# Compressible quantum matter with vanishing Drude weight

Ahmed Abouelkomsan  Department of Physics, Massachusetts Institute of
Technology, Cambridge, Massachusetts 02139, USA  Nisarga Paul  Department of
Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA  Ady Stern  Department of Condensed Matter Physics, Weizmann
Institute of Science, Rehovot 7610001, Israel  Liang Fu  Department of
Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts
02139, USA

###### Abstract

We explore the possibility of quantum states of matter that are compressible
but have vanishing DC conductivity in the absence of disorder. We show that
the composite Fermi liquid emerging from strong interaction in a generic Chern
band has zero Drude weight, in stark contrast to normal Fermi liquids. Our
work establishes the absence of Drude weight as the defining property of the
composite Fermi liquid phase, which distinguishes it from the Fermi liquid or
other types of non-Fermi liquids. Our findings point to a possibly wide class
of gapless quantum phases with unexpected transport and optical properties.

Compressibility and conductivity are two fundamental properties of quantum
states of matter. Compressibility is a thermodynamic property that describes
the response of charge density to a change in the chemical potential. It is
closely related to the charge gap: as long as the range of inter-particle
interaction is not too long, gapless states are compressible while gapped
states are incompressible. On the other hand, DC conductivity is a transport
property that distinguishes conducting and insulating states. Although
conceptually different, compressibility and conductivity often appear
concurrently: band insulators are incompressible, while metals are
compressible.

In this work, we demonstrate the existence of compressible quantum liquids
with electric transport properties that differ drastically from normal metals.
As a diagnostic for electrical conductivity in the absence of disorder, we
turn to the Drude weight which characterizes the response of a given system
under adiabatic flux insertion, physically amounting to applying a DC electric
field [1, 2]. A finite Drude weight is expected when level crossings between
the ground state and excited states appear under flux insertion, leading to
energy absorption. We present theoretical argument and numerical evidence for
the absence of Drude weight in certain strongly correlated quantum liquids.
These “Drudeless metals” are sharply distinguished from ordinary metals which
become perfect conductors in the clean limit, as well as Anderson insulators
which are compressible but insulating because of disorder-induced
localization.

Our study is motivated by the recent prediction of a non-Fermi liquid state in
twisted bilayer semiconductor $t$MoTe2 at zero magnetic field [3, 4]. This
state, dubbed “anomalous composite Fermi liquid” (ACFL), arises from strong
electron-electron interaction in a partially filled Chern band and serves as
the parent state of the fractional quantum anomalous Hall states observed in
$t$MoTe2 and rhombohedral graphene moiré superlattice at Jain sequence filling
factors $\nu=n/(2n\pm 1)$ [5, 6, 7, 8, 9]. We show that the ACFL represents a
novel compressible phase of matter with vanishing Drude weight, and is
therefore fundamentally distinct from the Fermi liquid (FL) state which has a
finite Drude weight. While the ACFL spontaneously breaks time reversal
symmetry and exhibits anomalous Hall effect, we also discuss the possibility
of Drudeless metals with time-reversal symmetry.

Compressible states in Chern band. Our main setting is the Chern band, a
fertile platform for various strongly interacting quantum phases of matter.
Generally speaking, the ground state of partially-filled Chern bands depends
on the bandwidth, the band gap to neighboring bands, the band wavefunctions
and electron-electron interactions. Large experimental and theoretical efforts
have focused on studying the possible incompressible states that could arise
in moiré Chern bands, especially when interactions are strong. Notable
examples include fractional Chern insulators, charge density waves and various
other ordered states [10, 11, 12, 13, 14, 15, 16, 17, 18, 19].

On the other hand, less is understood about compressible states in partially
filled Chern bands. The most familiar one is the Fermi liquid (FL) which is
formed when the bandwidth is very large compared to the interaction strength
or, even more exotically, could arise due to strong interaction-induced
kinetic energies [20, 21, 22]. In the opposite limit when the bandwidth is
narrow, a composite Fermi liquid may be realized at even-denominator filling
factors, which shares many similarities with the celebrated CFL in the lowest
Landau level (LLL) [23]. Given that both FL and CFL states can be realized in
a Chern band system, a natural question is how these two compressible states
can be distinguished. As we shall show in this work, in the clean limit, the
composite Fermi liquid and Fermi liquid are sharply distinguished by the Drude
weight.

For concreteness, we consider Chern bands in twisted semiconductor bilayers
$t$MoTe2 and $t$WSe2, described by a two-layer continuum model [24, 25]. Under
certain conditions, this model can be approximately mapped onto Landau levels
subject to a periodic potential [26, 27]. Motivated by this mapping, we study
a periodically modulated Landau level [28, 29, 30, 31] as the simplest case of
a Chern band. Depending on its bandwidth, which is controlled with the
periodic potential, the system at half filling can be tuned from the composite
Fermi liquid to the Fermi liquid, allowing a systematic study for the two
gapless phases.

We consider a periodic potential which takes the form
$V(\mathbf{r})=\sum_{\mathbf{g}}V_{\mathbf{g}}e^{i\mathbf{g}\cdot\mathbf{r}}$
where $\\{\mathbf{g}\\}$ are the relevant harmonics. We focus on the simplest
case where the unit cell of the periodic potential encloses one flux quanta,
so that the Landau level is broadened by the potential but does not split. We
further assume that strength of the potential $|V_{\mathbf{g}}|$ is small
relative to the cyclotron gap ($|V_{\mathbf{g}}|/\hbar\omega_{c}\ll 1$) so it
doesn’t mix different Landau levels. Under these assumptions, the $n$th flat
Landau level acquires a energy-momentum dispersion [30, 32]
$E_{n}(\mathbf{k})=\hbar\omega_{c}(n+\frac{1}{2})+\tilde{E}_{n}(\mathbf{k})$
with

