# Quantum spin ice in three-dimensional Rydberg atom arrays

Jeet Shah  Joint Quantum Institute, Department of Physics, University of
Maryland, College Park, MD 20742, USA  Condensed Matter Theory Center,
Department of Physics, University of Maryland, College Park, MD 20742, USA
Gautam Nambiar  Joint Quantum Institute, Department of Physics, University of
Maryland, College Park, MD 20742, USA  Alexey V. Gorshkov  Joint Quantum
Institute, Department of Physics, University of Maryland, College Park, MD
20742, USA  Joint Center for Quantum Information and Computer Science,
NIST/University of Maryland, College Park, MD 20742, USA  Victor Galitski
Joint Quantum Institute, Department of Physics, University of Maryland,
College Park, MD 20742, USA  Center for Computational Quantum Physics, The
Flatiron Institute, New York, NY 10010, USA

(August 28, 2024)

###### Abstract

Quantum spin liquids are exotic phases of matter whose low-energy physics is
described as the deconfined phase of an emergent gauge theory. With recent
theory proposals and an experiment showing preliminary signs of
$\mathbb{Z}_{2}$ topological order [G. Semeghini et al., Science 374, 1242
(2021)], Rydberg atom arrays have emerged as a promising platform to realize a
quantum spin liquid. In this work, we propose a way to realize a $U(1)$
quantum spin liquid in three spatial dimensions, described by the deconfined
phase of $U(1)$ gauge theory in a pyrochlore lattice Rydberg atom array. We
study the ground state phase diagram of the proposed Rydberg system as a
function of experimentally relevant parameters. Within our calculation, we
find that by tuning the Rabi frequency, one can access both the confinement-
deconfinement transition driven by a proliferation of “magnetic” monopoles and
the Higgs transition driven by a proliferation of “electric” charges of the
emergent gauge theory. We suggest experimental probes for distinguishing the
deconfined phase from ordered phases. This work serves as a proposal to access
a confinement-deconfinement transition in three spatial dimensions on a
Rydberg-based quantum simulator.

##  I Introduction

When the classical part of a many-body Hamiltonian is frustrated, quantum
fluctuations can break the degeneracy in interesting ways. An exotic form of
such breaking was pointed out by Anderson [1] where the ground state is a
superposition of several almost-degenerate states, and the excitations are
“fractional”[2]. Broadly, a common feature tying together such systems called
quantum spin liquids is that, at low energies, they can be described as lying
in a deconfined phase of an emergent gauge theory. The fractional excitations
are the “charge”-like and “flux/monopole”-like excitations of this gauge
theory. When these fractional excitations get confined, they cease to be
important for the low-energy physics, and the system becomes ordered. From
this point of view, transitions from a spin liquid to conventional ordered
phases are understood as a confinement-deconfinement transition, driven by a
proliferation of “flux/monopole”-like excitations, or a Higgs transition,
driven by a proliferation of “charge”-like excitations [3, 4, 5, 6]. Gauge
theories and their phase transitions are of fundamental importance in physics
[7, 8, 9, 10]. The prospect of this physics emerging in many-body systems
provides an important motivation for studying quantum spin liquids. They are
also interesting due to their possible role in the physics of strongly
correlated materials [11] and possible application in quantum computing [12,
13].

Traditionally, the main search space for spin liquids has comprised of solid
state systems. While consistent progress has been made [2, 14], conclusive
evidence for spin liquids is still lacking in these systems. One reason is
that the same feature that makes spin liquids interesting—being characterized
by non-local order parameters—also makes them hard to detect. Meanwhile, over
the past decade, Rydberg atom arrays have emerged as a promising platform for
engineering interacting Hamiltonians [15, 16, 17, 18, 19, 20, 21, 22, 23, 24,
25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. Rydberg states have large
principal quantum number $n$ $(\sim 20-100)$, and the van der Waals
interaction between them scales as $n^{11}$. The strong tunable interactions,
along with the ability to customize the lattice of atoms, locally control
qubits, and take wavefunction snapshots, make Rydberg atom arrays a
competitive platform to explore quantum many-body physics. Following theory
proposals [37, 38], promising signs of $\mathbb{Z}_{2}$ topological order have
been observed experimentally on this platform [26]. This has sparked a lot of
activity over the past two years in the general direction of proposing ways to
realize exotic states on quantum devices using analogue quantum simulation
[39, 40, 41, 42], digital quantum simulation [43], and projective measurements
[44, 45].

Our work is a proposal for realizing a $U(1)$ quantum spin liquid, described
by the deconfined phase of a compact $U(1)$ gauge theory on three-dimensional
Rydberg atom arrays, with an eye towards accessing the confinement-
deconfinement transition. It was shown by Polyakov [46, 47] that compact
$U(1)$ gauge theory in 2+1 dimensions is always in the confined phase in the
thermodynamic limit due to a proliferation of monopole events. Therefore we
turn to 3+1 dimensions, where Polyakov argued [47] for the existence of both
deconfined and confined phases separated by a transition driven by monopole
excitations. The deconfined phase consists of gapless “photons”, gapped
“monopoles” and gapped “charge” excitations. In the early 2000s, lattice
models of spins [48] and dimers [49] on corner-sharing polyhedra were
constructed that were strongly argued to realize this phase—a $U(1)$ spin
liquid, using perturbation theory, solvable limits [48] and later Quantum
Monte Carlo simulations [50, 51]. Our work is based on a spin model with easy-
axis antiferromagnetic interactions introduced by Hermele et al. [48] on the
pyrochlore lattice consisting of corner-sharing tetrahedra (see Fig. 1).

Figure 1: (a) The pyrochlore lattice. White circles denote atoms in the ground
state, while black circles denote atoms in the Rydberg state. The
configuration shown satisfies
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=2$
on each tetrahedron. The label $\bm{x}$ is used to denote the sites of the
pyrochlore lattice. (b) The diamond lattice. It is the bipartite lattice
formed by the centers of the tetrahedra marked by green (A sublattice) and
blue (B sublattice) dots. $\mathbf{e}_{\mu}$ for $\mu\in\\{0,1,2,3\\}$ label
the vectors joining an A site to its neighboring B sites. The label
$\mathbf{r}$ is used to denote the sites of the diamond lattice. (c) The red
links are the edges of the lattice dual to the diamond lattice shown in (b).
This lattice is also a diamond lattice, and we refer to it as the “dual
diamond lattice” in this paper to distinguish it from the “diamond lattice” in
(b). The sites of the dual diamond lattice are centers of the “polyhedra”
formed by four puckered hexagons of the diamond lattice.
$\bm{\mathsf{u}}_{\mu}$ for $\mu\in\\{0,1,2,3\\}$ label the vectors joining an
A site to its neighboring B sites on the dual diamond lattice. The label
$\bm{\mathsf{r}}$ [notice the difference in the font as compared to
$\mathbf{r}$ in (b)] is used to denote the sites of the dual diamond lattice.

The classical Ising limit of this model is the widely studied classical spin
ice [52, 53, 54, 55, 56], which has a large residual entropy at low
temperatures similar to water-ice [57]. This is because the ground states form
an exponentially degenerate set of states obeying the “ice rule” (see Sec.
II). The quantum model in Ref. [48] has also been a subject of intense study
in the context of pyrochlore materials like
$\text{Yb}_{2}\text{Ti}_{2}\text{O}_{7}$ and
$\text{Er}_{2}\text{Ti}_{2}\text{O}_{7}$ as potential quantum spin ice
(another name for the $U(1)$ spin liquid) candidates [58].

