#  XY* transition and extraordinary boundary criticality from fractional
exciton condensation in quantum Hall bilayer

Ya-Hui Zhang<EMAIL_ADDRESS>Department of Physics and Astronomy, Johns
Hopkins University, Baltimore, Maryland 21218, USA  Zheng Zhu
<EMAIL_ADDRESS>Kavli Institute for Theoretical Sciences, University of
Chinese Academy of Sciences, Beijing 100190, China  CAS Center for Excellence
in Topological Quantum Computation, University of Chinese Academy of Sciences,
Beijing, 100190, China  Ashvin Vishwanath<EMAIL_ADDRESS>Department of Physics, Harvard University, Cambridge MA 02138.

###### Abstract

XY* transitions represent one of the simplest examples of unconventional
quantum criticality, in which fractionally charged excitations condense into a
superfluid, and display novel features that combine quantum criticality and
fractionalization. Nevertheless their experimental realization is challenging.
Here we propose to study the XY* transition in quantum Hall bilayers at
filling $(\nu_{1},\nu_{2})=(\frac{1}{3},\frac{2}{3})$ where the exciton
condensate (EC) phase plays the role of the superfluid. Supported by exact
diagonalization calculation, we argue that there is a continuous transition
between an EC phase at small bilayer separation to a pair of decoupled
fractional quantum Hall states, at large separation. The transition is driven
by condensation of a fractional exciton, a bound state of Laughlin
quasiparticle and quasihole, and is in the XY* universality class. The
fractionalization is manifested by unusual properties including a large
anomalous exponent and fractional universal conductivity, which can be
conveniently measured through inter-layer tunneling and counter-flow
transport, respectively. We also show that the edge is likely to realize the
newly predicted extra-ordinary log boundary criticality. Our work highlights
the promise of quantum Hall bilayers as an ideal platform for exploring exotic
bulk and boundary critical behaviors, that are amenable to immediate
experimental exploration in dual-gated bilayer systems. The XY* critical
theory can be generalized to a bilayer system with an arbitrary Abelian state
in one layer and its particle-hole partner in the other layer. Therefore we
anticipate many distinct XY* transitions corresponding to the different
Laughlin states and Jain sequences in the single layer case.

###### pacs:

Valid PACS appear here

##  I Introduction

The study of quantum phase transitions and universal critical behaviors is one
of the major focuses in condensed matter physicsSachdev (1999); Sondhi  _et
al._ (1997). Although many quantum critical points (QCP) are described by the
well-established Landau-Ginzburg theory, exceptions arise due to
fractionalization beyond the conventional symmetry breaking order. One
category is the deconfined quantum critical points(DQCPs) between two
different symmetry breaking phasesSenthil  _et al._ (2004). Another category
is phase transitions between one phase with fractionalization or topological
order and another conventional phase. One simple example is the XY*
transition, initially discussed between a Z2 topological ordered insulator (or
quantum spin liquid) and a superfluid (or XY ferromagnetism) phaseChubukov
_et al._ (1994); Isakov  _et al._ (2012); Wang  _et al._ (2021a); Schuler  _et
al._ (2022). The critical theory of such a transition is well
understoodChubukov  _et al._ (1994); Schuler  _et al._ (2022) and its
existence in lattice models has been numerically verifiedIsakov  _et al._
(2012); Wang  _et al._ (2021b). However, experimental observation of the XY*
transition is still elusive. Given that even the unambiguous experimental
realization of a Z2 spin liquid phase is a great challenge, and that recent
progress in synthetic quantum systems target topological order in the absence
of global $U(1)$ symmetry Verresen  _et al._ (2021); Satzinger  _et al._
(2021); Semeghini  _et al._ (2021), the experimental study of an XY* QCP
adjacent to a quantum spin liquid phase remains challenging for the near
future.

Here we turn to quantum Hall systems, where fractionalization itself has been
well established at fractional fillingsStormer  _et al._ (1999). It is natural
to imagine that experimental realization of a QCP with fractionalization in
quantum Hall systems is easier, though such a possibility has not been well
explored except on plateau transitionsSondhi  _et al._ (1997). Here we
consider the quantum Hall bilayer system with the electron gases in two layer
separated by an insulating barrier, giving rise to two separate Landau levels
coupled together through the Coulomb repulsion Sarma and Pinczuk (2008);
Halperin (1983); Eisenstein and MacDonald (2004); Eisenstein (2014a). The
fillings in the two layers $\nu_{1},\nu_{2}$ can be controlled separately. In
addition, one can tune $d/l_{B}$ experimentally to study the possible phase
transitions. Here $d$ is the distance between the two layers and $l_{B}$ is
the magnetic length. At small $d/l_{B}$, it is known that the ground state is
an exciton condensation phase Moon  _et al._ (1995); Yang  _et al._ (1996);
Eisenstein and MacDonald (2004); Eisenstein (2014a); Liu  _et al._ (2017); Li
_et al._ (2017) for the whole line of $\nu_{1}+\nu_{2}=1$. There have been
many theoretical discussions on other possible phases at larger $d/l_{B}$ at
$(\nu_{1},\nu_{2})=(\frac{1}{2},\frac{1}{2})$Bonesteel  _et al._ (1996); Kim
_et al._ (2001); Schliemann  _et al._ (2001); Stern and Halperin (2002); Simon
_et al._ (2003); Sheng  _et al._ (2003); Park (2004); Shibata and Yoshioka
(2006); Möller  _et al._ (2008, 2009); Milovanović and Papić (2009); Alicea
_et al._ (2009); Papić and Milovanović (2012); Sodemann  _et al._ (2017);
Isobe and Fu (2017); Zhu  _et al._ (2017); Lian and Zhang (2018); Wagner  _et
al._ (2021).

Recently, the evolution under tuning $d/l_{B}$ was experimentally investigated
for this filling $(\nu_{1},\nu_{2})=(\frac{1}{2},\frac{1}{2})$Liu  _et al._
(2022). There one only finds a crossover between Bose-Einstein-condensation
(BEC) regime to Bardeen–Cooper–Schrieffer (BCS) regime all within a single
exciton condensation (EC) phase. There has been theoretical discussion of
superfluid to insulator transition at $(1/2,1/2)$ fillingYang (2001), but we
are not aware of any experimental observation so far. In contrast, for filling
$(\frac{1}{3},\frac{2}{3})$ (or relatedly
$(\nu_{1},\nu_{2})=(\frac{1}{3},-\frac{1}{3})$, where by $\nu<0$ we mean the
system is hole doped relative to the charge neutrality), a phase transition is
bound to happen. At small $d/l_{B}$ the ground state is still an exciton
condensation phase. In the large $d/l_{B}$ limit, the two layers decouple and
the top layer is in the $\nu=\frac{1}{3}$ Laughlin state Laughlin (1983),
while the bottom layer is in the $\nu=-\frac{1}{3}$ (or $\nu=\frac{2}{3}$)
Laughlin state. This large $d/l_{B}$ phase can be viewed as a fractional
quantum spin Hall insulator (FQSH) with a K matrix $K=\begin{pmatrix}3&0\\\
0&-3\end{pmatrix}$ if we view the layer as a pseudospin. Given the recent
experimental progress in tuning $d/l_{B}$ at
$(\nu_{1},\nu_{2})=(\frac{1}{2},\frac{1}{2})$, experimental measurements at
filling $(\nu_{1},\nu_{2})=(\frac{1}{3},-\frac{1}{3})$ or
$(\nu_{1},\nu_{2})=(\frac{1}{3},\frac{2}{3})$) should be straightforward.
Note, while conceptually one can think of changing the separation $d$, in
experiment one can tune the ratio $d/l_{B}$ more conveniently by
simultaneously changing the magnetic field and density to keep the filling
constant. Thus the transition is well within experimental reachZeng (2021)!
Actually there already exists some experimental evidence of a direct
transition between the exciton superfluid and FQSH phase at filling
$(\nu_{1},\nu_{2})=(\frac{1}{3},\frac{2}{3})$)Champagne  _et al._ (2008) in
GaAs quantum well system. However, the nature of the phase transition is not
clear from the existing experimental data. A previous theoretical work already
studied similar transition in a model with a hard-core interaction and
suggested the transition is in the XY universality classChen and Yang (2012).
Here we will provide numerical evidence for a continuous transition in a
realistic model with Coulomb interactions and also point out that this is a
XY* transition and propose experimental signatures of the distinction compared
to the simple superfluid to Mott insulator transition in the conventional XY
universality class.

We performed exact diagonalization (ED) Haldane (1985) for the Coulomb coupled
quantum Hall bilayer at filling $(\nu_{1},\nu_{2})=(\frac{1}{3},\frac{2}{3})$
and found a direct transition between the EC phase at small $d/l_{B}$ and the
FQSH phase at large $d/l_{B}$. The transition appears continuous in the finite
size calculation, suggesting the possibility of a continuous quantum critical
point (QCP). Motivated by the numerical calculation, we propose a critical
theory between the EC and FQSH phases in the universality class of XY*
transition. Starting from the FQSH phase, the Laughlin electron and Laughlin
hole in the two layers bind to form a fractional exciton, with bosonic
statistics whose condensation then confines all the anyons and leads to the EC
phase at small $d/l_{B}$. The critical theory is described by the superfluid
to insulator transition of the fractional exciton, which carries an exciton
charge $1/3$ compared to the ordinary exciton. We also discuss the realization
of an extra-ordinary boundary criticalityMetlitski (2022) in the edge at this
QCP. The XY* transition here can be easily generalized to the case with an
arbitrary Abelian FQHE phase in one layer and its particle-hole partner in the
other layer. Thus we anticipate many different XY* transitions in the
$(D,d/l_{B})$ parameter space, where $D$ is the displacement field to tune the
exciton density.