$\tilde{E}_{n}(\mathbf{k})=\sum_{\mathbf{g}}V_{\mathbf{g}}\eta_{\mathbf{g}}e^{-il^{2}(\mathbf{g}\wedge\mathbf{k})}e^{-l^{2}|\mathbf{g}|^{2}/4}L_{n}(\frac{l^{2}|\mathbf{g}|^{2}}{2})$
(1)

where $L_{n}(\mathbf{q})$ is the $n$th Laguerre polynomial and
$l=\sqrt{\hbar/eB}$ is the magnetic length. $\eta_{\mathbf{g}}$ is $1$ if
$\mathbf{g}/2$ is an allowed reciprocal lattice vector and $-1$ otherwise. We
focus on the lowest Landau level ($n=0$) and project the relevant terms onto
it giving rise to the following Hamiltonian

$H=\sum_{\mathbf{k}}\tilde{E}_{0}(\mathbf{k})c^{\dagger}_{\mathbf{k}}c_{\mathbf{k}}+\sum_{\mathbf{k}_{1},\mathbf{k}_{2},\mathbf{q}}U_{\mathbf{q}}e^{-l^{2}|\mathbf{q}|^{2}/2}c^{\dagger}_{\mathbf{k}_{1}+\mathbf{q}}c^{\dagger}_{\mathbf{k}_{2}-\mathbf{q}}c_{\mathbf{k}_{2}}c_{\mathbf{k}_{1}}$
(2)

where $c^{\dagger}_{\mathbf{k}}$ is the lowest Landau level (LLL) creation
operator with momentum $\mathbf{k}$ and we have dropped the constant factor
$\frac{1}{2}\hbar\omega_{c}$. In what follows, we consider a two-dimensional
long range Coulomb interaction
$U_{\mathbf{q}}=\frac{U_{0}a_{0}}{2S}\frac{2\pi}{|\mathbf{q}|}$ where
$U_{0}=\frac{e^{2}}{4\pi\epsilon\epsilon_{0}a_{0}}$ is the strength of the
interaction, $a_{0}$ is the period of the potential and $S$ is the area of the
system.

The Hamiltonian (2) defines a Chern band system. Without the periodic
potential ($V_{\mathbf{g}}=0$), it reduces back to the LLL problem. At half
filling in such a case, the interacting ($U_{0}\neq 0$) many-body ground state
is the composite Fermi liquid [23]. The composite fermion is constructed by
attaching $2$ flux quanta to each electron, which experiences an effective
magnetic field $B^{\ast}=B-2n\phi_{0}$ that is reduced from the external field
[33]. The FQH effect at Jain sequence filling factors $\nu=p/(2p+1)$ is then
mapped into the integer quantum Hall effect of composite fermions which fill
$p$ Landau levels. At half filling $\nu=1/2$, the effective magnetic field
seen by the CFs vanishes and the resulting state is a Fermi liquid of
composite fermions (CFL).

In the opposite limit when interactions are turned off ($U_{0}$ = 0) and only
the periodic potential is present ($V_{\mathbf{g}}\neq 0$), the ground state
at partial filling is a normal Fermi liquid (FL). The model (2) is therefore
the minimal model to study phase transitions between the FL and CFL states as
a function of the periodic potential strength relative to Coulomb interaction.

Drude Weight. In general, the longitudinal conductivity of a clean system at
low frequency can have a singular part of the form
$\sigma_{xx}(\omega)=D(\delta(\omega)+\frac{i}{\pi\omega})$, where the
coefficient of the delta function peak $D$ is known as the Drude weight [2].
$D$ can be extracted from the imaginary (non-dissipative) part of
$\sigma_{xx}$: $D=\pi\text{lim}_{\omega\rightarrow
0}\>\>\omega\>\text{Im}\>\sigma(\omega)$. The Drude weight is responsible for
optical absorption at low frequency. Gapless metallic states such as Fermi
liquids generally have a non-zero Drude weight $D\neq 0$, while gapped
insulating states with a finite spectral gap have zero Drude weight $D=0$
[34].

The absence of a charge gap implies finite compressibility
$\kappa=\frac{\partial n}{\partial\mu}$ with $\mu$ the chemical potential. In
a clean system, one might expect that the presence of gapless low-energy
charge excitations generally leads to finite Drude weight. However, as we
shall see, the composite Fermi liquid state in Chern bands is a compressible
state of matter ($\mathbf{\kappa}\neq 0$) but has zero Drude weight ($D=0$).

Note that in the absence of periodic potential, our Chern band reduces to a
Landau level in a Galilean invariant system, where the presence of Galilean
invariance dictates the longitudinal conductivity to be of the form
$\sigma_{\rm
LL}(\omega)\propto\frac{n}{m}(\delta(\omega-\omega_{c})+\delta(\omega+\omega_{c}))$
where $\omega_{c}=eB/m$ is the cyclotron frequency. Thus, the absence of
optical absorption at low frequency $\omega<\omega_{c}$ implies vanishing
Drude weight for all ground states in Landau Levels of Galilean invariant
systems[35], including the CFL state at half filling. In contrast, generic
Chern band systems lack Galilean invariance, which allows finite Drude weight.
Indeed, as we shall show explicitly, for a periodically modulated Landau level
defined by the Hamiltonian (2), the ground state at half filling is a metal
with finite Drude weight at weak interaction, and a CFL with vanishing Drude
weight at strong interaction.

As realized in the seminal work by Kohn [2], the Drude weight for generic
interacting Hamiltonians can be expressed as the second derivative of the
ground state energy with respect to twisted boundary conditions such that the
wavefunction satisfies $\psi(\mathbf{r}+\mathbf{L})=e^{i\phi}\psi(\mathbf{r})$
for $\phi\in[0,2\pi)$, which is equivalent to threading a flux through the
cycles of the system. The Drude weight is given by (in units of $\hbar=1$)

$D(\phi)=\pi e^{2}\dfrac{\partial^{2}E_{G}(\phi)}{\partial\phi^{2}}.$ (3)

It’s crucial that the derivative with respect to the flux in (3) is evaluated
first before the thermodynamic limit [34].