It was observed in Ref. [59] that the Hamiltonian in Ref. [48] can be viewed
as that of hard-core bosons hopping on an optical lattice with nearest-
neighbor repulsion, thus extending its relevance to the cold atom setting.
Ref. [60] studied a similar model of hard-core bosons hopping on a two-
dimensional checkerboard lattice. In Ref. [60], the atom’s internal state was
largely the ground state, but a dressing with Rydberg states was used to
engineer interactions between atoms. Later, Ref. [61] showed that dimer models
in two dimensions can be implemented on configurable Rydberg arrays—where the
atoms themselves are stationary but can internally be either in a ground state
or in a Rydberg state. In this setting, the atoms are driven with a laser (or
a pair of lasers making a two-photon transition) that is detuned from the
ground to Rydberg transition. The Rydberg interactions and the detuning define
a (frustrated) “classical” energy landscape. The laser driving induces quantum
fluctuations controlled by the Rabi frequency, leading (perturbatively) to
dimer moves or ring exchange terms that are required to deconfine a gauge
theory. The proposal [37] and experiment [26] mentioned above worked in the
same setting. Our work is also based on this setting in which the atom array
is configured in a 3D pyrochlore lattice.

In Sec. II, we explain our proposal. We show that within a window of laser
detunings, the classical landscape is identical to the set of ice rule obeying
states. Our Hamiltonian, when restricted to nearest-neighbor interactions, is
equivalent to the transverse-field Ising model on the pyrochlore lattice. In
the limit of small Rabi frequencies, it is perturbatively equivalent to the
model in [48], which was argued to have a spin liquid ground state. Away from
the perturbative limit, there is numerical evidence for a spin liquid phase
[62]. However, once we include the long-range $1/r^{6}$ interactions beyond
nearest-neighbor, the classical landscape is no longer degenerate, and it is a
priori unclear if the spin liquid survives as the ground state. We attempt to
answer this in Sec. III by comparing the energy of an ansatz wave function of
the spin liquid with that of an ordered state. Within our approximation, we
find that by dialing up the Rabi frequency, for fixed detuning and interaction
strength, one goes through a confinement-deconfinement transition from an ice
rule obeying ferromagnetic state into a deconfined spin liquid phase. Then, by
further increasing the Rabi frequency, one goes through a Higgs transition
from the spin liquid to a transverse-field-polarized state (see Sec. III.2).
While the analysis till this point focuses on the ground state, in Sec. III.3,
we comment on the role played by dynamical state-preparation in deciding the
nature of the state prepared in experiment. In Sec. IV, we present correlation
functions that distinguish the spin liquid from the confined phases, and
provide experimental protocols for measuring them. Finally, in Sec. V, we
present general discussions and conclusions.

##  II Proposal to realize a $U(1)$ quantum spin liquid using Rydberg atoms

In this section, we describe our proposal to realize a $U(1)$ Quantum Spin
Liquid (QSL) in Rydberg atom arrays. Consider a 3D Rydberg array in which the
atoms are positioned on the sites of the pyrochlore lattice [see Fig. 1(a)].
Each of the atoms can either be in the ground state $\ket{g}$ or in the
Rydberg state $\ket{r}$. In the rotating wave approximation and in a rotating
frame, the Hamiltonian is

$\displaystyle\hat{H}_{\text{ryd}}=$
$\displaystyle-\delta\sum_{i}\hat{n}_{i}+\frac{V}{2}\sum_{i\neq
j}\left(\frac{a}{|\bm{x}_{i}-\bm{x}_{j}|}\right)^{6}\hat{n}_{i}\hat{n}_{j}$
(1)
$\displaystyle+\frac{\Omega}{2}\sum_{i}(\hat{b}_{i}+\hat{b}_{i}^{\dagger}),$

where $\hat{b}_{i}=\ket{g_{i}}\bra{r_{i}}$,
$\hat{n}_{i}=\hat{b}_{i}^{\dagger}\hat{b}_{i}$, $\Omega$ is the Rabi
frequency, $\delta$ is the laser detuning, $V$ is the nearest-neighbor van der
Waals interaction strength, and $a$ is the distance between two neighboring
atoms. The summation $\sum_{i\neq j}$ is over distinct sites $i$ and $j$ of
the pyrochlore lattice (each pair is being counted twice), and $\sum_{i}$ is
over sites $i$. Below, we briefly describe the pyrochlore lattice.

The pyrochlore lattice is a face-centred cubic (FCC) lattice with a four-site
basis formed by the four vertices of an up-pointing tetrahedron. (Since each
lattice site belongs to one up-pointing tetrahedron and one down-pointing
tetrahedron, the down-pointing tetrahedra are formed automatically once we
create the up-pointing tetrahedra.) In Cartesian coordinates, the primitive
vectors of the FCC lattice are

$\displaystyle\mathbf{a}_{1}$ $\displaystyle=\sqrt{2}a(0,1,1),$ (2)
$\displaystyle\mathbf{a}_{2}$ $\displaystyle=\sqrt{2}a(1,0,1),$
$\displaystyle\mathbf{a}_{3}$ $\displaystyle=\sqrt{2}a(1,1,0).$

The pyrochlore lattice sites are physically located at
$\mathbf{r}+\mathbf{e}_{\mu}/2$ [and labeled $(\mathbf{r},\mu)$], where
$\mathbf{r}$ is an FCC lattice vector, and the vectors $\mathbf{e}_{\mu}$ for
$\mu\in\\{0,1,2,3\\}$ are defined as [see Fig. 1(b)]

$\displaystyle\mathbf{e}_{0}$
$\displaystyle=\frac{a}{\sqrt{2}}(1,1,1)=\frac{1}{4}(\mathbf{a}_{1}+\mathbf{a}_{2}+\mathbf{a}_{3}),$
(3) $\displaystyle\mathbf{e}_{1}$ $\displaystyle=\frac{a}{\sqrt{2}}(1,-1,-1),$
$\displaystyle\mathbf{e}_{2}$ $\displaystyle=\frac{a}{\sqrt{2}}(-1,1,-1),$
$\displaystyle\mathbf{e}_{3}$ $\displaystyle=\frac{a}{\sqrt{2}}(-1,-1,1).$

We map the two levels of the atoms to spins-1/2s:
$\ket{g}\rightarrow\ket{\downarrow}$, $\ket{r}\rightarrow\ket{\uparrow}$,
$\hat{n}_{i}\rightarrow\hat{S}_{i}^{z}+1/2$ and
$\hat{b}_{i}+\hat{b}_{i}^{\dagger}\rightarrow 2\hat{S}_{i}^{x}$. The term
$\hat{n}_{i}\hat{n}_{j}$ therefore maps to an $\hat{S}_{i}^{z}\hat{S}_{j}^{z}$
interaction in addition to a Zeeman term $\hat{S}_{i}^{z}$. Written in terms
of spins, the Hamiltonian, up to an additive constant, is

$\displaystyle\hat{H}_{\text{ryd}}=$
$\displaystyle-h\sum_{i}\hat{S}_{i}^{z}+\frac{V}{2}\sum_{i\neq
j}\left(\frac{a}{|\bm{x}_{i}-\bm{x}_{j}|}\right)^{6}\hat{S}_{i}^{z}\hat{S}_{j}^{z}$
(4) $\displaystyle+\Omega\sum_{i}\hat{S}_{i}^{x},$

where

$\displaystyle h=\delta-\frac{V}{2}\sum_{i\neq
0}\left(\frac{a}{|\bm{x}_{i}-\bm{x}_{0}|}\right)^{6},$ (5)

and is independent of the choice of $\bm{x}_{0}$ for an infinite lattice.
Evaluating this sum numerically for the pyrochlore lattice, we obtain
$h=\delta-3.46V$. It is useful to separate the total Hamiltonian, Eq. (4),
into three parts,
$\hat{H}_{\text{ryd}}=\hat{H}_{0}+\hat{H}_{\Omega}+\hat{H}_{\text{LR}}$, where

