Quantum order-by-disorder induced phase transition in Rydberg ladders with
staggered detuning

Madhumita Sarkar, Mainak Pal, Arnab Sen, and K. Sengupta

School of Physical Sciences, Indian Association for the Cultivation of
Science, Kolkata 700032, India

⋆<EMAIL_ADDRESS>
## Abstract

${}^{87}{\rm Rb}$ atoms are known to have long-lived Rydberg excited states
with controllable excitation amplitude (detuning) and strong repulsive van der
Waals interaction $V_{{\bf r}{\bf r^{\prime}}}$ between excited atoms at sites
${\bf r}$ and ${\bf r^{\prime}}$. Here we study such atoms in a two-leg ladder
geometry in the presence of both staggered and uniform detuning with
amplitudes $\Delta$ and $\lambda$ respectively. We show that when $V_{{\bf
rr^{\prime}}}\gg(\ll)\Delta,\lambda$ for $|{\bf r}-{\bf r^{\prime}}|=1(>1)$,
these ladders host a plateau for a wide range of $\lambda/\Delta$ where the
ground states are selected by a quantum order-by-disorder mechanism from a
macroscopically degenerate manifold of Fock states with fixed Rydberg
excitation density $1/4$. Our study further unravels the presence of an
emergent Ising transition stabilized via the order-by-disorder mechanism
inside the plateau. We identify the competing terms responsible for the
transition and estimate a critical detuning $\lambda_{c}/\Delta=1/3$ which
agrees well with exact-diagonalization based numerical studies. We also study
the fate of this transition for a realistic interaction potential $V_{{\bf
r}{\bf r^{\prime}}}=V_{0}/|{\bf r}-{\bf r^{\prime}}|^{6}$, demonstrate that it
survives for a wide range of $V_{0}$, and provide analytic estimate of
$\lambda_{c}$ as a function of $V_{0}$. This allows for the possibility of a
direct verification of this transition in standard experiments which we
discuss.

###### Contents

  1. 1 Introduction
  2. 2 Model Hamiltonian
  3. 3 Phases of $H_{\rm eff}$
  4. 4 Effect of vdW interaction
  5. 5 Discussion
  6. A Variational wavefunction approach for $\lambda\simeq\Delta$
  7. B $\ell_{0}$-leg ladder

##  1 Introduction

It is usually expected that fluctuations, thermal or quantum, in a generic
many-body system shall lead to suppression of order. However, a somewhat less
intuitive counterexample, namely, stabilization of order in a system with
competing interactions due to quantum or thermal fluctuations is now well-
accepted [1, 2, 3]. This phenomenon occurs due to the presence of
macroscopically degenerate manifold of classical ground states in such
systems; the presence of fluctuations then may lift this degeneracy leading to
a ground state with definite order. This mechanism is dubbed as order-by-
disorder [1]. Examples of this phenomenon is seen in a variety of quantum
many-body systems involving spins [4, 5, 6], bosons [7, 8, 9] and fermions
[10].

In recent years, ultracold atoms in optical lattice have proved to be
efficient emulators of several well-known model Hamiltonians in condensed
matter systems [11]. A primary example of this is the emulation of the Bose-
Hubbard model [12, 13, 14, 15] using ultracold bosonic atoms [16]. A study of
such bosons in their Mott phases and in the presence of a tilted lattice
generating an artificial electric field have led to the realization of a
translational symmetry broken ground state and an associated quantum phase
transition which belongs to the Ising universality class [17, 18, 19, 20, 21,
22, 23, 24]. More recently, another such ultracold atom system, namely, a
chain of ${{}^{87}}{\rm Rb}$ atoms supporting long-lived Rydberg excited
states has also been studied in detail. The amplitude of realizing a Rydberg
excitation at any site of such a chain can be controlled by changing the
detuning, i.e, the difference between the energy (frequency) of the external
laser and the energy gap between the ground and excited states of a Rydberg
atom [25, 26, 27, 28]. These atoms, in their Rydberg excited states,
experience strong repulsive van der Waals(vdW) interaction leading to a finite
Rydberg blockade radius [29, 30, 31, 32, 33]. Such a system leads to the
realization of symmetry broken phases separated by both Ising and non-Ising
quantum phase transitions [25, 26, 34, 35, 36, 37]. The out-of-equilibrium
dynamics of such systems leading to realization of the central role of quantum
scars in such dynamics have also been studied both theoretically [38, 39, 40,
41, 42, 43] and experimentally [26]. These studies have also been extended to
two-dimensional (2D) arrays of Rydberg atoms leading to the possibility of
realization of Kitaev spin liquids in these systems [44, 45, 46, 47, 48, 49].

In all of the above-mentioned studies [25, 26, 27, 28, 34, 35, 36, 37, 44, 45,
46, 47, 48, 49, 38, 39, 40, 41, 42, 43], detuning of the Rydberg atoms have
been assumed to be uniform. However, recently, chains of such Rydberg atoms in
the presence of an additional staggered detuning have also been studied
extensively [50, 51, 52, 53, 54, 55, 56, 57, 58, 59]. The main motivation
behind such studies stemmed from the possibility of realization of quantum
link models using ultracold atom systems [50, 51, 52, 53, 55, 56]; these
models are well-known to exhibit quantum confinement[60, 61] which is usually
difficult to realize in a generic condensed matter setup [57, 58, 62]. More
recently, the out-of-equilibrium dynamics of such models have also been
studied; it was found that they host several interesting phenomena such as
ultra-slow dynamics following a quench [57], dynamical freezing in the
presence of a periodic drive, and Floquet scars [59]. However, to the best of
our knowledge, the phases of coupled ladders of Rydberg atoms with staggered
detuning have not been studied in detail so far.

In this work, we study the phases of such coupled chains of Rydberg atoms in
the presence of both uniform and staggered detunings with amplitudes $\lambda$
and $\Delta$ respectively. In addition, such a system allows for a coupling
between the Rydberg excited and the ground states of the atoms with an
amplitude $w$ [25, 26]. Such coupled chains leads to a two-leg ladder geometry
as shown in Fig. 1(a). The corresponding blockade radius for the Rydberg
excitations, which prevents existence of two Rydberg excitation on nearest-
neighbor sites in such ladders, is also shown in Fig. 1(a). In our work, we
first concentrate in the regime where the vdW interaction between the Rydberg
atoms satisfy $V_{{\bf r}{\bf r^{\prime}}}\ll\lambda,\Delta,w$ outside the
blockade radius. In this regime, we find that for a wide range of
$\lambda/\Delta$ such a system shows a plateau where the Rydberg excitation
density, $n$, is fixed to $n=1/4$. The plateau destabilizes at
$\lambda/\Delta\simeq\pm 1$ leading to a change in $n$; we analyze this
behavior using a variational wavefunction method which agrees well with exact
diagonalization (ED) results.

Figure 1: (a) Schematic representation of a two-leg Rydberg ladder. The
circles represent sites on the Rydberg chain. The presence of a Rydberg
excitation on a site is denoted by the red circle; this precludes excitations
of nearby sites (denoted by blue circles). The corresponding Rydberg blockade
radius is schematically shown by the triangle. The other sites (grey circles)
may have Rydberg excitations since they would experience weaker repulsive vdW
interaction falling off as $1/|\vec{r}|^{6}$ with distance $|\vec{r}|$ from
the site with a Rydberg excited atom (red circle). (b) Schematic
representation of one of the states with Rydberg excitation density $n=1/4$
with excitations occurring at even sites of one of the ladders. This state is
one of the two possible classical Fock states which represents the ground
state for $\lambda>\lambda_{c}$. (c) A similar diagram for a state in the low-
energy manifold with $n=1/4$ whose superposition forms the ground state at
$\lambda<\lambda_{c}$ but not for $\lambda>\lambda_{c}$. See text for details.