Figure 1: (a) Illustration of quantum Hall bilayer, with an insulating layer
(blue) in between the two 2DEGs (b) schematic phase diagram in terms of
$(D,\,d)$ while fixing $\nu_{1}+\nu_{2}=0$. The displacement field $D$ is the
chemical potential for excitons. The distance $d$ tunes inter-layer Coulomb
interaction strength. We are interested in the quantum phase transition
(indicated by the blue arrow) tuned by $d$ at fixed exciton density
$(\nu_{1},\nu_{2})=(\frac{1}{3},-\frac{1}{3})$. SF denotes the exciton
condensation phase with $\langle c^{\dagger}_{1}c_{2}\rangle\neq 0$ and FQSH
denotes fractional quantum spin Hall insulator formed by two decoupled
Laughlin states with opposite chiralities.

##  II Model and symmetry

We consider the quantum Hall bilayer at filling $(\nu_{1},\nu_{2})=(x,-x)$,
illustrated in Fig. 1. Here $\nu_{a}=\frac{N_{e}}{N_{\Phi}}$, where $N_{\Phi}$
is the number of the magnetic flux in the system. $\nu_{2}=-x$ means that the
system is hole doped with hole density at $x$ per flux. We will be mainly
focused on $x=\frac{1}{3}$, but similar physics can happen for other rational
$x$ with an incompressible Abelian FQHE state in the decoupling limit. $x$
here is the exciton density and can be tuned through the displacement field
$D$, while the total filling $\nu_{1}+\nu_{2}$ is fixed to be $0$. Up to a
stacking of an integer quantum Hall state at the layer $2$, it is also
equivalent to consider the filling $(\nu_{1},\nu_{2})=(x,1-x)$ with
$\nu_{1}+\nu_{2}=1$. Thus we also consider the filling
$(\nu_{1},\nu_{2})=(\frac{1}{3},\frac{2}{3})$.

We have the Hamiltonian:

$\mathcal{H}=\frac{1}{2}\sum_{a,b=1,2}V_{ab}(\mathbf{q}):\rho_{a}(\mathbf{q})\rho_{b}(-\mathbf{q}):$
(1)

where $\rho_{a}(\mathbf{q})=\int
d^{2}\mathbf{q}\rho_{a}(\mathbf{r})e^{-i\mathbf{q}\cdot\mathbf{r}}$ and
$\rho_{a}(\mathbf{r})$ is the charge density at layer $a$ projected to the
lowest Landau level. We have the Coulomb interaction
$V_{11}(q)=V_{22}(q)=\frac{e^{2}}{\varepsilon q}$ and
$V_{12}(q)=V_{21}(q)=\frac{e^{2}}{\varepsilon q}e^{-qd}$. $d$ represents the
distance between two layers in the unit of magnetic length $l_{B}=\sqrt{\hbar
c/eB}$.

The Hamiltonian considered above has an anti-unitary symmetry
$\mathcal{M}\mathcal{C}\mathcal{T}$ for the quantum Hall bilayer at filling
$(\nu_{1},\nu_{2})=(x,-x)$. The symmetry is a combination of layer exchange
symmetry $\mathcal{M}$, charge conjugation $\mathcal{C}$ and time reversal
$\mathcal{T}$. We define electron operators in layer 1 and 2 as
$c_{1}(\mathbf{r})$ and $c_{2}(\mathbf{r})$. The symmetry
$\mathcal{M}\mathcal{C}\mathcal{T}$ acts as: $c_{1}(\mathbf{r})\rightarrow
c_{2}^{\dagger}(\mathbf{r}),c_{2}(\mathbf{r})\rightarrow
c_{1}^{\dagger}(\mathbf{r})$ combined with complex conjugate $\mathcal{K}$.
Under $\mathcal{M}\mathcal{C}\mathcal{T}$, we have
$\rho_{1}(\mathbf{r})\rightarrow-\rho_{2}(\mathbf{r})$ and
$\rho_{1}(\mathbf{q})\rightarrow-\rho_{2}(-\mathbf{q})$. One can check that
the Hamiltonian satisfies this symmetry.

FIG. 2: (a) The low-lying energy spectra as a function of layer distance
$d/l_{B}$ for the filling $(\nu_{1},\nu_{2})=(\frac{1}{3},-\frac{1}{3})$. Here
we only show the inequivalent momentum sectors. There are 9-fold degenerate
states at larger $d/l_{B}$ (b) The correlator
$\langle{{S^{y}_{i}}{S^{y}_{i+N_{\phi}/2}}}\rangle$ versus layer distances
$d/l_{B}$, the inset shows $\left\langle{{S^{y}_{i}}{S^{y}_{j}}}\right\rangle$
as a function of orbital distance $|i-j|$. Here, $i,j=1,\ldots,N_{\phi}$ are
orbital indices and the corresponding distance is $|i-j|L/N_{\phi}$. From top
to bottom, $d/l_{B}$ ranges from 0 to 2.4 with interval 0.4. (c) The fidelity
$F(d,\Delta d)$ as a function of the layer distances $d/l_{B}$ taking
different intervals of parameters $\Delta d/l_{B}$. Note that with decreased
$\Delta d$ the dip is weakened.

##  III Phase diagram.

We fix the filling to be $(\nu_{1},\nu_{2})=(\frac{1}{3},\frac{2}{3})$ and
study the phase diagram of tuning $d/l_{B}$ through ED. At small $d/l_{B}$,
the system is in an exciton condensation phase with an order parameter
$\langle c_{1}^{\dagger}c_{2}\rangle\neq 0$. Here $c_{1},c_{2}$ are
annihilation operator of electron in layer 1 and 2 respectively. At large
$d/l_{B}$, the two layers decouple and we have a fractional quantum spin Hall
(FQSH) phase(up to stacking an integer quantum Hall state at layer 2) if
viewing layers 1 and 2 as spin up and spin down. The question is whether there
is a direct phase transition or an intermediate phase in between.

Figure 2 (a) shows the flow of low-lying energies with layer distance
$d/l_{B}$. For simplicity, we set $l_{B}=1$ in the following discussions. We
use a torus geometry and the Landau gauge. The evolution of energy spectra
indicates a direct transition at $d_{c}\approx 1.7$. When $d>d_{c}$, we can
identify a 9-fold near degeneracy expected for decoupled $\nu=\pm 1/3$
Laughlin states in the two layers. When approaching $d_{c}$ from large $d$,
the topological order indicated by the ground state degeneracy disappears at
$d_{c}$.

The phase at $d<d_{c}$ is an exciton superfluid with order parameter $\langle
S^{y}(\mathbf{r})\rangle\neq 0$, where
$S^{y}(\mathbf{r})=i(c^{\dagger}_{1}(\mathbf{r})c_{2}(\mathbf{r})-h.c.)$. In
the lowest Landau level, we have operators $c_{1;m}$ and $c_{2;m}$ where $m$
is the Landau index and labels the position along $x$ direction in our gauge.
So we can define $S^{y}_{m}=i(c^{\dagger}_{1;m}c_{2;m}-h.c.)$, where
$m=1,2,...,N_{\Phi}$. Then we calculate the correlation function $\langle
S^{y}_{i}S^{y}_{j}\rangle$ which is a function of $|i-j|$. In the inset of
Fig. 2(b) we show that $\langle S^{y}_{i}S^{y}_{j}\rangle$ is almost a
constant with $|i-j|$ at small $d$, but decays fast at large $d$. In
particular, we can use $\langle S^{y}_{i}S^{y}_{i+{N_{\Phi}}/{2}}\rangle$ to
characterize the exciton condensation. In Fig. 2(b) it is clear that $\langle
S^{y}_{i}S^{y}_{i+{N_{\Phi}}/{2}}\rangle$ is non-zero at $d<d_{c}$ and almost
vanishes at $d>d_{c}$. When approaching $d_{c}$ from small $d$, the exciton
condensation order parameter disappears smoothly across $d_{c}$.

To further probe the nature of the transition at $d_{c}$, we compute the
ground state fidelity, which is defined by the wave function overlap between
the ground state at $d-\Delta d$ and $d$, i.e., $F(d,\Delta
d)=\left|{\left\langle\Psi(d-\Delta
d)\right|\left.\Psi(d)\right\rangle}\right|$. The fidelity has been shown to
be a good indicator to distinguish the continuous transition from the first-
order transition for both symmetry-breaking and topological phase transitions
Zanardi and Paunković (2006); Gu (2010). As shown in Fig. 2 (c), the ground-
state fidelity displays a single weak dip at the critical distance $d_{c}$
instead of showing a sudden jump. Meanwhile, the dip is further weakened with
the decrease of $\Delta d$. Thus the numerical evidence indicates the
transition might be continuous, though one cannot rule out a weak first-order
transition in a finite-size calculation. In the following, we will propose a
critical theory for this QCP in the universality class of XY*. The XY*
transition is well established to be continuous in other contexts, which
further supports the continuous transition scenario of the QCP at $d_{c}$ from
the theoretical side.

##  IV Field theory of an XY* transition

We turn to the filling $(\nu_{1},\nu_{2})=(\frac{1}{3},-\frac{1}{3})$ for
simplicity. The FQSH phase at the large d limit is described by the following
effective field theory:

$\mathcal{L}=-\frac{3}{4\pi}a_{1}da_{1}+\frac{3}{4\pi}a_{2}da_{2}+\frac{1}{2\pi}A_{1}da_{1}-\frac{1}{2\pi}A_{2}da_{2}$
(2)

where $a\,db$ is an abbreviation of
$\epsilon_{\mu\nu\sigma}a_{\mu}\partial_{\nu}b_{\sigma}$ . Here, $a_{1;\mu}$
and $a_{2;\mu}$ are emergent dynamical gauge fields while $A_{1;\mu}$ and
$A_{2;\mu}$ are probe fields of the two layers. For example
$\vec{E}_{a}=-\vec{\nabla}A_{a;0}-\frac{\partial\vec{A}_{a}}{\partial t}$ is
the electric field applied to layer $a$. Note in the experiment one can apply
$\vec{E}_{1}$ and $\vec{E}_{2}$ separately and measure currents in a layer
resolved fashion. We then define physical charge $(Q_{1},Q_{2})$ under
$(A_{1},A_{2})$. We can also label anyon excitations in terms of their charges
$l=(l_{1},l_{2})$ under $(a_{1},a_{2})$. The physical charge of the anyon $l$
is $(Q_{1},Q_{2})=(\frac{1}{3}l_{1},\frac{1}{3}l_{2})$. Its statistics is
$\theta=\frac{l_{1}^{2}-l_{2}^{2}}{3}\pi$. We also make a basis change to
define $A_{c;\mu}=\frac{A_{1;\mu}+A_{2;\mu}}{2}$ and
$A_{s;\mu}=A_{1;\mu}-A_{2;\mu}$. The corresponding charge is
$Q_{c}=Q_{1}+Q_{2}$ and $Q_{s}=\frac{Q_{1}-Q_{2}}{2}$. $Q_{s}$ is the layer
pseudospin viewed as a spin $1/2$, $S_{z}$ .

