# Interaction driven Metal-Insulator Transition with Charge Fractionalization

Yichen Xu  Department of Physics, University of California, Santa Barbara, CA
93106, USA  Xiao-Chuan Wu  Department of Physics, University of California,
Santa Barbara, CA 93106, USA  Meng-Xing Ye  Kavli Institute for Theoretical
Physics, University of California, Santa Barbara, CA 93106, USA  Zhu-Xi Luo
Kavli Institute for Theoretical Physics, University of California, Santa
Barbara, CA 93106, USA  Chao-Ming Jian  Department of Physics, Cornell
University, Ithaca, New York 14853, USA  Cenke Xu  Department of Physics,
University of California, Santa Barbara, CA 93106, USA

###### Abstract

It has been proposed that an extended version of the Hubbard model which
potentially hosts rich correlated physics may be well simulated by the
transition metal dichalcogenide (TMD) moiré heterostructures. Motivated by
recent reports of continuous metal-insulator transition (MIT) at half filling,
as well as correlated insulators at various fractional fillings in TMD moiré
heterostructures, we propose a theory for the potentially continuous MIT with
fractionalized electric charges. The charge fractionalization at the MIT will
lead to various experimental observable effects, such as a large resistivity
as well as large universal resistivity jump at the continuous MIT. These
predictions are different from previously proposed theory for interaction-
driven continuous MIT. Physics in phases near the MIT will also be discussed.

##  I Introduction

Many correlated phenomena have been observed in graphene based moiré systems,
such as high temperature superconductivity (compared with the bandwidth of the
moiré bands), correlated insulators Cao et al. (2018a, b); Chen et al.
(2019a); Yankowitz et al. (2019); Saito et al. (2020); Stepanov et al.
(2020a); Chen et al. (2019b); Liu et al. (2020); Cao et al. (2020a), and the
strange metal phase Cao et al. (2020b); Polshyn et al. (2019), etc. The most
fundamental reason for the emergence of these correlated physics is that the
slow modulating moiré potential leads to very narrow bandwidths Bistritzer and
MacDonald (2011); Lopes dos Santos et al. (2012). Great theoretical interests
and efforts have been devoted to the graphene based moiré systems Xu and
Balents (2018); Yuan and Fu (2018); Isobe et al. (2018); Koshino et al.
(2018); Thomson et al. (2018); Dodaro et al. (2018); Po et al. (2018); Kang
and Vafek (2018); Zou et al. (2018); You and Vishwanath (2018); Bultinck et
al. (2020a, b); Wu et al. (2018a); Lian et al. (2019); Lee et al. (2019);
Khalaf et al. (2021); Xu et al. (2020a); Fernandes and Venderbos (2020). But
the theoretical description and understanding of the graphene based moiré
systems may be complicated by the fact that in the noninteracting limit the
moiré mini bands can have various types of either robust or fragile nontrivial
topologies Chittari et al. (2019); Serlin et al. (2019); Zhang et al. (2019);
Chen et al. (2020a); Repellin and Senthil (2019); Stepanov et al. (2020b);
Chen et al. (2020b); Pierce et al. (2021); Wu et al. (2019a, b), although the
exact role of the band topology to the interacting physics at fractional
filling is not entirely clear. Hence similar narrow band systems with trivial
band topology and unambiguous concise theoretical framework would be highly
desirable. It was proposed that much of the correlated physics of the
transition metal dichalcogenide (TMD) moiré heterostructure can be captured by
an extended Hubbard model with an effective spin-1/2 electron on a triangular
moiré lattice Wu et al. (2018b)

$\displaystyle
H=\sum_{{\bm{r}},{\bm{r}}^{\prime},\alpha}-t_{{\bm{r}},{\bm{r}}^{\prime}}c^{\dagger}_{{\bm{r}},\alpha}c_{{\bm{r}}^{\prime},\alpha}+H.c.+\sum_{\bm{r}}Un_{{\bm{r}},\uparrow}n_{{\bm{r}},\downarrow}+\cdots$
(1)

The electron operator $c_{{\bm{r}},\alpha}$ is constructed by states within a
topologically trivial moiré mini band. Due to the strong spin-orbit coupling,
the spin and valley degrees of freedom are locked with each other in the TMD
moiré system. We will use $\alpha=\uparrow,\downarrow$ or $1,2$ to denote two
spin or equivalently two valley flavors. When a moiré band is partially
filled, the correlated physics within the partially filled moiré mini bands
may be well described by Eq. 1, which only contains half of the degrees of
freedom of a mini band in a graphene based moiré system. The ellipsis in Eq. 1
can include further neighbor hopping, “spin-orbit” coupling terms allowed by
symmetry Pan et al. (2020), and further neighbor interaction. Note the “spin-
orbit” coupling here refers to the hopping terms in Eq. 1 that depend on the
spin index $\alpha$ and should not be confused with the bare spin-orbit
coupling within the TMD system before the moiré superlattice is imposed. The
TMD moiré systems are hence considered as a simulator for the extended Hubbard
model on a triangular lattice Tang et al. (2019).

Like the graphene-based moiré systems, the TMD moiré heterostructure is a
platform for many correlated physics. This manuscript mainly concerns the
metal-insulator transition (MIT) driven by interaction. The MIT of the Hubbard
model on a triangular lattice has attracted much numerical efforts recently
Szasz et al. (2020); Szasz and Motruk (2021). The symmetry of the TMD moiré
heterostructure is different from the simplest version of the Hubbard model,
hence even richer physics can happen in the system. Continuous MIT has been
reported at half-filling of the moiré bands (electron filling $\nu=1/2$, or
one electron per moiré unit cell on average) in the TMD moiré system Li et al.
(2021); Ghiotto et al. (2021). The experimental tuning parameter of the MIT in
the TMD heterostructure is the displacement field, i.e. an out-of-plane
electric field, which tunes the width of the mini moiré bands, and hence the
ratio between the kinetic and interaction energies in the effective Hubbard
model. Correlated insulators have also been observed at various other
fractional electron fillings, though the nature of the MITs at these
fractional fillings have not been thoroughly inspected experimentally Regan et
al. (2020); Jin et al. (2021); Xu et al. (2020b); Huang et al. (2021). In this
manuscript we will mainly focus on $\nu=1/2$, but other fractional fillings
will also be briefly discussed.

The nature of an interaction driven MIT depends on the nature of the insulator
phase near the MIT. The Hubbard model on the triangular lattice has one site
per unit cell, which based on the generalized Lieb-Shultz-Matthis theorem Lieb
et al. (1961); Hastings (2004) demands that the insulator phase at half-
filling should not be a trivial incompressible (gapped) state which preserves
the translation symmetry. If the insulator phase has a semiclassical spin
order that breaks the translation symmetry, the evolution between the metal
and insulator could involve two transitions: at the first transition a
semiclassical spin order develops, which reduces the Fermi surface to several
Fermi pockets; and at the second transition the size of the Fermi pockets
shrink to zero, and the system enters an insulator phase. A more interesting
scenario of the MIT only involves one single transition Lee and Lee (2005);
Senthil (2008a, b), but then the insulator phase is not a semiclassical spin
order, instead it is a spin liquid state with a spinon Fermi surface. An
intuitive picture for this transition is that, at the MIT, the charge degrees
of freedom are gapped, but the spins still behave as if there is a “ghost”
Fermi surface. The spinon Fermi surface can lead to the Friedel oscillation
just like the metal phase Mross and Senthil (2011). The structure of the Fermi
surface does not change drastically across the transition.

In a purely two dimensional system, conductivity (or resistivity) is a
dimensionless quantity, hence it can take universal value at the order of
$e^{2}/h$ (or $h/e^{2}$) in various scenarios. For example, the Hall
conductivity of the quantum Hall state is precisely $\sigma_{H}=\nu e^{2}/h$;
the conductivity (or resistivity) at a $(2+1)d$ quantum critical point also
takes a universal value at the order of $e^{2}/h$ (or $h/e^{2}$) Cha et al.
(1991). One central prediction given by the theory above for interaction
driven continuous MIT is that, there is a universal resistivity jump at the
order of $\sim h/e^{2}$ at the MIT compared with the metal phase. In this
manuscript we will argue that the current experimental observations suggest
that the nature of the MIT in MoTe2/WSe2 moiré superlattice without twisting
Li et al. (2021) is beyond the theory that was discussed before Lee and Lee
(2005); Senthil (2008a, b), and we propose an alternative candidate theory of
MIT with further charge fractionalizations. We will discuss how the
alternative theory can potentially address the experimental puzzles, and more
predictions based on our theory will be made. Our assumption is that the MIT
in this system is indeed driven by electron-electron interaction (as was
suggested by Ref. Li et al., 2021); If the disorder plays the dominant role in
this system, the MIT may be described by the picture discussed in Ref. Spivak
et al., 2010.

The paper is organized as follows: In section II we introduce an alternative
parton construction for systems described by the extended Hubbard model with a
spin-orbit coupling, which naturally leads to charge fractionalization at the
interaction-driven MIT even at half-filling; we also give an intuitive
argument of physical effects caused by charge fractionalization at the MIT. In
section III, we will discuss the theory for MIT when the insulating phase
spontaneously breaks the translation symmetry. Section IV studies the theory
of MIT when the insulating phase has different types of topological orders. In
section V we discuss various experimental predictions based our theory, for
the MIT and also the phases nearby. We present the details of our theory in
the appendix, including the projective symmetry group, field theories, and
calculation of DC resistivity, etc.

##  II Two Parton constructions

The previous theory for the interaction-driven continuous MIT for correlated
electrons on frustrated lattices was based on a parton construction. The
parton construction splits the quantum number of an electron into a bosonic
parton which carries the electric charge, and a fermionic parton which carries
the spin. In the current manuscript we compare two different parton
constructions:

$\displaystyle\mathrm{I}:c_{{\bm{r}},\alpha}=b_{\bm{r}}f_{{\bm{r}},\alpha},\ \
\ \ \mathrm{II}:c_{{\bm{r}},\alpha}=b_{{\bm{r}},\alpha}f_{{\bm{r}},\alpha}.$
(2)

In parton construction-I only one species of charged bosonic parton $b$ is
introduced for electrons with both spin/valley flavors; while in parton
construction-II a separate charged bosonic parton $b_{\alpha}$ is introduced
for each spin/valley flavor. As we will see later, the two different parton
constructions will lead to very different observable effects. The
construction-I is the standard starting point of the theory of MIT that was
used in previous literature Lee and Lee (2005); Senthil (2008a, b);
construction-II is usually unfavorable for systems with a full spin SU(2)
invariance, because the parton construction itself breaks the spin rotation
symmetry. But the construction-II is a legitimate parton construction for the
system under study, whose band structure in general does not have full
rotation symmetry between the two spin/valley flavors.

The time-reversal symmetry of the microscopic TMD system relates the two
spin/valley flavors. But it is not enough to guarantee a full SU(2) rotation
symmetry between the two flavors. In fact, since the two flavors can be tied
to the two valleys of the TMD material, the trigonal warping of the TMD bands,
which takes opposite signs for the two different valleys, can lead to the
breaking of such an SU(2) rotation symmetry. To estimate the trigonal warping
effect in the Hubbard model, one can compare the $k^{2}$ term and the
$k_{x}^{3}-2k_{x}k^{2}_{y}$ term in the electron dispersion of one of the two
layers in the heterostructure expanded at one valley. Then the relative
strength of the trigonal warping compared to the SU(2)-invariant hopping in
Eq. 1 is given by the ratio between the lattice constant of the original TMD
material and that of the morié superlattice. In addition, the natural
microscopic origin of the interactions in the Hamiltonian Eq. 1 is the Coulomb
interaction between the electrons. The Coulomb interaction when projected to
the low-energy bands relevant to the moiré-scale physics is expected to
contain SU(2)-breaking interaction terms. The momentum conservation only
guarantees the valley U(1) symmetry. Assuming the unscreened Coulomb
interaction between electrons before the projection to the low-energy bands,
further neighbor interaction will appear in the extended Hubbard model. The
relative strength of the SU(2)-breaking interaction terms obtained from the
projection compared to the SU(2)-invariant interactions can again be estimated
by the ratio between the lattice constant of the original TMD material and the
moiré superlattice, as the Fourier transform of unscreened Coulomb interaction
in $2d$ space is $V_{q}\sim 1/q$.

The most important difference between these two parton theories resides in the
filling of the bosonic partons. Since each bosonic parton carries the same
electric charge as an electron, the total number of bosonic partons should
equal to the number of electrons. Hence at electron filling $\nu$ (meaning
$2\nu$ electrons per unit cell), the filling factor of boson $b$ in
construction-I is $\nu_{b}=2\nu$, i.e. $2\nu$ bosonic parton per unit cell; in
construction-II the filling factor of boson $b_{\alpha}$ has filling factor
$\nu^{\alpha}_{b}=\nu$ for each spin/valley flavor. Hence even with one
electron per site (half-filing or $\nu=1/2$ of the extended Hubbard model),
the bosonic parton in construction-II is already at half filling for each
spin/valley flavor. The half-filling will lead to nontrivial features inside
the Mott insulator phase, as well as at the MIT. Another more theoretical
difference is that, in construction-I there is one dynamical emergent ${\rm
U}(1)$ gauge field $a_{\mu}$ which couples to both $b$ and $f_{\alpha}$; while
in construction-II there are two dynamical ${\rm U}(1)$ gauge fields
$a_{\alpha,\mu}$, one for each spin/valley flavor.

The current experimental observation of the MIT in MoTe2/WSe2 moiré
heterostructure at $\nu=1/2$ also suggests that construction-II may be a more
desirable formalism in the system. In construction-I, the bosonic parton $b$
is at integer filling, and the MIT is naturally interpreted as a superfluid to
Mott insulator (SF-MI) transition of boson $b$. At the MIT, using the Ioffe-
Larkin rule Ioffe and Larkin (1989), the DC resistivity of system is
$\rho=\rho_{b}+\rho_{f}$, where $\rho_{b}$ and $\rho_{f}$ are the resistivity
contributed by the bosonic and fermionic partons respectively. $\rho_{f}$
caused by disorder or interaction such as the Umklapp process is a smooth
function of the tuning parameter, the drastic change of $\rho$ across the MIT
arises from $\rho_{b}$. In the metal phase, i.e. the “superfluid phase” of
$b$, $\rho_{b}$ is zero, and the total resistivity is just given by
$\rho_{f}$. Also, in the superfluid phase of $b$, the ${\rm U}(1)$ gauge field
$a_{\mu}$ that couples to both $b$ and $f_{\alpha}$ is rendered massive due to
the Higgs mechanism caused by the condensate of $b$. In the insulator phase,
$\rho_{b}$ and $\rho$ are both infinity, and the system enters a spin liquid
phase with a spinon Fermi surface of $f_{\alpha}$ that couples to the
dynamical ${\rm U}(1)$ gauge field $a_{\mu}$. The MIT which corresponds to the
condensation of $b$ belongs to the 3D XY universality class. The dynamical
gauge field $a_{\mu}$ is argued to be irrelevant at the transition due to the
overdamping of the gauge field that arises from the spinon Fermi surface
Senthil (2008a, b), and hence does not change the universality class of the
SF-MI transition of $b$.