The ground state of the system within the plateau involves macroscopic number
of degenerate classical Fock states having a Rydberg excitation in one of the
two sites in every even rung of the ladder (Fig. 1(b) and 1(c)) and is chosen
by a quantum order-by-disorder mechanism from these classical Fock states. For
$\lambda>\lambda_{c}$, the ground state breaks $\mathbb{Z}_{2}$ symmetry; it
consists of Fock states where all Rydberg excitations are localized in any one
of the two rungs of the ladder (Fig. 1(b)). In contrast, for
$\lambda<\lambda_{c}$, we find a unique ground state which constitutes a
macroscopic superposition of Fock states with fixed $n=1/4$. One such state is
schematically shown in Fig. 1(c). This necessitates the presence of a quantum
phase transition which belongs to the 2D Ising universality class at
$\lambda=\lambda_{c}$. We carry out a Schrieffer-Wolff transformation to
obtain an effective low-energy Hamiltonian which provides analytical insight
into this transition and identifies the competing terms responsible for it.
Our perturbative analysis, based on the effective Hamiltonian, estimates the
critical point to be $\lambda_{c}\simeq\Delta/3$ which shows an excellent
match with exact numerical results based on ED. Thus our analysis indicates
that the order-by-disorder mechanism necessarily leads to a quantum phase
transition separating the above-mentioned ground states. As we detail in the
appendix, our study naturally leads to a class of spin models involving
ladders with $\ell_{0}$ legs that realize ground states with broken
$\mathbb{Z}_{\ell_{0}}$ symmetry. For $\ell_{0}=3$, as we show both using ED
and analytic perturbation theory in the appendix, this provides a route to
realization of a quantum critical point belonging to the three-state Potts
universality class in 2D. To the best our knowledge, such emergent transitions
arising from an order-by-disorder mechanism have not been reported in the
context of Rydberg systems in the literature [25, 26, 27, 28, 34, 35, 36, 37,
44, 45, 46, 47, 48, 49].

Next, in an attempt to make contact with realistic experimental systems, we
discuss the fate of this transition for the two-leg ladder in the presence of
a realistic vdW interaction characterized by $V_{\bf r\bf
r^{\prime}}=V_{0}/|\bf r-\bf r^{\prime}|^{6}$. We discuss an experimentally
relevant regime where $V_{0}\gg\lambda,\Delta,w$ so that states with one or
more nearest-neighbor Rydberg excitations do not feature in the ground state
manifold. However, Rydberg excitations on second and higher neighboring sites
now experience a finite repulsive interaction. We show that the main effect of
having such a second and higher neighbor interaction is to shift
$\lambda_{c}/\Delta$ to a higher value; the Ising transition still persists
for a wide range of $V_{0}$ and $w$. We analyze the transition using a Van
Vleck perturbation theory, discuss the significance of the order-by-disorder
mechanism for its stability, and provide an estimate of $\lambda_{c}$ as a
function of $V_{0}$ and $w$. Finally, we discuss experiments which can test
our theory.

The plan of the rest of the paper is as follows. In Sec. 2, we define the
model Hamiltonian for Rydberg atoms on a two-leg ladder. Next, in Sec. 3, we
analyze the phases of the model by ignoring the second and higher neighboring
repulsive interaction between the Rydberg atoms. This is followed by Sec. 4
where we discuss the effect of finite $V_{0}$. Finally, we discuss our main
results, point out possible experiments which can test our theory, and
conclude in Sec. 5. We discuss our variational wavefunction results around
$\lambda\simeq\Delta$ in App. A and chart out the phases of ladders with
$\ell_{0}>2$ rungs, which could be more challenging to realize within current
experimental setups, in App. B.

##  2 Model Hamiltonian

In this section, we outline the model Hamiltonian for the Rydberg atoms. We
consider an arrangement of two Rydberg chains each having $2L$ sites shown
schematically in Fig. 1(a). The sites of these chains host $N=4L$ Rydberg
atoms whose low-energy effective Hamiltonian is written as [25, 26, 27, 28]

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{2L}\sum_{\ell=1}^{2}\left(w\sigma_{j,\ell}^{x}-\frac{1}{2}[\Delta(-1)^{j}+\lambda]\sigma_{j,\ell}^{z}\right)+\sum_{\bf
r\neq\bf r^{\prime}}V_{\bf r\bf r^{\prime}}\hat{n}_{\bf r}\hat{n}_{\bf
r^{\prime}},$ (1)

where $\sigma_{j,\ell}^{\alpha}$ for $\alpha=x,y,z$ denotes the usual Pauli
matrices on the site $j$ of the $\ell^{\rm th}$ chain and the lattice spacing
$a$ between the sites is set to unity so that the coordinate $\bf r$ of any
site is denoted by integers $j$ and $\ell$: ${\bf r}=(j,\ell)$. In Eq. 1,
$\lambda$ and $\Delta$ denote the amplitudes of uniform and staggered detuning
such that $\lambda+\Delta$ and $\lambda-\Delta$ represent the energy
differences between the energy of the applied external laser and the energy
gap between ground and excited Rydberg atomic levels on even and odd sites
respectively. Here $w>0$ denotes the coupling strength between the Rydberg
ground and excited states. In an experimental setup, this coupling is
controlled by two-photon processes having Rabi frequency $w/\hbar$ [25, 26,
27, 28]. Thus for any site with coordinate $\bf r$,

$\displaystyle\sigma_{\bf r}^{x}$ $\displaystyle\equiv$
$\displaystyle\sigma_{j,\ell}^{x}=(|G\rangle_{\bf r}\,{}_{\bf r}\langle
R|+{\rm h.c.}),\quad\hat{n}_{\bf
r}\equiv\hat{n}_{j,\ell}=(1+\sigma_{j,\ell}^{z})/2,$ (2)

where $|R\rangle_{\bf r}\equiv|\uparrow_{\bf r}\rangle$ and $|G\rangle_{\bf
r}\equiv|\downarrow_{\bf r}\rangle$ denotes excited and ground states of a
Rydberg atom on the site $\bf r$ respectively and $\hat{n}_{\bf r}$ is the
number operator corresponding to the Rydberg excitations. In the Rydberg
excited state, the atoms experience strong vdW repulsion which is modeled by
$V_{\bf r\bf r^{\prime}}$ given by

$\displaystyle V_{\bf r\bf r^{\prime}}$ $\displaystyle=$ $\displaystyle
V_{0}/|\bf r-\bf r^{\prime}|^{6},$ (3)

where $V_{0}$ is the interaction strength. It is well-known that in a typical
experiment, $V_{0}$ can be tuned to induce Rydberg blockade between
neighboring sites. For the rest of this paper, we shall assume that
$V_{0}\gg\Delta_{0},\lambda,w$ so that the neighboring sites of the two-rung
ladder have at most one Rydberg excited atom (Fig. 1(a)). In this limit, one
can split the Hamiltonian in two parts $H_{a}$ and $H_{b}$ given by

$\displaystyle H_{a}$ $\displaystyle=$ $\displaystyle\sum_{\bf r,\bf
r^{\prime}}V_{0}\hat{n}_{\bf r}\hat{n}_{\bf r^{\prime}}\delta_{|{\bf r}-{\bf
r^{\prime}}|-1},$ $\displaystyle H_{b}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{2L}\sum_{\ell=1}^{2}\left(w\sigma_{j,\ell}^{x}-\frac{1}{2}[\Delta(-1)^{j}-\lambda]\sigma_{j,\ell}^{z}\right)+\frac{V_{0}}{2}\sum_{\bf
r\bf r^{\prime}}\frac{n_{\bf r}\hat{n}_{\bf r^{\prime}}}{|\bf r-\bf
r^{\prime}|^{6}}(1-\delta_{|{\bf r}-{\bf r^{\prime}}|-1}).$ (4)