The elementary anyon is $l=(\pm 1,0)$ and $l=(0,\pm 1)$ with charge $\pm 1/3$
at each layer. When we decrease $d$, inter-layer Coulomb interaction
increases. Then an anyon with charge $1/3$ at layer 1 tends to bind with an
anyon with charge $-1/3$ at layer 2 into an exciton. When $d$ is further
decreased, the binding energy increases and this exciton of anyon can condense
and lead to the exciton condensation phase at small $d$. This fractional
exciton is labeled by $l=(1,-1)$ with physical charge
$(Q_{1},Q_{2})=(\frac{1}{3},-\frac{1}{3})$ or $Q_{c}=0,Q_{s}=\frac{1}{3}$. We
label the creation operator of this fractional exciton as $\varphi^{\dagger}$.
The condensation of $\varphi$ is captured by the following critical theory
(see the supplementary for derivation):

$\displaystyle\mathcal{L}_{c}$
$\displaystyle=|(\partial_{\mu}-i\frac{1}{3}A_{s;\mu})\varphi|^{2}-s|\varphi|^{2}-g|\varphi|^{4}+\frac{1}{6\pi}A_{c}dA_{s}$
(3)

When $s<0$, this is a superfluid phase of $A_{s}$. When $s>0$, we have the
correct response of $\frac{1}{6\pi}A_{c}dA_{s}$ for the FQSH phase. In
principle $\varphi$ is also coupled to a gauge field which however does not
affect the critical properties we discuss here due to a Chern Simons term (see
Appendix). Note further that tuning the transition at fixed layer density
eliminates the single time-derivative chemical potential term. Further, an
anti-unitary MCT symmetry (see Appendix) that interchanges layer, and performs
the combination of particle hole and time reversal symmetry ensures there is
no background flux for the $\varphi$. It is clear that the critical theory is
the usual ‘relativistic’ XY transition driven by the condensation of a boson
which carries charge $1/3$ under $A_{s}$. A counter-intuitive feature that is
shared with other XY* transitions is that despite the condensation of a
fractionally charged boson, the superfluid itself is conventional. One can
readily check that the only gauge invariant order parameter is the usual one
for integer charge, and all anyons are confined. Alternatively one can show
the vortex quantization is the conventional one despite the fractional charge,
as a result of attaching an anyon to the fundamental vortex Kivelson  _et al._
(1988). In the dual viewpoint, starting from the superfluid phase, the triple
vortex becomes gapless and condenses, leading to an insulator. But the
elementary vortex of the superfluid phase remains gapped across the QCP and
becomes the anyon in the FQSH phase (see the Appendix).

##  V Experimental signatures:

We then move to the possible experimental signatures of this unusual QCP. In
terms of $\varphi$, Eq. 3 is the standard critical theory for the XY
transition describing interaction tuned superfluid to Mott insulator
transition. The critical exponents for thermodynamic quantities are the same
as the XY transition. However, the critical boson $\varphi$ here is a non-
local field and does not correspond to the microscopic order parameter. Hence
the transition is usually called XY* to highlight its difference from the
conventional XY transition, which will be manifested in exciton correlation
function and conductivity.

Exciton correlation function: First, at the QCP, the critical boson has a
power law correlation function:
$\langle\varphi^{\dagger}(\mathbf{x})\varphi(\mathbf{y})\rangle\sim\frac{1}{|\mathbf{x}-\mathbf{y}|^{1+\eta}}$
with $\eta\approx 0.038$. However, the fractional exciton order $\varphi$ is
not measurable. The physical order parameter is the conventional exciton
operator $\Phi^{\dagger}=c_{1}^{\dagger}c_{2}$. It is a composite operator in
the critical theory: $\Phi=\varphi^{3}$ and its correlation function has a
large decaying exponent:
$\langle\Phi^{\dagger}(\mathbf{x})\Phi(\mathbf{y})\rangle\sim\frac{1}{|\mathbf{x}-\mathbf{y}|^{1+\eta^{*}}}$
with $\eta^{*}\approx 3.2$ estimated from the scaling dimension of the
$\varphi^{3}$ mode of the 3D XY universality class.Hasenbusch and Vicari
(2011). The same exponent appears in the correlation function along the time
direction, which leads to
$\langle\Phi^{\dagger}(\mathbf{r},\omega)\Phi(\mathbf{r},-\omega)\rangle\sim\omega^{\eta^{*}}$
at position $\mathbf{r}$.

Interestingly, this exponent can be measured through a local inter-layer
tunneling experiment at position $\mathbf{r}$. Considering a local tunneling
term $H^{\prime}=\Gamma\int
d^{2}\mathbf{r}\delta(\mathbf{r})c_{1}^{\dagger}(\mathbf{r})c_{2}(\mathbf{r})+h.c.$,
linear response theory derives
$I=2e\Gamma^{2}\text{Im}\chi_{R}(\omega=eV,\mathbf{r})$ with $I$ and $V$ as
current and voltage in the z direction. $\chi_{R}(\omega,\mathbf{r})$ is the
Fourier transformation of
$\chi_{R}(t,\mathbf{r})=-i\theta(t)\langle[\Phi^{\dagger}(\mathbf{r},t),\Phi(\mathbf{r},0)]\rangle$
in the time direction.Wen (1991). So we expect that $\frac{dI}{dV}\sim
V^{\eta^{*}-1}\approx V^{2.2}$ and is nonlinear to $V$ at zero temperature at
$d=d_{c}$. On the other hand, when $d<d_{c}$, we expect $\frac{dI}{dV}$ to
have a zero bias peakEisenstein (2014b) and when $d>d_{c}$ it should have a
threshold gap. Non-linear I-V curve is expected at the edge of FQHE phase with
fractional chargeWen (1991). Here we offer an example of the bulk tunneling at
the QCP and the large exponent $\eta^{*}$ is a manifestation of the fractional
charge carried by the critical boson. Sometimes it is more convenient to
measure a global tunnelingSpielman  _et al._ (2001) from the term
$H^{\prime}=\Gamma\int
d^{2}\mathbf{r}c_{1}^{\dagger}(\mathbf{r})c_{2}(\mathbf{r})+H.c.$. In this
case we expect $I=2e\Gamma^{2}\text{Im}\chi_{R}(\omega=eV,\mathbf{q}=0)$Stern
_et al._ (2001)111In this case, it is important to align the two graphene
layers so the inter-layer tunneling probes the spectral weight of the exciton
order parameter at momentum $\mathbf{q}=0$ and energy $\omega=eV$.. We have
$\frac{dI}{dV}\sim V^{\eta^{*}-3}\approx V^{0.2}$ at the critical point.

Universal conductivity: The XY criticality is known to exhibit a universal
conductivity. For our system, we define a $4\times 4$ conductivity tensor in
the direct current (DC) limit as
$\sigma=\begin{pmatrix}\sigma_{xx}&\sigma_{xy}\\\
-\sigma_{xy}&\sigma_{xx}\end{pmatrix}$, where $\sigma_{xx}$ and $\sigma_{xy}$
are both symmetric $2\times 2$ matrix in layer space.
$\sigma_{xx;ab}=\frac{J_{x;a}}{E_{x;b}}$, where $a,b=1,2$ labels the two
layers and $\sigma_{xy;ab}$ is defined similarly.

From Eq. 3 we get the longitudinal conductivity tensor at the QCP to be:

$\sigma_{xx}=\frac{\sigma_{b}}{9}\frac{e^{2}}{h}\begin{pmatrix}1&-1\\\
-1&1\end{pmatrix}$ (4)

where $\sigma_{b}$ is the universal conductivity for the ordinary XY
transition, a number of order one in units of $e^{2}/h$ . The factor of
$\frac{1}{9}$ is because the critical boson carries only $1/3$ of the ordinary
exciton. Thus the conductivity at this XY* transition is of order $\sim
0.1\frac{e^{2}}{h}$. Further, $\sigma_{xy}$ is purely from the background
Chern-Simons term in Eq. 3. For the filling
$(\nu_{1},\nu_{2})=(\frac{1}{3},-\frac{1}{3})$, we have
$\sigma_{xy}=\begin{pmatrix}\frac{1}{3}&0\\\
0&-\frac{1}{3}\end{pmatrix}\frac{e^{2}}{h}$. For
$(\nu_{1},\nu_{2})=(\frac{1}{3},\frac{2}{3})$, we have
$\sigma_{xy}=\begin{pmatrix}\frac{1}{3}&0\\\
0&\frac{2}{3}\end{pmatrix}\frac{e^{2}}{h}$.

The inverse of $\sigma$ gives the resistivity tensor:
$\rho=\begin{pmatrix}\rho_{xx}&-\rho_{yx}\\\
\rho_{yx}&\rho_{xx}\end{pmatrix}$. For
$(\nu_{1},\nu_{2})=(\frac{1}{3},-\frac{1}{3})$, we have
$\rho_{xx}=\frac{h}{e^{2}}\sigma_{b}\begin{pmatrix}1&1\\\ 1&1\end{pmatrix}$
and $\rho_{yx}=\frac{h}{e^{2}}\begin{pmatrix}3&0\\\ 0&-3\end{pmatrix}$. For
$(\nu_{1},\nu_{2})=(\frac{1}{3},\frac{2}{3})$, we have
$\rho_{xx}=\frac{h}{e^{2}}\frac{1}{4+\sigma_{b}^{2}}\begin{pmatrix}4\sigma_{b}&-2\sigma_{b}\\\
-2\sigma_{b}&\sigma_{b}\end{pmatrix}$ and
$\rho_{yx}=\frac{h}{e^{2}}\frac{1}{4+\sigma_{b}^{2}}\begin{pmatrix}12+\sigma_{b}^{2}&\sigma_{b}^{2}\\\
\sigma_{b}^{2}&6+\sigma_{b}^{2}\end{pmatrix}$.