$\displaystyle\hat{H}_{0}=$ $\displaystyle\frac{V}{2}\sum_{\langle
i,j\rangle}\hat{S}_{i}^{z}\hat{S}_{j}^{z}-h\sum_{i}\hat{S}_{i}^{z},$ (6)
$\displaystyle\hat{H}_{\Omega}=$
$\displaystyle\Omega\sum_{i}\hat{S}_{i}^{x},\text{ and
}\hat{H}_{\text{LR}}=\frac{V}{2}\sum_{i\neq
j}{\vphantom{\sum}}^{\prime}\left(\frac{a}{|\bm{x}_{i}-\bm{x}_{j}|}\right)^{6}\hat{S}_{i}^{z}\hat{S}_{j}^{z},$

where $\sum_{\langle i,j\rangle}$ is over nearest-neighbor pairs and
$\sum^{\prime}_{i\neq j}$ in $\hat{H}_{\text{LR}}$ is over the remaining pairs
that are not nearest-neighbor (in both $\sum$ and $\sum^{\prime}$, each pair
is counted twice). The coefficients of the second, third, and fourth nearest-
neighbor interactions are $V/27$, $V/64$, and $V/125$, respectively. Since
these are small in comparison to $V$, we will drop $\hat{H}_{\text{LR}}$ for
the rest of this section because doing so allows us to connect to some
previously known results [48, 51, 63]. We will study the effect of the long-
range van der Waals interaction $\hat{H}_{\text{LR}}$ in Sec. III.

Since the pyrochlore lattice is made of corner-sharing tetrahedra and since
all edges in a tetrahedron are of equal length, we see that $\hat{H}_{0}$ can
be written up to an additive constant as (for convenience, in the expression
below, we switch back to the hard-core boson notation)

$\hat{H}_{0}=\frac{V}{2}\sum_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}_{\mathbf{r}}}\left(\hat{n}_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}_{\mathbf{r}}}-\rho\right)^{2},$
(7)

where the sum is over all tetrahedra,
$\rho=\frac{1}{2}\left(4+\frac{h}{V}\right)=\frac{1}{2}\left(0.54+\frac{\delta}{V}\right)$,
and
$\hat{n}_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}_{\mathbf{r}}}=\sum_{i\in{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}_{\mathbf{r}}}}\hat{n}_{i}$
denotes the total number of atoms in the excited state on a given tetrahedron
$\mathchoice{\includegraphics[height=4.30554pt]{tetrahedron}}{\includegraphics[height=7.3194pt]{tetrahedron}}{\includegraphics[height=6.45831pt]{tetrahedron}}{\includegraphics[height=2.15277pt]{tetrahedron}}_{\mathbf{r}}$.
Minimizing $\hat{H}_{0}$ to obtain the classical ground state imposes a
constraint on
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}$
for each tetrahedron depending on the value of $\rho$:

$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=\begin{cases}0&\text{
if }\rho<1/2,\\\ \text{floor}\left(\rho+\frac{1}{2}\right)&\text{ if
}1/2<\rho<7/2,\\\ 4&\text{ if }7/2<\rho.\end{cases}$ (8)

The cases
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=0$
and
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=4$
are trivial, in that the classical ground state is unique. However, in the
remaining cases, namely
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=1,2,3$,
the classical ground state manifold is degenerate with exponentially (in
system size) many states in it. In the case
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=2$,
the number of configurations satisfying this constraint is approximately
$(3/2)^{N_{\text{tetrahedra}}}$ (where $N_{\text{tetrahedra}}$ is the number
of tetrahedra) [64]. This is based on an argument similar to the one given by
Pauling to explain the residual entropy of water-ice at zero temperature [57].
From now onward, we will refer to the condition
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=2$
as the “ice rule”. An ice rule obeying configuration is shown in Fig. 1(a). In
these non-trivial cases, the configurations with fixed
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}$
can be mapped to configurations of dimers on the bipartite diamond lattice
formed by the centers of tetrahedra of the pyrochlore lattice [Fig. 1(b)],
with exactly
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}$
many dimers touching each diamond site (see Fig. 2). The A and B sites of the
diamond lattice are located at $\mathbf{n}$ and $\mathbf{n}+\mathbf{e}_{0}$,
respectively, where $\mathbf{n}$ is an FCC lattice vector. For later use in
this paper, we also show the lattice dual to this diamond lattice in Fig. 1(c)
(also a diamond lattice, which we call the “dual diamond lattice”).

Figure 2: Mapping between Rydberg array configurations and dimer
configurations. A Rydberg atom (black dot) is mapped to the presence of a
dimer (orange bar), while a ground state atom (white dot) is mapped to the
absence of a dimer. (a), (b), and (c) show example dimer configurations
corresponding to
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=1$,
$2$, and $3$, respectively. In each case,
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}$
many dimers touch the center of each tetrahedron (the centers of the
tetrahedra form the diamond lattice).

An atom in the Rydberg state on site $i$ is mapped to a dimer on the
corresponding link of the diamond lattice, while an atom in the ground state
is mapped to no dimer. Such dimer models have been studied extensively in both
two and three dimensions [65, 48, 66, 67].

In the limit $\Omega\ll V$, $\hat{H}_{\Omega}$ leads to quantum fluctuations
that break the exponential degeneracy of the low-energy manifold. We will
study this effect perturbatively in the following section (Sec. II.1).
Classically, the energy gap between the degenerate ground state space and the
lowest excited states corresponding to two tetrahedra violating Eq. (8) by
either $+1$ or $-1$ is
$2V\times\text{min}\left(\\{\rho+1/2\\},1-\\{\rho+1/2\\}\right)$. Here,
$\\{x\\}\equiv x-\text{floor}(x)$ is the fractional part of $x$. It should be
noted that, in the borderline cases when $\rho=m+1/2$ with
$m\in\\{0,1,2,3\\}$, the energy gap closes and our perturbative analysis
cannot be used. We assume going forward that $\rho$ is away from these
borderline values.

###  II.1 Perturbation theory

We work in the limit $\Omega\ll V$ and treat $\hat{H}_{\Omega}$ as a
perturbation over $\hat{H}_{0}$, ignoring for now $\hat{H}_{LR}$ whose effects
will be considered later in Sec. III. We calculate the effective Hamiltonian
within the ground state manifold of $\hat{H}_{0}$ using the Schrieffer-Wolff
formulation of perturbation theory. For simplicity, we present the calculation
of the effective Hamiltonian only for
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=2$
here. The only difference between these three cases will be the Hilbert space
on which the Hamiltonian acts. Calculating, at $k^{\text{th}}$ order in
perturbation theory, the matrix element of the effective Hamiltonian between
two states $\ket{n}$ and $\ket{m}$ lying in the degenerate manifold involves
starting from $\ket{m}$, applying the perturbation $k$ times, and reaching the
state $\ket{n}$. Since $\hat{H}_{\Omega}$ changes the particle number by $\pm
1$, the corrections at all odd orders are zero. Hence, we need to consider
only the corrections at even orders.

