# Gauge theory description of Rydberg atom arrays with a tunable blockade
radius

Yanting Cheng  Institute of Theoretical Physics and Department of Physics,
University of Science and Technology Beijing, Beijing 100083, China  Chengshu
Li<EMAIL_ADDRESS>Institute for Advanced Study, Tsinghua University,
Beijing 100084, China

###### Abstract

We discuss a Rydberg atom chain with a tunable blockade radius from the gauge
theoretic perspective. When the blockade radius is one lattice spacing, this
system can be formulated in terms of the PXP model, and there is a
$\mathbb{Z}_{2}$ Ising phase transition known to be equivalent to a
confinement–deconfinement transition in a gauge theory, the lattice Schwinger
model. Further increasing the blockade radius, one can add a next-nearest
neighbor (NNN) interaction into the PXP model. We discuss the interpretation
of NNN interaction in terms of the gauge theory and how finite NNN interaction
alters the deconfinement behavior and propose a corresponding experimental
protocol. When the blockade radius reaches two lattice spacing, the model
reduces to the PPXPP model. A novel gauge theory equivalent to the PPXPP model
is formulated, and the phases in the two formulations are delineated. These
results are readily explored experimentally in Rydberg quantum simulators.

##  I Introduction

Gauge theories play a pivotal role in our understanding of the fundamental
interactions, culminating in the overarching standard model [1, 2]. On a more
emergent level, gauge theories also serve as an important tool for the
understanding of various condensed matter systems, from quantum spin liquids
[3, 4] to quantum Hall effects [5]. Among various developments of the monument
of gauge theories, the idea of regulating them on a lattice — known as the
lattice gauge theory (LGT) [6, 7] — provides both conceptual progress [8, 9,
10, 11, 12] and closer relation to condensed matter/quantum simulation [13,
14, 15, 16, 17, 18, 19, 20, 21, 22].

Recently, programmable Rydberg atom arrays attract considerable research
interest thanks to their relevance both as interesting many-body systems and
as potential quantum computing platforms [23, 24, 25]. Focusing on the former,
there are two major frontiers. The first is to explore the ground state
quantum phases and the transitions thereof [26, 27]. Highlights here include a
quantum spin liquid with the atoms located on the bonds of a kagome lattice
[28, 29, 30, 20, 31, 32, 33]. The second is to probe dynamical processes, a
surprising discovery along this line being the quantum scar states [34, 35,
36, 37, 38, 39, 40, 41] that violate the eigenstate thermalization hypothesis.
It turns out that the PXP model and its generalizations constitute a simple
yet powerful modelling incorporating the all-important blockade effect in
these systems. In this simplest case, the Rydberg blockade effect prescribes
that two nearest-neighbor atoms can not be simultaneously excited to the
Rydberg state, and the Hamiltonian reads

$\hat{H}=\sum_{i}\hat{P}_{i-1}\hat{X}_{i}\hat{P}_{i+1}-m\sum_{i}\hat{Z}_{i},$
(1)

where one uses spin up (down) to denote the Rydberg (ground) state,
$\hat{P}_{i}=(1-\hat{Z}_{i})/2$ is a projector to the ground state, and
$\hat{X},\hat{Z}$ are the Pauli X, Z operators. The first term couples the
ground state and the Rydberg state with a Rabi oscillation, and the second
term originates from detuning. Focusing on the ground state, there are two
phases — a disordered phase when $m\to-\infty$ where all the atoms are in the
ground state, and a $\mathbb{Z}_{2}$ translation symmetry breaking phase as
$m\to\infty$ where the atoms are alternatively excited to the Rydberg state.
The phase transition in between is of the Ising universality [42, 39].

A close interplay between the PXP models and the LGT becomes manifest thanks
to joint efforts from the condensed matter/cold atom physics and the high
energy physics communities [19, 20]. On one hand, the PXP models can be
reformulated as LGTs, enabling application of gauge theory concepts to the
understanding of the former. On the other hand, LGTs of interest can be recast
into the PXP models, allowing experimental simulations via the Rydberg
blockade. This is particularly interesting when one realizes that confinement,
a phenomenon from gauge theory that is of great significance yet hard to
directly measure, can be approached on a tabletop setting via this mapping
[43, 44]. The two phases mentioned above, then, correspond to confined and
deconfined phases in the LGT language.

