ${\bf\it
n}={\bf\it h}/\absolutevalue{{\bf\it h}}$.

We can verify our inequality (17) by explicitly calculating the spin static
structure factor with the Fourier transform of the spin density operator
$s_{{\bf\it q}}^{z}=(\hbar/2)\sum_{k}({c}^{\dagger}_{{\bf\it
k}\uparrow}{c}_{{\bf\it k}+{\bf\it q}\uparrow}-{c}^{\dagger}_{{\bf\it
k}\downarrow}{c}_{{\bf\it k}+{\bf\it q}\downarrow})=(\hbar/2)\sum_{{\bf\it
k}}{\Psi}^{\dagger}_{{\bf\it k}}{\Psi}_{{\bf\it k}+{\bf\it q}}$. The leading
order in $q$ of the spin static structure factor is given by $S^{s}_{{\bf\it
q}}=(\hbar^{2}/16\pi)q^{2}K+\dots$ with
$K=(1/8\pi)\int\differential^{2}{{\bf\it
k}}~{}\sum_{\alpha=x,y}\quantity(\partialderivative*{{\bf\it
n}}{k_{\alpha}})^{2}$. From a mathematical inequality
$(\partialderivative*{{\bf\it n}}{k_{x}})^{2}+(\partialderivative*{{\bf\it
n}}{k_{y}})^{2}\geq 2\absolutevalue{\quantity(\partialderivative*{{\bf\it
n}}{k_{x}})\crossproduct\quantity(\partialderivative*{{\bf\it n}}{k_{y}})}$,
one can explicitly confirm the inequality (17) holds.

##  Appendix E Details on calculations of twisted TMD Materials

The effective Hamiltonian for each valley and spin is given by [25]

$\displaystyle H=\begin{pmatrix}-\frac{\hbar^{2}({\bf\it
k}-{\bf\it\kappa}_{+})^{2}}{2m^{*}}+\Delta_{1}({\bf\it r})&\Delta_{T}({\bf\it
r})\\\ \Delta_{T}^{\dagger}({\bf\it r})&-\frac{\hbar^{2}({\bf\it
k}-{\bf\it\kappa}_{-})^{2}}{2m^{*}}+\Delta_{2}({\bf\it r})\end{pmatrix},$ (52)

where $\kappa_{\pm}=2\pi/a_{M}(-1/\sqrt{3},\pm 1/3)$ is the corner of the
moiré Brillouin zone, $a_{M}=a/\theta$ is the moiré lattice constant with the
twist angle $\theta$ and the monolayer lattice constant $a$.
$\Delta_{T}({\bf\it r})$ and $\Delta_{1,2}({\bf\it r})$ are spatially
modulated by the moiré superlattice of the following form:

$\displaystyle\Delta_{1,2}({\bf\it r})$
$\displaystyle=2V\sum_{j=1,3,5}\cos({\bf\it g}_{j}\dotproduct{\bf\it
r}\pm\psi),$ (53) $\displaystyle\Delta_{T}({\bf\it r})$
$\displaystyle=w(1+e^{-i{\bf\it g}_{2}\dotproduct{\bf\it r}}+e^{-i{\bf\it
g}_{3}\dotproduct{\bf\it r}}),$ (54)

where ${\bf\it g}_{1}=(4\pi/\sqrt{3}a_{M},0)$ is the moiré reciprocal lattice
vector and ${\bf\it g}_{j}$ are $(j-1)\pi/3$ counter-clockwise rotation of
${\bf\it g}_{1}$. The parameters for twisted WSe2 is given in Ref. [9] as:
$m^{*}=0.43m_{0},V=$9.0\text{\,}\mathrm{meV}$,\psi=$$,w=$18\text{\,}\mathrm{meV}$,a_{0}=$3.317\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$
with the bare electron mass $m_{0}$, while the parameters for MoTe2 is given
by [26]
$m^{*}=0.62m_{0},V=$11.2\text{\,}\mathrm{meV}$,\psi=$$,w=$13.3\text{\,}\mathrm{meV}$,a_{0}=$3.52\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$.

We calculated the static structure factor from which the Bragg peak divergence
is subtracted:

$\displaystyle S_{{\bf\it q}}^{\prime}$
$\displaystyle=\frac{1}{V}(\expectationvalue{\rho_{{\bf\it q}}\rho_{-{\bf\it
q}}}-\expectationvalue{\rho_{{\bf\it q}}}\expectationvalue{\rho_{-{\bf\it
q}}}).$ (55)

$S_{{\bf\it q}}^{\prime}$ reduces to the static structure factor $S_{{\bf\it
q}}$ defined in the main text if ${\bf\it q}$ does not coincide with the
reciprocal lattice vectors. $S_{{\bf\it q}}^{\prime}$ is calculated by
diagonalizing the Hamiltonian (52) in $k$-space to obtain the Bloch
wavefunction for each band and plugging them into the following expression for
$S_{{\bf\it q}}^{\prime}$:

$\displaystyle S_{{\bf\it q}}^{\prime}$ $\displaystyle=\int_{\rm
BZ}\frac{\differential^{d}{{\bf\it k}}}{(2\pi)^{d}}\Tr[P({\bf\it k})(P({\bf\it
k})-P({\bf\it k}+{\bf\it q}))],$ (56)

where $P({\bf\it k})=\sum_{n}^{\rm occ}\ket{u_{n{\bf\it k}}}\bra{u_{n{\bf\it
k}}}$ is the projection operator onto the occupied bands at wavevector
${\bf\it k}$ with $\ket{u_{n{\bf\it k}}}$ the cell-periodic Bloch wavefunction
for $n$-th band. In the diagonalization, we consider plane waves with
wavevector of the form ${\bf\it k}={\bf\it k}_{0}+n_{1}{\bf\it
g}_{1}+n_{3}{\bf\it g}_{3}$, where ${\bf\it k}_{0}$ is the wavevector in the
first Brillouin zone and $n_{1,3}$ is integer between $-3$ and $3$. The
integration over the Brillouin zone is approximated by the summation over
$30\times 30$ mesh in the first Brillouin zone. In our calculation, $P({\bf\it
k})$ is for the occupied bands of the holes, and for filling factor $\nu=-1$,
$P({\bf\it k})$ is the projection operator onto the highest valence band.