Acting with $\frac{\Omega}{2}(\hat{b}_{i}+\hat{b}_{i}^{\dagger})$ on an ice
rule obeying state creates two excited tetrahedra (whose common site is $i$),
which violate the constraint
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=2$.
Therefore, the only second-order process that takes us back to the ice
manifold (the degenerate manifold of the ice rule obeying states) is the one
in which two excited tetrahedra are created and annihilated, as illustrated in
Figs. 3(a) and (b).

Figure 3: (a) and (b) constitute a virtual process at second order in
perturbation theory in $\Omega/V$. Starting from (a) which is a configuration
that satisfies
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=2$
on all sites, $\hat{b}_{1}+\hat{b}_{1}^{\dagger}$ is applied giving (b). To
complete the second order process, $\hat{b}_{1}+\hat{b}_{1}^{\dagger}$ is
applied to (b) giving back (a). Tetrahedra for which
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}\neq
2$ are shaded in red. Sub-figures (a)–(g) constitute a sixth-order process in
the perturbation theory that contributes to the ring exchange term in the
effective Hamiltonian, Eq. (9). Starting from (a), the perturbation
$\hat{b}_{i}+\hat{b}_{i}^{\dagger}$ is applied sequentially on sites
$i=1,2,\ldots,6$. At the end of the six steps, a configuration with
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=2$
is obtained as shown in (g). Note that the configuration of the atoms on the
hexagon is flipped in (g) as compared to (a) thereby producing the effect of a
ring exchange. Other sixth-order processes where the perturbation is not
applied sequentially also contribute to Eq. (9), but are not shown here. (h)
Ring exchange process which appears in the effective Hamiltonian Eq. (9). A
flippable configuration is mapped to the complimentary flippable
configuration.

Since such processes are present for all the states of the ice manifold, they
contribute only a constant energy shift and can be ignored. The same is true
for the fourth-order processes. Now, the pyrochlore lattice has hexagonal
plaquettes, some of which are shown in Fig. 4.

Figure 4: Shaded in red are the four nonequivalent hexagonal plaquettes of the
pyrochlore lattice.

This allows for non-trivial processes to exist at sixth order. In fact, non-
trivial ring exchange over hexagonal plaquettes of the pyrochlore lattice is
obtained by the process shown in Figs. 3(a)–(g) (some sixth-order processes
also result in a constant energy shift which we neglect). A flippable
configuration—one in which atoms on a hexagonal plaquette are alternately in
the ground and Rydberg states—is mapped to the complementary flippable
configuration by the ring exchange process as illustrated in Fig. 3(h). Thus,
the effective Hamiltonian consists of ring exchange terms:

$\hat{H}_{\text{eff}}=-J_{\text{ring}}(\rho)\sum_{\hexagon}\ket{\raisebox{-2.84544pt}{
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
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}{}\pgfsys@moveto{0.0pt}{4.26773pt}\pgfsys@moveto{1.0pt}{4.26773pt}\pgfsys@curveto{1.0pt}{4.82002pt}{0.55229pt}{5.26773pt}{0.0pt}{5.26773pt}\pgfsys@curveto{-0.55229pt}{5.26773pt}{-1.0pt}{4.82002pt}{-1.0pt}{4.26773pt}\pgfsys@curveto{-1.0pt}{3.71544pt}{-0.55229pt}{3.26773pt}{0.0pt}{3.26773pt}\pgfsys@curveto{0.55229pt}{3.26773pt}{1.0pt}{3.71544pt}{1.0pt}{4.26773pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{4.26773pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@beginscope\pgfsys@invoke{
}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{1,1,1}\pgfsys@color@gray@fill{1}\pgfsys@invoke{
}{}\pgfsys@moveto{-3.69597pt}{-2.13387pt}\pgfsys@moveto{-2.69597pt}{-2.13387pt}\pgfsys@curveto{-2.69597pt}{-1.58157pt}{-3.14368pt}{-1.13387pt}{-3.69597pt}{-1.13387pt}\pgfsys@curveto{-4.24826pt}{-1.13387pt}{-4.69597pt}{-1.58157pt}{-4.69597pt}{-2.13387pt}\pgfsys@curveto{-4.69597pt}{-2.68616pt}{-4.24826pt}{-3.13387pt}{-3.69597pt}{-3.13387pt}\pgfsys@curveto{-3.14368pt}{-3.13387pt}{-2.69597pt}{-2.68616pt}{-2.69597pt}{-2.13387pt}\pgfsys@closepath\pgfsys@moveto{-3.69597pt}{-2.13387pt}\pgfsys@fillstroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}+\text{H.c.},$ (9)

where $J_{\text{ring}}(\rho)=\gamma(\rho)\Omega^{6}/V^{5}$, the sum is over
all hexagonal plaquettes of the pyrochlore lattice, and $\gamma(\rho)$ is a
dimensionless number obtained by summing over virtual processes and is plotted
as a function of $\rho$ in Fig. 5 .

Figure 5: Plot showing the variation of $\gamma(\rho)$ (which is the
proportionality constant in
$J_{\text{ring}}(\rho)=\gamma(\rho)\Omega^{6}/V^{5}$) as a function of $\rho$.
For $\rho=0.5,1.5,2.5,$ and $3.5$, the energy gap between the low-energy and
the high-energy sectors closes and $\gamma(\rho)$ diverges.

We note that, when $\rho$ is an integer, the value of $\gamma(\rho)$ is
$63/16$ and is the same as the one appearing in Refs. [68, 69]. Although the
effective Hamiltonian was derived here assuming
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=2$,
the effective Hamiltonian we obtain for
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=1,3$
is also given by Eq. (9).

In terms of dimers on the diamond lattice, the effective Hamiltonian Eq. (9)
corresponds to a kinetic energy of the dimers. It is well known that dimer
models can be made exactly solvable by adding a potential energy $V_{RK}$ for
the dimers and tuning to a special point $V_{RK}=J_{\text{ring}}$ called the
Rokhsar-Kivelson (RK) point [65]. The Hamiltonian with such a potential energy
term takes the form

$\displaystyle\hat{H}_{\text{dimer}}=$ $\displaystyle-
J_{\text{ring}}(\rho)\sum_{\hexagon}\ket{\raisebox{-2.84544pt}{
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The Rydberg system we are interested in [Eq. (9)] is obtained from Eq. (10) by
setting $V_{RK}=0$.

###  II.2 $U(1)$ quantum spin liquid—relation to Hermele-Fisher-Balents [48]

The Hamiltonian in Eq. (10) was also derived by Hermele, Fisher and Balents in
Ref. [48] starting from the Heisenberg model on the pyrochlore lattice and
taking the easy-axis limit where the Hamiltonian is

$\hat{H}_{\text{easy-axis}}=\frac{1}{2}\sum_{\langle
i,j\rangle}\left[J_{z}\hat{S}^{z}_{i}\hat{S}^{z}_{j}+J_{\perp}\left(\hat{S}^{x}_{i}\hat{S}^{x}_{j}+\hat{S}^{y}_{i}\hat{S}^{y}_{j}\right)\right],$
(11)

where $J_{z}\gg J_{\perp}>0$. When $J_{\perp}=0$, the ground state is
exponentially degenerate with
$S^{z}_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=0$
on each tetrahedron, which is equivalent to
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=2$.
The $J_{\perp}$ term was treated as a perturbation over the $J_{z}$ term, and
at third order, a ring exchange term identical to Eq. (9) was obtained.
Written in terms of the spins, the ring-exchange term is

$\hat{H}_{\text{eff}}=-J_{\text{ring}}\sum_{\hexagon}\hat{S}_{1}^{+}\hat{S}_{2}^{-}\hat{S}_{3}^{+}\hat{S}_{4}^{-}\hat{S}_{5}^{+}\hat{S}_{6}^{-}+\text{H.c.},$
(12)

where the sum is over hexagonal plaquettes of the pyrochlore lattice. The RK
potential term was added by hand in Ref. [48] giving Eq. (10).