In parton construction-I, at the MIT the bosonic parton contribution to the
resistivity $\rho_{b}$ is given by $\rho_{b}=Rh/e^{2}$, where $R$ is an
order-1 universal constant. In the order of limit $T\rightarrow 0$ before
$\omega\rightarrow 0$, $R$ is associated to the 3D XY universality class
Fisher et al. (1990), because the gauge field $a_{\mu}$ is irrelevant as
mentioned above. This universal conductivity at the 3D XY transition has been
studied through various analytical and numerical methods Cha et al. (1991);
Fazio and Zappalà (1996); Šmakov and Sørensen (2005); Witczak-Krempa et al.
(2014); Chen et al. (2014a); Chester et al. (2020); Haviland et al. (1989);
Liu et al. (1991); Lee and Ketterson (1990). At finite $T$ and zero frequency,
the gauge field $a_{\mu}$ can potentially enhance the value $R$ to
$R^{(0)}>R$, based on a large-$\bf{N}$ calculation in Ref. Witczak-Krempa et
al., 2012 ($\bf{N}$ is different from $N$ in our work). The evaluation in Ref.
Witczak-Krempa et al., 2012 gave $R^{(0)}\sim 7.92$, while we evaluate the
same quantity to be $R^{(0)}\sim 7.44$. Hence the prediction of the
construction-I is that, the DC resistivity of the system right at the MIT has
a universal jump compared with the resistivity at the metallic phase close to
the MIT Senthil (2008a, b), i.e. $\Delta\rho=\rho_{b}=R^{(0)}h/e^{2}$. With
moderate disorder, at the MIT $\rho_{b}$ of the bosonic parton is supposed to
dominate the resistivity $\rho_{f}$ of the fermionic parton $f_{\alpha}$,
hence the total resistivity $\rho=\rho_{b}+\rho_{f}$ should be close to
$\rho_{b}$.

In the experiment reported on the MoTe2/WSe2 moiré superlattice, deep inside
the metallic phase the value of resistivity suggests that disorder in the
system is not playing a dominant role, and the continuous MIT is mainly driven
by the interaction Li et al. (2021). However, the reported resistivity $\rho$
increases rapidly with the tuning parameter (the displacement field) near the
MIT. The bare value of $\rho$ near and at the MIT is significantly greater
than $h/e^{2}$ (and significantly larger than the computed value of
$\rho_{b}\sim R^{(0)}h/e^{2}$ mentioned above), and it is clearly beyond the
Mott-Ioffe-Regel limit, i.e. the system near and at the MIT is a “bad metal”
Emery and Kivelson (1995); Hussey et al. (2004). Hence the resistivity jump
$\Delta\rho_{b}$ at the MIT is likely significantly larger than $h/e^{2}$.
This suggests that the MIT is not a simple SF-MI transition of $b$, or in
other words $b$ should be “much less conductive” compared with what was
predicted in construction-I considered in previous literature. We will
demonstrate that construction-II can potentially address the large resistivity
at the MIT. The most basic picture is that, since $b_{1}$ and $b_{2}$ are both
at half-filling, the LSM theorem Lieb et al. (1961); Hastings (2004) dictates
that the Mott insulator phase of each flavor of boson cannot be a trivial
insulator, namely the Mott insulator must either be a density wave that
spontaneously breaks the translation symmetry, or have topological order. In
either case, The MIT is not a simple 3D XY transition, and the most prominent
feature of the transition is that, the bosonic parton number (or the electric
charge) must further fractionalize.

The MIT with charge fractionalization will be discussed in detail in the next
section using the dual vortex formalism, but the consequence of this charge
fractionalization can be understood from a rather intuitive picture. Suppose
$b$ fractionalizes into $N$ parts at the MIT, meaning the charge carriers at
the MIT have charge $e_{\ast}=e/N$, then each charge carrier will
approximately contribute a resistivity at the order of $h/e_{\ast}^{2}\sim
N^{2}h/e^{2}$ at the MIT; and if there are in total $N_{b}$ species of the
fractionalized charge carriers, at the MIT the bosonic parton will
approximately contribute resistivity

$\displaystyle\rho_{b}\sim\frac{N^{2}h}{N_{b}e^{2}}.$ (3)

There is a factor of $N_{b}$ in the denominator because intuitively the total
conductivity of $b$ will be a sum of the conductivity of each species of
fractionalized charge carriers, i.e.
$\sigma_{b}=\sum_{j=1}^{N_{b}}\sigma_{j}$, in the unit of $e^{2}/h$ (a more
rigorous rule of combining transport from different partons will be discussed
later). Hence when $N^{2}/N_{b}>1$, the construction-II with inevitable charge
fractionalization can serve as a natural explanation for the large
$\Delta\rho$ and bad metal observed at the MIT.

##  III Mott insulator with translation symmetry breaking

###  III.1 General Formalism

In this section we will discuss the MIT following the parton construction-II
discussed in the previous section. The MIT is still interpreted as the SF-MI
transition of both spin/valley flavors of the bosonic parton $b_{\alpha}$,
although as we discussed previously the insulator cannot be a trivial
incompressible state of $b_{\alpha}$. In the superfluid phase of $b_{\alpha}$,
both ${\rm U}(1)$ gauge fields $a_{1,\mu}$ and $a_{2,\mu}$ that couple to the
two flavors of partons are gapped out by the Higgs mechanism, and the system
enters a metal phase of the electrons; $b_{1}$ and $b_{2}$ must undergo the
SF-MI transition simultaneously, since the time-reversal or spatial reflection
symmetries both interchange the two flavors of partons due to the spin-valley
locking.

Figure 1: The triangular moiré lattice, and its dual honeycomb lattice. In the
parton construction-II, the bosonic parton $b_{\alpha}$ is at half-filling for
each spin/valley flavors, which becomes a $\pi-$flux of the dual gauge field
$A_{\mu}$ through the hexagon of the dual honeycomb lattice. Hence the vortex
$\psi$ defined on the dual honeycomb lattice does not have a uniform hopping
amplitude, the dashed links on the dual honeycomb lattice have negative
hopping amplitudes. The symmetry of the lattice will be realized as a
projective symmetry group. There are eight dual sites per unit cell (shaded
area) in this gauge choice. At each spin/valley flavor, there are translation
symmetries $T_{1,2}$, a rotation symmetry $R_{\frac{2\pi}{3}}$, and a product
of reflection $P_{x}(x\rightarrow-x)$ and time-reversal $\mathcal{T}$. We also
argue that $P_{y}$ is a symmetry of the system as long as there is no valley
mixing; and the six-fold rotation $R_{\pi/3}$ becomes a good approximate
symmetry of the Hubbard model in the case of long moiré lattice constant.

The dual vortex theory Peskin (1978); Dasgupta and Halperin (1981); Fisher and
Lee (1989) is the most convenient formalism that describes a transition
between a superfluid and a nontrivial insulator of a boson at fractional
filling. If we start with a boson $b$, after the boson-vortex duality, a
vortex of the superfluid phase of $b$ becomes a point particle that couples to
a dynamical ${\rm U}(1)$ gauge field $A_{\mu}$, which is the dual of the
Goldstone mode of the superfluid (not to be confused with the ${\rm U}(1)$
gauge field $a_{\mu}$ mentioned before that couples to the bosonic parton
$b$). In the dual picture, the superfluid phase of $b$ (which corresponds to
the metal phase of the electron) is the insulator phase of the vortex field;
while the Mott insulator phase of $b$ corresponds to the condensate of the
vortices, which “Higgses” the ${\rm U}(1)$ gauge field $A_{\mu}$, and drives
the boson $b$ into a gapped insulator phase. If at low energy there is only
one component of vortex field with gauge charge $1$ under $A_{\mu}$ (which
corresponds to integer filling of boson $b$), the insulator phase of $b$ is a
trivial insulator without any further symmetry breaking or topological order;
if there are more than one component of the vortex fields at low energy, or if
the vortex field carries multiple gauge charges of $A_{\mu}$, the insulator
must be of nontrivial nature.

For example, when $b$ has a fractional filling $\nu_{b}=1/q$ with integer $q$,
Ref. Balents et al., 2005; Burkov and Balents, 2005 studied the quantum phase
transition between the bosonic SF and various MIs with commensurate density
waves which spontaneously break the translation symmetry but have no
topological order. The study is naturally generalized to filling factor
$\nu_{b}=p/q$ with coprime integers $(p,q)$. We can use this formalism in our
system. Hereafter we focus on one spin/valley flavor $\alpha$, and the index
$\alpha$ will be hidden for conciseness. In this case the theory for the SF-MI
transition at one spin/valley flavor is:

$\displaystyle\mathcal{L}^{(1)}$ $\displaystyle=$
$\displaystyle\sum_{j=0}^{N-1}(|(\partial_{\mu}-\mathrm{i}A_{\mu})\psi_{j}|^{2}+r|\psi_{j}|^{2})+u(\sum_{j=0}^{N-1}|\psi_{j}|^{2})^{2}$
(6) $\displaystyle+\frac{\mathrm{i}}{2\pi}A\wedge
d(a+eA_{\mathrm{ext}})+\cdots$

Here $\psi_{j}$ with $j\in\\{0,\cdots N-1\\}$ are $N$ flavors of vortex fields
of the boson $b$ at low energy, and $A_{\mu}$ is the dual gauge field of boson
$b$: $\frac{1}{2\pi}dA=J_{b}$, where $J_{b}$ is the current of boson $b$.
$a_{\mu}$ is the gauge field that couples to both $b$ and $f$, and
$A_{\mathrm{ext}}$ is the external electromagnetic field. The reason there are
$N$ flavors of the vortex field is that, the vortex which is defined on a dual
honeycomb lattice will view the partially filled boson density as a fractional
background flux of the dual gauge field $A_{\mu}$ through each hexagon, and
the band structure of the vortex will have multiple minima in the momentum
space. The degeneracy of the multiple minima is protected by the symmetry of
the triangular lattice. $\psi_{j}$ transforms as a representation of the
projective symmetry group (PSG) of the lattice. Notice that since Eq. 6
describes one of the two spin/valley flavors, the PSG that constrains Eq. 6
should include translation, and $2\pi/3$ rotation of the lattice
($R_{\frac{2\pi}{3}}$). There is another more subtle symmetry
$P_{x}\mathcal{T}$ for each spin/valley flavor of the boson and vortex fields.
$P_{x}$ that takes $x\rightarrow-x$, and time-reversal $\mathcal{T}$ both
exchange the two spin/valley indices, but their product will act on the same
spin/valley species, and part of its role is to take momentum $k_{y}$ to
$-k_{y}$.

In the appendix we will argue that $P_{y}$ which takes $y$ to $-y$ within each
valley is also a good symmetry of the system, as long as valley mixing is
negligible. One consequence of the $P_{y}$ symmetry is that the expectation
value of gauge flux $da$ can be set to zero for the theory Eq. 6, or
equivalently the $P_{y}$ symmetry ensures that the “chemical potential” term
$\psi_{j}^{*}\partial_{\tau}\psi_{j}$ does not appear in Eq. 6, as $P_{y}$
transforms a vortex to anti-vortex: $\psi_{a}\rightarrow
U_{ab}\psi^{\ast}_{b}$. Also, with long moiré lattice constant, the trigonal
warping $k_{x}^{3}-3k_{x}k_{y}^{2}$ in each valley of the original BZ of the
system becomes less important compared with the leading order quadratic
dispersion expanded at each valley, hence the six-fold rotation $R_{\pi/3}$
becomes a good approximate symmetry of the effective Hubbard model with long
moiré lattice constant.

The theory in Eq. 6 also has an emergent particle-hole symmetry. The simplest
choice of the particle-hole symmetry is $\psi_{a}\rightarrow
U_{ab}\psi^{*}_{b}$, $A\rightarrow-A$, $a\rightarrow-a$ and $A_{\rm
ext}\rightarrow-A_{\rm ext}$. Although we used the same transformation matrix
$U_{ab}$ as $P_{y}$, this emergent particle-hole symmetry is different from
$P_{y}$ as it does not involve any spatial transformations. However, note that
any (spatially uniform) $P_{y}$-symmetric terms involving only the “matter
fields” $\psi_{j}$ must also preserve this emergent particle-hole symmetry.
Another potentially relevant particle-hole-symmetry-breaking perturbations
that needs to be examined is given by the finite density of the fluxes $dA$.
$dA$ is tied to the physical U(1) charge density (compared to the charge
density set by the fixed electron filling $\nu=1/2$) and hence should have a
vanishing spatial average. At the SF-MI transition point, the translation
symmetry of the theory Eq. 6 and the fact that $dA$ has a vanishing spatial
average guarantee that $dA$ has a vanishing expectation value everywhere,
which respects the particle-hole symmetry. Therefore, the particle-hole
symmetry is a valid emergent symmetry at the SF-MI critical point described by
Eq. 6. The same argument would also conclude the emergent particle-hole
symmetry at the ordinary SF-MI transition in the Bose-Hubbard model.

For parton construction-II, when the electron has filling $\nu=1/2$, both
$b_{1}$ and $b_{2}$ are at filling $\nu^{\alpha}_{b}=1/2$. For each flavor of
$b_{\alpha}$, the formalism in Ref. Burkov and Balents, 2005 would lead to a
dual vortex theory with $N=4$ components of vortex fields, i.e. there are four
degenerate minima of the vortex band structure in the momentum space for each
spin/valley index. This calculation is analogous to the frustrated Ising model
on the honeycomb lattice Moessner and Sondhi (2001); Xu and Sachdev (2009).
Using the gauge choice of Fig. 1, the four minima are located at the $K$ and
$K^{\prime}$ points of the reduced Brillouin zone (BZ), with two fold
degeneracy at each point.