In the limit, where $V_{0}$ is the largest energy scale in the problem, it is
possible to define a hierarchy of energies for the eigenstates based on the
number of nearest-neighbor Rydberg excitations [27]. This can be encoded via a
projection operator $T_{m}$ which satisfies $[H_{a},T_{m}]=mV_{0}T_{m}$. The
role of $T_{m}$ is to project the Hamiltonian into a sector of $m$-nearest
neighbor Rydberg excitation. In what follows, we shall be concerned with the
effective Hamiltonian in the sector of $m=0$ nearest-neighbor dipoles. A
straightforward calculation similar to the one carried out in Ref. [27] for a
Rydberg chain yields [17, 63]

$\displaystyle H_{\rm eff}^{V}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{2L}\sum_{\ell=1}^{2}w\tilde{\sigma}_{j,\ell}^{x}-\sum_{j=1}^{2L}\sum_{\ell=1}^{2}\frac{1}{2}[\Delta(-1)^{j}+\lambda]\sigma_{j,\ell}^{z}+\frac{V_{0}}{2}\sum_{\bf
r\bf r^{\prime}}\frac{n_{\bf r}\hat{n}_{\bf r^{\prime}}}{|\bf r-\bf
r^{\prime}|^{6}}(1-\delta_{|{\bf r}-{\bf r^{\prime}}|-1}),$
$\displaystyle\tilde{\sigma}^{x}_{j,\ell}$ $\displaystyle=$ $\displaystyle
P_{j-1,\ell}\left(\prod_{\ell^{\prime}\neq\ell}P_{j\ell^{\prime}}\sigma^{x}_{j,\ell}\right)P_{j+1,\ell},$
(5)

where we have used the fact that $H_{0}$ does not contribute in this sector
and $P_{j,\ell}=(1-\sigma^{z}_{j,\ell})/2$ is the local projection operator
which ensures the absence of nearest-neighbor Rydberg excitations. The higher-
order terms can be systematically computed involving $m\neq 0$ nearest-
neighbor Rydberg excitations but are unimportant for the regime that we are
interested. Furthermore, in the regime where $V_{0}\gg\lambda,\Delta\gg w\gg
V_{0}/(\sqrt{2})^{6}$, it is possible to neglect the last term in $H_{\rm
eff}$. In this regime one obtains the following generalized PXP
Hamiltonian[17, 63] on the $2$-leg ladder

$\displaystyle H_{\rm eff}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{2L}\sum_{\ell=1}^{2}w\tilde{\sigma}_{j,\ell}^{x}-\frac{1}{2}\sum_{j=1}^{2L}\sum_{\ell=1}^{2}[\Delta(-1)^{j}+\lambda]\sigma_{j,\ell}^{z}.$
(6)

In the next section, we shall obtain the ground state phase diagram of $H_{\rm
eff}$ given in Eq. 6. The effect of finite second and higher-neighboring
interaction shall be discussed, using $H_{\rm eff}^{V}$ (Eq. 5) in Sec. 4.

##  3 Phases of $H_{\rm eff}$

To understand the phase-diagram of $H_{\rm eff}$, we first set $w=0$. It is
then easy to see that for $\lambda<0$ and $|\lambda|\gg\Delta$, we have
$\langle\sigma_{j,\ell}^{z}\rangle=-1$ on all sites leading to
$n=\langle\sum_{\vec{r}}\hat{n}_{\vec{r}}\rangle/N=0$. Similarly for
$\lambda>0$ and $\lambda\gg\Delta$, the ground states hosts maximal number of
possible up-spins or Rydberg excitations. However, due to the constraint of
having at most one Rydberg excitation on neighboring sites, it can have such
excitations only on $N/2$ sites. This results in a net Rydberg excitation
density of $n=1/2$ and leads to a two-fold degenerate ground state. In between
for $-\Delta\leq\lambda\leq\Delta$, the ground state hosts one Rydberg
excitation on every even rung of the ladder and thus has $n=1/4$. This is due
to the fact that such excitations requires $\delta E_{\rm
even}=-(\lambda+\Delta)<0$ in this regime. In contrast, a Rydberg excitation
on an odd site costs $\delta E_{\rm odd}=\Delta-\lambda>0$. The ground state
manifold therefore has macroscopic $2^{L}$ fold classical degeneracy for
$w=0$. At $\lambda=\pm\Delta$, $n$ exhibits jumps for $w=0$ which constitute
first order transitions.

These discontinuous transitions become smooth crossovers due to quantum
fluctuations introduced by a finite $w$ (Fig. 2 (a)). This can be understood
using a variational wavefunction based analysis. Here we illustrate this for
$\lambda/\Delta\simeq-1$; a similar analysis can be carried out for
$\lambda\simeq\Delta$ and is shown in App. A. We start by noting that for
$|\lambda|\gg\Delta,w$ and $\lambda<0$, the ground state of $H_{\rm eff}$ is
given by

$\displaystyle|\psi_{1}\rangle=\prod_{j=1}^{L}|\downarrow_{2j-1,1}\downarrow_{2j-1,2};\downarrow_{2j,1}\downarrow_{2j,2}\rangle.$
(7)

In contrast, for $\Delta>|\lambda|$, it is clearly energetically favorable to
have a Rydberg excitation on even sites. A variational wavefunction
representing such a state is given by

$\displaystyle|\psi_{2}\rangle$ $\displaystyle=$
$\displaystyle\prod_{j=1}^{L}\left(\cos\phi|\downarrow_{2j-1,1}\downarrow_{2j-1,2};\uparrow_{2j,1}\downarrow_{2j,2}\rangle+\sin\phi|\downarrow_{2j-1,1}\downarrow_{2j-1,2};\downarrow_{2j,1}\uparrow_{2j,2}\rangle\right).$
(8)

Note that we have chosen the variational parameter $\phi$ to be independent of
position since we intend to carry out a mean-field analysis of the problem
here. Near $\lambda/\Delta=-1$, we construct a variational wavefunction
$|\psi_{v}\rangle=\cos\theta|\psi_{1}\rangle+\sin\theta|\psi_{2}(\phi)\rangle$
which leads to $E_{v}=\langle\psi_{v}|H|\psi_{v}\rangle$ given by

$\displaystyle E_{v}$ $\displaystyle=$
$\displaystyle(\lambda\cos^{2}\theta-\Delta\sin^{2}\theta)-\sqrt{2}w\sin
2\theta\cos(\phi-5\pi/4),$ (9)

where we have ignored an irrelevant constant term. The minimization of $E_{v}$
fixes

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\phi_{0}=5\pi/4,\quad\theta=\theta_{0}=\frac{1}{2}\arctan\left[\frac{-2\sqrt{2}w}{\lambda+\Delta}\right].$
(10)

Figure 2: (a): Plot of the Rydberg excitation density $n$ as a function of
$\lambda$ with $w=0$ (black dotted line) and $w=0.05$ (blue dots) for the two
leg-ladder indicating the plateau at with $n=1/4$ between $-1\leq\lambda\leq
1$ as obtained using ED. The red solid line indicates $n$ as computed using
variational wavefunction analysis around $\lambda=\pm 1$. The black circle at
$\lambda\sim 0.33$ shows the position of the Ising transition. (b): Plot of
$s_{x}$ as a function of $\lambda$ across the crossover at $\lambda\simeq-1$.
The solid red line indicate results from the variational wavefunction approach
while the blue dots indicates ED results corresponding to $N=20$. All energies
are scaled in units of $\Delta$. See text for details.