A finer model including longer-range interactions between Rydberg atoms reads

$\hat{H}=\sum_{i}\hat{X}_{i}-m\sum_{i}\hat{Z}_{i}+V_{0}\sum_{i<j}\frac{\hat{n}_{i}\hat{n}_{j}}{|r_{i}-r_{j}|^{6}},$
(2)

where $\hat{n}_{i}=(\hat{Z}_{i}+1)/2$ is the Rydberg state number operator at
site $i$ and one includes the $1/r^{6}$ van der Waals interaction. Throughout
the paper we set the lattice spacing to be 1 so $r_{i}=i$. The typical energy
scale $V_{0}$ can be translated to a typical length scale via
$r_{\mathrm{b}}=(V_{0}/2)^{1/6}$, known as the Rydberg blockade radius. This
length scale determines the range within which two atoms can not be
simultaneously excited. Eq. (2) reduces to the PXP model in Eq. (1) for
$r_{\mathrm{b}}\simeq 1$. Focusing on the case where $r_{\mathrm{b}}$
interpolates from 1 to 2, we keep only the next-nearest-neighbor (NNN)
interaction and the model simplifies to

$\hat{H}=\sum_{i}\hat{P}_{i-1}\hat{X}_{i}\hat{P}_{i+1}-m\sum_{i}\hat{Z}_{i}+V\sum_{i}\hat{n}_{i}\hat{n}_{i+2}.$
(3)

This PXP–$V$ model has been extensively studied in Refs. [45, 46, 47, 48];
also see Refs. [49, 50, 51] for relevant discussions. Three major phases of
disordered, $\mathbb{Z}_{2}$, and $\mathbb{Z}_{3}$ translation symmetry
breaking, together with small floating phases have been identified.

In this work, we generalize the PXP–LGT identification to the case with a
tunable blockade radius, or the PXP–$V$ model. We focus on two particular
parameter regimes. First, we ask how a finite $V$ modifies the confinement and
deconfinement behavior of the system. While the original protocol introduced
in Ref. [43] does not seem to give consistent results for finite $V$, closer
inspection shows that the underlying cause is kinetic instead of dynamic. We
therefore propose a new protocol to probe the confinement–deconfinement
transition that works for finite as well as vanishing $V$. Second, we zero in
on the limit $V\to\infty$, $|m|\ll V$, where the model reduces to that of
PPXPP, defined later in Eq. (11). Given the different nature of constraint, we
introduce a novel LGT that naturally incorporates the constraint and again
boasts confined and deconfined phases. In both cases, we perform numerical
simulations to support our analysis, and these proposals are readily carried
out experimentally in Rydberg quantum simulators.

##  II Gauge theory perspective of the PXP–$V$ model

Figure 1: Schematic of the mapping between the PXP–$V$ model and the LGTs,
where filled circles represent particles and empty circles represent holes.
(a) When the NNN interaction is small, one obtains a modified lattice
Schwinger model. (b) When the NNN interaction is strong enough, the Rydberg
blockade radius is effectively 2 and one gets a novel LGT.

In this section we discuss the behavior of the system from the perspective of
gauge theory with the Rydberg blockage radius interpolating from 1 to 2. As
mentioned above, this can be described by the PXP–$V$ model, Eq. (3). Before
studying the effect of the NNN interaction, we first briefly review the PXP
model, the lattice Schwinger model and the relation thereof. The PXP
Hamiltonian dictates that two nearest-neighbor atoms can not be simultaneously
excited to the Rydberg state, denoted by spin-up. In previous works, it is
proven that the PXP model is equivalent to the U(1) LGT,

$\displaystyle\hat{H}_{\mathrm{LGT}}=\sum_{l}\left[\hat{S}_{l,l+1}^{+}\hat{f}_{l}\hat{f}_{l+1}+h.c.+m\hat{n}_{l}^{f}\right]$
(4)

in a particular subspace to be clarified shortly [43, 44]. Note that we now
index the matter field fermions with $l$ and
$\hat{n}_{l}^{f}=\hat{f}_{l}^{\dagger}\hat{f}_{l}$, and the gauge field spins
are indexed with $(l,l+1)$, and $\hat{S}^{z}=\hat{Z}/2$. This U(1) LGT is
nothing but the lattice Schwinger model. The action of the theta term in
$1+1$D continuous quantum electrodynamics (QED), viz. the Schwinger model,
reads

$\displaystyle S_{\theta}=\int d^{2}x\frac{\theta}{2\pi}F_{01}.$ (5)