In practice, there exist multiple methods for computing the Drude weight in
finite systems (which become equivalent in the thermodynamic limit) [36]. A
straightforward approach is to evaluate the Drude weight at a fixed value of
the inserted flux $\phi=\phi_{0}$ and then extrapolate to the thermodynamic
limit. Alternatively, one may evaluate the Drude weight at flux
$\phi=\phi_{\text{min}}$ corresponding to where the ground state energy is
minimized. For this method, $\phi_{\text{min}}$ generally depends on the
system size which makes finite-size scaling challenging.

To mitigate the finite-size effect, we choose to average over different
boundary conditions. This has been shown to yield excellent results in small
systems [37, 36]. We compute the Drude weight by averaging over all fluxes in
one direction, ${\langle D\rangle}=\frac{1}{2\pi}\int^{2\pi}_{0}d\phi
D(\phi)$. It is important to note that by definition, the average Drude weight
${\langle D\rangle}$ vanishes when the ground state energy $E_{G}(\phi)$
changes smoothly with the flux. The average ${\langle D\rangle}$ can be
nonzero only when unavoided level crossings with excited states occur in the
energy spectrum $E(\phi)$. Indeed, this is the case for non-interacting
metals, where level crossings between single-particle states of different
momenta are induced upon varying the flux or equivalently shifting the single-
particle momenta $\mathbf{k}\rightarrow\mathbf{k}+\phi/L$.

Generally speaking, when level crossings are present, $E_{G}(\phi)$ shows
kinks as a function of $\phi$, thus allowing for a nonzero Drude weight
${\langle D\rangle}$. Specifically, at each level crossing, the first
derivative of energy with respect to the flux is discontinuous, thus giving
rise to a finite value of the integral ${\langle
D\rangle}\propto\sum_{p}(\frac{\partial
E_{0}}{\partial\phi}|_{p_{+}}-\frac{\partial E_{0}}{\partial\phi}|_{p_{-}})$
where $p$ runs over all level crossings and $\pm$ refers to evaluating the
first derivative from the left ($+$) or the right ($-$).

Figure 1: The energy dispersion $\tilde{E}_{0}(\mathbf{k})$ (equation (1))
along a cut in the Brillouin zone. The inset shows the Fermi surface contours
at half filling where the shaded region denotes occupied states in the non-
interacting limit.

In the absence of the periodic potential, our system reduces to the LLL. Here,
inserting the flux along the $x$ direction simply translates the Landau
orbitals in the Landau gauge along $y$, which does not affect the ground state
energy. Thus, it follows from Kohn’s formula (3) that the Drude weight is
zero. We now study the evolution of the Drude weight as the periodic potential
strength increases. For a weak potential, the first order correction to the
many-body ground state energy $E_{G}$ is given by $\delta E=\langle
G|V|G\rangle=\int d\mathbf{r}\rho(\mathbf{r})V(\mathbf{r})$. Here $|G\rangle$
is the unperturbed CFL ground state in the half-filled LLL in the absence of
the periodic potential, which has a uniform density
$\rho(\mathbf{r})=\rho_{0}$. This implies that $\delta E=0$, for all inserted
flux. Therefore, the Drude weight (3) remains zero to first order in the
periodic potential strength $V_{0}$.

This contrasts sharply with the noninteracting case ($U_{0}=0$), where the
Drude weight already appears at first order in $V_{0}$, as we show now. In
this case, the ground state at partial filling is a noninteracting band metal
with finite Drude weight given by [32]

$\displaystyle D_{\alpha\beta}=2\pi e^{2}\int_{\rm
BZ}\frac{d^{2}k}{(2\pi)^{2}}\theta(\mu-\tilde{E}_{0}(\mathbf{k}))m^{-1}_{\alpha\beta}(\mathbf{k})$
(4)

with $m^{-1}_{\alpha\beta}(\mathbf{k})$ the inverse effective mass given by

$\displaystyle
m^{-1}_{\alpha\beta}(\mathbf{k})=\frac{1}{\hbar^{2}}\frac{\partial^{2}\tilde{E}_{0}(\mathbf{k})}{\partial
k_{\alpha}k_{\beta}}$ (5)

where $\alpha,\beta=x,y$. The integral runs over the Brillouin zone, $\theta$
is a step function, and $\mu$ is the chemical potential. Since the band
dispersion $\tilde{E}_{0}(\mathbf{k})$ is induced by the periodic potential,
the Drude weight in the noninteracting limit appears at first order in
$V_{0}$.

Figure 2: (a)-(b) The many-body spectrum in the CFL phase (a) and the FL phase
(b). (c)-(d) Momentum space occupation number $n(\mathbf{k})$ of the many-body
ground states in (a) the CFL phase and (b) the FL phase. $V_{0}/U_{0}$ denotes
the ratio of the periodic potential strength to the interaction strength.
Calculations are done at zero flux $\phi=0$. $\kappa$ labels the total
momentum sectors as defined in the supplementary material [32].

To study an arbitrary ratio of periodic potential and interaction strengths,
we numerically solve the Hamiltonian (2) using exact diagonalization [32]. For
concreteness, we consider a $C_{6}$ symmetric potential with the lowest
harmonics only, $|V_{\mathbf{g}}|=-V_{0}$ for
$\mathbf{g}_{i}=\frac{4\pi}{\sqrt{3}a_{0}}(\cos 2\pi(i-1)/3,\sin 2\pi(i-1)/3)$
for $i=1,\dots,6$. $a_{0}$ is the potential period, chosen such that the unit
cell of the potential encompasses $2\pi$ flux. In Fig 1, we plot the single-
particle energy dispersion $\tilde{E}_{0}(\mathbf{k})$ given by Eq.(1).

