# Superconductor-Insulator Transition in TMD moiré systems and the
Deconfined Quantum Critical Point

Nayan Myerson-Jain  Cenke Xu  Department of Physics, University of California,
Santa Barbara, CA 93106

###### Abstract

We propose that the recently observed superconductor-insulator transition
(SIT) in a twisted bilayer transition metal dichalcogenides moiré system Xia
et al. (2024) at hole filling $\nu=1$ may be described by the deconfined
quantum critical point (DQCP), which was originally proposed for the
transition between the Néel order and the valence bond solid (VBS) order on
the square lattice Senthil et al. (2004a, b). The key symmetries involved in
the original DQCP include a $\mathrm{SO}(3)_{s}$ spin symmetry, as well as a
$C_{4}$ lattice rotation symmetry for the VBS order that is enlarged into a
$\mathrm{U}(1)_{v}$ symmetry near the DQCP. In the current SIT under
consideration, the counterpart of the $\mathrm{SO}(3)_{s}$ spin symmetry is an
approximate $\mathrm{SO}(3)_{v}$ symmetry that transforms between different
crystalline orders on the triangular lattice, and the role of the
$\mathrm{U}(1)_{v}$ symmetry is replaced by the ordinary
charge-$\mathrm{U}(1)_{e}$ symmetry. At the DQCP, the
$\mathrm{SO}(3)_{v}\times\mathrm{U}(1)_{e}$ may enlarge into an emergent SO(5)
symmetry. Under strain, the SIT is driven into either a prominent first order
transition, or an “easy-plane” DQCP, which is expected to have an emergent
O(4) symmetry.

##  I Introduction

The recent rapid experimental progress of searching for novel correlated
states of matter in moiré systems has generated continuous excitement in the
condensed matter community. Various exotic phases of matter, including the
fractional Chern insulator Spanton et al. (2018); Xie et al. (2021),
fractional quantum anomalous Hall effects Cai et al. (2023); Park et al.
(2023); Zeng et al. (2023); Lu et al. (2023); Xu et al. (2023), and even
signatures of fractional topological insulators Kang et al. (2024) have all
been reported. Most recently, superconductivity has also been reported in
twisted bilayer transition metal dichalcogenides (TMD) moiré systems at hole
filling $\nu=1$ (one hole per moiré unit cell) Xia et al. (2024); Guo et al.
(2024), and a continuous superconductor-insulator transition (SIT) tuned by
the displacement field was observed Xia et al. (2024). Given all the
remarkable unconventional phases identified in the TMD moiré systems, it is
natural to ask whether the quantum phase transitions in these systems can also
transcend the ordinary paradigm. The goal of this work is to propose that the
SIT observed in Ref. Xia et al., 2024 may be described by the deconfined
quantum critical point (DQCP) Senthil et al. (2004a, b), an archetypal example
of unconventional QCP beyond the standard paradigm.

The original DQCP was proposed as an unconventional quantum phase transition
between the Néel order with spontaneous symmetry breaking (SSB) of the SO(3)s
spin symmetry, and a valence bond solid (VBS) order which spontaneously breaks
the lattice $C_{4}$ rotation symmetry, and at the DQCP the $C_{4}$ symmetry is
enlarged into a $\mathrm{U}(1)_{v}$ emergent symmetry. The DQCP has been an
extremely active subfield in the theoretical and numerical condensed matter
community Senthil (2023), and it has very profound connections to many other
notions in theoretical physics, such as the ’t Hooft anomaly, emergent
symmetry Senthil and Fisher (2005); Sandvik (2007); Melko and Kaul (2008);
Nahum et al. (2015a, b); Sun et al. (2021), the symmetry protected topological
(SPT) phases Chen et al. (2013a, 2012) in one higher dimension Vishwanath and
Senthil (2013), the web of duality Wang et al. (2017); Senthil et al. (2019),
the “loss of conformality” Kaplan et al. (2009); Wang et al. (2017); Zhou et
al. (2024), and the intrinsically gapless SPT phases Thorngren et al. (2021);
Wen and Potter (2023). But the study on DQCP has been mostly theoretical and
numerical, except for a possible experimental realization of the DQCP with
certain anisotropy in SrCu2(BO3)2 Guo et al. (2020); Zhao et al. (2019); Cui
et al. (2023); Guo et al. (2023); Lee et al. (2019), which was expected to be
connected to the “easy-plane” DQCP Metlitski and Thorngren (2018). But the
experimental connection of the “full” DQCP with supposedly an emergent SO(5)
symmetry has been elusive. In this work we hope to bridge the gap of the
experimental connection of the DQCP.

The twisted bilayer TMD moiré system with hole filling can be modelled as a
two-orbital continuum model for each spin/valley flavor, one orbital from each
layer Wu et al. (2019); Yu et al. (2019); Devakul et al. (2021). The
tunnelling amplitude as well as the energy difference between the two layers
together form a three component vector $\mathbf{\Delta}(\mathbf{r})$, which
depends on the spatial coordinate $\mathbf{r}$ due to the twisting. The
configuration of $\mathbf{\Delta}(\mathbf{r})$ may have a nonzero integer
Skyrmion number within the moiré unit cell, which leads to a nonzero Chern
number for the moiré minibands for each spin/valley index. At small twisting
angle, the physics of the first two moiré minibands can be captured by the
Kane-Mele tight-binding model Kane and Mele (2005a, b) on the honeycomb moiré
lattice. A displacement field in the twisted bilayer TMD moiré system
explicitly breaks the exchange symmetry between the A/B sublattice (the
$C_{2y}$ symmetry) of the moiré honeycomb lattice, i.e. the holes prefer to
stay on (for example) the sublattice A of the moiré lattice, which forms a
triangular lattice. With larger twisting angle, at least one more sublattice
at the center of each hexagon is needed to reproduce the physics of the moiré
minibands Crépel and Millis (2024), but a displacement field still lifts the
degeneracy among the three sublattices. Although the superconductivity happens
when the holes are at half-filling of a pair of Chern bands with opposite
Chern numbers, the nontrivial topology of the Chern bands may not play a
crucial role at the observed superconductor-insulator transition, as the
superconductor could still be ordinary rather than a topological
superconductor. In fact, once the charge $\mathrm{U}(1)_{e}$ is broken, a
quantum spin Hall insulator can be smoothly connected to an ordinary
superconductor.