We note that $S_{{\bf\it q}}^{\prime}$ is not periodic in ${\bf\it q}$ even
for periodic systems. For example, $S_{{\bf\it q}}^{\prime}$ at finite
reciprocal lattice vector ${\bf\it G}$ is not equal to $S_{{\bf\it q}=0}$; the
former represents the density fluctuation with wavevector ${\bf\it G}$ and
finite in general, while the latter represents the density fluctuation with
wavevector ${\bf\it q}=0$, which is zero in charge conserved systems.

The quantum weight is calculated by integrating the quantum metric:

$\displaystyle K$ $\displaystyle=\int_{\rm BZ}\frac{\differential^{2}{{\bf\it
k}}}{2\pi}(g_{xx}({\bf\it k})+g_{yy}({\bf\it k})).$ (57)

where the quantum metric is calculated with the following formula:

$\displaystyle g_{\mu\nu}({\bf\it k})$ $\displaystyle=\real\sum_{n}^{\rm
occ}\sum_{m}^{\rm unocc}\frac{\matrixelement{u_{n{\bf\it
k}}}{\partial_{\mu}H}{u_{m{\bf\it k}}}\matrixelement{u_{m{\bf\it
k}}}{\partial_{\nu}H}{u_{n{\bf\it k}}}}{(\varepsilon_{n{\bf\it
k}}-\varepsilon_{m{\bf\it k}})^{2}}$ (58)

where $\varepsilon_{n{\bf\it k}}$ is the band dispersion for $n$-th band,
$\partial_{\mu}$ is the derivative with respect to $k_{\mu}$, and the
summation of $n$ (m) is over the occupied (unoccupied) bands. The integration
in Eq. (57) is executed with a code of Julia by hcubature function in
HCubature package with a relative tolerance of
$1\text{\times}{10}^{-4}\text{\,}$.

Figure 3: Structure factor of twisted MoTe2.

Results for other twist angles are shown in Fig. 3.

##  Appendix F Feynmann-Bijl formula for general systems with $U(1)$ symmetry

Here we derive the Feynman-Bijl formula for general $U(1)$ symmetric systems.
To find an upper bound on the energy gap, we consider constructing a state
$\ket{ex}$ orthogonal to the ground state $\ket{gs}$. As long as $\ket{ex}$
and $\ket{gs}$ are orthogonal to each other, the energy expectation value of
$\ket{ex}$ is always larger than or equal to the first excited state.
Therefore, we can find an upper bound on the energy gap by evaluating the
energy difference between $\ket{ex}$ and $\ket{gs}$. We choose the following
state as $\ket{ex}$:

$\displaystyle\ket{ex}$ $\displaystyle=(\rho_{-{\bf\it
q}}-\expectationvalue{\rho_{-{\bf\it q}}}_{gs})\ket{gs},$ (59)

where $\expectationvalue{\dots}_{gs}$ is the expectation value in the ground
state $\ket{gs}$ and $\rho_{{\bf\it q}}$ is the $U(1)$ charge density. The
orthogonality $\innerproduct{ex}{gs}=0$ is easily confirmed. Now, the energy
gap $\Delta$ has an upper bound:
$\Delta\leq\matrixelement{ex}{H}{ex}/\innerproduct{ex}{ex}-\expectationvalue{H}_{gs}$.
This can be rewritten as

$\displaystyle\Delta\leq\frac{\expectationvalue{\commutator{\rho_{{\bf\it
q}}}{\commutator{H}{\rho_{-{\bf\it q}}}}}_{gs}}{2VS_{{\bf\it q}}},$ (60)

with the structure factor $S_{{\bf\it q}}$ for the $U(1)$ charge in the ground
state.

For electronic systems, $U(1)$ charge $\rho$ can represent the electric charge
and the Hamiltonian $H$ is quadratic in the momentum ${\bf\it p}$ with kinetic
energy ${\bf\it p}^{2}/(2m)$ for each electron. Using a relation from the
continuity equation, $\commutator{H}{\rho_{-{\bf\it q}}}=-\hbar{\bf\it
q}\dotproduct{\bf\it j}_{-{\bf\it q}}$, and the commutation relation
$\commutator{\rho_{{\bf\it q}}}{{\bf\it j}_{-{\bf\it q}}}=\hbar{\bf\it q}N/m$
with the number of electron $N$ and the mass $m$, the numerator is reduced to
a simple form and the gap bound is obtained as

$\displaystyle\Delta\leq\frac{\hbar^{2}{\bf\it q}^{2}n}{2mS_{{\bf\it q}}}.$
(61)

This is the Feynman-Bijl formula used in Ref. [29]. The same formula as (60),
but with the projected density operator $\bar{\rho}_{{\bf\it q}}$ and the
projected Hamiltonian $\bar{H}$ instead of the full density operator
$\rho_{{\bf\it q}}$ and the full Hamiltonian $H$, is used in a study of
magneto-roton excitations in fractional quantum Hall systems [18].