To show the efficacy of this approach, we compare results obtained via
variational wavefunction method with exact numerics. To this end, we analyze
$H_{\rm eff}$ numerically using ED for $N\leq 40$ and using periodic boundary
condition along the chains. We compute the expectation values $\hat{n}$ and

$\displaystyle\hat{s}_{x}=\sum_{j=1}^{L}\sum_{\ell=1}^{2}\sigma_{2j,\ell}^{x}/N,$
(11)

from both the variational wavefunction and using exact numerics. The former
yields analytical expressions for $n$ and $s_{x}=\langle\hat{s}_{x}\rangle$
given by

$\displaystyle n$ $\displaystyle=$
$\displaystyle\frac{1}{8}\sin^{2}\theta_{0},\quad s_{x}=\frac{\sqrt{2}\sin
2\theta_{0}}{4}.$ (12)

The plots of these as a function of $\lambda/\Delta$ with $w/\Delta=0.05$ are
compared with their exact numerical counterparts in Fig. 2(a) and 2(b). The
plots show an excellent match; this shows that the crossover between phases
with $n=0$ to $n=1/4$ is well captured by our variational analysis. A similar
match is obtained for the crossover at $\lambda/\Delta\simeq 1$ as can be seen
from Fig. 2(a); this has been presented in detail in App. A. We also note here
that the extent of the plateau reduces with increasing $w$; we therefore work
in the regime $w\ll\lambda,\Delta$ in this work.

Figure 3: (a): Plot of $(\Delta E)N^{z}$ for a two-leg ladder as a function of
$\lambda$ for $w=0.05$ for several $N$ showing a crossing at
$\lambda=\lambda_{c}\simeq 0.333$ for $z=1$. (b) Plot of $(\Delta E)N^{z}$ as
a function of $N^{1/\nu}(\lambda-\lambda_{c})$ showing perfect scaling
collapse for $z=\nu=1$. (c): Plot of $SN^{2-z-\eta}$ as a function of
$\lambda$ showing a crossing at $\lambda_{c}\simeq 0.331$ for $\eta=0.25$. (d)
Plot of $SN^{2-z-\eta}$ as a function of $(\lambda-\lambda_{c})N^{1/\nu}$
showing perfect scaling collapse for $\nu=1$ and $\eta=0.25$. All energies are
scaled in units of $\Delta$. See text for details.

Next, we concentrate on the nature of the ground state of the system for
$-1\leq\lambda/\Delta\leq 1$. Here the ground state is chosen by a quantum
order-by-disorder mechanism from the manifold of $2^{L}$ classical Fock states
with $n=1/4$. Our numerical analysis reveals the presence of a quantum phase
transition inside the plateau with $n=1/4$ at $\lambda=\lambda_{c}$. We find
numerically that the ground state of the system at $\lambda>\lambda_{c}$ hosts
Rydberg excitations on even sites of any one of the two chains; it thus breaks
a $\mathbb{Z}_{2}$ symmetry as shown schematically in Fig. 1(b). In contrast
for $\lambda<\lambda_{c}$, the ground state is unique and constitute a
superposition of all Fock states with $n=1/4$ (Fig. 1(c)); it does not break
any symmetry. This necessitates a transition at $\lambda=\lambda_{c}$
characterized by an order parameter

$\displaystyle\hat{O}$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{L}\sum_{\ell=1}^{2}(-1)^{\ell}\sigma_{2j,\ell}^{z}.$
(13)

We note that the quantum phase transition is a result of the fact that the
order-by disorder mechanism chooses two different ordered states, one of which
breaks a discrete $\mathbb{Z}_{2}$ symmetry while the other does not.

Figure 4:  Schematic representation of the virtual processes leading to the
effective Hamiltonian $H^{\prime}$. The red (blue) circles represent atoms in
Rydberg excited (ground) states. (a): The process which favors all Rydberg
excitations to be on the even sites of one of the chains (chosen to be the top
chain ($\ell=1$) here) and leads to the first term in $H^{\prime}$. The
excited state constitutes one additional Rydberg excitation on an odd site of
the bottom chain as shown schematically. (b) The competing virtual process
which leads to the second term of $H^{\prime}$. The excited state, which
belongs to the high energy manifold, constitutes one additional Rydberg atom
in its ground state at an even site of the top chain as shown. See text for
details.

To check for the universality class of this transition, we carry out finite-
size scaling analysis for $N\leq 40$ on these ladders. We use the well-known
scaling relations [64, 17]

$\displaystyle\Delta E$ $\displaystyle=$ $\displaystyle
N^{-z}f\left[N^{1/\nu}(\lambda-\lambda_{c})\right],$ (14) $\displaystyle S$
$\displaystyle=$
$\displaystyle\frac{1}{N}\langle\hat{O}^{2}\rangle=N^{2-z-\eta}g\left[N^{1/\nu}(\lambda-\lambda_{c})\right],$

where $z$ and $\nu$ are the dynamical critical and the correlation length
exponents respectively, $\eta$ indicates the anomalous dimension of $\hat{O}$,
$S$ is the order-parameter correlation, $f$ and $g$ are scaling functions, and
$\Delta E$ denotes the energy gap between the ground and the first excited
states. Such a scaling analysis, shown in Fig. 3 indicates the presence of a
quantum phase transition at $\lambda_{c}\simeq 0.33\Delta$ with $z=\nu=1$ and
$\eta=1/4$; these exponents confirm that the universality class is the same as
the critical point of the classical 2D Ising model.

To obtain analytic insight into this transition, we construct a perturbative
effective Hamiltonian using a Schrieffer-Wolff transformation as follows.
First, we note that $w\ll|\lambda|,\Delta$ in the regime of interest. Using
Eq. 6, we therefore write $H_{\rm eff}=H_{0}+H_{1}$ where $H_{1}=w$
$\sum_{j,=1}^{2L}\sum_{\ell=1}^{2}\tilde{\sigma}_{j,\ell}^{x}$. We then use a
canonical transformation operator $S^{\prime}$ to write

$\displaystyle H^{\prime}=\exp[iS^{\prime}]H_{\rm
eff}\exp[-iS^{\prime}]=H_{0}+H_{1}+[iS^{\prime},H_{0}+H_{1}]+\frac{1}{2}[iS^{\prime},[iS^{\prime},H_{0}]]+...,$
(15)

where the ellipsis indicates higher order terms. Next, following standard
procedure, we eliminate all ${\rm O}(w)$ terms in $H^{\prime}$ which takes one
out of the low-energy manifold of states. This allows to determine
$S^{\prime}$ using the resultant condition $[iS^{\prime},H_{0}]=-H_{1}$
yielding

$\displaystyle
S^{\prime}=-w\sum_{j=1}^{2L}\sum_{\ell=1}^{2}(\Delta(-1)^{j}+\lambda)^{-1}\tilde{\sigma}_{j,\ell}^{y}.$
(16)

Using the expression of $S^{\prime}$, a straightforward calculation yields the
effective Hamiltonian $H^{\prime}=H_{0}$
$+[iS^{\prime},[iS^{\prime},H_{0}]]/2$. A subsequent projection to the
manifold of states with $n=1/4$ leads to [57, 65]

$\displaystyle H^{\prime}$ $\displaystyle=$
$\displaystyle\frac{-w^{2}}{\Delta-\lambda}\sum_{j=1}^{L}\Big{[}\sum_{\ell=1}^{2}P_{2j,\ell}P_{2j+2,\ell}+\frac{\alpha_{0}}{2}\sum_{\ell,\ell^{\prime}=1,2;\ell\neq\ell^{\prime}}P_{2j,\ell^{\prime}}(\sigma^{x}_{2j,\ell}\sigma^{x}_{2j,\ell^{\prime}}+\sigma^{y}_{2j,\ell}\sigma^{y}_{2j,\ell^{\prime}})\Big{]},$
(17)

where $\alpha_{0}=(\Delta-\lambda)/(\Delta+\lambda)$. The virtual processes
induced by $w$ which play a key role for realization of these two terms are
schematically shown in Fig. 4.