The above discussions are exactly at the QCP and zero temperature. In
practice, the experiments are always at finite temperature and one expects
critical scaling $\rho(T,\delta)=F(\frac{T}{\delta^{\nu z}})$, where $F$ is a
universal function and $\delta=d-d_{c}$ is the deviation from the critical
point. We have $z=1$ and $\nu\approx 0.67$ as the known critical exponents for
the XY transition. From collapsing the data of $(T,d-d_{c})$ one can
extrapolate the exponent $\nu z$ and the universal conductivity. Such a
scaling has been performed for the superconductor to insulator
transitionsHebard and Paalanen (1990); Yazdani and Kapitulnik (1995); Marković
_et al._ (1998); Bielejec and Wu (2002); Wang  _et al._ (2021b); Hen  _et al._
(2021).

Figure 3: (a) Illustration of the different behaviors of inter-layer tunneling
I-V curves between the bulk and edge at the critical point. (b) Hall
resistivity $R^{xy}_{11}$ and $R^{xy}_{21}$ (at units of $\frac{h}{e^{2}}$) in
the Hall bar geometry as tuning the distance $d$ for the filling
$(\nu_{1},\nu_{2})=(\frac{1}{3},\frac{2}{3})$. At $d_{c}$, because of the
extra-ordinary boundary criticality, the exciton still behaves like a
superfluid at the edge, which guarantees that $R^{yx}_{11}=R^{yx}_{12}=1$.
This is very different from the bulk values
$R^{yx}_{11}=\frac{12+\sigma_{b}^{2}}{4+\sigma_{b}^{2}}$ and
$R^{yx}_{12}=\frac{\sigma_{b}^{2}}{4+\sigma_{b}^{2}}$ at the QCP, which are at
intermediate values between the $d<d_{c}$ and $d>d_{c}$ phases.

##  VI Extra-ordinary log boundary criticality:

For the FQSH phase at $d>d_{c}$, there are helical edge modes. At $d<d_{c}$
the helical edge modes will be gapped out by the long-range exciton order.
Here let us decide the fate of these edge modes at the critical point. The
edge theory of the FQSH phase is:

$S_{0}=\int
dtdx\frac{1}{2\pi\tilde{\upsilon}_{F}\lambda}\big{(}(\partial_{t}\tilde{\theta})^{2}-\tilde{\upsilon}_{F}^{2}(\partial_{x}\tilde{\theta})^{2}\big{)}$
(5)

where $\tilde{\theta}$ represents the helical edge modes of the FQSH phase.
Here $e^{i\tilde{\theta}}$ creates a fractional exciton with charge $1/3$
under $A_{s}$ at the edge. $\lambda=\frac{4}{3}$ in the decoupling limit and
becomes smaller when including inter-layer repulsing at finite $d$. So we have
$\lambda<\frac{4}{3}$. At the QCP, it is further coupled to the bulk critical
boson through:

$S_{\text{boundary}}=S_{0}-s\int
dxdt(e^{i\tilde{\theta}}\varphi^{*}+e^{-i\tilde{\theta}}\varphi)$ (6)

We assume the anti-unitary layer exchange symmetry
$\mathcal{M}\mathcal{C}\mathcal{T}$ to guarantee that $e^{i\tilde{\theta}}$
carries zero momentum. Without the $\mathcal{M}\mathcal{C}\mathcal{T}$
symmetry, the above term is absent due to momentum mismatch and disorder needs
to be involved, which we leave to future analysis.

The scaling dimension of the coupling $s$ is
$[s]=2-\Delta_{\varphi}-\frac{1}{4}\lambda\approx 0.78-\frac{1}{4}\lambda>0$,
where we use $\Delta_{\phi}=1.22$ as the boundary scaling dimension of the
order parameter. So the coupling is relevant and flows to infinity (see the
supplementary). It is thus very likely that it flows to the extraordinary-log
boundary critical pointMetlitski (2022) recently proposed for the 3D XY
transition. At this fixed point, the exciton order is almost long-range
ordered at the edge:
$\langle\Phi^{\dagger}(x)\Phi(y)\rangle\sim\frac{1}{\log(|x-y|)^{\tilde{q}}}$.
This is in contrast to the large power-law decaying exponent for the
correlation function in the bulk, manifested in the inter-layer tunneling
$I-V$ curves as illustrated in Fig. 3(a). Besides, the exciton transport at
the edge is still superfluid like with an infinite conductance
$G=\frac{1}{9\lambda}$Giamarchi (2003) which should be infinite at zero
temperature dramatically different from the metallic bulk transport. As a
result, transport measurements in Hall bar geometry with edge and in the
Corbino geometry without edge are very different at the QCP. The flow of
$\frac{1}{\lambda}$ to zero is only logarithmic, so at finite temperature we
expect $G\sim\log\frac{1}{T}$, which may be tested in experiment.

So far we have focused on the filling
$(\nu_{1},\nu_{2})=(\frac{1}{3},-\frac{1}{3})$. For the filling
$(\nu_{1},\nu_{2})=(\frac{1}{3},\frac{2}{3})$, the bulk behavior is exactly
the same. But there is an additional edge mode from the integer quantum Hall
effect. At clean sample or weak disorder regime, this integer quantum Hall
edge model can not be hybridized with the FQSH edge modes and can be ignored.
Thus we expect the same extraordinary critical behavior. For example, at
filling $(\nu_{1},\nu_{2})=(\frac{1}{3},\frac{2}{3})$, we have shown that
$\rho^{yx}_{11}$ and $\rho^{yx}_{21}$ in the bulk are at certain fractional
values depending on the universal conductivity $\sigma_{b}$. However, because
of the extra-ordinary boundary behavior, we expect
$\rho^{yx}_{11}=\rho^{yx}_{21}=1$ in the Hall bar measurement, as illustrated
in Fig. 3(b). Distinction between edge and bulk transport can be a direct
verification of the proposed extra-ordinary boundary criticality. If there is
strong disorder, then the two edge modes in the layer 2 may be coupled
together and flow to the Kane-Fisher-Polchinski fixed pointKane  _et al._
(1994) for the filling $(\nu_{1},\nu_{2})=(\frac{1}{3},\frac{2}{3})$. If this
happens, we expect the coupling of the bulk exciton order parameter to the
edge to be irrelevant and we have an ordinary boundary critical
behaviorMetlitski (2022). It is interesting to study the transition between
extra-ordinary boundary criticality and the ordinary boundary criticality
tuned by the disorder strength, which we leave to future work.

##  VII  XY* transition for General Abelian states

In the previous sections we focus on the filling
$(\nu_{1},\nu_{2})=(\frac{1}{3},-\frac{1}{3})$. Here we point out that XY*
transition exist for any rational filling $(\nu_{1},\nu_{2})=(x,-x)$ as long
as there is an Abelian FQHE phase at the filling $x$.

Let us consider bilayer with an arbitrary Abelian FQHE state in one layer and
its particle-hole partner in the other layer. We still have the
$\mathcal{M}\mathcal{C}\mathcal{T}$ symmetry. Any Abelian FQHE phase can be
captured by a $K$ matrix with dimension $N$. The Low energy theory in the
decoupled limit is:

$\displaystyle\mathcal{L}$
$\displaystyle=-\frac{1}{4\pi}\mathbf{a}_{1}^{T}\mathbf{K}d\mathbf{a}_{1}+\frac{1}{4\pi}\mathbf{a}_{2}^{T}\mathbf{K}d\mathbf{a}_{2}$
$\displaystyle~{}~{}~{}+\frac{1}{2\pi}A_{1}\mathbf{q}^{T}d\mathbf{a}_{1}-\frac{1}{2\pi}A_{2}\mathbf{q}^{T}d\mathbf{a}_{2}$
(7)

where $\mathbf{K}$ is a $N\times N$ matrix. $\mathbf{q}$ is a $N\times 1$
vector. Similarly $\mathbf{a}_{1},\mathbf{a}_{2}$ are emergent U(1) gauge
fields with $N$ components in the two layers. As before $A_{1},A_{2}$ are
probe fields in the two layers with only one component. An Abelian FQHE phase
is specified by $\mathbf{K}$ and $\mathbf{q}$. The analysis below applies to
any Abelian FQHE phase.

The $\mathcal{M}\mathcal{C}\mathcal{T}$ transforms in the following way:
$\big{(}\mathbf{a}^{0}_{1}(t,\mathbf{r}),\vec{\mathbf{a}}_{1}(t,\mathbf{r})\big{)}\rightarrow\big{(}-\mathbf{a}^{0}_{2}(-t,\mathbf{r}),\vec{\mathbf{a}}_{2}(-t,\mathbf{r})\big{)}$,
$\big{(}A^{0}_{1}(t,\mathbf{r}),\vec{A}_{1}(t,\mathbf{r})\big{)}\rightarrow\big{(}-A^{0}_{2}(-t,\mathbf{r}),\vec{A}_{2}(-t,\mathbf{r})\big{)}$.