###  III.2 From $N=4$ to “$N=\infty$”

Ref. Burkov and Balents, 2005 considered a specific band structure of the
vortex, which only involved the nearest neighbor hopping of vortices on the
dual honeycomb lattice. But there is no fundamental reason that further
neighbor hopping of vortices should be excluded. Indeed, once we take into
account of further neighbor hopping, the dual vortex theory has a much richer
possibility. We have explored the phase diagram of the dual vortex theory up
to seventh neighbor hopping, and we obtained the phase diagram in Fig. 2$a$.
Further neighbor hopping of the vortex field can modify the band structure,
and lead to $N=6$ or $N=12$ components of vortex fields by choosing different
hopping amplitudes. The $N=6$ minima are located at three inequivalent $M$
points of the reduced BZ (Fig. 2), each $M$ point again has two-fold
degeneracy. The two-fold degeneracy at each $M$ point is protected by the
translation symmetry of the triangular moiré lattice only, which is required
by the LSM theorem. The shift of the vortex field minima from the $K$ points
to $M$ points is similar to what was discussed in the context of frustrated
quantum Ising models with further neighbor couplings Slagle and Xu (2014); Xu
and Balents (2011). With symmetries $T_{1,2}$, $R_{\frac{2\pi}{3}}$ and
$P_{x}\mathcal{T}$ at each spin/valley flavor, the degeneracy of the $N=6$
minima at the $M$ points are protected.

Figure 2: (a) The minima of the vortex band structure. With nearest neighbor
vortex hopping on Fig. 1, the minima locate at the $K$ and $K^{\prime}$ points
of the Brillouin zone, each $K$ point has two fold degeneracy; with further
neighbor hoppings, the minima can shift to the three $M$ points, still with
two fold degeneracy at each $M$ point. (b) The phase diagram of vortex modes
with seventh neighbor hopping $t_{7}=0.1t_{1}$, and by tuning $t_{2}$ there
are two regions in the phase diagram with $N=12$ vortex modes at low energy.
The 12 vortex modes are located either on the lines between $\Gamma$ and
$K/K^{\prime}$ or $\Gamma$ and $M$. (c) With only $t_{1}$ and $t_{2}$, there
is a large region of the phase diagram where there is a ring degeneracy of the
vortex band structure. (d) All the symmetries (including approximate
symmetries) of the system can protect up to 24 degenerate vortex modes, which
locate at 12 incommensurate momenta in the BZ.

There are two regions in the phase diagram in Fig. 2$b$ with $N=12$ modes of
vortex, two at each momentum. The six incommensurate momenta at the minima of
the vortex band structure can be located either on the lines between $\Gamma$
and $K/K^{\prime}$ or $\Gamma$ and $M$. With the $R_{\pi/3}$ symmetry that
becomes a good approximate symmetry with long moiré lattice constant, the
degeneracy of the $N=12$ vortex modes is protected. In principle, all the
symmetries together including $R_{\pi/3}$ can protect up to $N=24$ degenerate
minima, as shown in Fig. 2$d$.

For a theory with $N$ components of vortex fields, the electric charge carried
by the boson $b$ will fractionalize. Under the boson-vortex duality
$\frac{1}{2\pi}dA=J_{b}$, the boson number of $b$ becomes the flux number of
the dual gauge field $A_{\mu}$. The gauge flux of $A_{\mu}$ is trapped at the
vortex core of each field $\psi_{j}$ (we denote the vortex of $\psi_{j}$ as
$\varphi_{j}$). With $N$ components of the vortex fields, the vortex of each
$\psi_{j}$ field will carry $1/N$ flux quantum of the gauge field $A_{\mu}$,
hence the charge $e_{\ast}$ of each fractionalized charge carrier should be
$e/N$ at the MIT. And there are in total $N_{b}=2N$ species of the charge
carriers (the factor of 2 comes from the two spin/valley flavors).

With just $t_{1}$ and $t_{2}$ (first and second neighbor vortex hopping),
there is a large region of the parameter space where the minima of the vortex
band structure form a ring. This one dimensional ring degeneracy is not
protected by the symmetry of the system, but its effect may still be
observable for a finite energy range. A ring degeneracy is analogous to
$N=\infty$ in Eq. 6. Condensed matter systems with a ring degeneracy have
attracted considerable interests Wu et al. (2008); Wang et al. (2010); Zhang
and Chen (2021); Lake et al. (2021). By integrating out the vortices with ring
degeneracy, a “mass term” for the transverse component of $A_{\mu}$ is
generated in the infrared limit Lake et al. (2021) (in the limit of momentum
goes to zero before frequency), meaning the fluctuation of $A_{\mu}$ is highly
suppressed, which is consistent with the intuition of $N=\infty$.

The ellipsis in Eq. 6 includes other terms allowed by the PSG of the
triangular lattice, but break the enlarged flavor symmetry of the CPN-1 model
field theory. More details about PSG, extra terms in the Lagrangian, coupling
to fermionic parton $f_{\alpha}$ Musser et al. (2021), and the possible
valence bond solid orders with $N=6$ will be discussed in appendix A and B.
The exact fate of the critical theory in the infrared is complicated by these
extra perturbations. It was shown previously that nonlocal interactions can
drive a transition to a new fixed point Grover and Vishwanath (2012); Xu et
al. (2020c); Jian et al. (2021), and here nonlocal interactions arise from
coupling to the fermionic partons Musser et al. (2021). Hence the transition
may eventually flow to a CFT different from the CPN-1 theory in Eq. 6, or be
driven to a first order transition eventually. But as long as the first order
nature is not strong, the charge fractionalization and large resistivity to be
discussed in the next subsection is expected to hold at least for a
considerable energy/temperature window.

So far we have not paid much attention to the dynamical gauge fields $a_{\mu}$
in parton construction-I or $a_{\alpha,\mu}$ in construction-II shared by the
bosonic and fermionic partons, as the gauge coupling between $b$
($b_{\alpha}$) and the gauge field is irrelevant at the MIT with a background
spinon Fermi surface. Here we briefly discuss the fate of the spinon Fermi
surface in the insulator phase. When the bosonic parton $b$ is gapped, the
theory of spinon Fermi surface coupled with the dynamical ${\rm U}(1)$ gauge
field is a problem that has attracted a great deal of theoretical efforts
Polchinski (1994); Nayak and Wilczek (1994a, b); Lee (2009); Mross et al.
(2010); Metlitski and Sachdev (2010a, b). These studies mostly rely on a
“patch” theory approximation of the problem, which zooms in one or two patches
of the Fermi surface. Then an interacting fixed point with a nonzero gauge
coupling is found in the IR limit based on various analytical perturbative
expansion methods.

Previous studies have also shown that the non-Fermi liquid obtained through
coupling a Fermi surface to a dynamical bosonic field can be instable against
BCS pairing of fermions Metlitski et al. (2015); Wang and Chubukov (2015);
Mandal (2016); Lederer et al. (2015); Wang et al. (2016); Lederer et al.
(2017); Zou and Chowdhury (2020). If there is only one flavor of ${\rm U}(1)$
gauge field, the low energy interacting fixed point is expected to be robust
against this pairing instability, because the ${\rm U}(1)$ gauge field leads
to repulsive interaction between the spinons. However, when there are two
flavors of ${\rm U}(1)$ gauge fields, like the case in our parton
construction-II, the two ${\rm U}(1)$ gauge fields can lead to interflavor
spinon pairing instability Zou and Chowdhury (2020). This interflavor pairing
can still happen at the MIT. But depending on the microscopic parameters this
instability can happen at rather low energy scale.

###  III.3 Resistivity at the MIT

For low frequency and temperature, the resistivity of a system is usually
written as $\rho(x)$ with $x=\omega/T$. The DC conductivity at zero
temperature corresponds to $x=0$, i.e. the limit $\omega\rightarrow 0$ before
$T\rightarrow 0$. As we have mentioned, the interaction driven MIT has a jump
of resistivity at the MIT compared with the metal phase near MIT, and this
jump is given by the resistivity $\rho_{b}$ of the bosonic parton
$b_{\alpha}$. For a bosonic system with an emergent particle-hole symmetry in
the infrared, $\rho_{b}(x)$ with $x=0$ or $x=\infty$ have attracted most
studies. In general both $\rho_{b}(0)$ and $\rho_{b}(\infty)$ should be
universal numbers at the order of $\sim h/e^{2}$. The reason $\rho_{b}(0)$
could be finite even without considering disorder and Umklapp process is that,
with an emergent particle-hole symmetry in the infrared discussed in the
previous subsection, there is zero overlap between the electric current and
the conserved momentum density. The universal $\rho_{b}(0)$ was evaluated in
Ref. Witczak-Krempa et al., 2012 for the interaction-driven MIT without charge
fractionalization. The calculation therein was based on Boltzmann equation in
a theoretical large$-\bf{N}$ limit and eventually $\bf{N}$ was taken to 1 (we
remind the readers that the $\bf{N}$ introduced in Ref. Witczak-Krempa et al.,
2012 was for technical reasons, it is not to be confused with $N$ used in this
work).

We have generalized the computation in Ref. Witczak-Krempa et al., 2012 to our
case with $N-$components of vortex fields and charge fractionalization. To
proceed with the computation we need to turn on “easy plane” anisotropy to Eq.
6 and perform duality to the basis of fractional charge carriers $\varphi_{j}$
(Eq. 63). The $\varphi_{j}$ will be coupled to multiple gauge fields which are
the dual of the $\psi_{j}$ fields. Eventually the total resistivity
$\rho_{b}(0)$ is obtained through a generalized Ioffe-Larkin rule, which
combines the resistivity of each parton $\varphi_{j}$ into $\rho_{b}$:

$\displaystyle\rho_{b}=\frac{\hbar}{e^{2}}\left(\sum_{j=0}^{N-1}\rho_{b,j}\right).$
(7)

$\rho_{b,j}$ is the resistivity of each charge carrier $\varphi_{j}$ when its
charge is taken to be $1$. The detail of the computation is presented in the
appendix, and we summarize the results here. For $N$ flavors of vortices in
Eq. 6, the resistivity $\rho_{b}(0)$ at the MIT roughly increases linearly
with $N$, as was expected through the intuitive argument we gave before:

$\displaystyle\rho_{b}(0)=\Delta\rho=\left(R^{(0)}+R^{(1)}(N-1)\right)\frac{h}{e^{2}},$
(8)

where $R^{(0)}\sim 3.62$, $R^{(1)}\sim 1.68$. We would like to compare our
prediction with the previous theory of MIT without charge fractionalization.
In the previous theory, the DC resistivity jump is evaluated to be
$\Delta\rho\sim 7.92h/e^{2}$ Witczak-Krempa et al. (2012) (we reproduced this
calculation and our result at $N=N_{b}=1$ is $7.44h/e^{2}$). Eq. 3 suggests
that when $N\geq 4$, the resistivity jump in our case is indeed larger than
that predicted by the previous theory of MIT.

We would also like to discuss the AC resistivity $\rho_{b}(\infty)$. One way
to evaluate $\rho_{b}(\infty)$ is to again start with Eq. 63, and follow the
same strategy as the calculation of the DC resistivity. According to the
generalized Ioffe-Larkin rule, the AC resistivity contributed by each valley
is given by

$\displaystyle\rho_{b}=N\frac{1}{\sigma_{\varphi}}\frac{\hbar}{e^{2}},\ \ \
\sigma_{\varphi}=\lim_{\omega\rightarrow 0}\frac{1}{\mathrm{i}\omega}\langle
J^{\varphi}_{\omega}J^{\varphi}_{-\omega}\rangle_{\vec{p}=0},$ (9)

where $J^{\varphi}=\mathrm{i}\varphi^{\ast}_{j}\nabla\varphi_{j}+h.c.$ is the
current of the charge carrier $\varphi_{j}$. With the theoretical
large-$\bf{N}$ limit mentioned above, the effects of all the dynamical gauge
fields are suppressed, and $\varphi_{j}$ will contribute conductivity
$\sigma_{\varphi}(\infty)=\frac{1}{16}$ (contrary to DC transport,
$\sigma_{\varphi}(\infty)$ does not need collisions; the effects of dynamical
gauge fields can be included through the $1/{\bf N}$ expansion). Eventually
one would obtain resistivity from each valley

$\displaystyle\rho_{b}=\frac{8N}{\pi}\frac{h}{e^{2}},$ (10)

the final resistivity of the system is half of Eq. 10 due to the two
spin/valley flavors. With $N=1$, the transition should belong to the ordinary
3D XY universality class, and the value given by Eq. 10 is not far from what
was obtained through more sophisticated methods (see for instance Ref.
Witczak-Krempa et al., 2014; Šmakov and Sørensen, 2005; Chen et al., 2014a,
$\rho_{b}\sim 2.8h/e^{2}$). This should not be surprising as the 3D XY
universality class can be obtained perturbatively from the free boson theory.
In our current case with charge fractionalization, with $N\geq 4$, the total
AC resistivity which is half of the value in Eq. 10 is larger than the
universal resistivity at the 3D XY transition.

Another way to evaluate the resistivity of Eq. 6 is by integrating out
$\psi_{j}$ from Eq. 6, and an effective Lagrangian for $A_{\mu}$ is generated

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\sum_{p_{\mu}}\frac{Np}{16}\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)A_{\mu}(p)A_{\nu}(-p).$
(11)

This effective action is supposed to be accurate in the limit of
$N\rightarrow\infty$. The electric current carried by $b$ is
$J^{b}=\frac{e}{2\pi}dA$, hence the current-current correlation can be
extracted from the photon Green’s function based on the effective action Eq.
11:

$\displaystyle\rho_{b,N\rightarrow\infty}=\frac{\pi N}{8}\frac{h}{e^{2}}.$
(12)

Again the final resistivity of the system is half of Eq. 12 due to the two
spin/valley flavors. The evaluation Eq. 12 is still proportional to $N$ just
like Eq. 10. These two different evaluations discussed above give different
values for $N=N_{b}=1$, and compared with the known value of the universal
resistivity at the 3D XY transition, the evaluation in Eq. 10 is much more
favorable, though the evaluation Eq. 12 based on Eq. 11 is supposed to be
accurate with large $N$.

When there is a ring of degeneracy in the vortex band structure, as we
mentioned before the gauge field $A_{\mu}$ will acquire a “mass term” after
integrating out $\psi_{j}$ Lake et al. (2021). In this case the resistivity of
the system at the MIT will be infinity, as the dynamnics of $A_{\mu}$ is fully
suppressed by the mass term in the infrared. One can also integrate out the
action of $A_{\mu}$ with the mass term, and verify that the response theory of
$A_{\mathrm{ext}}$ is no different from that of an insulator in the infrared
limit. This is consistent with both Eq. 10,12 by naively taking $N$ to
infinity. In Ref. Lake et al., 2021 when the boson field has a ring
degeneracy, the phase is identified as a bose metal; this is because in Ref.
Lake et al., 2021 it is the boson with ring degeneracy that carries charges.
But in Eq. 6 the electric charge is carried by the flux of $A_{\mu}$.