Hermele et al. then go to the quantum rotor variables
$n_{\mathbf{r}\mathbf{r}^{\prime}}\in\mathbb{Z}$ and
$\theta_{\mathbf{r}\mathbf{r}^{\prime}}\in[-\pi,\pi)$, which live on the links
$\mathbf{r}\mathbf{r}^{\prime}$ of the diamond lattice (equivalently, sites of
the pyrochlore lattice) and satisfy the canonical commutation relations
$[\hat{n}_{\mathbf{r}\mathbf{r}^{\prime}},\hat{\theta}_{\mathbf{r}\mathbf{r}^{\prime}}]=i$:

$\hat{S}^{z}\rightarrow\hat{n}-\frac{1}{2},\quad\hat{S}^{\pm}\rightarrow
e^{\pm i\hat{\theta}}.$ (13)

The constraint $n=0\text{ or }1$ is imposed by adding a term to the
Hamiltonian that energetically penalizes states violating this constraint:

$\displaystyle\hat{H}_{\text{eff}}=$
$\displaystyle\frac{U}{2}\sum_{\mathbf{\langle\mathbf{r},\mathbf{r}^{\prime}\rangle}}\left(\hat{n}_{\mathbf{r}\mathbf{r}^{\prime}}-\frac{1}{2}\right)^{2}$
(14)
$\displaystyle-2J_{\text{ring}}\sum_{\hexagon}\cos(\hat{\theta}_{1}-\hat{\theta}_{2}+\hat{\theta}_{3}-\hat{\theta}_{4}+\hat{\theta}_{5}-\hat{\theta}_{6}),$

where the first sum is over all the links of the diamond lattice and the
second one is over the hexagonal plaquettes of the pyrochlore lattice. In the
limit $U\rightarrow\infty$, Eq. (14) reduces to the effective Hamiltonian Eq.
(12).

The local constraint,
$S_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}_{\mathbf{r}}}^{z}=0$
for each tetrahedron, gives a gauge structure to the effective Hamiltonian
where the gauge transformations are generated by
$\hat{S}_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}_{\mathbf{r}}}^{z}$.
The presence of this local symmetry motivated Hermele et al. to write Eq. (14)
as a lattice $U(1)$ gauge theory. The electric field and the vector potential
were defined as

$\hat{e}_{\mathbf{r}\mathbf{r}^{\prime}}=\pm\left(\hat{n}_{\mathbf{r}\mathbf{r}^{\prime}}-\frac{1}{2}\right),\quad\hat{a}_{\mathbf{r}\mathbf{r}^{\prime}}=\pm\hat{\theta}_{\mathbf{r}\mathbf{r}^{\prime}}.$
(15)

The positive (negative) sign is chosen if $\mathbf{r}$ belongs to $A$ ($B$)
sublattice of the diamond lattice. The Hamiltonian written in terms of the
electric field and the vector potential takes the form of a compact $U(1)$
lattice gauge theory [70, 46]:

$\hat{H}_{\text{eff}}=\frac{U}{2}\sum_{\langle\mathbf{r},\mathbf{r}^{\prime}\rangle}\hat{e}_{\mathbf{r}\mathbf{r}^{\prime}}^{2}-2J_{\text{ring}}\sum_{\hexagon}\cos\left((\text{curl
}\hat{a})_{\hexagon}\right),$ (16)

where the second summation is over hexagonal plaquettes of the diamond lattice
and

$\left(\text{curl
}\hat{a}\right)_{\hexagon}=\mathchoice{\mathbin{\ooalign{$\displaystyle\sum_{\mathbf{r},\mathbf{r}^{\prime}\in\hexagon}$\crcr$\displaystyle\leavevmode\hbox
to8.94pt{\vbox to8.94pt{\pgfpicture\makeatletter\hbox{\hskip
4.46791pt\lower-4.46791pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.26791pt}{0.0pt}\pgfsys@curveto{4.26791pt}{2.35713pt}{2.35713pt}{4.26791pt}{0.0pt}{4.26791pt}\pgfsys@curveto{-2.35713pt}{4.26791pt}{-4.26791pt}{2.35713pt}{-4.26791pt}{0.0pt}\pgfsys@curveto{-4.26791pt}{-2.35713pt}{-2.35713pt}{-4.26791pt}{0.0pt}{-4.26791pt}\pgfsys@curveto{2.35713pt}{-4.26791pt}{4.26791pt}{-2.35713pt}{4.26791pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$}}}{\mathbin{\ooalign{$\textstyle\sum_{\mathbf{r},\mathbf{r}^{\prime}\in\hexagon}$\crcr$\textstyle\leavevmode\hbox
to8.94pt{\vbox to8.94pt{\pgfpicture\makeatletter\hbox{\hskip
4.46791pt\lower-4.46791pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.26791pt}{0.0pt}\pgfsys@curveto{4.26791pt}{2.35713pt}{2.35713pt}{4.26791pt}{0.0pt}{4.26791pt}\pgfsys@curveto{-2.35713pt}{4.26791pt}{-4.26791pt}{2.35713pt}{-4.26791pt}{0.0pt}\pgfsys@curveto{-4.26791pt}{-2.35713pt}{-2.35713pt}{-4.26791pt}{0.0pt}{-4.26791pt}\pgfsys@curveto{2.35713pt}{-4.26791pt}{4.26791pt}{-2.35713pt}{4.26791pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$}}}{\mathbin{\ooalign{$\scriptstyle\sum_{\mathbf{r},\mathbf{r}^{\prime}\in\hexagon}$\crcr$\scriptstyle\leavevmode\hbox
to8.94pt{\vbox to8.94pt{\pgfpicture\makeatletter\hbox{\hskip
4.46791pt\lower-4.46791pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.26791pt}{0.0pt}\pgfsys@curveto{4.26791pt}{2.35713pt}{2.35713pt}{4.26791pt}{0.0pt}{4.26791pt}\pgfsys@curveto{-2.35713pt}{4.26791pt}{-4.26791pt}{2.35713pt}{-4.26791pt}{0.0pt}\pgfsys@curveto{-4.26791pt}{-2.35713pt}{-2.35713pt}{-4.26791pt}{0.0pt}{-4.26791pt}\pgfsys@curveto{2.35713pt}{-4.26791pt}{4.26791pt}{-2.35713pt}{4.26791pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$}}}{\mathbin{\ooalign{$\scriptscriptstyle\sum_{\mathbf{r},\mathbf{r}^{\prime}\in\hexagon}$\crcr$\scriptscriptstyle\leavevmode\hbox
to8.94pt{\vbox to8.94pt{\pgfpicture\makeatletter\hbox{\hskip
4.46791pt\lower-4.46791pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{4.26791pt}{0.0pt}\pgfsys@curveto{4.26791pt}{2.35713pt}{2.35713pt}{4.26791pt}{0.0pt}{4.26791pt}\pgfsys@curveto{-2.35713pt}{4.26791pt}{-4.26791pt}{2.35713pt}{-4.26791pt}{0.0pt}\pgfsys@curveto{-4.26791pt}{-2.35713pt}{-2.35713pt}{-4.26791pt}{0.0pt}{-4.26791pt}\pgfsys@curveto{2.35713pt}{-4.26791pt}{4.26791pt}{-2.35713pt}{4.26791pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$}}}\;\;\;\hat{a}_{\mathbf{r}\mathbf{r}^{\prime}},\vskip
6.0pt plus 2.0pt minus 2.0pt$ (17)