For weak to moderate values of the periodic potential ($V_{0}/U_{0}\leq 0.6$),
we find the system at half filling to be a CFL. In Fig 2(a), we show the many-
body energy spectrum for $V_{0}/U_{0}=0.2$. The structure of the low-lying
states follows momentum configurations which correspond to the most compact
Fermi sea of composite fermions [38, 39]; details can be found in the
Supplementary Material [32]. Moreover, each momentum sector hosts a pair of
nearly-degenerate ground states [3, 4], reminiscent of the center-of-mass
degeneracy of the CFL state in a perfectly clean LLL. When the strength of the
periodic potential increases, the system undergoes a transition to a normal
Fermi liquid phase. In Fig 2(b), the many-body spectrum at $V_{0}/U_{0}=1$
clearly shows a distinct set of low-lying states, which correspond to shell
filling of the single-particle energy levels $\tilde{E}_{0}(\mathbf{k})$.
Further evidence comes from momentum space occupation of the ground states
$n(\mathbf{k})$ in Fig 2(d) which shows a well-defined Fermi surface, in sharp
contrast to the CFL phase (Fig 2(c)) where the momentum space occupation is
relatively uniform.

Figure 3: Spectral flow of the low-lying energy states in (a) the composite
Fermi liquid phase and (b) the Fermi liquid phase. The colors denote different
total momentum sectors. A flux is inserted in one of the two identical cycles
of the cluster [32].

The distinction between both CFL and FL phases is further reflected on the
spectral flow of the many-body ground states. While both phases are gapless in
nature, we find very contrasting behavior under flux insertion. In a normal
Fermi liquid, low-energy excitations live on a sharply defined Fermi surface
in momentum space. Flux insertion amounts to shifting the single particle
momenta and results in multiple level crossings between different momentum
sectors as a function of the flux, as evident from Fig 3(b). Correspondingly,
the lowest energy ground state changes its momentum sector, leading to kinks
in $E_{G}(\phi)$.

In contrast, the spectrum of CFL phase exhibits a smooth spectral flow as
evident in Fig 3(a). The degenerate ground states in the different momentum
sectors vary smoothly with the flux. Remarkably, in the CFL phase we find that
the lowest energy state on the 28-site cluster stays in the same momentum
sector throughout the full range of the flux. In other words, there are no
level crossings or kinks in $E_{G}(\phi)$ 111It’s important to note that level
crossings can exist independent of the underlying phase. Indeed, the total
momentum changes as $\mathbf{K}\rightarrow\mathbf{K}+N_{p}\phi/L$ when flux
$\phi$ is inserted where $N_{p}$ is the number of particles. Depending on the
system geometry, the ground state total momenta can flow from one sector to
another. However, level crossings in the FL phase occur at generic flux values
not dictated by this relation while this is not the case for the CFL ground
states..

Figure 4: Drude weight averaged over twisted boundary conditions, ${\langle
D\rangle}=\frac{1}{2\pi}\int_{0}^{2\pi}D(\phi)d\phi$ as a function of the
ratio of the periodic potential strength to the interaction strength
$V_{0}/U_{0}$. $N_{s}$ denotes the number of sites in each cluster used [32].

We further calculate the Drude weight as a function of the ratio of periodic
potential strength to the interaction strength $V_{0}/U_{0}$. In Fig 4, we
plot the Drude weight calculated using the averaging method. In the CFL regime
(approximately $V_{0}/U_{0}\leq 0.6$), we find the Drude weight to (nearly)
vanish. Upon increasing $V_{0}/U_{0}$, we notice increasing values of the
Drude weight as the system undergoes the transition to the Fermi liquid regime
where the Drude weight is non-zero. While averaging over boundary conditions
clearly illustrates the difference between the two phases, our conclusions
remain the same if instead the Drude weight is calculated without averaging
[32], further supporting our conclusion that the Drude weight for generic CFL
states is vanishing.

As we have shown above, the periodically modulated LLL at half filling
provides perhaps the simplest setting for realizing CFL and FL phases, which
are distinguished by the Drude weight being zero or non-zero. Interestingly,
our numerical study of the Drude weight, shown in Fig.4, suggests a direct
transition between CFL and FL phases as a function of $V_{0}/U_{0}$.
Determining the nature of this (possibly continuous) phase transition [41, 42]
(Fig. 4) from finite systems studies can be challenging and requires careful
analysis which we defer to the future.

Discussion. In this paper, we have shown that composite Fermi liquids in Chern
bands have zero Drude weight. By introducing a minimal model consisting of
Landau levels with a periodic potential, Galilean invariance is broken, but
the vanishing Drude weight still persists in the CFL regime. We have provided
evidence for such a conclusion using a perturbative argument when the
potential strength is weak combined with numerical results from exact
diagonalization.

Our model Eq. (2) by construction is based on a dispersive band with band
wavefunctions inherited from the Landau levels. In systems where the Chern
band wavefunctions are different from those of the LLL (such as twisted
$\text{MoTe}_{2}$), we have also observed a vanishing Drude weight in the CFL
regime [32].

We have established that the absence of Drude weight is the defining property
of the composite Fermi liquid as a unique gapless phase distinct from other
types of non-Fermi liquids (which by definition have vanishing quasiparticle
weight). Our work therefore shows that the composite Fermi liquid state can be
experimentally established by measuring electronic compressibility and low-
frequency optical conductivity.

A vanishing Drude weight in the CFL implies that the DC conductivity
$\sigma_{xx}(\omega\rightarrow 0)$ doesn’t have any delta function
singularities but it doesn’t necessarily mean it will vanish. On this point,
we note that the absence of Drude weight is consistent with the form of low-
frequency conductivity $\sigma_{xx}\propto i\omega$ (instead of $1/i\omega$)
suggested by the Ioffe-Larkin rule for the composite Fermi liquid [43],
implying that $\sigma_{xx}(\omega\rightarrow 0)=0$. We leave the explicit
calculation of the full optical spectrum of CFL to the future.