Here $E=F_{01}$ is the $1+1$D electric field and the $\theta$ angle can be
regarded as a background electric field, which affects the quantization of the
electric field by $E=M+\frac{\theta}{2\pi}$ where $M$ is an integer. In the
lattice model, the value of the electric field is $\pm 1/2$ which means that
the topological angle $\theta$ for this model is $\pi$. The lattice Schwinger
model possesses local symmetries — for each site, the transformation
$\hat{f}_{l}\rightarrow e^{i\phi_{l}}\hat{f}_{l}$ with
$\hat{S}^{+}_{l-1,l}\rightarrow e^{-i\phi_{l}}\hat{S}^{+}_{l-1,l}$ and
$\hat{S}^{+}_{l,l+1}\rightarrow e^{-i\phi_{l}}\hat{S}^{+}_{l,l+1}$ leaves the
Hamiltonian invariant. As a result, one can define a local conserved quantity
at each site,

$\displaystyle\hat{G}_{l}=\hat{S}_{l-1,l}^{z}+\hat{S}_{l,l+1}^{z}+\hat{n}^{f}_{l},$
(6)

which is the Gauss’ law.

When $\hat{G}_{l}|\cdot\rangle=0$ for all $l$, the PXP model is equivalent to
the lattice Schwinger model. This can be seen intuitively from the Fig. 1(a).
For example, when the spin between site $l$ and site $l+1$ is spin-up, the
matter fields in site $l$ and site $l+1$ are empty according to
$\hat{G}_{l}|\cdot\rangle=0$. Then, because the matter field can not be
annihilated from zero, it is no longer possible to flip the spins
$\hat{S}_{l-1,l}$ and $\hat{S}_{l+1,l+2}$ to spin-up, which is the Rydberg
blockade effect in the PXP model.

Figure 2: (a) The phase diagram of the lattice Schwinger model. $\theta$ is
the topological angle which is defined by the quantization of the electric
field (see main text for more information). (b) Part of the phase diagram of
the PXP–$V$ model. (c) The protocol we propose to probe the
confinement–deconfinement transition in the PXP–$V$ model.

In the previous work [43], one of the authors studied the PXP model from the
gauge theory description by studying the phase diagram of the LGT which is
shown in the Fig. 2(a) [52]. The dashed line represents the deconfined phase,
and the complementary region is the confined phase. Roughly speaking, in the
confined phase pulling away two particles costs an energy proportional to the
distance in between and therefore not possible, while in the deconfined phase
one is able to separate two particles with a finite energy cost. The protocol
proposed to probe the confinement–deconfinement dichotomy is as follows. One
first prepares a pair of fermions/holes by flipping one particular spin
starting from the ground state. For example, when the ground state is that all
the spins are spin-down and all the matter sites are filled, we flip a spin to
spin-up as in Fig. 1(a) and then according to the Gauss’ law we create a pair
of holes next to the flipped spin. Then we use the correlation function

$\mathcal{G}(r,t)=\sum_{l}\braket{\Psi(t)}{(\hat{n}^{f}_{l}-\bar{n}^{f}_{l})(\hat{n}^{f}_{l+r}-\bar{n}^{f}_{l+r})}{\Psi(t)}$
(7)

to characterize the distance between the two fermions/holes we added. Here
$|\Psi(t)\rangle$ is the wavefunction after a spin is flipped and
$\bar{n}_{l}$ denotes the mean particle number at site $l$ in the ground state
before the spin is flipped. If the system is in the confined phase, the
distance between these two fermions/holes is always 1; therefore the peak of
the correlation function $\mathcal{G}(r,t)$ is always at $r=1$ for any $t$. If
the system is in the deconfined phase, these two fermions/holes are able to
get apart from each other; therefore the correlation function
$\mathcal{G}(r,t)$ becomes distributed on the whole system as time goes by. As
$m$ goes from $-\infty$ to $\infty$, Ref. [43] finds that $\mathcal{G}(r,t)$
changes from being localized to being extended. One thus establishes an
identification between the confinement–deconfinement transition and the
$\mathbb{Z}_{2}$ transition, and gets an experimental access to the former.

Now we wish to apply this scheme to the PXP–$V$ model, with the phase diagram
in focus here shown in Fig. 2(b). We first note that the gauge invariance is
intact with the NNN term since it only involves $\hat{Z}$ operators. If one
uses the aforementioned protocol literatim, however, numerical results do not
seem to exhibit deconfinement behavior where they are expected to, see Fig.
3(a, b). Here we set $V=1.5$ and $m=-1,4$ in the two subfigures. These two
parameter choices lie in the disordered and ordered phases, respectively (cf.
Fig. 2(b)), and therefore the corresponding $\mathcal{G}(r,t)$ are expected to
be localized and extended, contrary to the numerical results where both are
localized.