where
$\mathchoice{\mathbin{\ooalign{$\displaystyle\sum$\crcr$\displaystyle\leavevmode\hbox
to6.09pt{\vbox to6.09pt{\pgfpicture\makeatletter\hbox{\hskip
3.04527pt\lower-3.04527pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.84528pt}{0.0pt}\pgfsys@curveto{2.84528pt}{1.57141pt}{1.57141pt}{2.84528pt}{0.0pt}{2.84528pt}\pgfsys@curveto{-1.57141pt}{2.84528pt}{-2.84528pt}{1.57141pt}{-2.84528pt}{0.0pt}\pgfsys@curveto{-2.84528pt}{-1.57141pt}{-1.57141pt}{-2.84528pt}{0.0pt}{-2.84528pt}\pgfsys@curveto{1.57141pt}{-2.84528pt}{2.84528pt}{-1.57141pt}{2.84528pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$}}}{\mathbin{\ooalign{$\textstyle\sum$\crcr$\textstyle\leavevmode\hbox
to6.09pt{\vbox to6.09pt{\pgfpicture\makeatletter\hbox{\hskip
3.04527pt\lower-3.04527pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.84528pt}{0.0pt}\pgfsys@curveto{2.84528pt}{1.57141pt}{1.57141pt}{2.84528pt}{0.0pt}{2.84528pt}\pgfsys@curveto{-1.57141pt}{2.84528pt}{-2.84528pt}{1.57141pt}{-2.84528pt}{0.0pt}\pgfsys@curveto{-2.84528pt}{-1.57141pt}{-1.57141pt}{-2.84528pt}{0.0pt}{-2.84528pt}\pgfsys@curveto{1.57141pt}{-2.84528pt}{2.84528pt}{-1.57141pt}{2.84528pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$}}}{\mathbin{\ooalign{$\scriptstyle\sum$\crcr$\scriptstyle\leavevmode\hbox
to6.09pt{\vbox to6.09pt{\pgfpicture\makeatletter\hbox{\hskip
3.04527pt\lower-3.04527pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.84528pt}{0.0pt}\pgfsys@curveto{2.84528pt}{1.57141pt}{1.57141pt}{2.84528pt}{0.0pt}{2.84528pt}\pgfsys@curveto{-1.57141pt}{2.84528pt}{-2.84528pt}{1.57141pt}{-2.84528pt}{0.0pt}\pgfsys@curveto{-2.84528pt}{-1.57141pt}{-1.57141pt}{-2.84528pt}{0.0pt}{-2.84528pt}\pgfsys@curveto{1.57141pt}{-2.84528pt}{2.84528pt}{-1.57141pt}{2.84528pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$}}}{\mathbin{\ooalign{$\scriptscriptstyle\sum$\crcr$\scriptscriptstyle\leavevmode\hbox
to6.09pt{\vbox to6.09pt{\pgfpicture\makeatletter\hbox{\hskip
3.04527pt\lower-3.04527pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}{{}}{}{{{}} {}{}{}{}{}{}{}{}
}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{2.84528pt}{0.0pt}\pgfsys@curveto{2.84528pt}{1.57141pt}{1.57141pt}{2.84528pt}{0.0pt}{2.84528pt}\pgfsys@curveto{-1.57141pt}{2.84528pt}{-2.84528pt}{1.57141pt}{-2.84528pt}{0.0pt}\pgfsys@curveto{-2.84528pt}{-1.57141pt}{-1.57141pt}{-2.84528pt}{0.0pt}{-2.84528pt}\pgfsys@curveto{1.57141pt}{-2.84528pt}{2.84528pt}{-1.57141pt}{2.84528pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}$}}}_{\;\mathbf{r},\mathbf{r}^{\prime}\in\hexagon}$
is a sum along the directed bonds of a hexagonal plaquette of the diamond
lattice. The adjective “compact” refers to the vector potential
$\hat{a}_{\mathbf{r}\mathbf{r}^{\prime}}$ being an angular variable. There is
an important difference between the above gauge theory and the compact $U(1)$
gauge theory studied by Polyakov [46, 47, 71]—the gauge theory obtained by
Hermele et al. is an odd gauge theory, i.e., electric fields are half-
integers, $e_{\mathbf{r}\mathbf{r}^{\prime}}\in\mathbb{Z}+1/2$, while the
gauge theory studied by Polyakov was an even gauge theory, i.e., the electric
fields were integers, $e_{\mathbf{r}\mathbf{r}^{\prime}}\in\mathbb{Z}$.
Because of this difference, the phases of the two theories differ.

The phases of a gauge theory can be characterized by the interaction between
two externally added opposite electric charges separated by a distance $R$. If
the potential between charges goes to zero (or increases as at most $\log{R}$
in $2+1$D) as $R\rightarrow\infty$, then the gauge theory is in the deconfined
phase. On the other hand, if the potential increases linearly with $R$ or
faster, then these opposite charges cannot be separated, and the gauge theory
is in the confined phase. In the limit $U\rightarrow\infty$, the even gauge
theory was shown to be in the confined phase in Refs. [46, 70], while the odd
gauge theory can be in either the confined phase or the deconfined phase [48].
This can be understood intuitively as follows.

In the even gauge theory, in the limit $U\rightarrow\infty$, the electric
fields are forced to be 0, $e_{\mathbf{r}\mathbf{r}^{\prime}}=0$, to minimize
the energy in the absence of any external charges. However, in the presence of
two opposite external charges, the Gauss’s law requires that the electric
field can no longer be zero everywhere. The spreading of the electric field
is, however, penalized by the term
$\frac{U}{2}\sum_{\langle\mathbf{r},\mathbf{r^{\prime}}\rangle}\hat{e}_{\mathbf{r}\mathbf{r}^{\prime}}^{2}$.
This forces the electric field to be nonzero only in a narrow tube joining the
two charges, leading to a linearly rising potential between the two charges.
Thus, in the limit $U\rightarrow\infty$, the even gauge theory is in a
confined phase, and there is no deconfined phase in this limit. This
confinement of charges has been shown in Refs. [46, 47, 70, 72].

On the other hand, in an odd gauge theory, in the limit $U\rightarrow\infty$,
the electric field can take two values, $e_{\mathbf{r}\mathbf{r}^{\prime}}=\pm
1/2$. This results in frustration, i.e., allows for many configurations of the
electric field, so that the ground state in this limit is non-trivial. When
two external charges are introduced, the electric field is not necessarily
confined in a string between the charges, but can spread in space similar to
the familiar Coulomb-law field lines of a non-compact $U(1)$ gauge theory.
This suggests that it is possible for the odd gauge theory to be in the
deconfined phase even in the $U\rightarrow\infty$ limit. In fact, the odd
gauge theory on the pyrochlore lattice (14) is indeed in the deconfined phase
in the $U\rightarrow\infty$ limit [50, 51, 63].

Hermele et al. have shown that the dimer model with the Hamiltonian Eq. (10)
is described by the deconfined phase of the underlying compact $U(1)$ gauge
theory close to the RK point (for $V_{RK}$ smaller than $J_{\text{ring}}$ but
close to $J_{\text{ring}}$). This phase is the $U(1)$ quantum spin liquid. It
has three types of emergent excitations—gapless photons, gapped magnetic
monopoles and gapped fractionalized electric charges, also called as spinons.
The spinons are the tetrahedra which violate the constraint on
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}$,
Eq. (8).