Then we redefine $A_{c}=\frac{1}{2}(A_{1}+A_{2}),A_{s}=A_{1}-A_{2}$,
$\mathbf{a}_{c}=\mathbf{a}_{1}+\mathbf{a}_{2}$,
$\mathbf{a}_{s}=\mathbf{a}_{1}-\mathbf{a}_{2}$, the action for the decoupled
phase is

$\displaystyle\mathcal{L}$
$\displaystyle=-\frac{1}{4\pi}\mathbf{a}_{1}^{T}\mathbf{K}d\mathbf{a}_{1}+\frac{1}{4\pi}\mathbf{a}_{2}^{T}\mathbf{K}d\mathbf{a}_{2}$
$\displaystyle~{}~{}~{}+\frac{1}{2\pi}A_{c}\mathbf{q}^{T}d\mathbf{a}_{s}+\frac{1}{4\pi}A_{s}\mathbf{q}^{T}d\mathbf{a}_{c}$
(8)

Suppose the lowest charged excitation at each layer is generated by the vector
$\mathbf{l}_{0}$. Then $\mathbf{l}=(\mathbf{l}_{0},-\mathbf{l}_{0})^{T}$
generates a boson with $Q_{c}=0$ and
$Q_{s}=\mathbf{q}^{T}\mathbf{K}^{-1}\mathbf{l}_{0}$. Let us label this boson
as $\varphi$, then the critical theory is

$\displaystyle\mathcal{L}$
$\displaystyle=|(\partial_{\mu}-i\mathbf{l}_{0}^{T}(\mathbf{a}_{1;\mu}-\mathbf{a}_{2;\mu}))\varphi|^{2}-s|\varphi|^{2}-g|\varphi|^{4}$
$\displaystyle~{}~{}~{}-\frac{1}{4\pi}\mathbf{a}_{1}^{T}\mathbf{K}d\mathbf{a}_{1}+\frac{1}{4\pi}\mathbf{a}_{2}^{T}\mathbf{K}d\mathbf{a}_{2}$
$\displaystyle~{}~{}~{}+\frac{1}{2\pi}A_{c}\mathbf{q}^{T}d\mathbf{a}_{s}+\frac{1}{4\pi}A_{s}\mathbf{q}^{T}d\mathbf{a}_{c}$
(9)

which can be rewritten as:

$\displaystyle\mathcal{L}$
$\displaystyle=|(\partial_{\mu}-i\mathbf{l}_{0}^{T}\mathbf{a}_{s;\mu}))\varphi|^{2}-s|\varphi|^{2}-g|\varphi|^{4}$
$\displaystyle~{}~{}~{}-\frac{1}{4\pi}\mathbf{a}_{c}^{T}\mathbf{K}d\mathbf{a}_{s}+\frac{1}{2\pi}A_{c}\mathbf{q}^{T}d\mathbf{a}_{s}+\frac{1}{4\pi}A_{s}\mathbf{q}^{T}d\mathbf{a}_{c}$
(10)

With the assumption that $\det\mathbf{K}\neq 0$, we can integrate
$\mathbf{a}_{c}$, which locks $\mathbf{a}_{s}=\mathbf{K}^{-1}\mathbf{q}A_{s}$.
Then the final critical theory is:

$\displaystyle\mathcal{L}$
$\displaystyle=|(\partial_{\mu}-iQ_{s}A_{s;\mu}))\varphi|^{2}-s|\varphi|^{2}-g|\varphi|^{4}+\frac{\sigma^{cs}_{xy}}{2\pi}A_{c}dA_{s}$
(11)

where $Q_{s}=\mathbf{q}^{T}\mathbf{K}^{-1}\mathbf{l}_{0}$.
$\sigma^{cs}_{xy}=\mathbf{q}^{T}\mathbf{K}^{-1}\mathbf{q}$.

The above action clearly describes a XY* transition with a condensation of a
fractional exciton of exciton charge
$Q_{s}=\mathbf{q}^{T}\mathbf{K}^{-1}\mathbf{l}_{0}$. Similar to our discussion
for $(\nu_{1},\nu_{2})=(\frac{1}{3},-\frac{1}{3})$, there is a fractional
counter-flow conductivity:

$\sigma_{xx}=Q_{s}^{2}\sigma_{b}\frac{e^{2}}{h}\begin{pmatrix}1&1\\\
1&1\end{pmatrix}$ (12)

where $\sigma_{b}$ is again the universal conductivity of the usual XY
transition.

One simple example is $(\nu_{1},\nu_{2})=(\frac{1}{5},-\frac{1}{5})$. Then the
K matrix is one dimensional with $K=5$, $l=1$, $q=1$. We simply reach that
$Q_{s}=\frac{1}{5}$, indicating a fractional exciton with $1/5$ exciton charge
at the XY* transition. A more nontrivial example is to consider
$(\nu_{1},\nu_{2})=(\frac{2}{5},-\frac{2}{5})$ or equivalently
$(\frac{2}{5},\frac{3}{5})$. Now in the decoupled phase we should use
$K=\begin{pmatrix}3&1\\\ 1&2\end{pmatrix}$ and $q=(1,0)^{T}$. The smallest
charged anyon is generated by $\mathbf{l}_{0}=(1,1)^{T}$, with charge $1/5$
and statistics $\frac{3}{5}\pi$. In the bilayer setup
$\mathbf{l}=(\mathbf{l}_{0},-\mathbf{l}_{0})^{T}$ generates a bosonic
fractional exciton with charge $Q_{s}=\frac{1}{5}$. Again we expect a XY*
transition with $Q_{s}=\frac{1}{5}$. In both these cases the universal
counter-flow conductivity is $\frac{1}{25}\sigma_{b}e^{2}/h$. The physical
exciton order parameter is $\Phi=\varphi^{5}$ and should have an even larger
anomalous scaling dimension than the
$(\nu_{1},\nu_{2})=(\frac{1}{3},\frac{2}{3})$ case. For this case a finite
inter-layer tunneling destroys the superfluid phase, but the XY* QCP is stable
because $-\varphi^{5}-H.c.$ is known to be irrelevant for the XY transition.
In contrast, for the filling $(\nu_{1},\nu_{2})=(\frac{1}{3},\frac{2}{3})$, an
inter-layer tunneling term acts as $-\varphi^{3}-H.c.$ at low energy and
drives the QCP to be first order transition.

One direct evidence of the fractional exciton in the XY* transition is a
fractional universal conductivity. However, this requires the value of
$\sigma_{b}$, the universal conductivity of the ordinary XY transition.
Unfortunately there is no measurement or accurate prediction of the universal
conductivity $\sigma_{b}$ so far. However, in quantum Hall bilayer we can find
XY* transitions at different fillings $(x,-x)$ to independently measure both
$\sigma_{b}$ and $Q_{s}$. For example, there are a XY* transition at filling
$(\nu_{1},\nu_{2})=(\frac{1}{3},-\frac{1}{3})$ with $Q_{s}=\frac{1}{3}$, and
another XY* transition at filling
$(\nu_{1},\nu_{2})=(\frac{2}{5},-\frac{2}{5})$ with $Q_{s}=\frac{1}{5}$. A
clear prediction is that the ratio of the universal counter-flow conductivity
at the QCP of these two fillings should be $\frac{25}{9}$. There should be
many different XY* transitions corresponding to different rational fillings
$x=\frac{m}{2pm\pm 1}$ with the charge of the elementary anyon known from
well-established theory, thus one can even do scaling between the counter-flow
conductivity at the QCP and the expected value of $Q_{s}$ to test the picture
that the critical exciton boson is formed by a pair of anyons.

##  VIII Summary

In conclusion, we proposed a route to accessing the XY* quantum critical point
(QCP) by tuning magnetic field (and hence the $d/l_{B}$ ratio) while keeping
the filling fixed at $(\nu_{1},\nu_{2})=(\frac{1}{3},-\frac{1}{3})$ or
$(\nu_{1},\nu_{2})=(\frac{1}{3},\frac{2}{3})$ in a quantum Hall bilayer. At
such a QCP, two anyons from the FQSH phase in the $d>d_{c}$ side form a
fractional exciton and condense, leading to the exciton condensation phase in
the $d<d_{c}$ side. The fractional charge of the exciton is manifested in the
large anomalous exponent of the exciton correlation function and a $1/9$
factor in the universal conductivity. We also argue that the edge at this XY*
transition shows extra-ordinary boundary criticality behavior. At criticality,
the exciton behaves like a superfluid at the edge despite it showing metallic
transport in the bulk. The generalization of our theory to other fillings like
$(\nu_{1},\nu_{2})=(\frac{1}{5},-\frac{1}{5})$ is straightforward.
Interestingly, in that case the presence of weak interlayer tunneling, which
is expected to introduce a $(\lambda\varphi^{5}+{\rm h.c.})$ anisotropy which
is expected to be dangerously irrelevant Lou  _et al._ (2007), which will
remove the Goldstone mode in the exciton condensed phase, but will preserve
the critical properties. The relation between anyons in the topological phase
and vorticity on the ordered side has led to the proposal of interesting
‘memory’ effects in other contexts Senthil and Fisher (2001) which can also be
explored here. These directions will be worth exploring in the future. In
summary, our work suggests a new approach to studying exotic quantum phase
transitions with fractionalization and unusual boundary critical behaviors in
the highly tunable quantum Hall bilayer systems.

Note added After our manuscript appeared, we became aware of another
preprintWu  _et al._ (2023) discussing continuous quantum phase transition in
quantum Hall bilayer system, but in a different setup with different physics.

Acknowledgement We thank Max Metlitski for insightful comments on the
manuscript. YHZ thanks Cory Dean, Jia Li and Yihang Zeng for discussions on
related experiments in quantum Hall bilayer. Z.Z. thanks Zlatko Papić and
Songyang Pu for helpful discussions. YHZ is supported by a startup fund from
Johns Hopkins university. Z.Z. was supported by the National Natural Science
Foundation of China (Grant No.12074375),the Fundamental Research Funds for the
Central Universities, the Strategic Priority Research Program of CAS (Grant
No.XDB33000000). AV was supported by the Simons Collaboration on Ultra-Quantum
Matter, which is a grant from the Simons Foundation (651440, AV) and from NSF-
DMR 2220703.