##  IV Mott insulator with topological order

As we explained in the previous subsection, due to the fractional filling of
boson $b_{\alpha}$, the vortex dynamics is frustrated by the background
fractional flux through the hexagons. To drive the system into an insulator
phase, the vortex can either condense at multiple minima in the BZ as was
discussed in the previous section, or form a bound state that carries multiple
gauge charge of $A_{\mu}$ and become “blind” to the background flux. In parton
construction-II, with electron filling $\nu=1/2$, each flavor of boson is at
filling $\nu_{b}=1/2$. The double-vortex, i.e. bound state of two vortices, or
more generally the bound state of $N$ vortices with even integer $N$, no
longer see the background flux. Hence the $N$-vortex can condense at zero
momentum, and its condensate will drive the system into a $Z_{N}$ topological
order.

After the boson-vortex duality, the theory for the $N$-vortex condensation at
one of the two spin/valley flavors is

$\displaystyle\mathcal{L}^{(2)}$ $\displaystyle=$
$\displaystyle|(\partial_{\mu}-\mathrm{i}NA_{\mu})\psi|^{2}+r|\psi|^{2}+g|\psi|^{4}$
(13) $\displaystyle+$ $\displaystyle\frac{\mathrm{i}}{2\pi}A\wedge
d(a+eA_{\mathrm{ext}})+\cdots$ (15)

The condensate of $\psi$ will break the ${\rm U}(1)$ gauge field to a $Z_{N}$
gauge field, whose deconfined phase has a nontrivial $Z_{N}$ topological
order. In the $Z_{N}$ topological order as well as at the MIT, the charge
carrier is an anyon of the $Z_{N}$ topological order, and it carries charge
$e^{\ast}=e/N$. We still label the fractional charge carrier as $\varphi$.
$\varphi$ carries charge $e/N$, and is coupled to a $Z_{N}$ gauge field
originated from the $Z_{N}$ topological order discussed in the previous
paragraph.

In our case, in order to preserve the time-reversal symmetry, both spin/valley
flavors should form a $Z_{N}$ topological order simultaneously. Hence there is
one species of $\varphi_{\alpha}$ field for each spin/valley flavor. The MIT
can equally be described as the condensation of the $\varphi_{\alpha}$ field,
and since the $Z_{N}$ gauge field does not lead to singular correction in the
infrared, the condensation of $\varphi_{\alpha}$ is a 3D XY∗ transition, and
the transition for $N=2$ was discussed in Ref. Calabrese et al., 2003; Isakov
et al., 2007; Hastings, 2004; Senthil et al., 2004a, b; Wang et al., 2021. The
$b_{\alpha}$ field is now a composite operator of $\varphi_{\alpha}$. In the
condensate of $\varphi_{\alpha}$, the electron operator $c_{\alpha}$ is
related to the fermionic parton operator $f_{\alpha}$ through
$c_{\alpha}\sim\langle b_{\alpha}\rangle
f_{\alpha}\sim\langle\varphi^{N}_{\alpha}\rangle f_{\alpha}$. The coupling
between the two flavors of $\varphi_{\alpha}$, i.e. the coupling
$|\varphi_{1}|^{2}|\varphi_{2}|^{2}$ is irrelevant at the decoupled 3D XY∗
transition according to the known critical exponents of the 3D XY∗ transition.
There are also couplings such as
$|\varphi_{\alpha}|^{2}f^{\dagger}_{\alpha}f_{\alpha}$ allowed by all the
symmetries, but after formally integrating out the fermions, the generated
couplings for $\varphi_{\alpha}$ is also irrelevant at the two decoupled 3D
XY∗ universality class. The reason is that after formally integrating out the
fermions, terms such as
$\frac{|\omega|}{q}|\varphi_{\alpha}|^{2}_{\omega,\vec{q}}|\varphi_{\beta}|^{2}_{-\omega,-\vec{q}}$
can be generated, but this term is irrelevant knowing that the standard
critical exponent $\nu>2/3$ for the 3D XY∗ transition.

Following the large$-\bf{N}$ calculation discussed before, the DC resistivity
jump $\rho_{b}(0)$ would be $N^{2}/2$ times that of the previous theory
Witczak-Krempa et al. (2012), namely

$\displaystyle\rho_{b}(0)\sim R^{(2)}N^{2}\frac{h}{e^{2}},$ (16)

where $R^{(2)}=R^{(0)}/2\sim 3.7$ based on our evaluation. The AC resistivity
jump at the MIT is enhanced by the same factor compared with the previous
theory. We also note that the fractional universal conductivity at the
transition between the superfluid and a $Z_{2}$ topological order was observed
numerically in Ref. Wang et al., 2021.

Another set of natural topological orders a boson at fractional filling can
form are bosonic fractional quantum Hall (bFQH) states which are close
analogues to the bosonic Laughlin’s wave function. We would like to discuss
this possibility as a general exploration, although this state breaks the
$P_{y}$ symmetry (but it still preserves the product $P_{x}\mathcal{T}$
symmetry). If we interpret the half-filled boson at each site as a quantum
spin-1/2 system, this set of states are analogous to a chiral spin liquid
Kalmeyer and Laughlin (1987, 1989). The Chern-Simons theory for this set of
states at each valley reads

$\displaystyle\mathcal{L}_{\mathrm{cs}}=-\frac{\mathrm{i}k}{4\pi}A\wedge
dA+\frac{\mathrm{i}}{2\pi}A\wedge d(a+A_{\mathrm{ext}}),$ (17)

with an even integer $k$ and a dynamical Spinc U(1) gauge field $A$. The
topological order characterized by this theory is the SU$(k)_{1}$ topological
order. Here, the integer $k$ needs to be even so that this theory is
compatible with the LSM constraint imposed by the boson filling $1/2$ on the
lattice Cheng et al. (2016). This is because the boson filling 1/2 requires
the topological phase to contain an Abelian anyon that carries a fractional
charge 1/2 (modulo integer). There should be one such anyon per unit cell to
account for the boson filling 1/2 on the lattice. The fact that such an anyon
carries a fractional charge 1/2 implies that this anyon should generate under
fusion an Abelian group $\mathbb{Z}_{p}$ with $p$ an even number. Such a
fusion rule is incompatible with any odd value of $k$. Therefore, $k$ needs to
be even in the theory given by Eq. 17. The time-reversal of the TMD moiré
system demands that the bosonic parton $b_{\alpha}$ with opposite spin/valley
index $\alpha$ forms a pair of time-reversal conjugate bFQH states. Or in
other words if we take both spin/valley flavors together, this state is a
fractional topological insulator, like the state discussed in Ref. Levin and
Stern, 2009.

The MIT is now a direct transition between the bFQH state and the superfluid
of $b_{\alpha}$. When the even integer $k$ is $k=2n^{2}$ with odd integer $n$,
there is a natural theory for this direct continuous transition, and its
simplest version with $n=1$ was proposed in Ref. Barkeshli and McGreevy, 2014.
The transition is a 3D QED with two flavors of Dirac fermions coupled to the
dynamical ${\rm U}(1)$ Spinc gauge field $A_{\mu}$ (the dual of the Goldstone
mode of the boson superfluid) with a Chern-Simons term at level-$n^{2}$, and
the fermions have gauge charge-$n$:

$\displaystyle\mathcal{L}^{(3)}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{2}\bar{\chi}_{j}\gamma\cdot(\partial-\mathrm{i}nA)\chi_{j}+M\bar{\chi}_{j}\chi_{j}-\frac{\mathrm{i}n^{2}}{4\pi}A\wedge
dA$ (18) $\displaystyle+$ $\displaystyle\frac{\mathrm{i}}{2\pi}A\wedge
d(a+eA_{\mathrm{ext}})+\cdots$ (20)

In this theory, the fact that $A$ is a Spinc U(1) gauge field and that $n$ is
odd guarantee that this theory describes the phases of a boson. The ellipsis
in this Lagrangian includes other terms such as the Maxwell term of the gauge
field $A_{\mu}$. Please note that this equation is for one of the two
spin/valley flavors of the physical system. The mass $M$ of the Dirac fermions
is the tuning parameter of the transition. With one sign of the mass term,
after integrating out the Dirac fermions, the Spinc U(1) gauge field $A$ will
acquire a Chern-Simons term at level $-2n^{2}$, which describes the
SU$(k)_{1}$ topological order with $k=2n^{2}$. With the opposite sign of $M$,
there is no Chern-Simons term of the gauge field $A$ after integrating out the
Dirac fermions, and the Maxwell term of the gauge field $A$ is the dual
description of the superfluid phase. Hence by tuning $M$ the system undergoes
a transition between the $k=2n^{2}$ bFQH state and the superfluid state of $b$
(the metal phase of the original electron system).

The translation symmetry of the system actually guarantees that the two
flavors of Dirac fermions are degenerate in Eq. 20. If these two Dirac
fermions are not degenerate, an intermediate topological order is generated by
changing the sign of the mass of one of the Dirac fermions in Eq. 20. Then
after integrating out the fermions, the gauge field $A$ acquires a total CS
term with an odd level $-n^{2}$, which violates the LSM constraint imposed by
the boson filling 1/2. Therefore, the masses of the two flavors of the Dirac
fermions in Eq. 20 should be the same. In fact, for the simplest case with
$n=1$ ($k=2$), an explicit parton construction of this transition can be given
following the strategy in Ref. Barkeshli and McGreevy, 2014, and the two Dirac
fermions in Eq. 20 are two Dirac cones of a $\pi-$flux state of $\chi$ on the
triangular lattice. The degeneracy of these two Dirac fermions is protected by
the translation symmetry of the triangular lattice. From the parton formalism
one can also see that the boson $b$ is constructed as a product of the two
fermions $\chi_{i}$.

At the transition $M=0$, though it is difficult to compute the resistivity of
Eq. 20 exactly, the resistivity $\rho(x)$ should scale as $1/k$ with large
$k\sim n^{2}$, as after integrating out $\chi_{j}$ the entire effective action
of $A$ scales linearly as $k$. Then after integrating out $A$, the response
theory to $A_{\mathrm{ext}}$ is proportional to $1/k$.

##  V Summary of Predictions

So far we have discussed three different kinds of possible Mott insulators at
half filling of the extended Hubbard model, based on the parton construction-
II: (1) Mott insulators with translation symmetry breaking; (2) a $Z_{N}$
topological order at each spin/valley flavor with even integer $N\geq 2$; and
(3) a pair of conjugate bFQH states at two spin/valley flavors. For all
scenarios, we have evaluated the bosonic parton contribution to the
resistivity $\rho_{b}$ at the MIT, which is also the universal jump of
resistivity $\Delta\rho$. The predicted resistivity jump for the three
scenarios are summarized in the table below.

Nature of Insulator |  $\Delta\rho$, or $\rho_{b}$
---|---
(1) Density wave | $\rho_{b}(0)\sim(R^{(0)}+R^{(1)}(N-1))\frac{h}{e^{2}}$
(2) $Z_{N}$ TO each flavor | $\rho_{b}(0)=R^{(2)}N^{2}\frac{h}{e^{2}}$
(3) Conjugate bFQH | $\rho_{b}(x)\sim\frac{1}{k}\frac{h}{e^{2}}$

Another observable effect predicted by the previous theory of interaction-
driven MIT is the scaling of quasi-particle weight $\sqrt{Z}$ near the MIT
Senthil (2008a, b), where $\sqrt{Z}\sim r^{\beta_{1}}\sim|r|^{0.33}$. Our
theory also gives a different prediction of the quasi-particle weight compared
with the previous theory, and this is most conveniently evaluated for scenario
(2). In the metal phase but close to the MIT, the quasi-particle weight scales
as

$\displaystyle\sqrt{Z}\sim\langle\varphi^{N}_{\alpha}\rangle\sim|r|^{\beta_{N}},$
(21)

where $\beta_{N}=\nu\Delta_{N}$. $\nu\sim 0.67$ is the standard correlation
length exponent at the 3D XY∗ transition (it is the same as the 3D XY
transition) and $\Delta_{N}$ is the scaling dimension of $\varphi^{N}$ at the
3D XY transition. These exponents can be extracted from numerical simulation
on the 3D XY and XY∗ transitions. For example, when $N=2$, $\beta_{2}$ should
be close to $0.8$ Calabrese et al. (2003); Isakov et al. (2007, 2012), hence
$\sqrt{Z}\sim|r|^{0.8}$. The scaling of quasi-particle weight can be checked
in future experiments through the measurement of local density of states of
electrons.

For scenario (1), i.e. where the insulator has translation symmetry breaking,
the scaling of quasiplarticle weight can be estimated with large-$N$ in Eq. 6.
The boson creation operator $b^{\dagger}$ is a monopole operator of $A_{\mu}$
which creates a $2\pi$ gauge flux. With large-$N$ in Eq. 6 the monopole
operator has scaling dimension proportional to $N$ Pufu and Sachdev (2013);
Dyer et al. (2015), hence the critical exponent $\beta$ in the quasiparticle
weight $\sqrt{Z}\sim|r|^{\beta}$ is expected to be proportional to $N$. The
similar evaluation applies to Eq. 20, and the creation operator $b^{\dagger}$
has a scaling dimension proportional to $k$, which is also proportional to
$\sqrt{Z}$.

As we explained, our theory provides a natural explanation of the anomalously
large resistivity at the MIT. Another qualitative experimental feature
reported in Ref. Li et al., 2021 is that, the resistivity drops rapidly as a
function of temperature at the MIT where the charge gap vanishes. Our theory
also provides a natural explanation for the temperature dependence of the
critical resistivity. At zero temperature the bosonic chargeon parton $b$
fractionalizes into multiple partons with smaller charges, and these partons
will couple to extra gauge fields. These extra gauge fields will all confine
at finite temperature. Hence at finite temperature, there is a crossover from
transport with fractionalized charge to unfractionalized charge, which will
cause a significant drop of resistivity with increasing temperature.

In the following paragraphs we discuss physics in phases near the MIT, based
on our theory. These analysis can distinguish the three possible scenarios
discussed to this point. Let us first discuss the insulator phase at fixed
electron filling $\nu=1/2$. The scenario (3) describes a topological order
that is essentially a topological fractional quantum spin Hall insulator,
hence this insulator phase, if does exists, must have nonchiral gapless modes
localized at the boundary of the system. This nonchiral edge gapless modes
should lead to similar experimental phenomena as the experiments on quantum
spin Hall insulator König et al. (2007); but rather than edge conductance
$2e^{2}/h$, the edge conductance of the fractional quantum spin Hall insulator
should be $2e^{2}/(kh)$, which is twice of the edge conductance of the bFQH
state with CS level-$k$. Also, the edge conductance should be suppressed by
external magnetic field, also analogous to what was observed in Ref. König et
al., 2007.