Figure 3: (a, b) The correlation function $\mathcal{G}(r,t)$ with the initial
wavefunction prepared by the original protocol. (c, d) The same function
obtained from the new protocol. We set $V=1.5,m=-1$ in (a, c) and $V=1.5,m=4$
in (b, d). (e, f) The averaged correlation function $\overline{\mathcal{G}}$
(see main text for definition), with the location of phase transition marked
by the dashed lines, obtained from calculating the two-point correlation
function. We take $N=18$ and periodic boundary conditions in all calculations.

The key insight to solve this conundrum is the following identity

$\begin{split}&\hat{n}_{l}^{f}\hat{n}_{l+1}^{f}\\\
={}&(-\hat{S}_{l-1,l}^{z}-\hat{S}_{l,l+1}^{z})(-\hat{S}_{l,l+1}^{z}-\hat{S}_{l+1,l+2}^{z})\\\
={}&\hat{n}_{l-1,l}\hat{n}_{l+1,l+2}-\hat{n}_{l-1,l}-\hat{n}_{l,l+1}-\hat{n}_{l+1,l+2},\end{split}$
(8)

where $\hat{n}^{f}$ denotes matter field fermion number operator and $\hat{n}$
denotes Rydberg atom number operator as usual. The second step follows from
$\hat{n}_{l-1,l}\hat{n}_{l,l+1}=\hat{n}_{l,l+1}\hat{n}_{l+1,l+2}=0$. Using
this, we have the Hamiltonian in the LGT language,

$\begin{split}\hat{H}_{\mathrm{LGT-V}}=&\sum_{l}\Big{[}\hat{S}_{l,l+1}^{+}\hat{f}_{l}\hat{f}_{l+1}+h.c.+\Big{(}m-\frac{3V}{2}\Big{)}\hat{n}_{l}^{f}\\\
&+V\hat{n}_{l}^{f}\hat{n}_{l+1}^{f}\Big{]}.\end{split}$ (9)

Therefore, an NNN Rydberg interaction corresponds to a _nearest-neighbor_
interaction between the matter field particles, modulo terms that can be
absorbed in the detuning. When one prepares a nearest-neighbor fermion/hole
pair from flipping a spin, the new state has an interaction energy difference
of $\pm V$. Energy conservation then forbids the pair to deconfine simply
because particles far apart do not have the same interaction energy. Put
another way, the disappearance of deconfinement behavior noted above is rooted
in kinetic, instead of dynamic, rationale. By this we are referring to the
difference between the local interaction energy of $V$ and the proportional-
to-distance confining energy.

The analysis above immediately suggests a cure to the original protocol, i.e.,
to create a non-nearest-neighbor fermion/hole pair when preparing the initial
state. It turns out, as easily verified, that in all allowed cases the number
of consecutive holes must be even. Hence a minimal choice is to create a next-
next-nearest-neighbor fermion/hole pair. In the Rydberg/spin language, one can
flip spins at sites $l,l+2,l+1$, in this order, see Fig. 2(c). Note that spin
flippings at neighboring sites do not commute.

We perform numerical simulations of this protocol, vis-à-vis the previous one
for comparison. Two typical spacetime dependences of $\mathcal{G}(r,t)$ are
shown in Fig. 3(c, d). For $m$ smaller than the critical value,
$\mathcal{G}(r,t)$ remains localized as time evolves, while $\mathcal{G}(r,t)$
quickly spreads to the whole system range for large $m$. Note that for large
enough systems we expect a light-cone-like behavior, and the reflections seen
here are due to the finite sizes available to numeric simulations. To reduce
this effect and distill a quantitative measure for the
confinement–deconfinement transition, we introduce an averaged correlation
function

$\overline{\mathcal{G}}=\frac{\langle\mathcal{G}(r,t)\rangle_{r>4,t}}{\langle\mathcal{G}(r=1,t)\rangle_{t}}.$
(10)

That is, we average $\mathcal{G}(r,t)$ over $r>4$ and $t$ and normalize it
against $\mathcal{G}(r=1,t)$ averaged over $t$. The averaged correlation
functions from both protocols are shown in Fig. 3(e, f), where we also mark
the phase transition points by vertical dashed lines, the locations of which
pinpointed from the conventional two-point function. The inability of the
original protocol to detect the deconfinement behavior with NNN interaction is
clear from Fig. 3(e), where apart from the $V=0$ case $\overline{\mathcal{G}}$
remains small across the whole parameter range. With the new protocol, on the
other hand, we see in Fig. 3(f) that the onset of deconfinement matches the
phase transition point perfectly well. These results thus lend full support to
our analysis.