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##  Appendix A Numerical Details

We consider $\nu_{T}=1/3+2/3$ quantum Hall bilayers subject to a perpendicular
magnetic field on torus, which is spanned by length vectors $\mathbf{L_{x}}$
and $\mathbf{L_{y}}$, and thus the orbital number (or flux number) in each
layer $N_{\phi}$ is determined by the area of torus, i.e.,

$|\mathbf{L_{x}}\times\mathbf{L_{y}}|=2\pi N_{\phi}$ (13)

Here, the magnetic length $l_{B}\equiv\sqrt{\hbar c/eB}\equiv 1$ (the unit of
length). We choose Landau gauge $\mathbf{A}=(By,0,0)$ and consider the torus
with aspect ratio to be 1. The single-particle wave functions in the lowest
Landau level (LLL) as basis reads

$\psi_{k}(x,y)=\frac{1}{\sqrt{L_{x}\sqrt{\pi}}}\sum_{n=-\infty}^{+\infty}e^{[\mathrm{i}(k+nL_{y})x-(y+nL_{y}+k)^{2}/2]},$
(14)

where $k\equiv{2\pi j}/L_{x}$ with $j=0,1,...,N_{\phi}-1$ due to periodical
boundary condition along $x$ direction. The single particle states $\psi_{k}$
are centered at $y=-k$ with a distance $2\pi/L_{x}$ apart along $y$ direction,
while they are extended in $x$ direction. Then $N_{\phi}$ states can be mapped
into one-dimensional (1D) lattice with each site representing a single
particle orbital $\psi_{k}$. Then one can perform numerical simulation on such
1D lattice in momentum space with the number of sites equal to the number of
orbitals. The relationship between the area of the torus and the size of 1D
lattice is determined by Eq.13. In order to realize the numerical
diagonalization on a larger system size, one needs to reduce the dimension of
the Hamiltonian block by taking advantage of magnetic translational symmetries
along $x$ or/and $y$ directions. The symmetry analysis was first provided by
Haldane Haldane (1985) with introducing two translation operators,
$T_{\alpha}$ $(\alpha=1,2)$ with eigenvalues $e^{2\pi iK_{\lambda}/N_{\phi}}$
($\lambda=x,y$ and $K_{\lambda}=0,...,N_{\phi}-1$).$T_{1}$ corresponds to the
magnetic translation in $x$-direction, where
$K_{x}=\sum_{k=0}^{N_{\phi}-1}kn_{k}$ (mod $N_{\phi}$) is total momentum (in
the unit of $2\pi/L_{x}$) of electrons taken modulo $N_{\phi}$. $T_{2}$
translates the entire lattice configuration one step
$L_{y}/N_{\phi}=2\pi/L_{x}$ to the right along $y$-direction. Taking advantage
of one or both symmetries, one can numerically diagonalize the Hamiltonian
efficiently. Different from the sphere geometry, there is no orbital number
shift on torus and the states are uniquely determined by their filling factor.

In the present work, we consider the physical systems with two identical 2D
layers (with zero width) in the absence of electron interlayer tunneling while
spins of electrons are fully polarized due to strong magnetic fields. Such a
system can be described by the projected Coulomb interaction,

$V=\frac{1}{N_{\phi}}\sum\limits_{i<j,\alpha,\beta}{\sum\limits_{{\bf{q}},{\bf{q}}\neq
0}{{V_{\alpha\beta}}\left(q\right)}}{e^{-\frac{q^{2}}{2}}}L^{2}_{n}[\frac{q^{2}}{2}]{e^{i{\bf{q}}\cdot\left({{{\bf{R}}_{\alpha,i}}-{{\bf{R}}_{\beta,j}}}\right)}}.$
(15)

Here, $\alpha(\beta)=1,2$ denote two layers or, equivalently, two components
of a pseudospin-$1/2$. $q$ =$|{\bf{q}}|$ =$\sqrt{q_{x}^{2}+q_{y}^{2}}$,
$V_{11}(q)=V_{22}(q)={e^{2}}/({\varepsilon q})$ and
$V_{12}(q)=V_{21}(q)={e^{2}}/({\varepsilon q})\cdot e^{-qd}$ are the Fourier
transformations of the intralayer and interlayer Coulomb interactions,
respectively. $d$ represents the distance between two layers in the unit of
magnetic length $l_{B}$. $L_{n}(x)$ is the Laguerre polynomial with Landau
level index $n$ and $\bf{R}_{\alpha,i}$ is the guiding center coordinate of
the $i$-th electron in layer $\alpha$. Here we consider rectangular unit cells
with $L_{x}=L_{y}=L$ and set magnetic length $l_{B}\equiv\sqrt{\hbar c/eB}$ as
the unit of length, $e^{2}/{\varepsilon l_{B}}$ as the unit of energy.
Numerically, one needs to use the second-quantization form:

$V=\sum\limits_{{j_{1}}{j_{2}}{j_{3}}{j_{4}}}{{V^{\alpha\beta}_{{j_{1}}{j_{2}}{j_{3}}{j_{4}}}}c_{{j_{1}}}^{\dagger}c_{{j_{2}}}^{\dagger}{c_{{j_{3}}}}{c_{{j_{4}}}}},$
(16)

with

${V^{\alpha\beta}_{{j_{1}}{j_{2}}{j_{3}}{j_{4}}}}={{\delta^{\prime}}_{{j_{1}}+{j_{2}},{j_{3}}+{j_{4}}}}\frac{1}{{4\pi
N_{\phi}}}\sum\limits_{{\bf{q}},{\bf{q}}\neq
0}{{{\delta^{\prime}}_{{j_{1}}-{j_{4}},{{{q_{y}}{L_{y}}}\mathord{\left/{\vphantom{{{q_{y}}{L_{y}}}{2\pi}}}\right.\kern-1.2pt}{2\pi}}}}}V_{\alpha\beta}(q)\exp\left[{{{-{q^{2}}}\mathord{\left/{\vphantom{{-{q^{2}}}2}}\right.\kern-1.2pt}2}-i\left({{j_{1}}-{j_{3}}}\right){{{q_{x}}{L_{x}}}\mathord{\left/{\vphantom{{{q_{x}}{L_{x}}}{{N_{\phi}}}}}\right.\kern-1.2pt}{{N_{\phi}}}}}\right]L^{2}_{n=0}[-{q^{2}}/2].$
(17)

Here, the Kronecker delta with the prime means that the equation is defined
modulo $N_{\phi}$. We also consider a uniform and positive background charge
so that the Coulomb interaction at $q=0$ is canceled out.

##  Appendix B Symmetry of the model

Here we point out a symmetry $\mathcal{M}\mathcal{C}\mathcal{T}$ for the
quantum Hall bilayer at filling $(\nu_{1},\nu_{2})=(x,-x)$. Here
$\nu_{a}=\frac{N_{e}}{N_{\Phi}}$, where $N_{\Phi}$ is the number of the
magnetic flux in the system. $\nu_{2}<0$ means that the system is hole doped
with hole density at $x$ per flux. We will mainly be interested in the $x=1/3$
point.

We define electron operators in layer 1 and 2 as $c_{1}(\mathbf{r})$ and
$c_{2}(\mathbf{r})$. Because the layer 2 is hole doped, it is convenient to
use the hole operator $h_{2}(\mathbf{r})=c_{2}^{\dagger}(\mathbf{r})$. The
Hamiltonian is

$\mathcal{H}=\int
d^{2}\mathbf{r}c^{\dagger}_{1}(\mathbf{r})\frac{(-i\vec{\nabla}-e\vec{A}_{1})^{2}}{2m}c_{1}(\mathbf{r})+\int
d^{2}\mathbf{r}h^{\dagger}_{2}(\mathbf{r})\frac{(-i\vec{\nabla}+e\vec{A}_{2})^{2}}{2m}h_{2}(\mathbf{r})-D\int
d^{2}\mathbf{r}\big{(}c^{\dagger}_{1}(\mathbf{r})c_{1}(\mathbf{r})+h^{\dagger}_{2}(\mathbf{r})h_{2}(\mathbf{r})\big{)}+\mathcal{H}_{\text{int}}$
(18)

where $e$ is the electron charge which is negative. $\vec{A}_{1}(\mathbf{r})$
and $\vec{A}_{2}(\mathbf{r})$ are vector fields in the two layers. For quantum
Hall bilayer system, we have the magnetic field:
$\vec{A}_{a}(\mathbf{r})=\frac{1}{2}B\hat{z}\times\mathbf{r}+\delta\vec{A}_{a}(\mathbf{r})$.
Here $\delta\vec{A}_{a}(\mathbf{r})$ is the probing field in each layer
applied to measure the response of the system. $D$ is the displacement field
and can be viewed as the chemical potential to tune the exciton density $x$.

The interaction term is:

$\mathcal{H}_{\text{int}}=\frac{1}{2}\int d^{2}\mathbf{r}\int
d^{2}\mathbf{r}^{\prime}V_{ab}(|\mathbf{r}-\mathbf{r}^{\prime}|):\rho_{a}(\mathbf{r})\rho_{b}(\mathbf{r}^{\prime}):$
(19)

where

$\rho_{1}(\mathbf{r})=ec^{\dagger}_{1}(\mathbf{r})c_{1}(\mathbf{r})$ (20)

and

$\rho_{2}(\mathbf{r})=-eh^{\dagger}_{2}(\mathbf{r})h_{2}(\mathbf{r})$ (21)

The Coulomb interaction is in the form
$V_{ab}(\mathbf{r})=\frac{1}{(2\pi)^{2}}\int
d^{2}\mathbf{q}V_{ab}(\mathbf{q})e^{i\mathbf{q}\cdot\mathbf{r}}$ with
$V_{11}(q)=V_{22}(q)=\frac{e^{2}}{\varepsilon q}$ and
$V_{12}(q)=V_{21}(q)=\frac{e^{2}}{\varepsilon q}e^{-qd}$. $d$ represents the
distance between two layers in the unit of magnetic length $l_{B}=\sqrt{\hbar
c/eB}$.