The emergence of the phase transition at a critical value of $\lambda/\Delta$
can be understood by noting the competition between the two terms of
$H^{\prime}$ in Eq. 17. The first term, whose amplitude dominates for
$\Delta-\lambda\simeq w$, prefers maximum possible Rydberg excitation on even
sites of any one of the two chains; such a state is schematically shown in
Fig. 1(b). The resultant ground state thus breaks $\mathbb{Z}_{2}$ symmetry.
In contrast, the second term, which dominates when $\Delta+\lambda\simeq w$,
prefers a linear superposition of all states with one Rydberg excitation on
any chain of even sites of the ladders. The resultant ground state does not
break any discrete symmetry. This necessitates an intermediate quantum
critical point belonging to the 2D Ising universality class.

Figure 5: (a): Plot of $n_{0}$ as a function of $V_{0}$ for $w=0.1$ and
$\lambda=1$ showing $V_{c}\simeq 1.6$ for $N=12$ (blue squares) and $N=16$
(red circles). (b) Plot of $V_{c}$ as a function of $w$ for $\lambda=1$. (c)
Plot of $n$ as a function of $\lambda$ for $w=0.1$ and $V_{0}=2$ showing the
plateau for $n$ and the position of the transition (black circle).(d) The
eigenenergies $E_{m}$ of $H^{V}_{\rm eff}$, measured from the ground state
energy, as a function of $m$ (for $m\leq 100$) for $\lambda=1$, $w=0$ and
$N=24$ showing the near-degenerate manifold. The two degenerate ground states
at $w=0$ correspond to two Fock states where the Rydberg excitations occur
alternately on even sites of ladders $1$ and $2$. All energies are scaled in
units of $\Delta$. See text for details.

To estimate the position of this critical point, we start from the
$\mathbb{Z}_{2}$ symmetry broken ground state with all Rydberg excitations on
even sites of the first chain (Fig. 1(b)). The simplest excited state over
these ground states constitutes equal superposition of Fock states with one
Rydberg excitation on any one of two chains for a single even site $2j$; for
$j^{\prime}\neq 2j$, the Rydberg excitations occur on the first chain. This
excited state can be represented as

$\displaystyle|\psi_{\rm ex}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\Big{(}|\downarrow_{1,1}\downarrow_{1,2};\uparrow_{2,1}\downarrow_{2,2}....\uparrow_{2j,1}\downarrow_{2j,2}...;\uparrow_{2L,1}\downarrow_{2L,2}\rangle$
(18)
$\displaystyle+|\downarrow_{1,1}\downarrow_{1,2};\uparrow_{2,1}\downarrow_{2,2}....\downarrow_{2j,1}\uparrow_{2j,2}...;\uparrow_{2L,1}\downarrow_{2L,2}\rangle\Big{)},$

where
$|\psi_{G}\rangle=|\downarrow_{1,1}\downarrow_{1,2};\uparrow_{2,1}\downarrow_{2,2}....\uparrow_{2j,1}\downarrow_{2j,2}...;\uparrow_{2L,1}\downarrow_{2L,2}\rangle$
is the symmetry broken ground state.

The energy cost of creating such an excited state can be easily computed as
$\delta E_{\rm ex}=$
$\langle\psi_{\rm ex}|H^{\prime}|\psi_{\rm
ex}\rangle-\langle\psi_{G}|H^{\prime}|\psi_{G}\rangle$. A straightforward
calculation yields $\delta E_{\rm ex}=\delta E_{1}+\delta E_{2}$ where

$\displaystyle\delta E_{1}$ $\displaystyle=$
$\displaystyle\frac{w^{2}}{2(\Delta-\lambda)},\quad\delta
E_{2}=-\frac{w^{2}}{\Delta+\lambda}.$ (19)

We note that $\delta E_{1}$ originates from the first term of $H^{\prime}$
which prefers all the Rydberg excitations to be on the same chain. In
contrast, $\delta E_{2}$ comes from the second term which maximizes
superposition of up-spins on different sites of the even rungs. An approximate
estimate of the critical point can be obtained by equating these two energies;
this yields the value of $\lambda$ at which $\delta E_{\rm ex}=0$ so that the
$\mathbb{Z}_{2}$ symmetry broken ground state becomes unstable to low-energy
excitations. Such an estimate yields

$\displaystyle\lambda_{c}=\Delta/3.$ (20)

We note that this simple estimate matches remarkably well with that obtained
from ED using finite-sized scaling analysis of the energy gap:
$\lambda_{c}^{\rm ex}/\Delta=0.332$. This provides support for the fact that
the transition is indeed realized due to an order-by-disorder mechanism which
can be captured by the perturbative effective Hamiltonian $H^{\prime}$ (Eq.
17).

We also note that the position of the transition can also be obtained as
follows. First, we define pseudospin states on even sites of the ladder as
$|1_{2j}\rangle\equiv|\uparrow_{2j,1};\downarrow_{2j,2}\rangle$ and
$|-1_{2j}\rangle\equiv|\downarrow_{2j,1};\uparrow_{2j,2}\rangle$. In terms of
Pauli matrices, $\vec{s}_{j}$, which acts on the space of these pseudospins,
one can then identify $P_{2j,1(2)}=(1-(+)s_{2j}^{z})/2$. Using this
identification, the first term of $H_{\rm eff}$ (Eq. 37) can be written,
ignoring an irrelevant constant term, as

$\displaystyle\sum_{j=1}^{L}\sum_{\ell=1}^{2}P_{2j,\ell}P_{2j+2,\ell}$
$\displaystyle\to$
$\displaystyle\frac{1}{2}\sum_{j=1}^{L}s_{2j}^{z}s_{2j+2}^{z}$ (21)

We then note that the second term of $H_{\rm eff}$ (Eq. 37), acting on these
pseudospin state, yields $|1_{2j}\rangle\Leftrightarrow|-1_{2j}\rangle$. This
is due to the fact that the spins even sites of the ladder can not be both
spin-up or both spin-down; the first condition follows from the constraint,
while the second is a consequence of the ground state belonging to $n=1/4$
sector. Using this, one can show

$\displaystyle\frac{1}{2}\sum_{j=1}^{L}\sum_{\ell,\ell^{\prime}=1,2;\ell\neq\ell^{\prime}}P_{2j,\ell^{\prime}}(\sigma^{x}_{2j,\ell}\sigma^{x}_{2j,\ell^{\prime}}+\sigma^{y}_{2j,\ell}\sigma^{y}_{2j,\ell^{\prime}})$
$\displaystyle\to$ $\displaystyle\sum_{j=1}^{L}s_{2j}^{x}$ (22)

Using Eqs. 21 and 22, we find an effective Ising representation of $H_{\rm
eff}$ given by

$\displaystyle H_{\rm eff}^{I}$ $\displaystyle=$
$\displaystyle\frac{-w^{2}}{\Delta-\lambda}\sum_{j=1}^{L}\left(\frac{1}{2}s_{2j}^{z}s_{2j+2}^{z}+\alpha_{0}s^{x}_{2j}\right)$
(23)

From this, one can, using Kramers-Wannier duality, read off the position of
the critical point to be at $\alpha_{0}=1/2$ which also leads to
$\lambda_{c}=\Delta/3$.