While the CFL phase breaks time reversal symmetry either through an external
magnetic field or spontaneously [3, 4], an intriguing possibility is the
realization of two copies of a CFL related by time-reversal symmetry [44, 45]
which could be realizable in moiré systems having a time-reversed pair of
Chern bands. In the absence of any interaction between the two flavors, the
Drude weight of this time-reversal-symmetric phase vanishes in a similar
manner to a single CFL as shown above. We leave the detailed study of the full
problem to the future.

Finally, we have neglected disorder in our analysis. It’s known [23] that for
the CFL in the LLL, the DC longitudinal conductivity is linear in the
wavevector. Disorder can mix different wavevectors such that the DC
conductivity becomes finite. The same reasoning applies to generic CFL states
in Chern bands: such Drudeless metal acquires DC longitudinal conductivity in
the presence of disorder, which is inversely proportional to the mean free
path. A comprehensive study of the role of disorder is an interesting future
direction.

###### Acknowledgements.

We thank Aidan Reddy for helpful discussions. This work was supported by the
Air Force Office of Scientific Research (AFOSR) under Award No.
FA9550-22-1-0432 and benefited from computing resources provided by the MIT
SuperCloud and Lincoln Laboratory Supercomputing Center. A.A was supported by
the Knut and Alice Wallenberg Foundation (KAW 2022.0348). N.P. acknowledges
the KITP graduate fellowship. L. F. was partly supported by the Simons
Investigator Award from the Simons Foundation. A.S. was supported by grants
from the ERC under the European Union’s Horizon 2020 research and innovation
programme (Grant Agreements LEGOTOP No. 788715 ) and the DFG (CRC/Transregio
183, EI 519/71),

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##  Appendix A Supplemental Material

##  Appendix B Periodically modulated Landau levels

In this section, we introduce the model studied in the main text which
consists of Landau levels in addition to a periodic potential. Due to magnetic
translation symmetry of the quantum Hall problem, it is possible to choose a
unit cell defined by two basis vectors $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$
that encloses $2\pi$ flux, $|\mathbf{a}_{1}\wedge\mathbf{a}_{2}|=2\pi l^{2}$
where $l=\sqrt{\hbar/eB}$ is the magnetic length. Such a choice is not unique
and any $2\pi$ flux-enclosing unit cell describes the problem equivalently.
One then obtains Bloch-like wavefunctions as representations of the magnetic
translation symmetry algebra. Let us denote them by $|n,\mathbf{k}\rangle$
where $n$ is the LL index and $\mathbf{k}$ is the ”Bloch” momentum. In the
symmetric gauge, these wavefunctions are known to be the modified Weierstrass
sigma functions [46, 47, 48, 49]

We consider adding a periodic potential of the form
$V(\mathbf{r})=\sum_{\mathbf{g}}V_{\mathbf{g}}e^{i\mathbf{g}\cdot\mathbf{r}}$
where $\\{\mathbf{g}\\}$ are the relevant harmonics which set the periodicity
of the potential. Let’s denote the two basis vectors that define the unit cell
of the potential to be $\mathbf{R}_{1}$ and $\mathbf{R}_{2}$ such that
$V(\mathbf{r}+n\mathbf{R}_{1}+m\mathbf{R}_{2})=V(\mathbf{r})$ for integer $n$
and $m$. The reciprocal basis vectors are obtained from the relation ,
$\mathbf{R}_{i}\cdot\mathbf{g}_{j}=2\pi\delta{{}_{ij}}$. The harmonics of the
potential can be then expanded in these basis,
$\mathbf{g}=n\mathbf{g}_{1}+m\mathbf{g}_{2}$. We consider the case where the
unit cell of the periodic potential encloses $2\pi$ flux quanta,
$|\mathbf{R}_{1}\wedge\mathbf{R}_{2}|=2\pi l^{2}$. Since the choice of the
quantum Hall basis vectors $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$ is not
unique, we can always take $\mathbf{a}_{1}=\mathbf{R}_{1}$ and
$\mathbf{a}_{2}=\mathbf{R}_{2}$. The Hamiltonian is given by,

$H=\sum_{\mathbf{k}}\hbar\omega_{c}(n+\frac{1}{2})c^{\dagger}_{n\mathbf{k}}c_{n\mathbf{k}}+\sum_{\mathbf{k}_{1}\mathbf{k}_{2}n_{1}n_{2}}\sum_{\mathbf{g}}\langle
n_{1},\mathbf{k}_{1}|V_{\mathbf{g}}e^{i\mathbf{g}\cdot\mathbf{r}}|n_{2},\mathbf{k}_{2}\rangle
c^{\dagger}_{n_{1}\mathbf{k}_{1}}c_{n_{2}\mathbf{k}_{2}}$ (S1)

To evaluate the matrix element $\langle
n_{1},\mathbf{k}_{1}|V_{\mathbf{g}}e^{i\mathbf{g}\cdot\mathbf{r}}|n_{2},\mathbf{k}_{2}\rangle$,
we briefly review the algebra of the guiding center and relative coordinate
operators. The position operator $\mathbf{r}$ can be decomposed into two parts
$\mathbf{r}=\mathbf{R}+\bar{\mathbf{R}}$ with
$\mathbf{R}=\mathbf{r}+(\hat{z}\wedge\boldsymbol{\Pi})l^{2}$ is the guiding
center operator and $\bar{\mathbf{R}}=-(\hat{z}\wedge\boldsymbol{\Pi})l^{2}$
is the relative (cyclotron) coordinate operator. $\boldsymbol{\Pi}$ denotes
the canonical momentum operator ,
$\boldsymbol{\Pi}=-\boldsymbol{\partial}-e\mathbf{A}$ where $\mathbf{A}$ is
the vector potential such that the magnetic field is
$B=\boldsymbol{\nabla}\wedge\mathbf{A}$. The guiding center and relatve
coordinate operators satisfy the following commutation relations,
$[R_{a},R_{b}]=-i\epsilon_{ab}l^{2}$ ,
$[\bar{R}_{a},\bar{R}_{b}]=i\epsilon_{ab}l^{2}$ and $[R_{a},\bar{R}_{b}]=0$.