The insulating phase of scenario (1) and scenario (2) also lead to distinctive
predictions. In scenario (1), the electric charges are only deconfined at the
MIT, but still confined in the insulating phase, which has no topological
order. Hence the charge deconfinement of scenario (1) is analogous to the
original deconfined quantum critical point discussed in Ref. Senthil et al.,
2004a, b. The confinement of fractional charges in scenario (1) happens even
at zero temperature in the insulating phase. However, in scenario (2), the
insulator phase has a $Z_{N}$ topological order that supports deconfined
fractional charge at zero temperature even in the insulator phase. While at
finite temperature, the $Z_{N}$ gauge field will lead to confinement of
fractional charges with confinement length $\xi\sim\exp(\Delta_{m}/(2T))$,
where $\Delta_{m}$ is the gap of the fractionalized $Z_{N}$ gauge fluxes,
which is an anyon with nontrivial statistics with the fractional charges. If
we look at the insulator phase close to the MIT, the gap of the fractional
charge, i.e. the $e-$anyon of the $Z_{N}$ topological order is suppsosed to be
smaller than $\Delta_{m}$, hence at very low temperature the thermally
activated $e-$anyon has a much smaller distance $l_{e}$ with each other
compared with $\xi$. Then at low but finite temperature the transport is
governed by charge carriers with gap $\Delta_{e}$ and charge $e_{\ast}=e/N$.
The gap $\Delta_{e}$ can be extracted from fitting the low temperature
transport data versus temperature. However, if one measures the tunnelling gap
through tunnelling spectroscopy, since the external device can only inject a
single electron which fractionalizes into multiple $e-$anyons, the tunneling
gap should be approximately $N\Delta_{e}$. This contrast between tunneling gap
and the thermally activated transport gap happens in scenario (2) but not
scenario (1).

We also consider the metallic phase next to the insulator after charge doping,
and we will see the scenario (2) also leads to very nontrivial predictions due
to the deconfined nature of the $Z_{N}$ topological order. In scenario (2),
after some charge doping, we expect a metallic state with charge
fractionalization at low temperature. The bosonic charge carriers are coupled
to the $Z_{N}$ gauge field as well as the U(1) gauge field $a_{\mu}$ that are
shared with the fermionic partons $f_{\alpha}$. When the temperature is
increased, the $Z_{N}$ gauge field will confine, and due to the time-reversal
symmetry, the confine-deconfine crossover should happen for both spin/valley
flavors simultaneously. In the following, we shall only focus on one
spin/valley. According to the Ioffe-Larkin composition rule, the total
resistivity is composed of contributions from both bosonic and fermionic
partons $\rho=\sigma^{-1}=\sigma_{b}^{-1}+\sigma_{f}^{-1}$. Let us assume the
resistivity of both the bosonic and fermionic sectors are dominated by the
scattering with the gauge field $a_{\mu}$ (this of course assumes that the
momentum of the gauge field $a_{\mu}$ can relax through other mechanism such
as disorder). This scattering mechanism was first evaluated in Ref. Lee and
Nagaosa, 1992. The gauge-field propagator can be written as
$D(\omega,\bm{q})^{-1}=i\gamma\omega/q+\chi_{d}q^{2}$, where the $\omega/q$
term is due to the Landau damping from the fermi-surface, and the
“diamagnetic” $\chi_{d}$ is roughly a constant within the temperature window
of interest. The scattering rate can then be estimated using the imaginary
part of the boson/fermion self-energy:

$\displaystyle\textrm{Im}\Sigma_{b,f}(\omega,\bm{k})=\int_{0}^{\infty}d\omega^{\prime}\int\frac{d^{2}\bm{k}^{\prime}}{(2\pi)^{2}}(1+n_{b}(\omega^{\prime}))(1\pm
n_{b,f}(\omega_{\bm{k}^{\prime}}))$
$\displaystyle\frac{(k_{\alpha}+k_{\alpha}^{\prime})(k_{\beta}+k_{\beta}^{\prime})}{2m_{b,f}}\frac{\delta_{\alpha\beta}-q_{\alpha}q_{\beta}}{\bm{q}^{2}}\delta(\omega-\omega_{\bm{k}^{\prime}}-\omega^{\prime})\textrm{Im}D(\omega^{\prime},\bm{q}),$

where $\bm{q}=\bm{k}^{\prime}-\bm{k}$, $n_{b,f}(\omega)$ denotes the Bose-
Einstein (Fermi-Dirac) distribution function, and $m_{b,f}$ is the
boson/fermion mass. We must stress that the expression of $\Sigma_{b,f}$ is
valid for partons with gauge charge-1. When the $Z_{N}$ gauge field is
deconfined, each boson carries the gauge charge-$1/N$ of the gauge field
$a_{\mu}$, and therefore there is an additional factor $1/N^{2}$ in the self-
energy. The integral was evaluated in Ref. Lee and Nagaosa, 1992, and the
time-scale responsible for transport has an extra factor proportional to
$q^{2}$ in the integral. After taking these into account, we obtain the
“transport” scattering rate for boson/fermion

$\displaystyle\frac{1}{\tau_{f}}\sim
T^{4/3},\qquad\frac{1}{\tau_{b}}\approx\frac{k_{B}T}{m_{b}\chi_{d}}.$ (22)

Comparing $1/\tau_{b}$ and $1/\tau_{f}$, we can see that the resistivity is
dominated by the boson-gauge scattering at low temperature, and the bosonic
partons are in a disordered phase rather than a quasi long range order at
finite temperature due to their coupling to the dynamical gauge field
$a_{\mu}$. We take the Drude formula for the dilute Bose gas that we use to
model the bosonic partons at finite temperature:

$\displaystyle\rho\sim\frac{m_{b}}{n_{*}e_{*}^{2}}\frac{1}{\tau_{b}}\sim\frac{g_{*}^{2}}{n_{*}e_{*}^{2}}\frac{k_{B}T}{\chi_{d}},$
(23)

where $e_{*}=e/N$ and $g_{*}=1/N$ denote the electric and gauge charges of
bosons, and $n_{*}e_{*}$ is the doped physical electric charge density. Here,
we have assumed that the resistivity $\rho$ is dominated by the boson
contribution because ($i.$) the scattering rate of the boson is bigger
compared to the fermions at low temperature as shown in Eq. 22, and ($ii.$)
the bosons have much lower density at low charge doping compared to the
fermions which already has finite fermi surface at zero charge doping. In the
following discussion, we will work under these assumptions at least up to the
temperature scale $T_{c}$ around which the $Z_{N}$ gauge becomes fully
confined.

The $Z_{N}$ gauge field is fully confined when $\xi$ is at the same order as
the lattice constant; $i.e.$ $T>T_{c}\sim\Delta_{m}$. Here we assume that the
gauge field $a_{\mu}$ that is coupled to the fermionic parton is less prone to
confinement due to its coupling to the large density of gapless fermoins.
Above $T_{c}$, the charge carriers in the system carry charge-$e$. The
equation above still hold with the substitutions $e_{*}\rightarrow
e=Ne_{*},g_{*}\rightarrow g=Ng_{*},n_{*}\rightarrow n=n_{*}/N$. We expect
there is a crossover from the deconfined value of resistivity $\rho(T\sim 0)$
to the confined value $\rho(T\geq T_{c})$:

$\displaystyle\frac{(d\rho/dT)_{T\geq T_{c}}}{(d\rho/dT)_{T\sim 0}}\sim N,$
(24)

This is an observable effect of scenario (2) that can be experimentally
verified.

Contrary to the Ioffe-Larkin rule, the total thermal conductivity of the
system is a sum of the contribution from the bosonic parton, fermionic parton,
and also the gauge boson. With low charge doping away from $\nu=1/2$, we
expect the fermionic partons dominates the thermal transport according to Ref.
Nave and Lee, 2007: $\kappa_{f}\sim T^{1/3}$. As we discussed above, in
scenario (2) The low-temperature charge transport is dominated by the boson
contribution $\sigma_{b}\sim 1/T$, while the thermal transport is dominated by
the fermion contribution $\kappa_{f}\sim T^{1/3}$. Due to the crossover of
charge transport at finite temperature caused by the confinement of the
$Z_{N}$ gauge field in scenario (2), there is also an observable prediction
one can make for the Lorentz number
$L=\kappa/(T\sigma)\approx\kappa_{f}/(T\sigma_{b})$:

$\displaystyle\frac{(L/T^{1/3})_{T\geq T_{c}}}{(L/T^{1/3})_{T\sim 0}}\sim N.$
(25)

##  VI Summary, Discussion, and Other fractional fillings

In this work we proposed a theory for a potentially continuous metal-insulator
transition for the extended Hubbard model on the triangular lattice at half-
filling (one electron per unit cell). The extended Hubbard model is simulated
by the TMD moiré systems. We introduce a different parton construction from
the previous literature, which leads to a series of observable predictions. We
demonstrated that our theory is more favorable given the current experiments
on the heterobilayer TMD moiré systems. Although our theory was motivated by
the recent experiments on MoTe2/WSe2 moiré superlattice Li et al. (2021), we
envision our theory can have broad application given the recent rapid
progresses in synthesizing pure two dimensional systems.

The moiré potential in the MoTe2/WSe2 moiré superlattice with no twisting is
formed due to the mismatch of the lattice constants of the two layers. There
is another experiment on MIT in twisted WSe2 Ghiotto et al. (2021). The
situation in twisted WSe2 seems rather different from MoTe2/WSe2 moiré
superlattice. Inside the “insulator phase”, the resistivity $\rho(T)$ at some
displacement fields first increases with decreasing temperature, and
eventually the plot seems to saturate at a finite value, which is much lower
than the resistivity observed in the MoTe2/WSe2 moiré superlattice near the
MIT. Hence the MIT of twisted WSe2 could be of a different nature, between the
metallic phase and the insulator phase, there could be an intermediate phase
with an order at nonzero momentum and reduced size of electron Fermi pockets.

Correlated insulators at other fractional fillings $\nu=p/q$ have been
reported in various TMD moiré systems Regan et al. (2020); Jin et al. (2021);
Xu et al. (2020b); Huang et al. (2021). Although the nature of the MIT at
these fillings has not been looked into carefully, here we briefly discuss the
theory for the possible continuous MIT at general fractional filling
$\nu=p/q$. As long as $q>2$, even for parton construction-I, the bosonic
parton $b$ will have fractional filling, and hence the insulator phase of $b$
cannot be a trivial incompressible state without translation symmetry breaking
or topological order. Here we would like to acknowledge that charge
fractionalization for interacting electron system at fractional electron
number per unit cell was discussed in previous literature Chen et al. (2014b),
using similar formalism as the parton construction-I. At electron filling
$\nu=1/q$, the boson filling $\nu_{b}=2/q$; if we only consider nearest
neighbor hopping of the vortex, the insulator has commensurate density wave
that spontaneously breaks the translation symmetry, and the MIT is described
by Eq. 6 with $N=q$ for odd integer $q$; $N=q/2$ for $q=4k+2$; and $N=q$ for
$q=4k$. The electron charge will further fractionalize at the continuous MIT.
In parton construction-I, there are in total $N$ species of the charge
carriers each carrying electric charge $e^{\ast}=e/N$. Hence the estimate of
$\rho_{b}$ is $\rho_{b}\sim Nh/e^{2}$.

For parton construction-II, with electron filling $\nu=1/q$, the boson filling
for each spin/valley flavor is $\nu_{b}=1/q$. Again, if only nearest neighbor
hopping of the vortices is considered, the MIT is described by Eq. 6 with
$N=q$ for odd integer $q$; $N=2q$ for even integer $q$. The field theory
describing the MIT is two copies of Eq. 6: $\psi_{j}$, $A_{\mu}$ and $a_{\mu}$
should all carry a spin index $\alpha$. There are in total $N_{b}=2N$ species
of the charge carriers each carrying electric charge $e^{\ast}=e/N$. Hence the
estimate of $\rho_{b}$ is $\rho_{b}\sim Nh/(2e^{2})$. If we consider further
neighbor hopping like section III, the charge carriers may carry even smaller
fractional charge, and hence larger $\rho_{b}$.

The authors thank L. Balents, Sung-Sik Lee, C. Nayak, and T. Senthil, Kevin
Slagle for very helpful discussions. C.X. is supported by NSF Grant No.
DMR-1920434, and the Simons Investigator program; Z.L. is supported by the
Simons Collaborations on Ultra-Quantum Matter, grant 651440 (LB); M.Y. was
supported in part by the Gordon and Betty Moore Foundation through Grant
GBMF8690 to UCSB, and by the NSF Grant No. PHY-1748958.

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##  Appendix A Field theories for $N=6$ and $N=12$ of scenario (1)

In the next section we will derive the projective symmetry group
transformation for the low energy vortex modes of scenario (1). For $N=6$,
with symmetries $R_{2\pi/3}$, translation, $P_{x}\mathcal{T}$, and $P_{y}$,
the PSG-invariant interactions between the vortex fields $\psi_{a}$ beyond Eq.
6 take the following form:

$\displaystyle\mathcal{L}^{(1)\prime}[\psi_{a}]=$ $\displaystyle
u_{1}\sum_{a=0}^{2}(|\psi_{2a}|^{2}+|\psi_{2a+1}|^{2})^{2}+u_{2}\left(\sum_{a=0}^{5}|\psi_{a}|^{2}\right)^{2}$
(26)
$\displaystyle+v_{1}\left(\sum_{a=0}^{5}\psi_{a}^{2}\right)\left(\sum_{a=0}^{5}(\psi_{a}^{*})^{2}\right)+v_{2}\sum_{a=0}^{2}(\psi_{2a}^{2}+\psi_{2a+1}^{2})((\psi_{2a}^{*})^{2}+(\psi_{2a+1}^{*})^{2})$
$\displaystyle+w_{1}\sum_{a=0}^{2}(|\psi_{2a}|^{2}-|\psi_{2a+1}|^{2})(\psi_{2a+2}\psi^{*}_{2a+3}+\psi^{*}_{2a+2}\psi_{2a+3})+w_{2}\left\\{\sum_{a=0}^{2}(\psi_{2a}^{2}-\psi_{2a+1}^{2})\psi^{*}_{2a+2}\psi^{*}_{2a+3}+c.c.\right\\}+\dots$

Here the dots stand for terms higher than the quartic order. The parameters
$\\{u_{1},u_{2},v_{1},v_{2},w_{1},w_{2}\\}$ in (26) are all real, and the
index $a$ for $\psi_{a}$ is regarded as cyclic modulo $6$.