Finally, we note that the Hamiltonian Eq. (9) provides an explanation of how
the phase boundary tilts in Fig. 2(b). The idea is that part of the detuning
term can be combined with the NNN interaction term to give an equivalent
nearest-neighbor matter field interaction, hence effectively reduces $m$ by an
amount $\sim 3V/2$. One therefore needs to go to a larger critical $m_{c}$ to
arrive at the phase transition. The predicted $\Delta m_{c}\sim 3\Delta V/2$
becomes exact for large $V$ [45]. This perspective complements the
Rydberg/spin model interpretation that a finite positive $V$ costs an extra
energy for the staggering spin configuration and therefore prefers the
disordered state at a given $m$.

##  III Gauge theory description of the PPXPP model

In increasing $V$, a new phase, namely the $\mathbb{Z}_{3}$ ordered phase
comes out, while the $\mathbb{Z}_{2}$ phase is pushed to $m=\infty$. It turns
out that the correlation function $\mathcal{G}(r,t)$ fails to capture this new
piece of physics. To make progress, we focus on the $V\to\infty$ limit where
the PXP–$V$ model reduces to the PPXPP model,

$\displaystyle\hat{H}_{\mathrm{PPXPP}}=\sum_{i}\hat{P}_{i-2}\hat{P}_{i-1}\hat{X}_{i}\hat{P}_{i+1}\hat{P}_{i+2}-m\sum_{i}\hat{Z}_{i}.$
(11)

That is, any two atoms within a range of 3 can not be excited to the Rydberg
state simultaneously. The two limiting cases are easy to understand. When
$m\to-\infty$, all the atoms are in the ground state and we are in the
disordered phase. When $m\to\infty$, the translational symmetry is
spontaneously broken to a $\mathbb{Z}_{3}$ phase. There also exists a small
critical, incommensurate floating phase [47], but we will focus on the two
major phases in this section.

Figure 4: (a, b) Schematic for creation of particles/holes in the LGT
corresponding to the PPXPP model in the $m\to\pm\infty$ limits. (c, d) The
correlation function $\mathcal{G}(r,t)$. We set $m=-1$ in (c) and $m=4$ in
(c). (e) The averaged correlation function $\overline{\mathcal{G}}$. The very
narrow floating phase is plotted in grey, with its width approximately equal
to the linewidth. We take $N=24$ and periodic boundary conditions in all
calculations.

We wish to similarly relate this model to an LGT. Apparently, the lattice
Schwinger model fails to accommodate the constraint, but the same philosophy
still works. We consider the following LGT,

$\displaystyle\hat{H}_{\mathrm{LGT}}^{\mathrm{PPXPP}}=\sum_{l}\left[\hat{S}_{l}^{+}\hat{b}_{l-1}\hat{b}_{l}\hat{b}_{l+1}+h.c.+\frac{2}{3}m\hat{n}_{l}^{b}\right],$
(12)

which is equivalent to the PPXPP Hamiltonian in a particular subspace which
will be clarified as follows. As shown in Fig. 1(b), now the matter fields and
the gauge fields are living on the same lattice sites. For each site, there is
a spin with a hard-core boson mode. The Hamiltonian Eq. (12) possesses a local
gauge symmetry that $\hat{b}_{l}\rightarrow\hat{b}_{l}e^{i\phi_{l}}$ together
with $\hat{S}_{l-1}^{+}\rightarrow\hat{S}_{l-1}^{+}e^{-i\phi_{l}}$,
$\hat{S}_{l}^{+}\rightarrow\hat{S}_{l}^{+}e^{-i\phi_{l}}$ and
$\hat{S}_{l+1}^{+}\rightarrow\hat{S}_{l+1}^{+}e^{-i\phi_{l}}$ leaves the
Hamiltonian invariant. The conserved quantity corresponding to this local
symmetry is

$\displaystyle\hat{G}_{l}^{\mathrm{PPXPP}}=\hat{S}_{i-1}^{z}+\hat{S}_{i}^{z}+\hat{S}_{i+1}^{z}+\hat{n}_{i}^{b}+\frac{1}{2}.$
(13)

This lattice gauge model is equivalent to the PPXPP model when
$\hat{G}_{l}^{\textrm{PPXPP}}|\cdot\rangle=0$ for all $l$. Once a spin at site
$i$ is flipped from spin-down to spin-up, three bosons are annihilated at site
$i-1$, $i$ and $i+1$. Then the spins at site $i-2$, $i-1$, $i+1$ and $i+2$ can
not be flipped to spin up any more, automatically satisfying the constraint in
the PPXPP model. We note that the trilinear form of the matter field dictates
use of bosons instead of fermions as prescribed by the superselection rule.