Before ending this section, we would like to point out that a similar model
ladders with $\ell_{0}>2$ would lead to spin models which hosts
$\mathbb{Z}_{\ell_{0}}$ symmetry broken ground states leading to realization
of non-Ising quantum critical points. We show in the appendix, by an analysis
similar to that carried out above, that such transition belong to 2D three-
state Potts universality class for $\ell_{0}=3$. However, realization of
$\ell_{0}>2$-leg ladders using Rydberg atoms chains with vdW interactions
whose low-energy behavior is governed by $H_{\rm eff}$ may be difficult; we
discuss this further in the appendix.

##  4 Effect of vdW interaction

In this section, we shall discuss the fate of the Ising transition for a
realistic vdW interaction given by Eq. 3. To this end, we concentrate in the
regime where $V_{0}$ is large enough so that there is at most one Rydberg
excitation in any pair of neighboring sites. To estimate the minimal
$V_{0}\equiv V_{c}$ required to satisfy this criteria, we diagonalize $H$ (Eq.
1) for a two-leg ladder with $N\leq 16$ and Hilbert space dimension
${\mathcal{D}}\leq 2^{16}$ to obtain its ground state $|\Psi_{G}\rangle$. We
then compute

$\displaystyle n_{0}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\sum_{\bf r\bf
r^{\prime}}\langle\Psi_{G}|n_{\bf r}n_{\bf r^{\prime}}\delta_{|{\bf r}-{\bf
r^{\prime}}|-1}|\Psi_{G}\rangle$ (24)

as a function of $V_{0}$ for a fixed $w$, $\lambda$, and $\Delta$. This allows
us to estimate $V_{c}(w/\Delta,\lambda/\Delta)$ as the minimum value of
$V_{0}$ for which $n_{0}\leq\epsilon_{0}\simeq 0.001$. This is shown in Fig.
5(a) where $n_{0}$ is plotted as a function of $V$ for $\lambda/\Delta=1$ and
$w/\Delta=0.1$ for $N=12,16$. We note that $V_{c}$ is independent of $N$; thus
one can safely use $V_{0}>V_{c}$ to access the constrained Hilbert space for
larger system sizes. A plot of $V_{c}/\Delta$ obtained from this procedure for
$N=16$ as a function of $w/\Delta$ for $\lambda/\Delta=1$ is shown in Fig.
5(b). We find that $V_{c}$ increases with increasing $w$; this can be easily
understood from the fact that increasing $w$ makes the freezing of spin on any
site energetically more costly.

Next, we fix $V_{0}/\Delta=2$ and $w/\Delta=0.1$ so that the ground state does
not have more than one Rydberg excitation on neighboring sites. We then
diagonalize $H_{\rm eff}^{V}$ (Eq. 5) for $N=40$, obtain the ground state, and
compute $n$ as a function of $\lambda/\Delta$ for $w=0$ and $0.1\Delta$. Our
results, shown in Fig. 5(c), indicate the presence of a wide range of
$\lambda/\Delta$ with $n=1/4$; for $w/\Delta=0.1$, $n$ starts deviating from
$1/4$ around $\lambda/\Delta\simeq 1.1$ and reaches its maximal value $1/2$
around $\lambda_{u}/\Delta\sim 2$. Our numerical, finite-size scaling
analysis, also finds a Ising critical point around $\lambda=\lambda_{c}$ such
that $\lambda_{c}/\Delta\simeq 1.12$ at the edge of the plateau. For
$\lambda_{c}<\lambda<\lambda_{u}$, the system has a double degenerate ground
state; it chose a single configuration only via spontaneous symmetry breaking.
This shows that the presence a finite second and higher neighbor interaction
does not necessarily obliterate the Ising transition; however the transition
is shifted to a higher value of $\lambda/\Delta$. We note that this is
possible since the residual interaction term in $H_{\rm eff}^{V}$ does not
lift the degeneracy between the two ${\mathbb{Z}}_{2}$ symmetry broken ground
states. Importantly, the presence of $V$ leads to a near-degenerate low-energy
manifold with ${\rm O}(L)$ states having $n\sim 1/4$; this is to be contrasted
with the exact degenerate ground state manifold of $2^{L}$ states when the
next nearest-neighbor interactions are neglected. This is shown in Fig. 5(d)
for $\lambda/\Delta=1.1$ and $w=0$. The ground states for $w=0$ shown in Fig.
5(d) correspond to two Fock states where Rydberg excitations occur
alternatively on even sites of ladders $1$ and $2$. We also note that the
presence of a finite $w$, whose magnitude is comparable to the width of the
nearly-degenerate low-energy manifold, leads to a different ground state which
constitutes a significant admixture of states within this manifold. The plot
also indicates that one needs a finite but small $w$ so that states with
$n\neq 1/4$ (higher excited states in Fig. 5(d)) are not strongly mixed with
near degenerate ${\rm O}(L)$ manifold of low-lying states.

Figure 6: (a): Exact numerics confirming the presence of the Ising transition
for $V_{0}=2.0$ and $w=0.1$ (a): Plot of $(\Delta E)N^{z}$ for a two-leg
ladder as a function of $\lambda$ for several $N$ showing a crossing at
$\lambda=\lambda_{c}\simeq 1.121$ for $z=1$. (b) Plot of $(\Delta E)N^{z}$as a
function of $N^{1/\nu}(\lambda-\lambda_{c})$ showing perfect scaling collapse
for $z=\nu=1$. (c): Plot of $SN^{2-z-\eta}$ as a function of $\lambda$ showing
a crossing at $\lambda_{c}\simeq 1.118$ for $\eta=0.25$. (d) Plot of
$SN^{2-z-\eta}$ as a function of $(\lambda-\lambda_{c})N^{1/\nu}$ showing
scaling collapse for $\nu=1$, $\lambda_{c}\simeq 1.118$, and $\eta=0.25$. All
energies are scaled in units of $\Delta$. See text for details.

Next, we carry out finite-size scaling analysis near the transition. The
results are indicated in Fig. 6. Our analysis reveals the presence of a
quantum critical point at $\lambda_{c}\simeq 1.12\Delta$ for $V_{0}=2\Delta$
and $w=0.1\Delta$. The Ising universality class of this transition is
confirmed from finite-size scaling as shown in Fig. 6; our results indicate
$z=\nu=1$ and are consistent with $\eta=0.25$ even though the scaling collapse
of $S$ suggests significant finite-size scaling corrections for the system
sizes accessed using ED. The variation of $\lambda_{c}$ with $V_{0}>V_{c}$ and
$w$ is shown in Fig. 7(a). We note that the transition exists for a wide range
of $V_{0}$ and $w$; $\lambda_{c}$ varies linearly with both $V_{0}$ and $w$
within this range. Moreover, we find that in contrast to its counterpart in
$H_{\rm eff}$, $\lambda_{c}$ depends on $w$ as shown in the inset of Fig.
7(a).