The guiding center operators are the generators of the magnetic translation
algebra, $t(\mathbf{q})=e^{i\mathbf{q}\cdot\mathbf{R}}$ obeying the following
commutation relation

$t(\mathbf{q}_{1})t(\mathbf{q}_{2})=e^{il^{2}\mathbf{q}_{1}\wedge\mathbf{q_{2}}}t(\mathbf{q}_{2})t(\mathbf{q}_{1})$
(S2)

To satisfy the above algebra, the action of $t(\mathbf{q})$ on
$|n,\mathbf{k}\rangle$ is given by

$t(\mathbf{q})|n,\mathbf{k}\rangle=e^{\frac{i}{2}l^{2}\mathbf{q}\wedge\mathbf{k}}|n,\mathbf{k}+\mathbf{q}\rangle$
(S3)

The wavefunction $|n,\mathbf{k}\rangle$ in the symmetric gauge satisfy the
following Brillouin zone boundary condition [49]

$t(\mathbf{g}_{i})|n,\mathbf{k}\rangle=(-1)e^{il^{2}\mathbf{g}_{i}\wedge\mathbf{k}}|n,\mathbf{k}\rangle$
(S4)

for $i=1,2$. Combining (S2) and (S4), we arrive to the following condition

$|n,\mathbf{k}+\mathbf{g}_{i}\rangle=-e^{\frac{i}{2}l^{2}\mathbf{g}_{i}\wedge\mathbf{k}}|n,\mathbf{k}\rangle$
(S5)

The relative coordinate operator $\bar{\mathbf{R}}$ moves a state different
Landau levels. Explicitly,
$\bar{R}_{x}=(\bar{a}^{\dagger}+\bar{a})/(\sqrt{2})$ and
$\bar{R}_{y}=(\bar{a}^{\dagger}-\bar{a})/(\sqrt{2}i)$ where
$\bar{a}^{\dagger}|n,\mathbf{k}\rangle=\sqrt{n+1}|n+1,\mathbf{k}\rangle$.
Utilizing this and the condition (S5), we arrive at the following expression
after lengthy algebra

$\langle
n_{1},\mathbf{k}_{1}|e^{i\mathbf{g}\cdot\mathbf{r}}|n_{2},\mathbf{k}_{2}\rangle=G_{n_{1}n_{2}}(\mathbf{g})F({\mathbf{k}_{1},\mathbf{k}_{2}},\mathbf{g})$
(S6)

where

$\displaystyle
G_{n_{1},n_{2}}(\mathbf{g})=e^{-l^{2}|\mathbf{g}|^{2}/4}\sqrt{\dfrac{n_{<}!}{n_{>}!}}L_{n_{<}}^{|n_{2}-n_{1}|}\bigg{(}\dfrac{|\mathbf{g}|^{2}l^{2}}{2}\bigg{)}\times\begin{cases}\big{(}\dfrac{-il\bar{z}}{\sqrt{2}}\big{)}^{|n_{2}-n_{1}|}&n_{2}>n_{1}\\\
\big{(}\dfrac{ilz}{\sqrt{2}}\big{)}^{|n_{2}-n_{1}|}&n_{1}\geq
n_{2}\end{cases}$ (S7) $\displaystyle
F(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{g})=\eta_{\mathbf{g}}\delta_{\mathbf{k}_{1},\mathbf{k}_{2}}e^{-il^{2}(\mathbf{g}\wedge\mathbf{k})}$
(S8)

where $n_{>}={\rm max}(n_{1},n_{2})$ and $n_{<}={\rm min}(n_{1},n_{2})$,
$L_{a}^{b}(x)$ is the generalized Laguerre polynomial, $z=g_{x}+ig_{y}$ and
$\eta_{\mathbf{g}}=1$ if $\mathbf{g}/2$ is allowed reciprocal lattice vector
and $-1$ otherwise. Assuming the potential strength is weak compared to the
cyclotron frequency, $|V_{\mathbf{g}}|<<\hbar\omega_{c}$, we can neglect any
Landau level mixing in (S1) such that each Landau level acquires a dispersion
$\tilde{E}_{n}(\mathbf{k})=\sum_{\mathbf{g}}\langle
n,\mathbf{k}|V_{\mathbf{g}}e^{i\mathbf{g}\cdot\mathbf{r}}|n,\mathbf{k}\rangle$
given by equation (1) in the main text which we rewrite below

$\tilde{E}_{n}(\mathbf{k})=\sum_{\mathbf{g}}V_{\mathbf{g}}\eta_{\mathbf{g}}e^{-il^{2}(\mathbf{g}\wedge\mathbf{k})}e^{-l^{2}|\mathbf{g}|^{2}/4}L_{n}(\frac{l^{2}|\mathbf{g}|^{2}}{2})$
(S9)

##  Appendix C Non-interacting Drude weight

In the non-interacting limit, it’s straight forward to see that there is a
finite Drude weight in a partially filled Landau level in the presence of a
periodic potential even though the Drude weight vanishes in the absence of
periodic potential due to Galilean invariance. We consider first the case
where there is no periodic potential. The Drude weight can be written [50] as

$D_{\alpha\beta}=(ne^{2}/m)\delta_{\alpha\beta}+S\sum_{n\neq
m}f_{nm}\frac{[j_{\alpha}]_{nm}[j_{\beta}]_{mn}}{E_{n}-E_{m}}$ (S10)

where $f_{nm}=f_{n}-f_{m}$ and $f_{n}=(e^{\beta(E_{n}-\mu)}+1)^{-1}$.
$\mathbf{j}=-e\boldsymbol{\Pi}/m$ is the current operator in the quantum Hall
problem with uniform magnetic field. The second term can be simplified using
the matrix representation of the current operator