If the holes are gapped on both sides of the SIT, one can just focus on the
bosonic sector of the system, if the goal is to understand the universality
class of the SIT. It was argued (for example in Ref. Balents et al., 2005)
that the universal low energy features of certain SITs can be understood as
superfluid-insulator transitions of interacting bosons. Given the observations
in the last paragraph, and the “strong pairing” nature of the superconductor
phase Xia et al. (2024), we may consider the SIT as a superfluid-insulator
transition of hard core bosons on an effective triangular lattice at half-
filling, i.e. on average half-boson per site. The hard core boson carries the
same quantum number as a Cooper pair. The SIT can also be modelled as the low
energy bosonic sector of interacting spin-1/2 fermions on the effective
triangular lattice, also at half-filling. A direct and continuous SIT in this
case must be beyond the ordinary Wilson-Fisher paradigm, at least in the limit
with no disorder. In fact half-filled bosons (or spin-1/2 fermions) on a
triangular lattice is prohibited from having a trivial insulator (gapped and
nondegenerate), due to the Lieb-Shultz-Mattis theorem Lieb et al. (1961);
Hastings (2004); Oshikawa (2000). More specifically, the insulator must either
spontaneously break certain symmetry, or it must form a gapless or topological
spin liquid.

Figure 1: The moiré lattice ($a$), and the effective triangular lattice ($b$)
for the study of the SIT. The mean field state of the fermionic parton
$\psi_{\alpha}$ sees a $\pi$-flux on half of the triangle plaquettes ($b$).

If the insulator has a $Z_{2}$ topological order, there is a previously known
mechanism for the SIT. In this scenario, the insulator phase in the phase
diagram is a $Z_{2}$ spin liquid Read and Sachdev (1991); Wen (1991, 2002);
Moessner and Sondhi (2001a). A $Z_{2}$ spin liquid on the triangular lattice
does not need to break any lattice symmetry. The simplest $Z_{2}$ spin liquid
can be constructed with a parton formalism, $c_{j,\alpha}=b_{j}f_{j,\alpha}$:
$c_{j,\alpha}$ corresponds to the physical fermion ($e.g.$ electron or hole),
$b_{j}$ and $f_{j,\alpha}$ are the bosonic and fermionic partons respectively,
$\alpha=\uparrow,\downarrow$ is the spin index, and $j$ labels the sites of
the triangular lattice. Such a $Z_{2}$ spin liquid corresponds to a mean field
state of $f_{\alpha}$ with the following mean field Hamiltonian:
$H_{\mathrm{MF}}=\sum_{\langle
i,j\rangle}-t_{ij}f^{\dagger}_{i,\alpha}f_{j,\alpha}+\Delta_{ij}(\epsilon_{\alpha\beta}f_{i,\alpha}f_{j,\beta})+h.c.$.
The parton construction mentioned above grants $f_{\alpha}$ a local
$\mathrm{U}(1)_{g}$ degrees of freedom, but the nonzero pairing amplitude
$\Delta_{ij}$ breaks the $\mathrm{U}(1)_{g}$ gauge symmetry down to $Z_{2}$,
hence the spin liquid phase becomes a $Z_{2}$ spin liquid. The bosonic parton
$b$ carries both the physical $\mathrm{U}(1)_{e}$ charge, and also the $Z_{2}$
gauge charge. The SIT in this case corresponds to condensing $b$. The
condensation of $b$ would suppress the $Z_{2}$ gauge field due to the Higgs
mechanism, and identify $c_{\alpha}$ and $f_{\alpha}$. The mean field
Hamiltonian of $f_{\alpha}$ then becomes the BdG Hamiltonian of the physical
fermion $c_{\alpha}$ Grover et al. (2010). It was also shown numerically that
a $Z_{2}$ spin liquid is indeed in proximity with a superconductor Jiang and
Kivelson (2021). Recently this mechanism of SIT has been explored in the
current context of twisted bilayer TMD moiré system with a three-orbital model
Kim et al. (2024).

Figure 2: Comparison between the original DQCP $(a)$ and the DQCP proposed in
this work $(b)$. In the original DQCP, the transition is between an AF Néel
order which has spontaneous symmetry breaking (SSB) of the SO(3)s spin
symmetry, and a VBS order that spontaneously breaks the $C_{4}$ rotation
symmetry of the square lattice. Here, the $C_{4}$ rotation symmetry enlarges
into an emergent $\mathrm{U}(1)_{v}$ symmetry near the DQCP. In our current
case, the counterpart of the Néel order is a crystalline order on the
triangular lattice, which is expected to enjoy an approximate
$\mathrm{SO}(3)_{v}$ symmetry and the role of the $\mathrm{U}(1)_{v}$ symmetry
is played by the ordinary charge $\mathrm{U}(1)_{e}$ symmetry. The vertical
axis in $(a)$ represents the term that breaks the $\mathrm{U}(1)_{v}$ symmetry
down to $C_{4}$, which is dangerously irrelevant as was discussed in Ref.
Senthil et al., 2004a, b. In our current case, when ${\cal L}_{3}$ in Eq. 21
is ignored, the crystalline orders on the triangular lattice will have an
emergent $\mathrm{SO}(3)_{v}$ symmetry at the SIT. The vertical axis of $(b)$
corresponds to Eq. 20 which breaks $\mathrm{SO}(3)_{v}$ and is dangerously
irrelevant at the DQCP. The operator ${\cal L}_{3}$ is a weakly relevant
perturbation at the DQCP.

In this work we propose a new candidate theory for the observed SIT. We will
demonstrate that the observed SIT could actually be a deconfined quantum
critical point (DQCP). In our current system, though there is no SO(3)s spin
symmetry, there is an approximate $\mathrm{SO}(3)_{v}$ symmetry that connects
different crystalline orders, which plays the role as the $\mathrm{SO}(3)_{s}$
symmetry of the original DQCP. The $\mathrm{U}(1)_{v}$ symmetry is replaced by
the charge $\mathrm{U}(1)_{e}$ symmetry (Fig. 2). We will also briefly discuss
a second new candidate theory for the SIT, which is described by an $N_{f}=2$
QED, and it is dual to the easy-plane DQCP. The theory-II applies when the
supercondcutor phase coexists with certain commensurate density wave.

Our proposal is partly motivated by the fact that experimentally the
insulating phase is “competing” with the superconductor in the sense that when
the SC is suppressed by (for example) temperature, the insulating phase
emerges Xia et al. (2024). And the DQCP is precisely a theory describing the
transition between two competing orders. The experimental signatures of our
proposal will be discussed in more detail in section IV.