Figure 7: (a): Plot of $\lambda_{c}$ as a function of $V_{0}$ for $w_{0}=0.1$
showing the presence of the transition for a wide range of $V_{0}>V_{c}$. The
inset shows a plot of $\lambda_{c}$ as a function of $w$ for $V_{0}=2$. For
both plots, the blue dots represent exact results obtained using ED while the
red lines shows the analytical result obtained using Eq. 29. (b) A plot of
$(\Delta E_{p})N^{z}$ as a function of $\lambda$ for different $N$ showing a
crossing at $\lambda_{c}\simeq 1.104$ for $z=1$. The inset shows a plot
$(\Delta E_{p})N^{z}$ as a function of $(\lambda-\lambda_{c})N^{1/\nu}$
showing data collapse for $z=\nu=1$. These plots indicate that the Van-Vleck
perturbation theory captures the essence of the Ising transition. All energies
are scaled in units of $\Delta$. See text for details.

To obtain an analytic estimate of the shift in the position of the transition,
we use a Van-Vleck perturbation theory to construct an effective low-energy
Hamiltonian. To this end, we write $H^{V}_{\rm eff}=H_{0}^{V}+H_{1}$ where
$H_{1}=w\sum_{j=1}^{2L}\sum_{\ell=1}^{2}\tilde{\sigma}^{x}_{j,\ell}$ and
$H_{0}^{V}$can be read off from Eq. 5. We first identify the low-energy
manifold of states corresponding to $H^{V}_{0}$; these states are represented
as $|m\rangle$. Next, we construct a canonical transformation $S^{\prime}$
leading to an effective Hamiltonian

$\displaystyle H^{\prime}_{V}$ $\displaystyle=$ $\displaystyle
e^{iS^{\prime}}H_{\rm
eff}^{V}e^{-iS^{\prime}}=H_{0}^{V}+H_{1}+[iS^{\prime},H_{0}+H_{1}]+\frac{1}{2}[iS^{\prime},[iS^{\prime},H_{0}^{V}]]+..,$
(25)

where the ellipsis indicate higher order terms in $w$ which we neglect. Note
that the presence of the interaction term in $H_{0}^{V}$ makes it difficult to
obtain a closed form operator expression for $S^{\prime}$; however, its matrix
elements can still be analytically expressed. To determine $S^{\prime}$, we
again demand that it is chosen such that all ${\rm O}(w)$ terms, which takes
one out the low-energy manifold of states $|m\rangle$, are eliminated; this
leads to matrix elements of $S^{\prime}$ as

$\displaystyle\langle n|iS^{\prime}|m\rangle$ $\displaystyle=$
$\displaystyle\frac{\langle
n|H_{1}|m\rangle}{E_{n}^{0}-E_{m}^{0}},\quad\langle
m|iS^{\prime}|m^{\prime}\rangle=0,$ (26)

where $|n\rangle$ denotes eigenstates of $H_{0}^{V}$ which are not part of the
low-energy manifold and $E_{n}^{0}$ are the corresponding eigenvalues. We also
note that the second relation in Eq. 26 is automatically satisfied when $w$ is
the large compared to energy width of the low-energy manifold (see Fig. 5(d));
this is the regime which we shall be interested in for the rest of this
section. One can then substitutes Eq. 26 in Eq. 25 and obtain the matrix
elements of $H^{\prime}_{V}$ within the low-energy manifold

$\displaystyle\langle m|H^{\prime}_{V}|m^{\prime}\rangle$ $\displaystyle=$
$\displaystyle E_{m}^{0}\delta_{mm^{\prime}}-\frac{w^{2}}{2}\sum_{n}\langle
m|\sum_{j=1}^{2L}\sum_{\ell=1}^{2}\tilde{\sigma}_{j,\ell}^{x}|n\rangle\langle
n|\sum_{j=1}^{2L}\sum_{\ell=1}^{2}\tilde{\sigma}_{j,\ell}^{x}|m^{\prime}\rangle$
(27)
$\displaystyle\times\left(\frac{1}{E_{n}^{0}-E_{m}^{0}}+\frac{1}{E_{n}^{0}-E_{m^{\prime}}^{0}}\right),$

where the sum over $n$ indicates sum over states outside the low-energy
manifold. Here $E_{m}^{0}$ denotes the energy of the state $|m\rangle$ due to
$H_{0}^{V}$ while the terms $\sim w^{2}$ results from second-order virtual
processes due to $H_{1}$. We identify the low-energy manifold of states
numerically and diagonalize $\langle m|H^{\prime}_{V}|m^{\prime}\rangle$ for
several $N$. The resultant energy gap between the ground and the first excited
state is denoted by $\Delta E_{p}$; a plot of $(\Delta E_{p})N^{z}$ as a
function of $\lambda$ is shown in Fig. 7(b) for $z=1$. We find that it
indicates the presence of quantum phase transition belonging to Ising
universality at $\lambda_{c}\simeq 1.104\Delta$; the inset of Fig. 7(b) shows
the plot of $(\Delta E_{p})N^{z}$ as a function of
$(\lambda-\lambda_{c})N^{1/\nu}$ showing scaling collapse for
$\lambda_{c}\simeq 1.104$ for $z=\nu=1$. These results agree remarkably well
with ED and shows that the essence of the Ising transition is well captured by
the perturbation theory. We note that when $V_{{\bf r}{\bf r^{\prime}}}$ for
$|{\bf r}-{\bf r^{\prime}}|>1$ are neglected, $E_{m}^{0}$ becomes a $m$
independent constant leading to an exactly degenerate manifold with all states
having $n=1/4$. The presence of a finite $V_{{\bf r}{\bf r^{\prime}}}$ for
$|{\bf r}-{\bf r^{\prime}}|>1$ lifts this degeneracy and also allows the
ground state manifold near the transition to have a small admixture of states
with $n>1/4$ in the presence of finite $w$ near the transition.

The picture of the transition that emerges from the above analysis is as
follows. We note that Fig. 5(d) shows clear indication of near-degenerate low-
energy manifold of states. The presence of a finite $w$ leads to a ground
state which is an admixture of these low-energy nearly-degenerate manifold of
states arising out of a quantum order-by-disorder mechanism. This leads to a
unique ground state below a critical $\lambda=\lambda_{c}$; in contrast, for
$\lambda>\lambda_{c}$, the ground state breaks a ${\mathbb{Z}}_{2}$ symmetry
which necessitates the presence of an Ising critical point at $\lambda_{c}$.

To obtain an analytical estimate of the position of the critical point, we now
approximate the ordered ground state for $\lambda>\lambda_{c}$ to be given by
$|\psi_{G}\rangle$. This is not strictly correct since numerically we find
that the ordered state has a small admixture of states with higher $n$.
However, since this admixture is small, the estimate we obtain using this
approximation is expected to be qualitatively correct. We then create an
excited state given by $|\psi_{\rm ex}\rangle$ (Eq. 18). Since both
$|\psi_{G}\rangle$ and $|\psi_{\rm ex}\rangle$ are part of low-energy manifold
of states, we use Eq. 27 to compute $\langle H\rangle$ for them. A
straightforward computation, keeping terms up to fourth nearest-neighbors in
interaction ($|{\bf r}-{\bf r^{\prime}}|\leq\sqrt{5}$), shows that