$[j_{\alpha}]_{cq;dq^{\prime}}=e\delta_{qq^{\prime}}(\sqrt{c+1}\delta_{c+1,d}+s\sqrt{c}\delta_{c-1,d})/(\sqrt{2s}\ell
mA)$ (S11)

where $s=\pm 1$ for $\alpha=x,y$ and $q$ is the intra-Landau level quantum
number (e.g. momentum as defined in the previous section). We then obtain

$\displaystyle D_{\alpha\alpha}$
$\displaystyle=\frac{ne^{2}}{m}\delta_{\alpha\beta}+\sum_{c\neq
d,q,q^{\prime}}f_{cd}\frac{[j_{\alpha}]_{cq;dq^{\prime}}[j_{\beta}]_{dq^{\prime};cq}}{E_{c}-E_{d}}$
(S12)
$\displaystyle=\frac{ne^{2}}{m}\delta_{\alpha\beta}+\frac{e^{2}}{2s\ell^{2}m^{2}A}\delta_{\alpha\beta}\sum_{c\neq
d,q,q^{\prime}}f_{cd}\delta_{qq^{\prime}}\frac{sd\delta_{c+1,d}+sc\delta_{c-1,d}}{E_{c}-E_{d}}$
$\displaystyle=\frac{ne^{2}}{m}\delta_{\alpha\beta}+\frac{e^{2}}{2\ell^{2}m^{2}A}\delta_{\alpha\beta}\left(\sum_{d,q}f_{d-1,d}\frac{d}{-\omega_{c}}+\sum_{c,q}f_{c,c-1}\frac{c}{\omega_{c}}\right)$
$\displaystyle=(\frac{ne^{2}}{m}-\frac{ne^{2}}{m})\delta_{\alpha\beta}=0.$

This is the familiar fact that longitudinal conductivity vanishes for Landau
levels at any chemical potential or temperature. Indeed, this is a consequence
of the $f$-sum rule and Kohn’s theorem [35], which dictates that the total
optical absorption in Landau levels is saturated by the absorption peak at
$\omega=\omega_{c}$.

We now turn to the case where there is a non-zero periodic potential. We have
shown in the previous section that this gives rise to an dispersion (S9). For
simplicity, let’s consider the lowest modulated Landau level at partial
filling. In the absence of interactions, the Drude weight in this case can be
expressed as

$\displaystyle D_{\alpha\beta}=2\pi e^{2}\int_{\rm
BZ}\frac{d^{2}k}{(2\pi)^{2}}\theta(\mu-\tilde{E}^{0}(\mathbf{k}))m^{-1}_{\alpha\beta}(\mathbf{k})$
(S13)

with $m^{-1}_{\alpha\beta}(\mathbf{k})$ the inverse effective mass given by

$\displaystyle
m^{-1}_{\alpha\beta}(\mathbf{k})=\frac{1}{\hbar^{2}}\frac{\partial^{2}\tilde{E}^{0}(\mathbf{k})}{\partial
k_{\alpha}k_{\beta}}$ (S14)

where $\alpha,\beta=x,y$. The integral runs over the Brillouin zone, $\theta$
is a step function, and $\mu$ is the chemical potential. Consider a periodic
potential with $D_{6}$ periodicity that is a sum of the lowest harmonics .
Using equation (S9), this is simplified to

$\tilde{E}^{0}(\mathbf{k})=-V_{0}e^{-l^{2}|\mathbf{g}_{0}|^{2}/4}\sum_{i=1,3,5}\cos(l^{2}\mathbf{g}_{i}\wedge\mathbf{k})$
(S15)

where $\mathbf{g}_{i}=|\mathbf{g}_{0}|(\cos[(i-1)\pi/3],\sin[(i-1)\pi/3])$ and
$|\mathbf{g}_{0}|=4\pi/(\sqrt{3}a_{0})$ with $a_{0}$ is the lattice constant
of the periodic potential. The integral in (S13) is generally non-zero unless
the Landau level is fully filled therefore we conclude that the addition of a
periodic potential that breaks Galilean invariance gives rise to a finite
Drude weight in the non-interacting limit.

Figure S1: Drude weight for periodically modulated Landau Levels (equation (2)
in the main text) evaluated at $\phi=0.0$ in (a) and (b) and $\phi=\phi_{\rm
min}$ in (c) and (d) corresponding to the flux that minimizes the energy.

##  Appendix D Additional results for the Drude weight

In the main text, we have calculated the Drude weight for interacting
periodically modulated Landau levels using the averaging method (c.f Fig. 4.
in the main text). Here we provide additional evidence that the Drude weight
vanishes for the periodically modulated Landau level model. In Fig. S1, we
show the Drude weight calculated at a zero flux $\phi=0$ and at flux
$\phi=\phi_{\rm min}$ which minimizes the ground state energy. In the CFL
phase (for example when $V_{0}/U_{0}=0.2$), the Drude weight monotonically
decreases as a function in system size as shown in Fig. S1(a) and Fig. S1(c)
suggesting that it vanishes in the thermodynamic limit. In the Fermi liquid
phase, we find the Drude weight to exhibit an opposite behavior as shown in
Fig. S1(b) and Fig. S1(d) suggesting it remains finite in the thermodynamic
limit. However, care should be exercised in interpolating due to the small
number of data points (We only considered systems that have equal aspect
ratio). The averaging method on the other hand mitigates all these subtleties
and provide a clear approach to the thermodynamic limit.

Figure S2: Drude weight for $\rm{MoTe_{2}}$ [17] at twist angle
$\theta=2.0^{\circ}$. (a) The Drude weight calculated at the flux that
minimizes the energy dispersion. (b) The average Drude weight ${\langle
D\rangle}$ as a function of the inverse dielectric constant. $N_{s}$ denotes
the number of sites.