##  II A new candidate theory for the SIT: the DQCP

As was mentioned in the last section, for the purpose of understanding the
universality of the SIT, we model the SIT as hard-core bosons, or (pseudo)
spin-1/2 degrees of freedom on the effective triangular lattice. The symmetry
allowed effective Hamiltonian of the pseudo spin-1/2 degrees of freedom takes
the following form

$\displaystyle H_{\text{eff}}=\sum_{\langle
i,j\rangle}J_{z}\tilde{S}^{z}_{i}\tilde{S}^{z}_{j}+J_{xy}(\tilde{S}^{x}_{i}\tilde{S}^{x}_{j}+\tilde{S}^{y}_{i}\tilde{S}^{y}_{j})+\cdots$
(1)

Here $\tilde{S}^{z}_{i}=n-1/2$, where $n$ is the density of Cooper pair; and
$\tilde{S}^{x}_{i}$, $\tilde{S}^{y}_{i}$ are the real and imaginary parts of
the Cooper pair operator. Note that there is a global constraint that
$\sum_{i}\tilde{S}^{z}_{i}=0$, due to the fixed half-filling of the charge
density. Then one can introduce the fermionic partons using the standard
formalism Wen (2002):

$\displaystyle\tilde{S}^{a}_{i}=\frac{1}{2}f^{\dagger}_{i,\alpha}\sigma^{a}_{\alpha\beta}f_{i,\beta},$
(2)

and construct states of Cooper pair $\tilde{S}^{a}_{i}$ using states of
$f_{\alpha}$. This parton formalism actually has a SU(2)g invariance, which
becomes explicit when one performs a particle-hole transformation of
$f_{\downarrow}$:

$\displaystyle\psi_{i,1}=f_{i,\uparrow},\ \ \
\psi_{i,2}=f^{\dagger}_{i,\downarrow}.$ (3)

Then the states of $\psi_{i,\alpha}$ could have a maximal SU(2)g gauge
invariance Wen (2002).

The effective (pseudo) spin-1/2 model corresponds to the low energy bosonic
sector of interacting fermions $c_{j,\alpha}$ on the same triangular lattice.
Hence alternatively, one can also formulate the parton theory directly
starting with $c_{j,\alpha}$, which is a formalism quite broadly used in the
studies of cuprates high temperature superconductor Sachdev et al. (2009);
Sachdev (2009); Moon and Sachdev (2009); Qi and Sachdev (2010); Moon and
Sachdev (2011); Huo et al. (2013); Chowdhury and Sachdev (2014, 2015); Bonetti
and Metzner (2022), and other exotic quantum liquid states Xu and Sachdev
(2010); Xu (2011); Zou and Chowdhury (2020):

$\displaystyle c_{j,\alpha}=Z_{j,\alpha\beta}\psi_{j,\beta},$ (4)

where $j$ again labels the sites of the effective triangular moiré lattice.
$Z_{\alpha\beta}$ is a $2\times 2$ SU(2) matrix field, whose left and right
transformations correspond to the spin symmetry, and a SU(2)g gauge
transformation:

$\displaystyle\mathrm{SU}(2)_{s}:Z\rightarrow U_{s}Z,$ (5) (6)
$\displaystyle\mathrm{SU}(2)_{g}:Z\rightarrow ZU^{\dagger}_{g},\ \
\psi\rightarrow U_{g}\psi,$ (7)

where $U_{s}$ and $U_{g}$ are both SU(2) matrices. Here we stress that it is
not necessary for the system to have a full spin SU(2)s symmetry, as we are
going to consider a phase diagram where $Z_{\alpha\beta}$ is gapped. Indeed,
the TMD moiré system has no spin SU(2)s symmetry due to the spin-orbit
coupling. In addition to the spin and gauge transformations, $\psi_{\alpha}$
also carries the charge $\mathrm{U}(1)_{e}$ symmetry:

$\displaystyle\mathrm{U}(1)_{e}:c_{\alpha}\rightarrow
e^{\mathrm{i}\theta}c_{\alpha},\ \ \ \psi_{\alpha}\rightarrow
e^{\mathrm{i}\theta}\psi_{\alpha}.$ (8)

The analysis of a parton formalism always starts with the mean field state of
the partons. In this formalism, the most general mean field Hamiltonian for
$\psi_{j,\alpha}$ takes the following form

$\displaystyle H_{\mathrm{MF}}=\sum_{\langle
i,j\rangle}-t_{ij,\alpha\beta}\psi^{\dagger}_{i,\alpha}\psi_{j,\beta}+h.c.$
(9)

where

$\displaystyle t_{ij,\alpha\beta}\sim\langle
Z^{\dagger}_{i,\alpha\sigma}t^{c}_{ij,\sigma\sigma^{\prime}}Z_{j,\sigma^{\prime}\beta}\rangle,$
(10)

and $t^{c}_{ij,\sigma\sigma^{\prime}}$ is the hopping amplitude of
$c_{j,\alpha}$ on the effective triangular lattice, which may depend on the
spin index $\sigma$ and $\sigma^{\prime}$.

The goal of this work is not to analyze which mean field state has the most
favorable energy, but to investigate the state that can potentially lead to an
interesting nontrivial SIT. We consider a mean field state with
$t_{ij,\alpha\beta}=\pm t\delta_{\alpha\beta}$, and the sign $\pm 1$
distribution on the triangular lattice leads to a staggered $\pi$-flux state
for the mean field band structure of $\psi_{\alpha}$ (Fig. 1). This mean field
state preserves the whole $\mathrm{SU}(2)_{g}$ gauge symmetry. Here we note
that, the staggered $\pi-$flux state discussed here is different from the
standard $\pi-$flux spin liquid on the triangular lattice often studied in the
literature Lu (2016); Song et al. (2020, 2019), which has $\mathrm{U}(1)_{g}$
rather than $\mathrm{SU}(2)_{g}$ gauge invariance. In fact, the state
discussed here would be “undesirable” in the context of spin liquid, as this
state would break the $\mathrm{SU}(2)_{s}$ spin symmetry of the effective
spin-1/2 operators $\tilde{S}^{a}_{i}$. But again, in our current system there
is no $\mathrm{SU}(2)_{s}$ symmetry to begin with, hence the staggered
$\pi-$flux state discussed here is legitimate.