###  II.3 Previous numerical work

In this section, we summarize some of the known work on the dimer model with
the Hamiltonian Eq. (10) and on the nearest-neighbor transverse-field Ising
model on the pyrochlore lattice.

Using quantum Monte Carlo simulations, Refs. [51] and [63] studied the range
of $V_{RK}$ [see Eq. (10)] over which the $U(1)$ spin liquid exists. They
found that the spin liquid is present in the range
$-0.5J_{\text{ring}}<V_{RK}<J_{\text{ring}}$ for the dimer model with
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=2$
and in the range $0.77J_{\text{ring}}<V_{RK}<J_{\text{ring}}$ for the dimer
model with
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=1$.
The dimer model with
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=3$
is equivalent to the one with
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=1$
by a particle-hole transformation. These numerical results are summarized in
Fig. 6.

Figure 6: For $\rho\in\left(3/2,5/2\right)$, corresponding to
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=2$,
the system is in the $U(1)$ spin liquid phase at $V_{RK}=0$ [51]. On the other
hand, for $\rho\in\left(1/2,3/2\right)$ and $\rho\in\left(5/2,7/2\right)$,
corresponding to
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=1$
and $3$, respectively, the system is in an ordered phase at $V_{RK}=0$ [63].
Note that for $\rho=1/2$, $3/2$, and $5/2$, the perturbation theory described
in Sec. II.1 does not apply, and we cannot comment on the phase of the system.

While a theory proposal to realize the RK potential exists [73], the RK
potential is a six-body term for the pyrochlore lattice and is difficult to
engineer experimentally. Thus, we focus on the case where $V_{RK}=0$. From
Fig. 6, we see that to obtain a spin liquid phase for $V_{RK}=0$, one must
have
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=2$,
which corresponds to $3/2<\rho<5/2$. In the cases
$n_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}}=1$
and $3$, the system is in an ordered state when $V_{RK}=0$. Hence, in
conclusion, assuming the long-range interactions $\hat{H}_{LR}$ can be
ignored, we expect that, in the limit $\Omega\ll V$, the Rydberg system will
be in a $U(1)$ quantum spin liquid phase for $3/2<\rho<5/2$.

When $\rho=2$, or equivalently $h=0$, and the long-range interactions
$\hat{H}_{LR}$ are ignored, the Hamiltonian of the system
$\hat{H}_{0}+\hat{H}_{\Omega}$ in Eq. (6) is the transverse field Ising model
on the pyrochlore lattice. For $\Omega\ll V$, we know from the perturbative
analysis of Sec. II.1 and Ref. [63] that the system is in the $U(1)$ quantum
spin liquid phase. For large $\Omega/V$, where perturbation theory cannot be
applied, Ref. [62] found using quantum Monte Carlo calculations that the
$U(1)$ spin liquid exists in the region $\Omega<0.55(5)V$, while, for
$\Omega>0.55(5)V$, the system is in a transverse-field-polarized (TFP) phase,
which extends to $\Omega/V\rightarrow\infty$ where the ground state is
polarized in the $x$-direction. This transition was also studied in Ref. [74]
using perturbation theory, where a transition was found at $\Omega\approx
0.6V$.

The effects of adding a third nearest-neighbor interaction, $V_{3NN}$, to the
dimer model were considered in Ref. [75]. It was found that the quantum spin
liquid transitioned into an ordered state (antiferromagnet [76]) at
$V_{3NN}\approx J_{\text{ring}}$. Thus non-nearest-neighbor interactions can
destabilize the quantum spin liquid. In fact, in a 2D model with neutral atoms
located on the bonds of a kagome lattice (same as the sites of a ruby
lattice), a spin liquid ground state was found if the interactions were short-
ranged using DMRG on cylinders [37, 26]. However, with the full long-range van
der Waals interactions, the spin liquid ceased to exist [37, 26]. Thus it is
important to consider the effects of long-range interactions. In the following
section, we will study the phase diagram of Hamiltonian (6) in the presence of
long-range interactions, using approximate methods.

##  III Phase Diagram—Approximate Methods

The goal of this section is to study the ground state phase diagram of
Hamiltonian (4) for $\delta=3.46V$ (which corresponds to $\rho=2$) including
long-range interactions $\hat{H}_{\text{LR}}$.

###  III.1 Confinement-deconfinement transition—Monte Carlo assisted
perturbation theory

Consider the full Hamiltonian
$\hat{H}=\hat{H}_{0}+\hat{H}_{\Omega}+\hat{H}_{\text{LR}}$ from Eq. (6) in the
case $\rho=2$ [see Eq. (7)]:

$\displaystyle\hat{H}_{0}=$
$\displaystyle\frac{V}{2}\sum_{\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}_{\mathbf{r}}}\left(\sum_{i\in\mathchoice{\includegraphics[height=3.01389pt]{tetrahedron}}{\includegraphics[height=5.12358pt]{tetrahedron}}{\includegraphics[height=4.52083pt]{tetrahedron}}{\includegraphics[height=1.50694pt]{tetrahedron}}_{\mathbf{r}}}\hat{S}^{z}_{i}\right)^{2},$
(18) $\displaystyle\hat{H}_{\Omega}=$
$\displaystyle\Omega\sum_{i}\hat{S}_{i}^{x},\text{ and
}\hat{H}_{\text{LR}}=\frac{V}{2}\sum_{i\neq
j}{\vphantom{\sum}}^{\prime}\left(\frac{a}{|\bm{x}_{i}-\bm{x}_{j}|}\right)^{6}\hat{S}_{i}^{z}\hat{S}_{j}^{z}.$

The long-range interaction $\hat{H}_{\text{LR}}$ splits the exponential
degeneracy of the ice manifold, and selects one configuration diagonal in the
$\hat{S}^{z}$ basis as the ground state of $\hat{H}_{0}+\hat{H}_{\text{LR}}$,
which we call the “ordered state”. On the other hand, $\hat{H}_{\Omega}$
prefers superpositions of ice rule obeying states, the $U(1)$ quantum spin
liquid (QSL) being one such superposition. Further, we also note that quantum
fluctuations around the “ordered state” due to $\hat{H}_{\Omega}$ may also
lead to a change in its energy relative to the QSL. It is this competition
between kinetic energy and long-range interactions that we will study in this
section.

We first show that the ground state in the classical limit $\Omega=0$ is the
zero-momentum state satisfying the ice rule which we call the “ice
ferromagnet”. We assume that, as one increases $\Omega$, there is no phase
transition to a different ordered state before the putative transition to a
QSL. In order to determine whether a QSL phase exists and, if yes, at what
$\Omega$ the transition to the QSL occurs, one needs to compare the energies
of ansatz wavefunctions of the QSL and the ordered state. When $\Omega\neq 0$,
such wavefunctions would necessarily involve configurations that violate the
ice rule. We incorporate the effect of nonzero $\Omega$ on the wavefunction
using perturbation theory. Our strategy is as follows. We treat
$\hat{H}_{1}\equiv\hat{H}_{\Omega}+\hat{H}_{\text{LR}}$, i.e., both the laser
driving term and the long-range interactions, as a perturbation to
$\hat{H}_{0}$ (unlike Sec. II.1, where we dropped $\hat{H}_{\text{LR}}$). We
perturbatively find an effective Hamiltonian $\hat{H}_{\text{eff}}$ acting on
the low-energy ice manifold. We then compare the expectation value of
$\hat{H}_{\text{eff}}$ in candidate wavefunctions that live entirely in this
low-energy space. Since a QSL wavefunction is a linear superposition of
exponentially (in system size) many ice rule obeying states, we calculate
$\expectationvalue{\hat{H}_{\text{eff}}}$ numerically using classical Monte
Carlo sampling.