##  Appendix E Drude weight for twisted $\text{MoTe}_{2}$

In this section, we study the Drude weight for the recently predicted
anomalous composite Fermi liquid phase in twisted $\rm{MoTe_{2}}$ [3, 4]. We
find a very similar behavior to the problem of periodically modulated Landau
levels discussed in the main text. We focus on the first valence band with a
Chern number $C=1$ [17]. We consider Coulomb interaction of the form
$V(\mathbf{q})=\frac{e^{2}}{\epsilon\epsilon_{0}}\frac{2\pi}{|\mathbf{q}|}$,
projected onto this band and perform exact diagonalization at half filling.
Previous studies [3] have shown evidence for a composite Fermi liquid phase
for $\epsilon=10$. We find that the CFL phase is stabilized for even weaker
interaction ($\epsilon^{-1}\approx 0.02$). As shown in Fig. S2, we find the
average Drude weight to vanish (Fig. S2(b)) when in the system is in the
composite Fermi liquid phase.

Figure S3: The momentum grids corresponding to the finite systems used in this
work. Each momentum point is labeled by an index $\kappa$.

##  Appendix F Exact Diagonalization and Drude weight calculation

In this paper, we use exact diagonalization (ED) on finite systems to study
the interacting Hamiltonian. The finite systems are are spanned by two
vectors,

$\displaystyle\mathbf{L}_{1}=m_{1}\mathbf{a}_{1}+n_{1}\mathbf{a}_{2}$ (S16)
$\displaystyle\mathbf{L}_{2}=m_{2}\mathbf{a}_{1}+n_{2}\mathbf{a}_{2}$

where $\mathbf{a}_{1}$ and $\mathbf{a}_{2}$ are two real space basis and
$m_{1},n_{1},m_{2},n_{2}$ are integers. We choose
$\mathbf{a}_{1}=a_{0}(\sqrt{3}/2,-1/2)$, $\mathbf{a}_{2}=a_{0}(0,1)$ where
$a_{0}$ is the lattice constant. This defines two reciprocal lattice vectors
$\mathbf{g}_{1}=\frac{4\pi}{\sqrt{3}a_{0}}(1,0)$ and
$\mathbf{g}_{2}=\frac{4\pi}{\sqrt{3}a_{0}}(1/2,\sqrt{3}/2)$. We work directly
in momentum space. The momentum points live on a grid given by

$\mathbf{k}=(2\pi\alpha+\phi_{1})\>\mathbf{b}_{1}+(2\pi\beta+\phi_{2})\>\mathbf{b}_{2}$
(S17)

where $\mathbf{b}_{1}$ and $\mathbf{b}_{2}$ are the basis vectors of the
momentum grid, defined as $\mathbf{b}_{i}\cdot\mathbf{L}_{j}=\delta_{ij}$ and
$\alpha,\beta$ are integers. $(\phi_{1},\phi_{2})\in[0,2\pi)$ are the flux
inserted through the cycles of the torus corresponding to generic twisted
boundary conditions.

In Fig. S3, we show the finite systems used in this work. We have only
considered finite systems that are fully $C_{6}$ symmetric and have equal
aspect ratio for a faithful representation of the models we studied.

For a given system, the interacting Hamiltonian is diagonalized and the low-
lying energy states are obtained as usual. We consider only threading a flux
$\phi$ in one direction, say in the $\mathbf{L}_{1}$ cycle. Due to the equal
aspect ratio of the clusters used and the invariance of the Hamilonian under
$C_{3}$ rotation, identical results are obtained when the flux is inserted in
the other direction instead.

In our work, we evaluate the Drude weight which is proportional to the second
derivative of the ground state energy with respect to the flux,
$\frac{\partial^{2}E_{0}(\phi)}{\partial\phi^{2}}$ where $E_{0}(\phi)$ is the
ground state of the system obtained from the ED at flux $\phi$. To evaluate a
single $\frac{\partial^{2}E_{0}(\phi)}{\partial\phi^{2}}$ , we use numerical
finite-difference method
$\frac{\partial^{2}E_{0}(\phi)}{\partial\phi^{2}}\approx(E_{0}(\phi+\delta\phi)-2E_{0}(\phi)+E_{0}(\phi-\delta\phi))/(\delta\phi^{2})$
where $\delta\phi$ is chosen to be $\delta\phi=2\pi/(20)$

To evaluate the average Drude weight obtained by integrating
$\frac{1}{2\pi}\int
d^{2}\phi\frac{\partial^{2}E_{0}(\phi)}{\partial\phi^{2}}$, we notice first
that if $E_{0}(\phi)$ is a smooth function of $\phi$ with no cusps, the
integral vanishes identically since it’s an integral of a smooth and periodic
function over the periodicity interval. The only way the integral can have a
finite value is when there exist cusps (discontinuities in the first
derivative) due to level crossings. In such a case, we identify all the cusp
points and the integral is calculated as

$\int
d^{2}\phi\frac{\partial^{2}E_{0}(\phi)}{\partial\phi^{2}}=\sum_{p}(\frac{\partial
E_{0}(\phi)}{\partial\phi}|_{p_{+}}-\frac{\partial
E_{0}(\phi)}{\partial\phi}|_{p_{-}})$

where $p$ runs over all cusp points and $\pm$ refers to evaluating the first
derivative from the left ($+$) or the right ($-$). By isolating all the cusp
points, the energy $E_{0}(\phi)$ as a function of flux is a piecewise function
defined over different intervals where in each interval $E_{0}(\phi)$ is
smooth. We then fit $E_{0}(\phi)$ to a second order polynomial of $\phi$ and
extract the first derivative $\partial E_{0}(\phi)/(\partial\phi)$ from the
fitted polynomial. We find this fitting procedure to give more accurate
results than evaluating the first derivative around the cusps using finite-
difference.