It is known that a $\pi$-flux state on the triangular lattice has two Dirac
points in the Brillouin zone, for each component $\alpha=1,2$. The index
$\alpha$ is a “color” index which couples to a SU(2)g gauge field. Hence at
low energy, the physics of the mean field state discussed above is expected to
be described by the following Lagrangian:

$\displaystyle{\cal
L}_{\mathrm{QCD}}=\sum_{v=1,2}\bar{\psi}_{\alpha,v}\gamma_{\mu}(\partial_{\mu}\delta_{\alpha\beta}-\sum_{l=1}^{3}\mathrm{i}a^{l}_{\mu}\tau^{l}_{\alpha\beta})\psi_{\beta,v}+\cdots,$
(11)

where $v=1,2$ labels the two Dirac valleys. We choose convention for the gamma
matrices, $\gamma_{\mu}=(-\sigma^{2},\sigma^{3},\sigma^{1})$ where
$\vec{\sigma}$ are Pauli matrices that carry only Dirac spinor indices (as
opposed to $\vec{\tau}$ which are labeled by color indices). We note that the
two Dirac-valleys discussed here are not the standard valleys associated with
the moiré superlattice, but are merely labels for two degenerate Dirac cones
in the spectrum of the staggered $\pi$-flux state.

Figure 3: The approximate $\mathrm{SO}(3)_{v}$ symmetry. Due to the existence
of the symmetry allowed term Eq. 20, the three component vector
$\hat{{\bm{V}}}$ has eight favored degenerate directions (for $u>0$),
corresponding to the corners of a cube. When ${\cal L}_{3}$ is absent, the
discrete symmetry of $\hat{{\bm{V}}}$ will enlarge into an emergent SO(3)v at
the DQCP, analogous to the $C_{4}$ symmetry of the VBS order on the square
lattice becomes an emergent $\mathrm{U}(1)_{v}$ symmetry at the original DQCP.
With the presence of ${\cal L}_{3}$, the vector $\hat{{\bm{V}}}$ prefers to
point towards four corners of a tetrahedron.

The theory in Eq. 11 is $N_{f}=2$ QCD with an SU(2)g gauge field, where the
ellipses in Eq. 11 include other terms allowed by the symmetry, such as the
Maxwell term of the SU(2)g gauge field, and gauge invariant four-fermion or
higher order terms. As we will show later, one particular combination of the
four-fermion terms will be important, as it is the tuning parameter of the
SIT.

Eq. 11 is the same field theory as the renowned $\pi$-flux spin liquid state
of the spin-1/2 frustrated quantum magnet on the square lattice. However, we
stress that here $\psi$ carries the charge $\mathrm{U}(1)_{e}$ symmetry, but
no spin quantum number. As was discussed in detail in Ref. Wang et al., 2017,
Eq. 11 is one of many possible representations of the deconfined quantum
critical point (DQCP), which was originally proposed as a potential direct
continuous quantum phase transition between the antiferromagnetic Néel order
and the Valence bond solid (VBS) order of a spin-1/2 frustrated quantum magnet
on the square lattice Senthil et al. (2004a, b). In the original DQCP, the key
symmetries include a SO(3)s spin symmetry, and a $C_{4}$ rotation symmetry of
the VBS order, which will be enlarged into a $\mathrm{U}(1)_{v}$ symmetry near
the DQCP. Furthermore, it was proposed theoretically Senthil and Fisher (2005)
and also observed numerically that Nahum et al. (2015a, b), the
$\mathrm{SO}(3)_{s}\times\mathrm{U}(1)_{v}$ symmetry further enlarge into an
emergent SO(5) symmetry in the infrared at the DQCP.

In fact, the SO(5) symmetry is the maximal possible continuous global symmetry
of Eq. 11, which is most clearly revealed in the Majorana fermion basis. When
we write the Dirac fermion $\psi$ as $\psi=\chi_{1}+\mathrm{i}\chi_{2}$ with
Majorana fermions $\chi_{1,2}$, another two-component flavor index was
introduced, and there are in total eight components of Majorana fermions
including the gauged color space, and the valley space. The maximal continuous
transformation on the eight Majorana fermions is SO(8), which includes the
SU(2)g gauge symmetry as its subgroup. Then, the maximal global symmetry
should be a subgroup of SO(8) which commutes with SU(2)g, and this maximal
global symmetry is SO(5). This symmetry analysis was discussed in detail in
Ref. Ran and gang Wen (2006); Xu (2008).

At first glance, it seems rather unlikely that the current system under study
can realize the DQCP, as the microscopic symmetry of the TMD moiré system is
too low. For example, there is no spin SO(3)s symmetry to begin with. However,
in the following we will demonstrate that, what plays the role of the
$\mathrm{SO}(3)_{s}$ symmetry of the originally proposed DQCP, is actually an
approximate $\mathrm{SO}(3)_{v}$ symmetry that transforms among different
crystalline orders on the triangular lattice; and the $\mathrm{U}(1)_{v}$
symmetry of the original DQCP is replaced by the charge $\mathrm{U}(1)_{e}$
symmetry.

In fact, it was observed long back that, the crystalline orders on the
triangular lattice may very well enjoy a far larger emergent symmetry near
various critical points or critical phases Moessner and Sondhi (2001b); Xu and
Sachdev (2009); Jian et al. (2018); Zou et al. (2021), compared with the
microscopic lattice symmetry. In order to demonstrate this approximate
$\mathrm{SO}(3)_{v}$ symmetry in our current set-up, let us consider the
following three-component vector of crystalline order:

$\displaystyle\hat{{\bm{V}}}\sim\bar{\psi}_{v}\vec{\mu}_{vv^{\prime}}\psi_{v^{\prime}}.$
(12)

$\vec{\mu}$ are the three Pauli matrices that operate on the Dirac-valley
space. With a displacement field, the current moiré system has the translation
symmetries, time-reversal symmetry, as well as a $C_{3}$ rotation symmetry of
the triangular lattice. Under these symmetries, the vector $\hat{{\bm{V}}}$
transforms as 111The symmetry of the effective triangular moiré lattice is
smaller than the full symmetry of a standard triangular lattice. For the
symmetries under consideration here one can use the projective symmetry group
(PSG) derived for the standard $\pi-$flux spin liquid state Lu (2016); Song et
al. (2020, 2019).