In addition to the quartic terms, the gauge invariant density wave order
parameter can couple to the Fermi surface of the fermionic partons, and
quartic terms of $\psi_{a}$ with singularity in the frequency space can be
generated as was pointed out by Ref. Musser et al., 2021, such as
$|\omega||S_{\omega,q}|^{2}$, where $S_{\omega,q}$ is a bilinear of
$\psi_{a}$. This coupling only arises for scenario (1). For scenario (2)
discussed in the main text, the 3D XY∗ fixed point should be stable against
symmetry allowed perturbations; the field theory Eq. 20 is also stable against
coupling to the fermionic parton Fermi surface.

Although we do not aim to give a full discussion of the fate of the infrared
limit of scenario (1), in the current work we establish the formalism for this
problem that one can use in the future. As we explained in the previous
paragraph, after integrating out the fermion that is connected by the finite
momentum of the density wave order parameter, a term is generated
$\sim|\omega||S_{\omega,q}|^{2}$, where $S=\psi^{\dagger}T\psi$ and $T$ is an
$N\times N$ matrix. One can introduce a new field $\Phi$ through the Hubbard-
Stratonovich transformation, and $\psi_{a}$ will interact with the $\Phi$
field Kaul and Sachdev (2008). We start with the first line of Eq. 26. The
field theory Eq. 6 with $u_{1}$ and $u_{2}$ in Eq. 26 can be reformulated by
introducing multiple Lagrange multipliers $\lambda_{i}$:

$\displaystyle\mathcal{L}^{(1)}$ $\displaystyle=$
$\displaystyle\sum_{a=0}^{N-1}|(\partial-\mathrm{i}A)\psi_{a}|^{2}+\mathrm{i}\sum_{i=1}^{N_{1}}\lambda_{i}\left(\sum_{\tau=1}^{N_{2}}|\psi_{\tau,i}|^{2}\right)+\mathrm{i}\Phi\psi^{\dagger}T\psi;$
(33)
$\displaystyle\langle\lambda_{i}(\vec{q})\lambda_{i^{\prime}}(-\vec{q})\rangle=\frac{8}{N_{2}}|q|\delta_{i,i^{\prime}},$
$\displaystyle\langle
A_{\mu}(\vec{q})A_{\nu}(-\vec{q})\rangle=\frac{16}{N}\left(\frac{\delta_{\mu\nu}-q_{\mu}q_{\nu}/q^{2}}{|q|}\right),$
$\displaystyle\langle\Phi(\vec{q})\Phi(-\vec{q})\rangle=g|\omega|.$

Here $N=N_{1}N_{2}$, and for the real system with $N=6$, $N_{1}=3$ and
$N_{2}=2$. Introducing $\lambda_{i}$ for each index $i$ physically means that
we are investigating the theory near the point with a SU$(N_{2})$ symmetry for
each index $i$, rather than the original CPN-1 theory with a large SU$(N)$
flavor symmetry. This is analogous to the “easy-plane bosonic QED3” considered
in Ref. Benvenuti and Khachatryan, 2019. The actions of $\lambda_{i}$ and the
transverse component of gauge field $A$ are generated by integrating out the
fields $\psi_{a}$. One possible way to proceed with the calculation is that,
we can fixed $N_{1}$, and take $1/N_{2}$ as a small parameter. When $g$ is the
same order of $1/N_{2}$, the interaction between $\psi_{a}$ and the $\Phi$
field will lead to the contribution comparable with that arising from coupling
to $\lambda_{i}$ and $A$. The calculation would be analogous to the one
formulated in Ref. Jian et al., 2021, where the nonlocal interaction on top of
a bosonic QED flows to a new fixed point. One can evaluate the scaling
behaviors (such as relevance/irrelevance in the IR) of the $v$ and $w$ terms
in the second and third lines in Eq. 33 at this new fixed point. By exploring
the parameter space of $g$, $1/N_{2}$, and different choice of matrix $T$, it
is possible to identify a finite region where Eq. 33 corresponds to a stable
fixed point where the $v$ and $w$ terms in Eq. 26 are irrelevant.

The same strategy can be applied to the situation with $N=12$. With long moiré
lattice constants, the 6-fold rotation $R_{\pi/3}$ also becomes a good
approximate symmetry. Together with $R_{\pi/3}$, the quartic terms in the
field theory for $N=12$ (please refer to the phase diagram in Fig. 2) are:

$\displaystyle\mathcal{L}^{(1)\prime}[\psi_{\sigma,\tau,i}]$ $\displaystyle=$
$\displaystyle
u_{1}\sum_{\sigma,i}\left(\sum_{\tau}|\psi_{\sigma,\tau,i}|^{2}\right)^{2}+u_{2}\left(\sum_{\sigma\tau
i}|\psi_{\sigma\tau i}|^{2}\right)^{2}$ (34) $\displaystyle+$ $\displaystyle
v_{1}\sum_{\sigma,i\neq
i^{\prime}}\left(\sum_{\tau}|\psi_{\sigma,\tau,i}|^{2}\right)\left(\sum_{\tau^{\prime}}|\psi_{\sigma,\tau^{\prime},i^{\prime}}|^{2}\right)+v_{2}\sum_{i}\left(\sum_{\tau}|\psi_{+,\tau,i}|^{2}\right)\left(\sum_{\tau^{\prime}}|\psi_{-,\tau^{\prime},i}|^{2}\right)$
(36) $\displaystyle+$ $\displaystyle
w_{1}\left|\sum_{i,\tau}\psi_{+,\tau,i}\psi_{-,\tau,i}\right|^{2}+\mathrm{i}w_{2}\left(\sum_{i,\tau,\tau^{\prime}}\psi^{*}_{+,\tau,i+1}\psi^{*}_{-,\tau,i+1}\psi_{+,\tau^{\prime},i}\psi_{-,\tau^{\prime},i}-h.c.\right)$
(38)

Here the 12 modes are labelled by $\psi_{\sigma,\tau,i}$ in which $\tau=\pm$
labels two degenerate modes at the same momentum, $\sigma=\pm$ labels two sets
of momenta that are each connected by $R_{2\pi/3}$, and $i=0,1,2\mod 3$ labels
these three momenta within each set.

We can again start with the first line of Eq. 38, and introduce Lagrange
multiplier $\lambda_{\sigma,i}$ which couples to the $\psi_{a}$ fields as
$\sum_{\tau=1}^{N_{2}}\lambda_{\sigma,i}|\psi_{\sigma,\tau,i}|^{2}$. Notice
that we have generalized $\tau$ to $1\cdots N_{2}$. Then the Hubbard-
Stratonovich transformation can introduce new fields that couple to $\psi_{a}$
to account for the singular terms generated through interacting with the Fermi
surface. A combined perturbation theory of $1/N_{2}$ and $g$ can again
determine the relevance/irrelevance of the second and third lines of Eq. 38.
In particular, the two terms in the second line of Eq. 38 are indeed
irrelevant with large-$N_{2}$, as the scaling dimension of
$\sum_{\tau}|\psi_{\sigma,\tau,i}|^{2}$ is $2$ with large-$N_{2}$.

##  Appendix B The PSG transformation for $N=6$ in scenario (1)

Under the boson-vortex duality, the dual vortex theory on the hexagonal
lattice takes the form

$H=\sum_{\langle
ij\rangle}-t_{ij}\phi^{*}_{i}\phi_{j}+H^{\prime}_{\phi}+V_{\phi}+\dots,\ \ \
t_{ij}=te^{-\mathrm{i}A_{ij}}$ (39)

Here $H^{\prime}_{\phi}$ describes hopping terms between further neighbors.
The potential $V_{\phi}$ includes a quadratic term $\sum_{i}r|\phi_{i}|^{2}$
which tunes through the phase transition.

Figure 3: Crystal symmetry of the triangular lattice, the nearest neighbor
hopping amplitudes of the vortices, and the unit cell after taking into
account of the sign of $t_{ij}$.

When $t_{ij}$ is nonzero only for nearest neighbor links on the dual honeycomb
lattice, and its takes positive sign on the solid links and negative sign on
the dashed links in Fig. 1 due to the $\pi$ flux of $A_{\mu}$ through each
hexagon, there are four minima of the vortex band structure in the Brillouin
zone (Fig. 2). We label the four minimum modes from 0 to 3, each have momentum
$(k_{x},k_{y})$ equal to

$\mathbf{Q}_{0,1}=\mathbf{K}=\left(\frac{2\pi}{3\sqrt{3}},0\right),\ \ \
\mathbf{Q}_{2,3}=\mathbf{K}^{\prime}=\left(\frac{\pi}{3\sqrt{3}},\frac{\pi}{3}\right).$
(40)

With further neighbor vortex hopping (please refer to the phase diagram in
Fig. 2), the minima of the vortex band structure can shift to the $M$ points,
similar to Ref. Slagle and Xu, 2014. When the degenerate minima are shifted to
the $M$ points (Fig. 2), the six corresponding momenta are

$\mathbf{Q}_{0,1}=\left(\frac{\pi}{2\sqrt{3}},-\frac{\pi}{6}\right),\
\mathbf{Q}_{2,3}=\left(\frac{\pi}{2\sqrt{3}},\frac{\pi}{6}\right),\
\mathbf{Q}_{4,5}=\left(0,\frac{\pi}{3}\right).$ (41)

Similar to the four minima case, the vortex field can be expanded using these
six modes as

$\phi_{n,\mathbf{r}}\sim\sum_{a=0}^{5}\psi_{a}v_{a,n}e^{\mathrm{i}\mathbf{Q}_{a}\cdot\mathbf{r}}.$
(42)

The coefficients $v_{a,n}$ are solved from the band structure.

The symmetries of the theory for one single valley must include translation
$T_{1},T_{2}$, three-fold rotation $R_{2\pi/3}$, $P_{x}\mathcal{T}$. These
transformations do not mix the two valleys. In the following we derive the PSG
matrices of these symmetries. We first need the form of the transformations
when acting on the 8 sites in each unit cell:

$\displaystyle
T_{1,2}(\phi_{n,\mathbf{k}})=\sum_{m}(t_{1,2})_{nm}\phi_{m,\mathbf{k}},\
t_{1}=\begin{pmatrix}0&0&0&0&0&0&1&0\\\ 0&0&0&0&0&0&0&1\\\ 0&0&0&-1&0&0&0&0\\\
0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0\\\ 0&0&0&0&-1&0&0&0\\\ -1&0&0&0&0&0&0&0\\\
0&-1&0&0&0&0&0&0\\\ \end{pmatrix},\ t_{2}=\begin{pmatrix}0&0&1&0&0&0&0&0\\\
0&0&0&0&1&0&0&0\\\ -1&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&-1&0\\\ 0&-1&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&1\\\ 0&0&0&1&0&0&0&0\\\ 0&0&0&0&0&-1&0&0\\\ \end{pmatrix}$ (43)
$\displaystyle
R_{2\pi/3}(\phi_{n,\mathbf{k}})=(r_{\pi/3})_{nm}\phi_{m,R_{2\pi/3}\mathbf{k}},\
r_{2\pi/3}=\begin{pmatrix}1&0&0&0&0&0&0&0\\\ 0&0&0&0&1&0&0&0\\\
0&0&0&0&0&0&1&0\\\ 0&0&1&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0\\\ 0&1&0&0&0&0&0&0\\\
0&0&0&1&0&0&0&0\\\ 0&0&0&0&0&0&0&1\\\ \end{pmatrix},$ (44) $\displaystyle
P_{x}\mathcal{T}(\phi_{n,\mathbf{k}})=(p_{x}t)_{nm}\phi_{m,-P_{x}\mathbf{k}},\
(p_{x}t)_{ab}=\begin{pmatrix}1&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0\\\
0&0&0&-1&0&0&0&0\\\ 0&0&-1&0&0&0&0&0\\\ 0&0&0&0&0&1&0&0\\\ 0&0&0&0&1&0&0&0\\\
0&0&0&0&0&0&-1&0\\\ 0&0&0&0&0&0&0&-1\end{pmatrix},$ (45)

Besides these symmetries, here we argue that, if the system does have an
effective Hubbard model description with two local Wannier orbitals per unit
cell (one for each valley), $P_{y}$ is also a good symmetry of the Hubbard
model, as long as the valley mixing is negligible, which is a justified
assumption with long wavelength moiré potential modulation. Let us first
assume there is no valley mixing, then for each valley the band structure of
the moiré mini band is described by a tight binding model with one orbital per
site on the moiré triangular lattice. The hopping amplitude $t(\theta)$ along
angle $\theta$ must satisfy the following relations based on the explicit
$P_{x}\mathcal{T}$ and translation symmetry:

$\displaystyle t(\theta)=t^{\ast}(\pi-\theta),\ \ \
t^{\ast}(\theta)=t(\pi+\theta),$ (46)

we can easily show that $t(\theta)=t(-\theta)$, namely the system should have
a $P_{y}$ symmetry.

However, when there is valley mixing, $t$ becomes a $2\times 2$ matrix with
off-diagonal terms that mix two valleys. A $2\times 2$ hopping matrix $t$
should satisfy four symmetries, $P_{x}$, $\mathcal{T}$, translation, and
$R_{2\pi/3}$ rotation. A natural choice of $P_{x}$ and $\mathcal{T}$ on $t$ is

$\displaystyle P_{x}:t(\theta)\rightarrow\sigma^{x}t(\pi-\theta)\sigma^{x};\ \
\
\mathcal{T}:t(\theta)\rightarrow(\mathrm{i}\sigma^{y})t^{\ast}(-\mathrm{i}\sigma^{y});$
(47)

and the translation symmetry plus hermicity demands
$t^{\dagger}(\theta)=t(\pi+\theta)$. $P_{y}$ does not change the valley
indices; if $P_{y}$ takes $t(\theta)$ to $t(-\theta)$, there exists a valley
mixing term $t(\theta)\sim\mathrm{i}\sigma^{x}\sin(3\theta)$ that preserves
all the symmetries mentioned above, but break $P_{y}$; while if $P_{y}$ takes
$t(\theta)$ to $\sigma^{z}t(-\theta)\sigma^{z}$ this term is
$t(\theta)\sim\mathrm{i}\sigma^{y}\cos(3\theta)$.