Let us ignore the probing field $\delta\vec{A}_{a}(\mathbf{r})$ for now. Then
the Hamiltonian has the following $\mathcal{M}\mathcal{C}\mathcal{T}$ anti-
unitary symmetry:

$\mathcal{M}\mathcal{C}\mathcal{T}:c_{1}(\mathbf{r})\leftrightarrow
h_{2}(\mathbf{r})$ (22)

Note that this is an anti-unitary and need to include a complex conjugate
$\mathcal{K}$. Under $\mathcal{M}\mathcal{C}\mathcal{T}$,
$\rho_{1}(\mathbf{r})\rightarrow-\rho_{2}(\mathbf{r})$ and
$\rho_{1}(\mathbf{q})\rightarrow-\rho_{2}(-\mathbf{q})$. Under
$\mathcal{M}\mathcal{C}\mathcal{T}$, an electron in layer 1 is transformed to
a hole in layer 2. We can also include the vector potential
$\delta\vec{A}_{a}(\mathbf{r})$ and also the electric potential term $\delta
H=-\rho_{1}(\mathbf{r})A^{0}_{1}(\mathbf{r})-\rho_{2}(\mathbf{r})A^{0}_{2}(\mathbf{r})$.
Thus under $\mathcal{M}\mathcal{C}\mathcal{T}$, we have
$A^{0}_{1}(\mathbf{r})\rightarrow-A^{0}_{2}(\mathbf{r})$ and
$\delta\vec{A}_{1}(\mathbf{r})\rightarrow\delta\vec{A}_{2}(\mathbf{r})$. In
the following we use $\vec{A}_{a}(\mathbf{r})$ to denote
$\delta\vec{A}_{a}(\mathbf{r})$. The space time coordinates transform as
$(t,\mathbf{r})\rightarrow(-t,\mathbf{r})$ under
$\mathcal{M}\mathcal{C}\mathcal{T}$.

One can diagonalize the kinetic part and project the interaction in the lowest
Landau levels. Within the lowest Landau level, the Hamiltonian is

$\mathcal{H}=\frac{1}{2}\sum_{a,b=1,2}V_{ab}(\mathbf{q}):\rho_{a}(\mathbf{q})\rho_{a}(-\mathbf{q}):$
(23)

Here

$\rho_{a}(\mathbf{q})=\int
d^{2}\mathbf{q}\rho_{a}(\mathbf{r})e^{-i\mathbf{q}\cdot\mathbf{r}}$ (24)

with $\rho_{a}(\mathbf{r})$ as the charge density operator projected to the
lowest Landau level:

$\rho_{1}(\mathbf{r})=e\sum_{m,n}c^{\dagger}_{1;m}c_{1;n}\varphi^{*}_{+;m}(\mathbf{r})\varphi_{+;n}(\mathbf{r})$
(25)

and

$\rho_{2}(\mathbf{r})=-e\sum_{m,n}h^{\dagger}_{2;m}h_{2;n}\varphi^{*}_{-;m}(\mathbf{r})\varphi_{-;n}(\mathbf{r})$
(26)

In the above $\varphi_{+,m}(\mathbf{r})$ and $\varphi_{-;m}$ are the
wavefunction of the state labeled by the Landau index $m$ for electron and
hole respectively. We have
$\varphi_{-;m}(\mathbf{r})=\varphi_{+;m}(\mathbf{r})^{*}$.

Under the symmetry $\mathcal{M}\mathcal{C}\mathcal{T}$, $c_{1;m}\rightarrow
h_{2;m}$ and $\rho_{1}(\mathbf{r})\rightarrow-\rho_{2}(\mathbf{r})$ still hold
in the Hamiltonian projected to the lowest Landau level.

##  Appendix C Relation to the FQSH to Paired-Superfluid Transition and an
Estimate for $d_{c}$

Thus far we have phrased the discussion in terms of exciton condensation on
the $(\nu_{1},\,\nu_{2})=(1/3,\,2/3)$ insulator. Now consider performing a
particle hole conjugation on Layer 2, as described above i.e.
$h_{2}(r)=c^{\dagger}_{2}(r)$. This leads to the two layers now having the
same density, but experiencing opposite magnetic fields as in Eqn. 18. This is
noting but the fractional version of the Quantum Spin Hall effect (FQSH) Levin
and Stern (2009). Furthermore, as a result of the particle-hole transformation
in the bottom layer, the repulsive inter-layer interaction now becomes an
attractive interaction, and leads to the formation of bound states between the
layers of Laughlin quasiparticles, i.e. Laughlin Cooper pairs of charge
$2e/3$. The condensation of these fractional Cooper pairs leads to a paired
superconductor Ghaemi  _et al._ (2012) (which is smoothly connected to the
paired condensate of electrons). Note, the Laughlin Cooper pair is a boson and
has mutual statistics with all the other anyons in the problem, hence its
condensation leads to a conventional superconductor.

The following simple energetic argument gives an estimate for the critical
distance $d_{c}$. Consider creating a Laughlin quasi-electron+quasi-hole in
one layer and a corresponding pair in the other layer. The energy for each
pair is just the gap: $\Delta_{1/3}\approx 0.1e^{2}/\epsilon l_{B}$ Morf and
Halperin (1986). Now consider the strength of the attractive interaction
between quasi-particle and quasi-hole in the two layers $\Delta
E={e^{*}}^{2}/\epsilon d$, where $e^{*}=e/3$. A simple estimate for the
critical point is when the binding energy overcomes the cost of creating the
quasiparticles and hence: ${e^{*}}^{2}/\epsilon d=\Delta_{1/3}$ or
$\frac{d_{c}^{*}}{l_{B}}=1.1$. In practice the excitons will have a dispersion
that further lowers their energy and we would expect that the true transition
occurs earlier i.e. $d_{c}>d_{c}^{*}$. Our numerical calculations gave
$d_{c}\approx 1.7$. Similarly one can estimate the critical distance for the
$(\nu_{1},\,\nu_{2})=(1/5,\,4/5)$ using $\Delta_{5}=0.024e^{2}/l_{B}$
Bonesteel (1995) and $e^{*}=e/5$.

##  Appendix D Critical theory

We consider the filling $(\nu_{1},\nu_{2})=(\frac{1}{3},-\frac{1}{3})$. In the
FQSH phase at large $d/l_{B}$, the effective low energy theory is captured by
a Chern-Simons theory with K matrix $K=\begin{pmatrix}3&0\\\
0&-3\end{pmatrix}$. There are two emergent gauge fields $a_{1;\mu},a_{2;\mu}$
whose charges are labeled as $l=(l_{1},l_{2})$. We consider a fractional
exciton labeled by $l=(1,1)$. It carries physical charge $Q_{c}=0$ and
physical spin $Q_{s}=\frac{1}{3}$. The transition between the FQSH phase and
the exciton condensation phase is described by the condensation of this
fractional exciton whose creation operator is labeled as $\varphi^{\dagger}$.
Then the critical theory is:

$\displaystyle\mathcal{L}$
$\displaystyle=|(\partial_{\mu}-i(a_{1;\mu}-a_{2;\mu}))\varphi|^{2}-s|\varphi|^{2}-g|\varphi|^{4}-\frac{3}{4\pi}a_{1}da_{1}+\frac{3}{4\pi}a_{2}da_{2}+\frac{1}{2\pi}A_{c}d(a_{1}-a_{2})+\frac{1}{4\pi}A_{s}d(a_{1}+a_{2})$
(27)

where $s$ is the tuning parameter. $A_{c;\mu}$ and $A_{s;\mu}$ are probing
field coupled to the total charge and the spin. The
$\mathcal{M}\mathcal{C}\mathcal{T}$ symmetry requires that $\vec{A}_{s}=0$.
Physically the magnetic fields of the two layers are the same and exciton does
not feel any net magnetic field. From symmetry analysis, one can in principle
include a term $\varphi^{*}(i\partial_{t}-A_{s;0})\varphi$. This term is fine
tuned to be zero because we are considering an interaction driven transition
through $d/l_{B}$ at fixed exciton density. On the other hand, one can also
tune a FQSH to SF transition through tuning the displacement field, which
corresponds to a chemical potential tuned transition. The dynamical exponent
for such a transition will be $z=2$ and is not our interest.

We then make a redefinition: $a_{c}=a_{1}+a_{2},a_{s}=a_{1}-a_{2}$. The above
actions change to:

$\displaystyle\mathcal{L}$
$\displaystyle=|(\partial_{\mu}-ia_{s;\mu})\varphi|^{2}-s|\varphi|^{2}-g|\varphi|^{4}-\frac{3}{4\pi}a_{c}da_{s}+\frac{1}{2\pi}A_{c}da_{s}+\frac{1}{4\pi}A_{s}da_{c}$
(28)

The redefinition of the gauge field changes the charge quantization rules. The
charge under $a_{c}$ is labeled as $q_{c}$ and the charge of $a_{s}$ is
$q_{s}$. We have the charge transformation: $q_{c}=\frac{q_{1}+q_{2}}{2}$ and
$q_{s}=\frac{q_{1}-q_{2}}{2}$, where $q_{1},q_{2}$ are charges under
$a_{1},a_{2}$. The elementary charge configuration is $(q_{1},q_{2})=(\pm
1,0),(0,\pm 1)$. Thus in terms of $(q_{c},q_{s})$, the elementary charge is
$(q_{c},q_{s})=(\pm\frac{1}{2},\pm\frac{1}{2})$. In the FQSH phase with
$\varphi$ gapped, one can check that for the excitation
$(q_{c},q_{s})=(\pm\frac{1}{2},\pm\frac{1}{2})$, it has statistics
$\theta=\pm\frac{\pi}{3}$, physical charge $Q_{c}=\pm\frac{1}{3}$ and physical
spin $Q_{s}=\pm\frac{1}{6}$. This is exactly the elementary anyon on layer 1
or layer 2 in the FQSH phase. When $\varphi$ is condensed, $a_{s}$ is higgsed
and we are left with the superfluid action
$\mathcal{L}_{SF}=\frac{1}{4\pi}A_{s}da_{c}$. We can see that the vortex
charge $q_{v}$ of the superfluid is $q_{v}=2q_{c}$. Then the elementary anyon
with $(q_{c},q_{s})=\pm(\frac{1}{2},\frac{1}{2})$ in the FQSH phase now
becomes the elementary vortex with $q_{v}=\pm 1$ in the superfluid phase. Of
course now it costs infinite energy due to the coupling to the gapless gauge
field $a_{c}$ which represents the Goldstone mode of the superfluid. From this
analysis one can see that the elementary anyon in the FQSH phase becomes the
vortex of the superfluid in the exciton condensation (EC) phase. Its energy
cost is infinite in the EC phase and finite in the FQSH phase. But it remains
gapped across the phase transition. It is known that the elementary vortex
become gapless and then condensed in the usual superfluid to insulator
transition. Later in the dual theory we will see that at the XY* critical
point what becomes gapless is a triple vortex, while the vortexes with
$|q_{v}|=1,2$ remain gapped.