$\displaystyle T_{1}$ $\displaystyle:$
$\displaystyle(\hat{V}_{1},\hat{V}_{2},\hat{V}_{3})\rightarrow(-\hat{V}_{1},-\hat{V}_{2},\hat{V}_{3}),$
(13) $\displaystyle T_{2}$ $\displaystyle:$
$\displaystyle(\hat{V}_{1},\hat{V}_{2},\hat{V}_{3})\rightarrow(-\hat{V}_{1},\hat{V}_{2},-\hat{V}_{3}),$
(15) $\displaystyle C_{3}$ $\displaystyle:$
$\displaystyle(\hat{V}_{1},\hat{V}_{2},\hat{V}_{3})\rightarrow(-\hat{V}_{3},\hat{V}_{1},-\hat{V}_{2}),$
(17) $\displaystyle{\cal T}$ $\displaystyle:$
$\displaystyle(\hat{V}_{1},\hat{V}_{2},\hat{V}_{3})\rightarrow(\hat{V}_{1},\hat{V}_{2},\hat{V}_{3}).$
(19)

Here, $\hat{V}_{i}$ are density waves at the three $M$ points of the moiré
Brillouin zone: $K_{1}=(\pi/\sqrt{3},-\pi)$, $K_{2}=(\pi/\sqrt{3},\pi)$,
$K_{3}=(2\pi/\sqrt{3},0)$ (Fig. 2$(b)$). Though these symmetries are smaller
than the full symmetry of a triangular lattice, they already guarantee that at
the quadratic level of $\hat{{\bm{V}}}$, the only symmetry allowed term is
$|\hat{{\bm{V}}}|^{2}$, which has a $\mathrm{SO}(3)_{v}$ symmetry, and it
should be part of the Lagrangian in Eq. 11. Higher order terms of
$\hat{{\bm{V}}}$ may break the $\mathrm{SO}(3)_{v}$ symmetry, but they are at
least sixth order of $\psi$. For example the following term

$\displaystyle{\cal L}_{4}\sim
u(|\hat{V}_{1}|^{4}+|\hat{V}_{2}|^{4}+|\hat{V}_{3}|^{4})$ (20)

is allowed by the microscopic symmetry, but it is a product of eight fermions,
and is expected to be highly irrelevant for the $N_{f}=2$ QCD Eq. 11. Another
term which is the product of six fermions is also allowed by symmetry:

$\displaystyle{\cal L}_{3}\sim u^{\prime}\hat{V}_{1}\hat{V}_{2}\hat{V}_{3}.$
(21)

We will discuss the role of ${\cal L}_{3}$ at the DQCP later.

Figure 4: ($a$). The originally expected RG flow of $g$ in Eq. 28. There is a
full SO(5) symmetry along the axis $r=0$, and there are two fixed points while
increasing $g$ from zero, the one with larger $g$ corresponds to the DQCP. The
tuning parameter $r$ in Eq. 31 tunes the transition between the superconductor
and the crystalline insulator. ($b$). A possible scenario of the RG flow based
on up-to-date understanding. The two fixed points in $(a)$ annihilate and
become a pair of nearby complex fixed points, causing $g$ to flow very slowly
in the dashed region. This “walking” RG flow leads to an approximate conformal
symmetry.

The $\mathrm{U}(1)_{v}$ symmetry of the original DQCP now becomes the charge
$\mathrm{U}(1)_{e}$ symmetry, and the order parameter of the
$\mathrm{U}(1)_{e}$ symmetry are the real and imaginary parts of the inter-
valley gauge singlet $s-$wave superconductor Cooper pair:

$\displaystyle\hat{\Delta}\sim\epsilon_{\alpha\beta}\psi^{t}_{\alpha
v}\gamma_{0}\mu^{2}_{vv^{\prime}}\psi_{\beta v^{\prime}}.$ (22)

The crystalline symmetries in Eq. 19, the $\mathrm{U}(1)_{e}$ symmetry, and
time-reversal together guarantee that all fermion-bilinear terms are not
allowed in Eq. 11, except for one: the chemical potential for charge density
$\mu\bar{\psi}\gamma_{0}\psi$. But the chemical potential is tuned to zero in
the system, due to the fixed half-filling of charge density.

To make further connections with DQCP, we define a five-component unit vector
${\bm{n}}=(n_{1},\cdots n_{5})$ with $|{\bm{n}}|^{2}=1$, and couple it to the
fermion bilinear operators as

$\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle{\cal
L}_{\mathrm{QCD}}+\frac{1}{g}(\partial_{\mu}{\bm{n}})^{2}$ (23)
$\displaystyle+$ $\displaystyle
m\bigg{(}\sum_{a=1}^{3}n_{a}\hat{V}_{a}+n_{4}\mathrm{Re}[\hat{\Delta}]+n_{5}\mathrm{Im}[\hat{\Delta}]\bigg{)}.$
(25)

Following the procedure of Ref. Abanov and Wiegmann, 2000, integrating out the
fermions would lead to the following low energy effective nonlinear sigma
model (NLSM) field theory for ${\bm{n}}$ with a Wess-Zumino-Witten term at
level-1:

$\displaystyle{\cal L}_{\mathrm{eff}}$ $\displaystyle=$
$\displaystyle\frac{1}{g}(\partial_{\mu}{\bm{n}})^{2}$ (26) $\displaystyle+$
$\displaystyle\frac{2\pi\mathrm{i}}{\Omega_{4}}\int_{0}^{1}du\epsilon_{abcde}n^{a}\partial_{x}n^{b}\partial_{y}n^{c}\partial_{\tau}n^{d}\partial_{u}n^{e}.$
(28)

The NLSM above is another “classic” description of the DQCP which well
captures the intertwinement of the order parameters involved in the DQCP
Senthil and Fisher (2005).