####  III.1.1 Expression for $\hat{H}_{\text{eff}}$

We perform a Schrieffer-Wolff transformation

$\hat{\tilde{H}}=\hat{U}_{S}\hat{H}\hat{U}_{S}^{\dagger}=\hat{U}_{S}\left(\hat{H}_{0}+\hat{H}_{\Omega}+\hat{H}_{\text{LR}}\right)\hat{U}_{S}^{\dagger},$
(19)

for a unitary $\hat{U}_{S}=e^{\hat{S}}$, where $\hat{S}$ is an anti-hermitian
operator chosen to make $\hat{\tilde{H}}$ block-diagonal in the (degenerate)
eigenbasis of $\hat{H}_{0}$, i.e.,

$\hat{\tilde{H}}=\hat{\mathcal{P}}\hat{\tilde{H}}\hat{\mathcal{P}}+(1-\hat{\mathcal{P}})\hat{\tilde{H}}(1-\hat{\mathcal{P}}),$
(20)

where $\hat{\mathcal{P}}$ projects onto the ice manifold. In the remainder of
this paper, we will restrict ourselves to the low-energy sector and therefore
only consider the
$\hat{H}_{\text{eff}}\equiv\hat{\mathcal{P}}\hat{\tilde{H}}\hat{\mathcal{P}}$
term above. We calculate $\hat{H}_{\text{eff}}$ perturbatively in
$\hat{H}_{1}=\hat{H}_{\Omega}+\hat{H}_{\text{LR}}$ (see Appendix B of Ref.
[77] for general expressions of $\hat{H}_{\text{eff}}$). As we saw in Sec.
II.1, if we consider only $\hat{H}_{\Omega}$ as the perturbation, then the
first non-trivial term appearing in $\hat{H}_{\text{eff}}$ is
$-J_{\text{ring}}\sum_{\hexagon}\ket{\raisebox{-2.84544pt}{ \leavevmode\hbox
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$J_{\text{ring}}=\frac{63}{16}\frac{\Omega^{6}}{V^{5}}+\Theta\left(\frac{\Omega^{8}}{V^{7}}\right).$
(21)

Since we are performing perturbation theory in two operators
$\hat{H}_{\Omega}$ and $\hat{H}_{LR}$, each of them comes with its own small
parameter. Since the perturbative expansion will involve polynomials in these
two small parameters, there is some arbitrariness in deciding how to compare
the two parameters relative to each other and thus in how to truncate the
expansion. In our calculation, we follow an operational scheme of keeping all
the terms up to sixth order in $\hat{H}_{\Omega}+\hat{H}_{\text{LR}}$.
Following this truncation scheme, we get (up to additive constants)

$\displaystyle\hat{H}_{\text{eff}}\approx$ $\displaystyle-
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$\displaystyle+\left(1-\frac{\Omega^{2}}{V^{2}}-\frac{61}{18}\frac{\Omega^{4}}{V^{4}}\right)\hat{H}_{\text{LR}}$
$\displaystyle-\frac{\Omega^{2}}{V}\left(\hat{W}^{(2)}_{\text{LR}}+\hat{W}^{(3)}_{\text{LR}}+\hat{W}^{(4)}_{\text{LR}}\right)$
$\displaystyle-\frac{\Omega^{4}}{V^{3}}\left(\frac{152}{27}\hat{W}^{(2)}_{\text{LR}}-\hat{L}^{(2)}_{\text{LR}}+\hat{M}^{(2)}_{\text{LR}}\right),$

where

$\displaystyle\hat{W}^{(2)}_{\text{LR}}$
$\displaystyle\equiv\frac{1}{4}\sum_{j}\sum_{\begin{subarray}{c}k_{1}\neq j\\\
k_{2}\neq
j\end{subarray}}v_{j,k_{1}}v_{j,k_{2}}\hat{S}^{z}_{k_{1}}\hat{S}^{z}_{k_{2}},$
(23) $\displaystyle\hat{L}^{(2)}_{\text{LR}}$
$\displaystyle\equiv\frac{109}{432}\sum_{\begin{subarray}{c}j_{1}\neq k_{1}\\\
j_{2}\neq k_{2}\end{subarray}}\delta_{\langle
j_{1},j_{2}\rangle}v_{j_{1},k_{1}}v_{j_{2},k_{2}}\hat{S}^{z}_{k_{1}}\hat{S}^{z}_{k_{2}},$
(24) $\displaystyle\hat{M}^{(2)}_{\text{LR}}$
$\displaystyle\equiv\frac{20}{27}\sum_{\begin{subarray}{c}j_{1}\neq k_{1}\\\
j_{2}\neq k_{2}\end{subarray}}\delta_{\langle
j_{1},j_{2}\rangle}v_{j_{1},k_{1}}v_{j_{2},k_{2}}\hat{S}^{z}_{j_{1}}\hat{S}^{z}_{k_{1}}\hat{S}^{z}_{j_{2}}\hat{S}^{z}_{k_{2}},$
(25) $\displaystyle\hat{W}^{(3)}_{\text{LR}}$
$\displaystyle\equiv\frac{1}{2}\sum_{j}\sum_{\begin{subarray}{c}k_{1}\neq j\\\
k_{2}\neq j\\\ k_{3}\neq
j\end{subarray}}v_{j,k_{1}}v_{j,k_{2}}v_{j,k_{3}}\hat{S}^{z}_{k_{1}}\hat{S}^{z}_{k_{2}}\hat{S}^{z}_{k_{3}}\hat{S}^{z}_{j},$
(26) $\displaystyle\hat{W}^{(4)}_{\text{LR}}$
$\displaystyle\equiv\frac{1}{4}\sum_{j}\sum_{\begin{subarray}{c}k_{1}\neq j\\\
k_{2}\neq j\\\ k_{3}\neq j\\\ k_{4}\neq
j\end{subarray}}v_{j,k_{1}}v_{j,k_{2}}v_{j,k_{3}}v_{j,k_{4}}\hat{S}^{z}_{k_{1}}\hat{S}^{z}_{k_{2}}\hat{S}^{z}_{k_{3}}\hat{S}^{z}_{k_{4}},$
(27) and $\displaystyle
v_{i,j}\equiv\begin{cases}\frac{a^{6}}{\left|\bm{x}_{i}-\bm{x}_{j}\right|^{6}}\text{
if }\bm{x}_{i},\bm{x}_{j}\text{ are not nearest neighbors,}\\\
0\;\;\;\;\;\;\;\;\;\;\;\text{ otherwise.}\end{cases}$ (28)

In the above equations, $\delta_{\langle i,j\rangle}$ enforces $i$ and $j$ to
be nearest neighbors.

The expectation value of the Hamiltonian (6) in a given state $\ket{\Psi}$ is

$\matrixelement{\Psi}{\hat{H}}{\Psi}=\left(\bra{\Psi}\hat{U}_{S}^{\dagger}\right)\left(\hat{U}_{S}\hat{H}\hat{U}_{S}^{\dagger}\right)\left(\hat{U}_{S}\ket{\Psi}\right).$
(29)

Suppose $\hat{U}_{S}\ket{\Psi}$ (i.e., $\ket{\Psi}$ transformed by the