At the critical point, the topological property of the anyon does not matter.
So we can integrate $a_{c}$ which simply locks $a_{s}=\frac{1}{3}A_{s}$. Then
we reach the final critical theory:

$\displaystyle\mathcal{L}_{c}$
$\displaystyle=|(\partial_{\mu}-i\frac{1}{3}A_{s;\mu})\varphi|^{2}-s|\varphi|^{2}-g|\varphi|^{4}+\frac{1}{6\pi}A_{c}dA_{s}$
(29)

When $s<0$, this is a superfluid phase of $A_{s}$. When $s>0$, we have the
correct response of $\frac{1}{6\pi}A_{c}dA_{s}$ for the FQSH phase, though we
have lost the information about the anyons by integrating $a_{c}$.

###  D.1 Dual theory

It is known that the XY critical theory such as in Eq. 29 has a dual theory.
Here we derive the dual critical theory. We start from Eq. 28 and apply the
standard particle vortex duality for $\varphi$, then we obtain a dual critical
theory as:

$\displaystyle\mathcal{L}$
$\displaystyle=|(\partial_{\mu}-ib_{\mu})\tilde{\varphi}|^{2}-r|\tilde{\varphi}|^{2}-g|\tilde{\varphi}|^{4}-\frac{3}{4\pi}a_{s}da_{c}+\frac{1}{2\pi}bda_{s}+\frac{1}{2\pi}A_{c}da_{s}+\frac{1}{4\pi}A_{s}da_{c}$
(30)

Integrating $a_{s}$, we locks $b=\frac{3}{2}a_{c}-A_{c}$. With a redefinition
$a=\frac{1}{2}a_{c}-\frac{1}{3}A_{c}$, we get:

$\mathcal{L}=|(\partial_{\mu}-i3a_{\mu})\tilde{\varphi}|^{2}-r|\tilde{\varphi}|^{2}-g|\tilde{\varphi}|^{4}+\frac{1}{2\pi}A_{s}da+\frac{1}{6\pi}A_{s}dA_{c}\\\
$

which is exactly the dual theory of Eq. 29. We choose the normalization of the
gauge field $a$ so we have the usual coupling $\frac{1}{2\pi}A_{s}da$. When
$\tilde{\varphi}$ is gapped in the $r>0$ side, this describes a superfluid
phase of $A_{s}$. Here $\tilde{\varphi}$ here carries vortex charge $q_{v}=3$
of the superfluid phase. Therefore at the QCP the triple vortex instead of the
elementary vortex of the superfluid becomes gapless. Condensation of this
triple vortex kills the superfluid phase and leads to an insulator. The
elementary vortex remains gapped across the QCP and will become the anyon in
the FQHE insulator after the superfluid is gone.

##  Appendix E Extra-ordinary boundary criticality

We start from the edge theory for the FQSH phase at $d>d_{c}$. For now let us
consider the filling $(\nu_{1},\nu_{2})=(\frac{1}{3},-\frac{1}{3})$. From the
K matrix $K=\begin{pmatrix}3&0\\\ 0&-3\end{pmatrix}$ we can write down the
effective action for the helical edge modes:

$\displaystyle S$ $\displaystyle=\int
dtdx\frac{3}{4\pi}\partial_{t}\varphi_{1}\partial_{x}\varphi_{1}-\frac{3}{4\pi}\partial_{t}\varphi_{2}\partial_{x}\varphi_{2}-\frac{3}{4\pi}\upsilon_{F}(\partial_{x}\varphi_{1})^{2}-\frac{3}{4\pi}\upsilon_{F}(\partial_{x}\varphi_{2})^{2}-g\frac{3}{2\pi}\upsilon_{F}(\partial_{x}\varphi_{1})(\partial_{x}\varphi_{2})$
(31)

where $\varphi_{1}$ and $\varphi_{2}$ represent the edge mode in layer 1 and 2
respectively. The $g<0$ term is from inter-layer repulsion. Note in our
current convention the density operators are
$\rho_{1}=e\frac{1}{2\pi}\partial_{x}\varphi_{1}$ and
$\rho_{2}=-e\frac{1}{2\pi}\partial_{x}\varphi_{2}$.

We have the commutation relations:

$[\varphi_{1}(x),\partial_{y}\varphi_{1}(y)]=i\frac{2\pi}{3}\delta(x-y)$ (32)

and

$[\varphi_{2}(x),\partial_{y}\varphi_{2}(y)]=-i\frac{2\pi}{3}\delta(x-y)$ (33)

Next we do a linear combination and define:

$\displaystyle\varphi_{1}(x)$
$\displaystyle=\frac{1}{\sqrt{3}}(\phi(x)+\theta(x))$
$\displaystyle\varphi_{2}(x)$
$\displaystyle=\frac{1}{\sqrt{3}}(\phi(x)-\theta(x))$

so

$[\theta(x),\partial_{y}\varphi(y)]=i\pi\delta(x-y)$ (35)

This leads to the action:

$S=\int
dtdx\frac{1}{\pi}\partial_{t}\phi\partial_{x}\theta-\frac{\tilde{\upsilon}_{F}}{2\pi}\big{(}K(\partial_{x}\theta)^{2}+\frac{1}{K}(\partial_{x}\phi)^{2}\big{)}$
(36)

where $\tilde{\upsilon}_{F}=\sqrt{1-g^{2}}\upsilon_{F}$ and
$K=\frac{1-g}{1+g}$. So we have $K>1$.

We can also integrate $\phi$ to reach

$S=\int
dtdx\frac{K}{2\pi\tilde{\upsilon}_{F}}\big{(}(\partial_{t}\theta)^{2}-\tilde{\upsilon}_{F}^{2}(\partial_{x}\theta)^{2}\big{)}$
(37)

This is the standard action for the Luttinger liquid, but the operator mapping
is different. Especially the fractional exciton $l=(1,-1)$ now corresponds to
$\psi^{\dagger}\sim
e^{i\varphi_{R}}e^{-i\varphi_{L}}=e^{i\frac{2}{\sqrt{3}}\theta(x)}$. It’s
scaling dimension is $[\psi]=\frac{1}{3K}<\frac{1}{3}$. We can make a
redefinition $\tilde{\theta}=\frac{2}{\sqrt{3}}\theta$, so the action is:

$S_{0}=\int
dtdx\frac{1}{2\pi\tilde{\upsilon}_{F}\lambda}\big{(}(\partial_{t}\tilde{\theta})^{2}-\tilde{\upsilon}_{F}^{2}(\partial_{x}\tilde{\theta})^{2}\big{)}$
(38)

with $\lambda=\frac{4}{3K}$. In this convention $e^{i\tilde{\theta}}$ creates
a fractional exciton with charge $1/3$ under $A_{s}$.

At the QCP, the boundary is described by the following action:

$S_{\text{boundary}}=\int
dtdx\frac{1}{2\pi\tilde{\upsilon}_{F}\lambda}\big{(}(\partial_{t}\tilde{\theta})^{2}-\tilde{\upsilon}_{F}^{2}(\partial_{x}\tilde{\theta})^{2}\big{)}-s\int
dxdt(e^{i\tilde{\theta}}\varphi^{*}+e^{-i\tilde{\theta}}\varphi)$ (39)

Following Ref. Metlitski, 2022, we obtain the renormalization flow equation:

$\displaystyle\frac{ds}{dl}$
$\displaystyle=(2-\Delta_{\varphi}-\frac{1}{4}\lambda)s$
$\displaystyle\frac{d\lambda}{dl}$ $\displaystyle=-\pi^{2}s^{2}\lambda^{2}$
(40)

Given that $\Delta_{\varphi}\approx 1.219$ and initially
$\lambda<\frac{4}{3}$, we will have $s$ flows to infinite and $\lambda$ flows
to zero when the RG flow $l$ approaches infinite.

The ordinary exciton order creation operator is $e^{i3\tilde{\theta}}$. In the
FQSH phase its correlation function is:

$\langle
e^{i3\tilde{\theta}(x)}e^{-i3\tilde{\theta}(y)}\rangle\sim\frac{1}{|x-y|^{9\lambda}}$
(41)

Therefore at the QCP, because $\lambda$ flows to zero, the exponent of the
above correlation function also flows to zero. In practice, it should have a
log singularity Metlitski (2022) at the QCP:

$\langle
e^{i3\tilde{\theta}(x)}e^{-i3\tilde{\theta}(0)}\rangle\sim\frac{1}{(\log
x)^{q}}\ ,x\rightarrow\infty$ (42)

This is so-called extra-ordinary-log boundary critical behavior. One can see
that the exciton order has an almost long-range order at the edge, despite
that its correlation function has a large decaying exponent in the bulk.

The exciton current $A_{s}$ couples in the following way:
$\partial_{\mu}\tilde{\theta}\rightarrow(\partial_{\mu}-i\frac{1}{3}A_{s;\mu})\tilde{\theta}$.
In the FQSH phase it is known the conductance under $A_{s}$ is
$G=\frac{1}{9}\frac{1}{\lambda}\frac{e^{2}}{h}$Giamarchi (2003). At the QCP,
the conductance $G$ then flows to infinite. This is expected because the
exciton has almost long-range order, so its transport should still be
superfluid-like.

We also want to comment on the local density of states (DoS) of one layer
probed by the scanning tunneling microscope (STM). Consider the layer 1, the
single electron creation operator is: $c_{1}^{\dagger}(x)\sim
e^{i3\varphi_{1}(x)}=e^{i\sqrt{3}(\phi(x)+\theta(x))}$. Then we expect the STM
of the layer 1 has $\frac{dI}{dV}\sim V^{\alpha}$ with
$\alpha=\frac{3}{2}(K+\frac{1}{K})-1$. In the decoupled phase at large
$d/l_{b}$ we have $\alpha\approx 2$ as $K\approx 1$. However, at the QCP,
$\alpha$ goes to infinite in the extra-ordinary criticality.