The relation between the NLSM, the $N_{f}=2$ QCD, and the DQCP is shown in the
proposed RG flow diagram of $g$ of Eq. 28. The originally proposed RG flow of
$g$ is that (Fig. 4($a$)), if one starts with small $g$ where the NLSM is in
its ordered phase, increasing $g$ is expected to first drive an order-disorder
phase transition; and due to the existence of the WZW term in Eq. 28, the
disordered phase was expected to be controlled by a stable fixed point with
SO(5) symmetry, which corresponds to the $N_{f}=2$ QCD and is also the DQCP
tuned to the critical point. The tuning parameter of the DQCP, which tunes the
Néel-VBS transition in the original DQCP, and the SIT transition in our
current context, corresponds to the following term:

$\displaystyle\delta{\cal L}$ $\displaystyle\sim$ $\displaystyle
r(n_{1}^{2}+n_{2}^{2}+n_{3}^{2}-n_{4}^{2}-n_{5}^{2})$ (29) $\displaystyle\sim$
$\displaystyle
r(|\hat{{\bm{V}}}|^{2}-|\hat{\Delta}|^{2})=r(|\bar{\psi}\vec{\mu}\psi|^{2}-|\psi^{t}\gamma_{0}\mu^{2}\epsilon\psi|^{2}).$
(31)

The first and second line of Eq. 31 are supposed to be the tuning parameters
of the DQCP in Eq. 28 and Eq. 11 respectively.

To investigate the details of the crystalline order ($r<0$ in Eq. 31), we need
to analyze the role of the higher order term Eq. 20 in the crystalline ordered
phase. This term is irrelevant at the DQCP, but it will play important role in
the phase with $r<0$, i.e. $u$ is actually “dangerously irrelevant”. For $r<0$
and $u>0$, the system would have an eight fold degenerate crystalline order,
corresponding to

$\displaystyle\langle\hat{{\bm{V}}}\rangle$ $\displaystyle\sim$ $\displaystyle
V(\pm 1,\pm 1,\pm 1);$ (32)

while for $u<0$, there are six degenerate ground states with

$\displaystyle\langle\hat{{\bm{V}}}\rangle$ $\displaystyle\sim$ $\displaystyle
V(\pm 1,0,0),\ V(0,\pm 1,0),\ V(0,0,\pm 1).$ (33)

A nonzero ${\cal L}_{3}$ in Eq. 21 would lift the degeneracy, and favor
${\bm{\hat{V}}}$ to point along four diagonal directions of a cube, which form
a tetrahedron (Fig. 3).

Here, we would also like to clarify the nature of the DQCP, which has been an
extensively studied subject in the last two decades. As we explained before,
the originally expected RG flow of the coupling constant $g$ of the NLSM is
shown in Fig. 4($a$) Ma and Wang (2020); Nahum (2020), with two fixed points
one of which is the DQCP and also the $N_{f}=2$ QCD. But since the early days
of the numerical simulation of the Néel-VBS DQCP, which is supposedly realized
in the $J-Q$ model on the square lattice Sandvik (2007); Melko and Kaul
(2008), people have noticed that though many aspects of the transition appear
to be continuous, there are always various difficulties fitting the data with
the standard $(2+1)d$ CFT. For example, the critical exponents are drifting
with the system size Sandvik (2010); Shao et al. (2016); Nahum et al. (2015a,
b), and the exponents extracted seem incompatible with the bounds established
by the conformal Bootstrap Poland et al. (2019); Nakayama and Ohtsuki (2016);
Li (2022). Also, nonlocal probes such as entanglement entropy at the DQCP are
found to be inconsistent with a $(2+1)d$ CFT Zhao et al. (2022); Song et al.
(2024a, b); Deng et al. (2024); D’Emidio and Sandvik (2024). In Fig. 4($b$) we
sketch one of the possible scenarios of the RG flow of $g$ based on the up-to-
date understanding: the two fixed points in Fig. 4$(a)$ in fact annihilate and
become a pair of nearby complex fixed points, causing $g$ to flow very slowly
in the dashed region Wang et al. (2017). This slow RG flow due to the
annihilation of real fixed points, and emergence of complex fixed points is
referred to as the “loss of conformality” Kaplan et al. (2009). But with a
large and finite system size, we are still expected to see critical behavior
due to the slow RG flow of $g$, even though eventually the DQCP is an
extremely weak first order transition Chen et al. (2013b); Takahashi et al.
(2024). In fact, the approximate conformal symmetry has been well-observed in
the fuzzy sphere realization Zhou et al. (2024) of Eq. 28, even though there
is a weakly relevant SO(5) singlet. We also note that, recently another
possible scenario for DQCP was proposed, that there exists a nearby multi-
critical point with a full SO(5) symmetry Chester and Su (2024); Takahashi et
al. (2024).

Now let us discuss the role of ${\cal L}_{3}$ in Eq. 21 at the DQCP. At first
glance, this term is a product of six fermions, which also appears highly
irrelevant. However, a more careful investigation shows that, assuming the
DQCP has an emergent SO(5) symmetry, then ${\cal L}_{3}$ belongs to the
symmetric rank-3 tensor representation of the SO(5) which is in the same
representation of the triple-monopole in the CP1 formalism of the DQCP and the
triple-monopole is allowed by symmetry for the Néel-VBS DQCP realized on a
honeycomb lattice. Both the Monte-Carlo study and the fuzzy sphere realization
of the DQCP suggest that the triple-monopole of the CP1 representation of the
DQCP is weakly relevant Sreejith and Powell (2015); Zhou et al. (2024); Chen
et al. (2024). For example, the Monte Carlo determines that the triple-
monopole has scaling dimension 2.80(3) Sreejith and Powell (2015). Hence
${\cal L}_{3}$ should also be a weakly relevant perturbation at the DQCP.
Nevertheless, previous numerical works suggest that even with this weakly
relevant perturbation, the DQCP may still behave like a continuous transition
over a large window of length scale. For example, signatures of DQCP were also
found at the Néel-VBS transition on the honeycomb lattice Ganesh et al.
(2013); Pujari et al. (2013).

##  III Signatures of the DQCP

In this section we discuss the experimental signatures of the proposed
candidate theory for SIT. In particular, we would like to distinguish the
current theory of SIT from the previously known theory of SIT between the
superconductor and a $Z_{2}$ spin liquid state discussed in the introduction.

The most obvious difference between these two theories is that, the SC-
to-$Z_{2}$ spin liquid theory should be insensitive to the lattice symmetry
breaking. In particular, if the $Z_{2}$ spin liquid and the superconductor are
both gapped, the nature of the transition should belong to the $3D$ XY∗
transition, whose universality class does not change when the lattice symmetry
is weakly broken. But our current DQCP theory is more sensitive to the lattice
symmetry. For example, let us consider an external strain field that breaks
the $C_{3}$ rotation symmetry. This strain field acts like a term
$-s|V_{3}|^{2}$ in the Lagrangian, which favors $V_{3}$ over $V_{1}$ and
$V_{2}$ if $s>0$ and the DQCP now has an “easy-axis” anisotropy. In the NLSM
description of the DQCP, this term is $-s(n_{3})^{2}$, and is a strongly
relevant perturbation.