As mentioned before, there are two phases in the PPXPP model in the two limits
of $m$. To manifest the corresponding LGT content, we again apply the scheme
introduced in Ref. [43]. In this case it turns out unnecessary to modify the
original protocol, and flipping one single spin suffices. See Fig. 4(a, b) for
an illustration of this process in the $m\to\pm\infty$ limits, where we add
three holes/particles into the system simultaneously by flipping a spin. The
numerical results of $\mathcal{G}(r,t)$, also defined with Eq. (7) albeit with
redefined matter fields, are shown in Fig. 4(c, d). This correlation function
also captures the distance between the holes/particles we add. Again, we see a
clear distinction between confinement and deconfinement behavior as $m$
changes: for $m=-1$ (Fig. 4(c)), $\mathcal{G}(r,t)$ remains localized at $r=1$
for all $t$, while for $m=4$ (Fig. 4(d)), $\mathcal{G}(r,t)$ becomes
distributed over the whole system. Similar to the PXP–$V$ case, we obtain a
single indicator from $\mathcal{G}(r,t)$ by defining an averaged correlation
function

$\overline{\mathcal{G}}=\frac{\langle\mathcal{G}(r=11,t)\rangle_{t}}{\langle\mathcal{G}(r=1,t)\rangle_{t}},$
(14)

averaged only over time now which suffices for our purpose. Here $r=11$ is the
maximal $r$ with non-trivial signals, see Fig. 4(d). Nevertheless, in
experiments we expect any reasonably large $r$ to work as well. The
correlation function $\overline{\mathcal{G}}$ is plotted in Fig. 4(e). A very
narrow floating phase [47], with its width approximately equal to that of the
grey vertical line, separates the disordered and the $\mathbb{Z}_{3}$ phases.
We see a consistent alignment between the onset of deconfinement behavior and
the entrance to the ordered phase. This indicates that the disordered
($\mathbb{Z}_{3}$ ordered) phase can be identified with the (de)confined
phase, a result similar to the PXP(–$V$) case.

##  IV Conclusion

In this work, based on previous works relating the PXP model describing a
Rydberg atom chain with an LGT, we consider systems with a tunable Rydberg
blockade radius. Such a system is conveniently modelled by the PXP–$V$ model,
a subject extensively studied in the literature. We focus on two particular
parameter regimes, one with small finite $V$ added to the PXP model and the
other in the limit of the PPXPP model. In the first scenario we show how to
modify the previously proposed protocol to incorporate the finite $V$ effect
by elucidating the rationale of the latter. With the new protocol the
confinement–deconfinement transition again becomes manifest. A gauge theoretic
explanation of the phase boundary is also discussed, predicting a shifting of
critical detuning of $\Delta m_{c}\sim 3\Delta V/2$. In the second scenario a
novel LGT is formulated to automatically satisfy the PPXPP constraint. We
establish an identification of the disordered ($\mathbb{Z}_{3}$ ordered) phase
and the (de)confined phase. Both analyses are corroborated by numerical
calculations and are expected to be of immediate experimental relevance.

While in Ref. [45] the authors propose a direct chiral transition from the
$\mathbb{Z}_{3}$ phase to the disordered phase in the parameter regime above
the integrable Potts point, the authors of Ref. [48], based on extensive
numerical calculation, argue that an intermediate floating phase exists in
between the two phases for large enough $V$. Calculations in Ref. [47] suggest
that this floating phase extends all the way to the $V\to\infty$ limit, the
transition in focus here in the PPXPP model. A relation between this phase and
the gauge theory hailed here is left for future investigation.

## Acknowledgement

We thank Hui Zhai and Shang Liu for helpful discussions. The project is
supported by NSFC under Grant No. 12204034 and Grant No. 11874083, Fundamental
Research Funds for the Central Universities (No.FRF-TP-22-101A1) and China
Postdoctoral Science Foundation (Grant No. 2022M711868). C. L. is also
supported by the International Postdoctoral Exchange Fellowship Program and
the Shuimu Tsinghua Scholar Program.

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