$P_{y}$ acts on the $\phi$ bosons as

$\displaystyle
P_{y}(\phi_{n,\mathbf{k}})=(p_{y})_{nm}\phi^{*}_{m,-P_{y}\mathbf{k}},\
(p_{y})_{ab}=\begin{pmatrix}0&0&0&0&0&0&0&1\\\ 0&0&0&0&0&0&1&0\\\
0&0&0&0&1&0&0&0\\\ 0&0&0&0&0&1&0&0\\\ 0&0&1&0&0&0&0&0\\\ 0&0&0&1&0&0&0&0\\\
0&1&0&0&0&0&0&0\\\ 1&0&0&0&0&0&0&0\end{pmatrix}.$ (48)

Furthermore, in the case with long moiré lattice constant, we additionally
have the six-fold rotation $R_{\pi/3}$

$\displaystyle
R_{\pi/3}(\phi_{n,\mathbf{k}})=(r_{\pi/3})_{nm}\phi_{m,R_{\pi/3}\mathbf{k}},\
(r_{\pi/3})_{ab}=\begin{pmatrix}0&0&0&0&0&0&0&-1\\\ 0&0&-1&0&0&0&0&0\\\
0&0&0&0&1&0&0&0\\\ 0&1&0&0&0&0&0&0\\\ 0&0&0&0&0&0&-1&0\\\ 0&0&0&-1&0&0&0&0\\\
0&0&0&0&0&1&0&0\\\ 1&0&0&0&0&0&0&0\\\ \end{pmatrix}.$ (49)

In the position space, the transformation rules can be summarized as

$G(\phi_{n,\mathbf{r}})=\sum_{m=1}^{8}g_{n,m}\phi_{m,\mathbf{r}^{\prime}_{m}}$
(50)

in which $\mathbf{r}^{\prime}_{m}$ is the center of the unit cell of field
$\phi_{m}$ which is obtained by certain site in the original unit cell
(centered at $\mathbf{r}$) after transformation under symmetry operation $G$.
For example, under $T_{1}$,
$\mathbf{r}^{\prime}_{7}=\mathbf{r}^{\prime}_{8}=\mathbf{r}+2\mathbf{a}_{2}$,
because sites 1 and 2 at unit cell $\mathbf{r}$ are transformed into sites 7
and 8 in the nearby enlarged unit cell which is centered at
$\mathbf{r}+2\mathbf{a}_{2}$. In general, we can write the transformation as
$\mathbf{r}_{m}^{\prime}=G\mathbf{r}+\vec{\Delta}_{G,m}$, in which
$\vec{\Delta}_{G,m}$ is a constant that does not depend on $\mathbf{r}$, and
$G\mathbf{r}$ is the coordinate of the center of the unit cell after spacial
symmetry $G$.

Now we plug in the low energy expansions of $\phi_{n_{\mathbf{k}}}$ around the
minima into the equation, which yields

$\sum_{a=0}^{N-1}G(\psi_{a})v_{a,n}e^{\mathrm{i}\mathbf{Q}_{a}\cdot\mathbf{r}}=\sum_{a=0}^{N-1}\sum_{m=1}^{8}\psi_{a}g_{nm}v_{m,a}e^{\mathrm{i}\mathbf{Q}_{a}\cdot\mathbf{r}_{m}^{\prime}}.$
(51)

The relation can be viewed as a vector identity with $n$ being the vector
index on both sides. Because all the vectors $v_{a,n}(a=0,\dots,N-1)$ are
orthogonal to each other, we can multiply the conjugated vector $v^{*}_{b,n}$
on both sides and sum over $n$:

$G(\psi_{b})e^{\mathrm{i}\mathbf{Q}_{b}\cdot\mathbf{r}}=\sum_{a=0}^{N-1}\sum_{m,n=1}^{8}\psi_{a}v^{*}_{b,n}g_{n,m}v_{a,m}e^{\mathrm{i}\mathbf{Q}_{a}\cdot\mathbf{r}_{m}^{\prime}}.$
(52)

For this equation to hold for all $\mathbf{r}$, the RHS needs to have the same
momentum. This requires $\mathbf{Q}_{b}=G^{-1}\mathbf{Q}_{a}$, which can only
be satisfied by two possible choices of $a$ (recall that in the convention of
eight-site unit cell, each momentum $\mathbf{Q}_{a}$ always has two fold
degeneracy for all $N$), denoted by $a_{1}$ and $a_{2}$. Thus we eventually
have

$G(\psi_{b})=\sum_{m,n=1}^{8}v^{\dagger}_{b,n}g_{nm}v_{a_{1},m}e^{\mathrm{i}\mathbf{Q}_{a_{1}}\cdot\vec{\Delta}_{G,m}}\times\psi_{a_{1}}+\sum_{m,n=1}^{8}v^{\dagger}_{b,n}g_{nm}v_{a_{2},m}e^{\mathrm{i}\mathbf{Q}_{a_{2}}\cdot\vec{\Delta}_{G,m}}\times\psi_{a_{2}}$
(53)

The final results can be organized into $N\times N$ matrices. For $N=6$, the
transformations read

$\displaystyle T_{1,2}(\psi_{a})=(\mathfrak{t}_{1,2})_{ab}\psi_{b},\
(\mathfrak{t}_{1})_{ab}=\begin{pmatrix}-1&0&0&0&0&0\\\ 0&1&0&0&0&0\\\
0&0&0&1&0&0\\\ 0&0&1&0&0&0\\\ 0&0&0&0&0&1\\\ 0&0&0&0&-1&0\\\ \end{pmatrix},\
(\mathfrak{t}_{2})_{ab}=\begin{pmatrix}0&1&0&0&0&0\\\ -1&0&0&0&0&0\\\
0&0&1&0&0&0\\\ 0&0&0&-1&0&0\\\ 0&0&0&0&0&-1\\\ 0&0&0&0&-1&0\\\ \end{pmatrix},$
(54) $\displaystyle
R_{2\pi/3}(\psi_{a})=(\mathfrak{R}_{2\pi/3})_{ab}\psi_{b},\
(\mathfrak{R}_{2\pi/3})_{ab}=\begin{pmatrix}0&0&0&0&1&0\\\ 0&0&0&0&0&1\\\
-1&0&0&0&0&0\\\ 0&-1&0&0&0&0\\\ 0&0&1&0&0&0\\\ 0&0&0&1&0&0\\\ \end{pmatrix},$
(55) $\displaystyle
P_{x}\mathcal{T}(\psi_{a})=(\mathfrak{P}_{x}\mathfrak{T})_{ab}\psi_{b},\
(\mathfrak{P}_{x}\mathfrak{T})_{ab}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&1&-1&0&0\\\
0&0&-1&-1&0&0\\\ 1&-1&0&0&0&0\\\ -1&-1&0&0&0&0\\\ 0&0&0&0&1&-1\\\
0&0&0&0&-1&-1\\\ \end{pmatrix}.$ (56) $\displaystyle
P_{y}(\psi_{a})=(\mathfrak{P}_{y})_{ab}\psi^{*}_{b},\
(\mathfrak{P}_{y})_{ab}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&1&1&0&0\\\
0&0&1&-1&0&0\\\ 1&1&0&0&0&0\\\ 1&-1&0&0&0&0\\\ 0&0&0&0&1&1\\\ 0&0&0&0&1&-1\\\
\end{pmatrix}.$ (57) $\displaystyle
R_{\pi/3}(\psi_{a})=(\mathfrak{R}_{\pi/3})_{ab}\psi_{b},\
(\mathfrak{R}_{\pi/3})_{ab}=\begin{pmatrix}0&0&0&-1&0&0\\\ 0&0&1&0&0&0\\\
0&0&0&0&0&1\\\ 0&0&0&0&-1&0\\\ 0&1&0&0&0&0\\\ -1&0&0&0&0&0\\\ \end{pmatrix}.$
(58)

Figure 4: Some possible density wave patterns of the original boson that
correspond to different condensate of $\psi_{a}$ with $a=0,\cdots 5$. The left
and right patterns correspond to $\vec{\Psi}\sim(1,0,0,0,0,0)$ and
$\vec{\Psi}\sim(0,1/\sqrt{2},1/2,-1/2,0,0)$ respectively.

Deep inside the vortex condensate phase with $r\ll 0$ in equation Eq. 6, the
vector $\vec{\Psi}=(\psi_{0},\psi_{1},\psi_{2},\psi_{3},\psi_{4},\psi_{5})$
can have different condensates depending on the parameters in Eq. 26. We
assume $u>0$ such that the sum $\sum_{a=0}^{5}|\psi_{a}|^{2}$ can be viewed as
a constraint and all the other quartic terms as perturbations. Without loss of
generality we set $\sum_{a=0}^{5}|\psi_{a}|^{2}=1$. The two figures in Eq. 4
illustrate the density waves of the bosonic parton centered at the bonds and
the sites on the moiré triangular lattice that correspond to two different
possible condensates of $\vec{\Psi}$. The density on the bond $l$ is inferred
from $t_{ij}\langle\phi_{i}^{*}\phi_{j}\rangle$, with $ij$ being the link on
the dual honeycomb lattice that is dual to $l$, and $t_{ij}$ takes the sign
according to the gauge convention of Fig. 1. The operator
$t_{ij}\langle\phi_{i}^{*}\phi_{j}\rangle$ is the energy density in terms of
vortex fields, and the modulation of this operator should correspond to the
valence bond solid of the original bosonic parton. We also consider an
operator centered on site $p$ of the original lattice (plaquette of the dual
lattice): $\sum_{\langle ij\rangle\in
p}t_{ij}\langle\phi_{i}^{*}\phi_{j}\rangle,$ with the summation over the links
that surround the plaquette $p$ on the dual honeycomb lattice, whose center
hosts the site $p$ of the original moiré triangular lattice. In both cases,
$\langle\phi_{i}^{*}\phi_{j}\rangle$ is evaluated using Eq. 42 and the value
of $\vec{\Psi}$ which minimizes the quartic energy. The left pattern in Eq. 4
is a rather common valence bond solid configuration for either spin-1/2 system
or hard core boson on the triangular lattice. If one started with the
construction-I of the parton construction, the discussion in this section
corresponds to the original electron system with an average 1/2 electron per
unit cell (the filling considered in Ref. Jin et al., 2021); while for
construction-II, the discussion here applies to one electron per unit cell,
and the analysis in this section corresponds to one of the two spin/valley
flavors of the system.

##  Appendix C Dual of the vortex theory

Here we derive the Lagrangian written in terms of the fractionally charged
bosonic partons for scenario (1). We start with Eq. 6 in our paper:

$\displaystyle\mathcal{L}^{(1)}=\sum_{j=0}^{N-1}|(\partial_{\mu}-\mathrm{i}A_{\mu})\psi_{j}|^{2}+r|\psi_{j}|^{2}+\frac{\mathrm{i}}{2\pi}A\wedge
d(a+eA_{\mathrm{ext}})+\cdots$ (59)

To facilitate the calculation of the DC resistivity which will be discussed in
the next subsection, we need to “dual back” to the charge-carriers, which
requires deforming Eq. 59 with an easy-plane anisotropy
$\sum_{j}|\psi_{j}|^{4}$. The bosonic fractional charge carriers $\varphi_{j}$
are the vortices of the vortex fields $\psi_{j}$. We first take the standard
duality for $\psi_{j}$, and Eq. 59 becomes:

$\displaystyle\mathcal{L}^{(1)}=\sum_{j=0}^{N-1}|(\partial-\mathrm{i}\tilde{A}_{j})\varphi_{j}|^{2}+\tilde{r}|\varphi_{j}|^{2}+\frac{\mathrm{i}}{2\pi}\tilde{A}_{j}\wedge
dA+\frac{\mathrm{i}}{2\pi}A\wedge d(a+eA_{\mathrm{ext}})+\cdots$ (60)

The basic duality relation is that the current of $\psi_{j}$, $i.e.$
$J_{\psi_{j}}\sim d\tilde{A}_{j}$. Now integrating out $A$ would lead to the
following constraint for the rest of the gauge fields:

$\displaystyle\sum_{j}\tilde{A}_{j}-a-eA_{\mathrm{ext}}=0.$ (61)

From this constraint we can take $\tilde{A}_{j}$ as

$\displaystyle\tilde{A}_{j}=\tilde{a}_{j}+\frac{1}{N}a+\frac{e}{N}A_{\mathrm{ext}},\
\ \ \sum_{j}\tilde{a}_{j}=0.$ (62)

Hence the dual of the dual theory becomes

$\displaystyle\mathcal{L}^{(1)}=\sum_{j=0}^{N-1}|(\partial-\mathrm{i}\tilde{a}_{j}-\mathrm{i}\frac{1}{N}a-\mathrm{i}\frac{e}{N}A_{\mathrm{ext}})\varphi_{j}|^{2}+\tilde{r}|\varphi_{j}|^{2}+\cdots.$
(63)

The gauge fields $\tilde{a}_{j}$ are still subject to the constraint
$\sum_{j}\tilde{a}_{j}=0$. $\varphi_{j}$ carries $e/N$ charge of external EM
gauge field; it also carries charge $1/N$ of gauge field $a$ which is shared
with the fermionic parton $f_{\alpha}$.

For scenario (2) the theory in terms of fractional parton $\varphi$ is much
simpler: there is only one flavor of $\varphi$ for each valley, and there is
no extra continuous gauge fields $\tilde{a}$ besides gauge field $a$:
Following the calculation in Ref. Witczak-Krempa et al., 2012, one can
generalize this one flavor of $\varphi$ in each valley to an $\bf{N}$
component of bosons:

$\displaystyle\mathcal{L}^{(2)}=\sum_{l=1}^{\bf{N}}|(\partial-\mathrm{i}\frac{1}{N}a-\mathrm{i}\frac{e}{N}A_{\mathrm{ext}})\varphi^{l}|^{2}+\mathrm{i}\lambda|\varphi^{l}|^{2}+\cdots$
(64)

and the bosons will scatter with both gauge field $a$ and field $\lambda$
which is introduced as a Lagrange multiplier. The fact that $\varphi^{l}$
carries charge $1/N$ of gauge field $a$ does not change the scattering rate
through the large-$\bf{N}$ calculation, as the gauge charge cancels out in the
calculation of scattering rate through the large-$\bf{N}$ approach. Compared
with scenario (2), in scenario (1) the parton $\varphi_{j}$ is also coupled to
extra gauge fields $\tilde{a}_{j}$, which will lead to extra scattering to the
charge carriers.

When computing the resistivity, especially the DC resistivity of scenario (1),
we also rely on a large$-{\bf N}$ generalization, namely we need to introduce
an extra $l=1\cdots{\bf N}$ index for each component of fractional charge
field: $\varphi_{j}^{l}$.

##  Appendix D DC resistivity jump in scenario (1)

In this section we present a detailed computation of the DC resistivity jump
in the scenario (1) of MIT, i.e. the scenario when the insulator has a density
wave. We start with Eq. 63. The resistivity jump at the MIT is given by the
universal resistivity of the bosonic sector of the system $\rho_{b}$ at the
MIT. First of all, one can prove a generalized Ioffe-Larkin rule, which
combines the resistivity of each parton $\varphi_{j}$ into $\rho_{b}$:

$\displaystyle\rho_{b}=\frac{\hbar}{e^{2}}\left(\sum_{j=0}^{N-1}\rho_{b,j}\right),$
(65)

where $\rho_{b,j}$ is the resistivity of each parton $\varphi_{j}$, when the
charge of $\varphi_{j}$ is taken to be 1. This generalized Ioffe-Larkin rule
can be proven by formally integrating out $\varphi_{j}$, gauge fields
$\tilde{a}_{j}$ and $a$ from Eq. 63, and eventually arrive at a response
function of $A_{\mathrm{ext}}$. At each level of the path integral, we keep a
quadratic form of the action, i.e. the random phase approximation. This Ioffe-
Larkin rule is independent of the assignment of electric charges on each
parton.