Figure 5: The phase diagram around the DQCP under strain. The parameter $s$ in
Eq. 34 is tuned by the strain. A positive $s$ will shift the transition point,
and drive the DQCP into a first order transition between the SC and an
insulator with the $V_{3}$ order; a negative $s$ will drive the DQCP into the
easy-plane DQCP between the SC and an insulator with the $V_{1},V_{2}$ order.

A relevant perturbation on a critical point can lead to several possible
consequences. At the DQCP, leveraging the SO(5) emergent symmetry, one can
conclude that, with a positive $s$, the DQCP becomes a first order transition
between the SC and an insulator with the order of $V_{3}$, and the transition
point is shifted from $r=0$. This analysis was given in Ref. Wang et al., 2017
in the context of the original DQCP on the square lattice. In the current
case, let us turn on both the tuning parameter $r$ of the DQCP, and the extra
strain which breaks the $C_{3}$ rotation symmetry:

$\displaystyle\delta{\cal L}^{\prime}=-s(n_{3})^{2}-r(n_{4}^{2}+n_{5}^{2}).$
(34)

Along the axis $s=r>0$, there is an emergent SO(3) symmetry that rotates among
$\vec{n}=(n_{3},n_{4},n_{5})$, and the system would develop an order of
$\vec{n}$ which spontaneously breaks the emergent SO(3). Now the line $r=s$
becomes the transition line, and tuning $r$ away from $s$ would drive a
prominent first order transition between the order of $(n_{4},n_{5})$ (the
superconductor) and the order of $n_{3}$, i.e. an insulator with the
crystalline order of $\langle V_{3}\rangle\neq 0$. Hence with the strain, one
should see hysteresis when scanning through the SIT.

Let us also consider a strain along the opposite direction, i.e. adding a term
$-s(n_{3})^{2}$ with negative $s$ in the Lagrangian. As was discussed
thoroughly in the literature of the DQCP, this term would drive the DQCP to an
easy-plane DQCP. The easy-plane DQCP is still a direct transition between the
SC and the crystalline insulator with $V_{1}$ or $V_{2}$ order. Again, though
the easy-plane DQCP should still be a very weak first order transition Qin et
al. (2017); Zhao et al. (2024); Serna and Nahum (2019); Sato et al. (2017), it
may appear continuous for a large length scale. And very nontrivial physics
can still occur at the easy-plane DQCP, including an emergent O(4) symmetry
Qin et al. (2017); Karthik and Narayanan (2017); Sato et al. (2017); Serna and
Nahum (2019); Zhao et al. (2019). Although ${\cal L}_{3}$ in Eq. 21 is a
weakly relevant perturbation at the full DQCP, to the best knowledge of the
authors, it is unclear whether ${\cal L}_{3}$ is relevant or irrelevant at the
easy-plane DQCP. The predicted phase diagram under strain in sketched in Fig.
5.

Another difference between the DQCP and the SC-to-Z2 spin liquid transition
theory is that, our current theory of DQCP is a theory of “competing orders”.
Indeed, the experiments suggest that, when the SC is suppressed, the
insulating state emerges. In fact, in our theory the SC and the crystalline
phases not only compete with each other but also have a mutual ’t Hooft
anomaly, in the sense that the defect of one order parameter should carry the
quantum number of the other. For example, the vortex of the SC carries the
fractionalized quantum number of $\hat{{\bm{V}}}$, which should have nonzero
lattice momentum. Hence if one creates a vortex inside the superconductor, the
vortex core should have crystalline order.

##  IV The second candidate theory

In this section we propose a second candidate theory for the observed SIT, but
the nature of the SC phase and the crystalline-insulator are different from
the previous theory. To facilitate the discussion, we take a different parton
construction from Eq. 4:

$\displaystyle c_{j,\uparrow}=b_{j}\psi_{j,1},\ \ \
c_{j,\downarrow}=b_{j}^{\dagger}\psi_{j,2}.$ (35)

We still consider mean field state of $\psi_{\alpha}$ with the same form of
mean field Hamiltonian as Eq. 9, with $t_{ij,\alpha\beta}=\pm
t\delta_{\alpha\beta}$. The sign of $t_{ij,\alpha\beta}$ still constitutes a
staggered $\pi-$flux shown in Fig. 1($b$), except now the gauge field
fluctuation is $\mathrm{U}(1)_{g}$ rather than $\mathrm{SU}(2)_{g}$. The low
energy physics of this state is given by $N_{f}=4$ QED, where the U(1) gauge
field couples via $\tau^{3}$ in Eq. 11. One can also obtain the same theory by
condensing a Higgs field in Eq. 11 that breaks the $\mathrm{SU}(2)_{g}$ gauge
group down to $\mathrm{U}(1)_{g}$.

Now, we consider the adding the following extra fermion bilinear terms to
$N_{f}=4$ QED:

$\displaystyle\delta{\cal
L}=m_{1}\bar{\psi}\tau^{3}\psi+m_{2}\bar{\psi}\mu^{3}\psi.$ (36)

The first term $m_{1}\bar{\psi}\tau^{3}\psi$ is actually allowed by the
symmetry and gauge invariance, and hence it should exist on top of the
$N_{f}=4$ QED. The second term $m_{2}\bar{\psi}\mu^{3}\psi$ is precisely
$\hat{V}_{3}$, and it breaks part of the lattice symmetry as it carries finite
momentum.

We assume that $m_{1}$ and $m_{2}$ are both finite and positive, and we
consider the tuning parameter $\delta m=m_{1}-m_{2}$. Near the critical point
$\delta m=0$, of the four Dirac fermions, the two with quantum numbers
$\tau^{3}\mu^{3}=+1$ are well-gapped, while the other two with
$\tau^{3}\mu^{3}=-1$ have low energy and mass $\delta m$. Hence, the low
energy theory describing this transition is $N_{f}=2$ QED:

$\displaystyle{\cal
L}_{\mathrm{QED}}=\bar{\psi}_{\alpha}\gamma_{\mu}(\partial_{\mu}\delta_{\alpha\beta}-\mathrm{i}a_{\mu}\tau^{3}_{\alpha\beta})\psi_{\beta}+\delta
m(\bar{\psi}\tau^{3}\psi)+\cdots$ (37)

Eq. 37 has “half” of the fermion modes of Eq. 11: the fermion in Eq. 37 with
$\tau^{3}=+1$ corresponds to the Dirac fermion in Eq. 11 with $\tau^{3}=+1$,
$\mu^{3}=-1$; and the fermion in Eq. 37 with $\tau^{3}=-1$ correspond to the
Dirac fermion with $\tau^{3}=-1$, $\mu^{3}=+1$ in Eq. 11.

The two phases in the phase diagram with positive and negative $\delta m$ are
identified as the SC phase and a crystalline-insulator phase respectively.
When $\delta m>0$, i.e. $m_{1}>m_{2}>0$, the gauge flux of $a_{\mu}$ will have
the charge of $\psi$. More precisely, it carries charge-$2e$ under the
$U(1)_{e}$ symmetry due to the dual quantum spin Hall effect (i.e., the spin-
flux carries electric charge). The gauge flux of $a_{\mu}$ is generically a
conserved quantity, and the photon phase of $a_{\mu}$ is dual to the
condensate of gauge fluxes. This condensate is equivalently a charge-$2e$
superconductor due to the dual quantum spin Hall effect. Hence we identify the
phase with $\delta m>0$ as a superconductor phase. In fact, the superconductor
phase arises here for essentially the same reason as Skyrmion condensation
induced superconductivity, when the Skyrmion carries electric charges Grover
and Senthil (2008); Khalaf et al. (2021); Liu et al. (2019, 2023). It was also
shown before that a “gauged” quantum spin Hall state can lead to inplane Néel
order and superconductor Ran et al. (2008).

The critical point of Eq. 37 is located at $\delta m=0$, which is $N_{f}=2$
QED. It was shown that $N_{f}=2$ QED is self-dual Xu and You (2015); Hsin and
Seiberg (2016); Mross et al. (2016), and hence it was conjectured to possess
an O(4) emergent symmetry in the infrared. This emergent O(4) symmetry was
seen numerically Qin et al. (2017); Karthik and Narayanan (2017); Sato et al.
(2017); Serna and Nahum (2019). It was also shown that the $N_{f}=2$ QED is
dual to the easy-plane Abelian Higgs model with two flavors of scalar fields
(also referred to as the easy-plane noncompact CP1 model) Potter et al.
(2017); Wang et al. (2017), which describes the DQCP with an easy-plane
anisotropy.

Figure 6: The phase diagram of the second candidate theory of the SIT, which
corresponds the $N_{f}=2$ QED, and it is also dual to the easy-plane DQCP.

To identify more details of the SC phase, we need to analyze the monopole
quantum number of the $\mathrm{U}(1)$ gauge field $a_{\mu}$, with $\delta
m>0$. Most importantly, we would like to know whether the Cooper pair
condenses at zero momentum, or finite momentum. This not an easy task, but
results from previous studies that thoroughly analyzed the monopole quantum
numbers of the staggered $\pi$-flux Dirac spin liquid on the triangular
lattice may be used. It was shown that Song et al. (2020, 2019), with a
quantum spin Hall mass term, $m_{1}\bar{\psi}\tau^{3}\psi$, the $\pi-$flux
state would have nonzero staggered charge density $\bar{\psi}\gamma_{0}\psi$
at the center of each triangle of the lattice. This charge distribution leads
to finite momentum carried by the monopole operator of the gauge field
considered therein. However, in our current case, we argue that since the
gauge field is generated by $\tau^{3}$, the extra charge density
$\bar{\psi}\gamma_{0}\psi$ should be “invisible” to the gauge fluxes. We
conjecture here that the superconductivity induced by the dual-QSH effect due
to the QSH mass term $m_{1}$ and the gauge fluctuation of $a_{\mu}\tau^{3}$
would still order at zero momentum.

Of course, we assumed that there is a nonzero $m_{2}$ throughout the phase
diagram, and a nonzero $m_{2}$ leads to a density wave order in the
superconductor. Hence in this second theory, the superconductor has coexisting
Cooper pair condensate and a density wave order at momentum
$K_{3}=(2\pi/\sqrt{3},0)$.

When $\delta m<0$, i.e. $m_{1}<m_{2}$, there is no dual quantum spin Hall
effect and the superconductor disappears. However, the monopole of the
$\mathrm{U}(1)$ gauge field will still carry crystalline symmetry quantum
number due to the fermion modes trapped by the monopole. In this case, there
is further crystalline symmetry breaking due to the flux condensate of the
$\mathrm{U}(1)$ gauge field. Further analysis analogous to Ref. Song et al.,
2020, 2019 is demanded to fully determine the crystalline order in this case.

##  V Summary and discussion

In this work we propose that the superconductor-to-insulator transition
observed recently in the twisted bilayer TMD moiré system may be described by
the DQCP, which is a subject that has attracted enormous attention and efforts
in multiple disciplines of physics in the last two decades. We propose that
the crystalline order on the effective triangular lattice has an approximate
$\mathrm{SO}(3)_{v}$ symmetry, which is the counterpart of the
$\mathrm{SO}(3)_{s}$ spin symmetry of the original DQCP; and the charge
$\mathrm{U}(1)_{e}$ symmetry plays the role as the emergent
$\mathrm{U}(1)_{v}$ symmetry that rotates between VBS orders in the original
DQCP. We also demonstrated that the DQCP may be driven into either a prominent
first order transition, or an easy-plane DQCP under strain.

Physics discussed in this work may also have applications in other systems,
especially when the system can be modelled by strongly interacting bosons on
the triangular lattice at half-filling. We note here that our theory is
different from the previously discussed superfluid-insulator transition for
bosons on the triangular lattice Burkov and Balents (2005). Potential
connections between these two formalisms will be explored in the future.

We also briefly discussed a second candidate theory for the SIT, where the
superconductor is induced by a “dual” quantum spin Hall effect, meaning the
spin gauge field fluctuation would lead to charge condensate, i.e. a
superconductor. To fully identify the nature of the superconductivity, and the
crystalline order of the insulator phase of the second theory, we need
detailed analysis of the monopole quantum numbers of the gauge field, which we
defer to future study.

The authors thank Matthew Fisher, Chao-Ming Jian, Xue-Yang Song for very
helpful discussions. C.X. acknowledges support from the Simons foundation
through the Simons investigator program.

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