To compute $\rho_{b}$, we formulate the quantum Boltzmann equation (QBE) for
the $\varphi_{j}$ fields of a given valley. The computation follows that for
$\rho_{b}$ at the MIT without charge fractionalization Witczak-Krempa et al.
(2012), where the gauge field dynamics needs to be modified due to the charge
fractionalization, which we explain in detail below for comparison. Note that
$\rho_{b}$ can be finite without momentum relaxation due to the emergent
particle-hole symmetry. Furthermore, the two-in two-out scatterings among the
$\varphi_{j}$ fields are enough to relax the current and generate finite DC
resistivity. For simplicity, we consider the scattering between the
$\varphi_{j}$ and emergent gauge fields in Eq. (63), where the gauge fields
are in thermal equilibrium and their dynamics is acquired due to the coupling
with the matter fields $\varphi_{j}$ and $f$. Here, we argue that treating the
gauge fields as in thermal equilibrium is a legitimate approximation. First,
the gauge field $a$ couples to the spinon field $f$, which is sensitive to
impurities and relaxes momentum fast. Second, diagrammatically, the two-in
two-out scatterings between the $\varphi_{j}$ fields that give finite DC
resistivity can be captured by the $\varphi_{j}$ scattering with the emergent
gauge fields.

To simplify the computation of the gauge field dynamics, it is convenient to
express Eq. 63 in terms of the gauge field $\tilde{A}_{j}$ (Eq. 61), together
with the effective action for the spinon field, the dual theory reads

$\displaystyle\mathcal{L}^{(1)}=\sum_{j=0}^{N-1}|(\partial-\mathrm{i}\tilde{A}_{j})\varphi_{j}|^{2}+\tilde{r}|\varphi_{j}|^{2}+\bar{f}\left(\partial_{\tau}-\mu-\mathrm{i}\sum_{j=0}^{N-1}\tilde{A}_{j,0}+\mathrm{i}eA_{{\rm
ext},0}+\frac{1}{2m}(\nabla-\mathrm{i}\sum_{j=0}^{N-1}\tilde{\bm{A}}_{j}+\mathrm{i}e\bm{A}_{\rm
ext})^{2}\right)f+\cdots.$ (66)

Integrating out $\varphi_{j}$ and $f$ fields, the gauge field propagators read

$\displaystyle
D^{(\tilde{A})}_{ij}=-\mathrm{i}\langle{\rm{T}}_{t}\tilde{A}_{i}\tilde{A}_{j}\rangle=\begin{cases}\frac{\Pi^{J}_{b}+(N-1)\Pi^{J}_{f}}{(\Pi_{b}^{J})^{2}+N\Pi^{J}_{b}\Pi^{J}_{f}}&\text{
if }i=j\\\ \frac{-\Pi^{J}_{f}}{(\Pi_{b}^{J})^{2}+N\Pi^{J}_{b}\Pi_{f}}&\text{
if }i\neq j\\\ \end{cases},$ (67)

where $\Pi_{b}^{J},\Pi_{f}^{J}$ is the current-current correlation function
for $\varphi_{j}$ and $f$ fields, respectively.

For a controlled systematic calculation of transport, we introduce a large
number of (complex) rotor and spinon flavors ${\bf N}$ with the constraint
$\sum_{l=1}^{\bf N}|\varphi_{j}^{l}|^{2}=1$ for all $j=0,1,...,N-1$, and only
the $l=1$ component couples to $A_{\rm ext}$. The ${\bf N}=1$ limit will be
taken at the end. The effective action for the extended model becomes

$\displaystyle\mathcal{L}=$
$\displaystyle\sum_{j=0}^{N-1}\left(\sum_{l=1}^{{\bf
N}}|(\partial-\mathrm{i}\tilde{A}_{j})\varphi_{j}^{l}|^{2}+\mathrm{i}\lambda_{j}(\sum_{l=1}^{{\bf
N}}|\varphi_{j}^{l}|^{2}-1)+\frac{1}{2g^{2}}(\epsilon_{\mu\nu\lambda}\partial_{\nu}\tilde{A}_{j,\lambda})^{2}\right)$
$\displaystyle+\sum_{l=1}^{\bf
N}\bar{f}_{l}\left(\partial_{\tau}-\mu-\mathrm{i}\sum_{j=0}^{N-1}\tilde{A}_{j,0}+\mathrm{i}eA_{{\rm
ext},0}\delta_{l,1}+\frac{1}{2m}(\nabla-\mathrm{i}\sum_{j=0}^{N-1}\tilde{\bm{A}}_{j}+\mathrm{i}e\bm{A}_{\rm
ext}\delta_{l,1})^{2}\right)f_{l}+\cdots.$ (68)

Using the Fourier expansion for the electrically charged rotor
$\varphi_{j}^{l=1}$ in terms of the holons (+) and doublons (-),

$\varphi_{j}^{l=1}=\int_{\bm{k}}\alpha_{+,j}(t,\bm{k})e^{\mathrm{i}\bm{k}\cdot\bm{x}}+\alpha_{-,j}(t,\bm{k})e^{-\mathrm{i}\bm{k}\cdot\bm{x}},$
(69)

the conductivity $\sigma_{b,j}=\rho_{b,j}^{-1}$ can be obtained as

$\displaystyle\sigma_{b,j}=\langle J_{x,j}\rangle/E_{x},\quad\langle
J_{x,j}\rangle=\int_{\bm{k}}\sum_{s=\pm}s\frac{\bm{k}}{\epsilon_{\bm{k}}}f_{s,j}(t,\bm{k}),$
(70)

where we define the distribution for holon ($s=+$) and doublon ($s=-$) as
$f_{s,j}=\langle\alpha_{s,j}^{\dagger}(t,\bm{k})\alpha_{s,j}(t,\bm{k})\rangle$,
and they satisfy the QBE as

$\displaystyle(\partial_{t}+s\bm{E}\cdot\partial_{\bm{k}})f_{s,j}(t,\bm{k})=\frac{1}{2{\bf
N}}(I_{\lambda_{j}}[f_{\pm,j}]+I_{\tilde{A}_{j}}[f_{\pm,j}]).$ (71)

Note that the gauge choice in Eq. 66 ensures that $f_{s,j}$ are decoupled and
equal for different $j$ within the approximation that $\tilde{A}_{j}$ is in
thermal equilibrium, so the subindex $j$ will be dropped unless there is
ambiguity. The RHS of Eq. 71 reads

$\displaystyle\mathrm{RHS}=$ $\displaystyle\frac{1}{2{\bf
N}}\int_{0}^{\infty}\frac{\mathrm{d}\Omega}{\pi}\int\frac{\mathrm{d}^{2}\bm{q}}{(2\pi)^{2}}\\{\tau_{\lambda}\mathop{\mathrm{Im}}D^{(\lambda)}(\Omega,\bm{q})+\tau_{\tilde{A}}\mathop{\mathrm{Im}}D^{(\tilde{A})}_{ii}(\Omega,\bm{q})\\}$
$\displaystyle\times\\{\frac{2\pi\delta(\epsilon_{\bm{k}}-\epsilon_{\bm{k}+\bm{q}}+\Omega)}{4\epsilon_{\bm{k}}\epsilon_{\bm{k}+\bm{q}}}[f_{s}(t,\bm{k})(1+f_{s}(t,\bm{k}+\bm{q}))n_{\bm{q}}(\Omega)-(1+f_{s}(t,\bm{k}))f_{s}(t,\bm{k}+\bm{q})(1+n_{\bm{q}}(\Omega))]$
$\displaystyle+\qquad\frac{2\pi\delta(\epsilon_{\bm{k}}-\epsilon_{\bm{k}+\bm{q}}-\Omega)}{4\epsilon_{\bm{k}}\epsilon_{\bm{k}+\bm{q}}}[f_{s}(t,\bm{k})(1+f_{s}(t,\bm{k}+\bm{q}))(1+n_{\bm{q}}(\Omega))-(1+f_{s}(t,\bm{k}))f_{s}(t,\bm{k}+\bm{q})n_{\bm{q}}(\Omega)]$
$\displaystyle+\qquad\frac{2\pi\delta(-\epsilon_{\bm{k}}-\epsilon_{\bm{k}+\bm{q}}+\Omega)}{4\epsilon_{\bm{k}}\epsilon_{\bm{k}+\bm{q}}}[f_{s}(t,\bm{k})f_{s}(t,\bm{k}+\bm{q})(1+n_{\bm{q}}(\Omega))-(1+f_{s}(t,\bm{k}))(1+f_{s}(t,\bm{k}+\bm{q}))n_{\bm{q}}(\Omega)]\\},$
(72)

where $\tau_{\lambda}=-1$ and
$\tau_{\tilde{A}}=(2\bm{k}\times\hat{\bm{q}})^{2}$ come from the bare vertex
functions.

$N$ | 1 | 2 | 3 | 4 | 5 | 6 | … | $\infty$
---|---|---|---|---|---|---|---|---
$\sigma_{b,j}(e^{2}/\hbar)$ | 0.021 | 0.029 | 0.034 | 0.036 | 0.038 | 0.039 |  | 0.047
$\rho_{b}(h/e^{2})$ | 3.72 | 5.41 | 7.09 | 8.76 | 10.44 | 12.11 |  | $\left(3.62+1.68(N-1)\right)$
Table 1:  Rotor conductivity ($\sigma_{b,j}$) and resistivity jump $\rho_{b}$
at the MIT with fractionally charged bosonic parton $e_{\ast}=e/N$.

$\mathop{\mathrm{Im}}D^{(\lambda)},\mathop{\mathrm{Im}}D^{(\tilde{A})}$
physically denote the density of states of the emergent fields that scatter
with $\varphi$, which are broad in the $(\Omega,\bm{q})$ space due to the
couplings with the $\varphi$ fields. Below, we ignore the bare dynamics.
$D^{(\lambda),(\tilde{A})}$ in the large-$\bf{N}$ limit reads

$\displaystyle D^{(\lambda)}(\Omega,\bm{q})$
$\displaystyle=\frac{1}{\Pi_{b}},$ $\displaystyle
D^{(\tilde{A})}_{ii}(\Omega,\bm{q})$
$\displaystyle=\frac{\Pi^{J}_{b}+(N-1)\Pi^{J}_{f}}{(\Pi_{b}^{J})^{2}+N\Pi^{J}_{b}\Pi^{J}_{f}}=\frac{N-1}{N}\frac{1}{\Pi_{b}^{J}}+\frac{1}{N}\frac{1}{\Pi_{b}^{J}+N\Pi_{f}^{J}},$
(73)

where $D^{(\tilde{A})}_{ii}$ reduces to the MIT without charge
fractionalization as discussed in Ref. Witczak-Krempa et al., 2012 when $N=1$.
For $N>1$, as only the linear combination of $\tilde{A}_{j}$, i.e.
$\sum_{j=0}^{N-1}\tilde{A}_{j}$ couples to the spinon field $f$ and is Landau
damped, there is a factor $\frac{1}{N}$ for the Landau damped component of the
gauge field propagator $D^{(\tilde{A})}_{ii}$, which may also be understood as
the $a$ component of gauge field in Eq. (63). The rest part is not Landau
damped, and is determined solely by $\Pi^{J}_{b}$. Note that as
$\mathop{\mathrm{Im}}\Pi_{f}^{J}\gg\mathop{\mathrm{Im}}\Pi_{b}^{J}$ in the
limit $\mu\gg T$, the Landau damped component can be approximated as
$\frac{1}{N}\frac{1}{\Pi_{b}^{J}+N\Pi_{f}^{J}}\approx\frac{1}{N}\frac{1}{\Pi_{b}^{J}(\Omega=0,\bm{q})+N\Pi_{f}^{J}(\Omega,\bm{q})}$,
and be treated in the same way as Ref. Witczak-Krempa et al., 2012 for the
gauge field $a$. On the other hand, the first term in $D^{(\tilde{A})}_{ii}$
should be determined for generic $\Omega,\bm{q}$. Using the standard
expression for polarizations $\Pi$,

$\displaystyle\Pi_{b}(\Omega,\bm{q})$
$\displaystyle=\frac{T}{2}\sum_{m}\int_{\bm{k}}\tau_{\lambda}\frac{1}{(\nu_{m}+\Omega_{n})^{2}+\epsilon_{\bm{k}+\bm{q}}^{2}}\frac{1}{\nu_{m}^{2}+\epsilon_{\bm{k}}^{2}}|_{\mathrm{i}\Omega_{n}\rightarrow\Omega+\mathrm{i}\delta}$
$\displaystyle\Pi_{b}^{J}(\Omega,\bm{q})$
$\displaystyle=\frac{T}{2}\sum_{m}\int_{\bm{k}}\tau_{\tilde{A}}\frac{1}{(\nu_{m}+\Omega_{n})^{2}+\epsilon_{\bm{k}+\bm{q}}^{2}}\frac{1}{\nu_{m}^{2}+\epsilon_{\bm{k}}^{2}}|_{\mathrm{i}\Omega_{n}\rightarrow\Omega+\mathrm{i}\delta}$
$\displaystyle\Pi_{f}^{J}(\Omega,\bm{q})$
$\displaystyle=-\frac{T}{2}\sum_{m}\int_{\bm{k}}\frac{(2\bm{k}\times\hat{\bm{q}})^{2}}{(2m)^{2}}\frac{1}{\mathrm{i}(\omega_{m}+\Omega_{n})-\xi_{\bm{k}+\bm{q}}}\frac{1}{\mathrm{i}\omega_{m}-\xi_{\bm{k}}}|_{\mathrm{i}\Omega_{n}\rightarrow\Omega+\mathrm{i}\delta},$
(74)

Eq. (71) can be solved self-consistently. In Tab. 1, we show $\sigma_{b,j}$
and the final resistivity $\rho_{b}=(N\sigma_{b,j}^{-1})/2$ at different $N$,
again the factor of $1/2$ arises from the two spin/valley flavors. $\rho_{b}$
increases roughly linearly with $N$, and the fit of the data points at
different $N$ gives

$\displaystyle\rho_{b}=\left(R^{(0)}+R^{(1)}(N-1)\right)\frac{h}{e^{2}}=\left(3.62+1.68(N-1)\right)\frac{h}{e^{2}}.$
(75)