$\displaystyle E_{\rm ex}^{V}$ $\displaystyle=$ $\displaystyle\langle\psi_{\rm
ex}|H^{\prime}_{v}|\psi_{\rm ex}\rangle=E_{G}^{V}+\frac{\delta
E_{1}^{V}}{2}+\delta E_{2}^{V},\quad
E_{G}^{V}=\langle\psi_{G}|H^{\prime}_{v}|\psi_{G}\rangle,$ $\displaystyle
E_{G}^{v}$ $\displaystyle=$ $\displaystyle
L\left(-(\Delta+\lambda)+\frac{V_{0}}{2^{6}}-\frac{w^{2}}{(\Delta-\lambda)+V_{0}/4}\right)+E_{0}$
$\displaystyle\delta E_{1}^{V}$ $\displaystyle\simeq$ $\displaystyle
2V_{0}n_{1}+\frac{w^{2}}{\frac{V_{0}}{4}+(\Delta-\lambda)},\quad\delta
E_{2}^{V}\simeq-\frac{w^{2}}{\Delta+\lambda},\quad
n_{1}=\left(\frac{1}{(\sqrt{5})^{6}}-\frac{1}{2^{6}}\right),$ (28)

where $E_{0}$ is the energy of the all-spin down state and we have neglected
terms ${\rm O}(w^{2}V_{0}$
$/(2^{6}(\Delta+\lambda)^{2}))$ and ${\rm
O}(w^{2}V_{0}/((\sqrt{5})^{6}(\Delta+\lambda)^{2}))$; these terms are always
small in the range of $V_{0}$ and $w$ we are interested in. The virtual
processes responsible for the $O(w^{2})$ term turns out to be same that in
Fig. 4; their amplitudes, however, change due to the presence of $V_{0}$. We
note that for $w=0$, $\delta E_{\rm ex}<0$ which means the ordered phase never
occurs in absence of the order-by-disorder mechanism realized via ${\rm
O}(w^{2})$ terms in $H^{\prime}_{V}$. Second, for $V_{0}=0$, $\delta
E_{1(2)}^{V}=\delta E_{1(2)}$ and we get back Eq. 19 for the excitation
energy. Equating $\delta E_{\rm ex}^{V}=0$, we finally find

$\displaystyle\lambda_{c}$ $\displaystyle=$
$\displaystyle\frac{V_{0}^{2}n_{1}+6w^{2}-\sqrt{(V_{0}n_{1}(8\Delta+V_{0})-2w^{2})^{2}+32w^{4}}}{8V_{0}n_{1}}.$
(29)

We note that $V_{0}=2\Delta$ and $w=0.1\Delta$, Eq. 29 yields
$\lambda_{c}\simeq 1.25\Delta$ which is close to the exact result obtained
from ED: $\lambda_{c}^{\rm exact}=1.12\Delta$. A plot of $\lambda_{c}$ as a
function of $V$ and $w$ shown in Fig. 7(a). Comparting the analytical result
(red solid line) with the numerical ones (blue dots), we find that the former
overestimates $\lambda_{c}$ by a small amount but provides similar dependence
of $\lambda_{c}$ on both $V$ and $w$. The numerical difference between the two
arises from the perturbative nature of the Van-Vleck result and also from the
approximate nature of the ordered ground state since we have neglected the
small admixture with states having $n>1/4$. We note that the presence of a
critical lower value of $w$ is clearly required for $\lambda_{c}<\lambda_{u}$;
for $w<w_{c}$, which can be estimated by equating $\lambda_{c}=\lambda_{u}$ in
Eq. 29, the transition does not occur.

Thus our analysis shows that the presence of a finite $V_{0}$ does not
obliterate the Ising transition. This allows for the possibility of concrete
realization of such a transition in experimentally relevant Rydberg atom
systems. Importantly, the perturbative analysis clearly indicates that the
transition is stabilized by a quantum order-by-disorder mechanism.

##  5 Discussion

In this work we have identified a quantum phase transition which is stabilized
by a quantum order-by-disorder mechanism in Rydberg ladders with staggered
detuning. Such a transition does not have any analogue in Rydberg chains
studied earlier. We have shown that this transition persists for a wide range
of vdW interaction strength between the Rydberg atoms. Our numerical studies
are supplemented by perturbative calculations which provides analytical
insight into the nature of the transition via identification of the competing
terms in the low-energy effective Hamiltonian of the system responsible for
it; moreover, they provide remarkably accurate estimate of the critical
detuning.

Our prediction can be tested using standard experiments on Rydberg atom
systems [25, 26]. The simplest geometry to consider would be the two-leg
ladder with $w\ll|\lambda|,\Delta$ and a finite $V_{0}$ within range shown in
Fig. 7(a). We envisage a situation where the detunings at the odd(even) sites
of the Rydberg chains, given by $\lambda_{\rm odd}(\lambda_{\rm even})$, are
[66]

$\displaystyle\lambda_{\rm even}$ $\displaystyle=$
$\displaystyle\lambda+\Delta,\quad\lambda_{\rm odd}=\lambda-\Delta.$ (30)

The Ising transition is then expected to occur, for example, for
$V_{0}=3\Delta$, at $\lambda_{c}\simeq 1.5\Delta$ which translates to
$\lambda_{\rm even}/\lambda_{\rm odd}\simeq 5$. These values can of course be
tuned by appropriate tuning of $V_{0}$ and $w$ within the specified range
discussed in Sec. 4. In the ordered phase, with $\lambda>\lambda_{c}$, all the
Rydberg excitations will preferentially happen on one of the two ladder
provided one breaks the $\mathbb{Z}_{2}$ symmetry by changing the detuning
slightly on any even site of the one of the chains. In contrast, the Rydberg
excitations in the same setting will be distributed on both chains for
$\lambda_{\rm even}/\lambda_{\rm odd}<5$. This change should be easily
detected by standard fluorescent imaging techniques used for detection of
Rydberg excitations in these systems [25, 26, 27, 28].

Finally, our analysis of the Rydberg ladder in the limit $V_{\bf r\bf
r^{\prime}}\ll w,\lambda,\Delta$ points towards an interesting class of spin
models where $\mathbb{Z}_{\ell_{0}}$ symmetry broken ground states and
associated quantum phase transitions are stabilized by an order-by-disorder
mechanisms. We note that this is in contrast to other spin models where a
critical point arises due to competition between a classical term and an
order-by-disorder induced term in the Hamiltonian [67]. For a ladder with
$\ell_{0}$ chains, as shown in App. B, a $\mathbb{Z}_{\ell_{0}}$ symmetry
broken ground state is realized for
$\lambda>\lambda_{c}=\Delta(\ell_{0}-1)/(\ell_{0}+1)$; in contrast, for
$\lambda<\lambda_{c}$, the ground state does not break any symmetry. This
leads to the realization of a critical point at $\lambda=\lambda_{c}$ which,
for example, belongs to three-state Potts university class for $\ell_{0}=3$.
However, unlike $\ell_{0}=2$, ladders with $\ell_{0}>2$ chains also require
periodic boundary conditions along the rung direction to stabilize non-Ising
transitions; this might be difficult to achieve in current experimental
setups. In conclusion we have studied the phase diagram of Rydberg ladders
with $\ell_{0}$ legs in the presence of staggered detuning. Our study
indicates that the low-energy behavior of such systems is described by class
of constrained spin models which support $\mathbb{Z}_{\ell_{0}}$ symmetry
broken ground states and associated emergent quantum criticality stabilized by
an order-by-disorder mechanism. For the two-leg ladder, we show, by
considering realistic vdW interactions, that the presence of such an
interaction do not obliterate the emergent Ising transition; this leads to the
possibility of its detection in standard experimental setup.

## Acknowledgement

MS thanks Roopayan Ghosh for discussions. KS thanks DST, India for support
through SERB project JCB/2021/000030.

##  Appendix A Variational wavefunction approach for $\lambda\simeq\Delta$

In this section, we shall address the ground state of the system around
$\lambda=\Delta$ and for $\lambda,\Delta\gg w$. To this end, we first note,
that for $\lambda\gg\Delta,w$, the ground state of the system prefers maximal
number of Rydberg excitations and can therefore be written as

$\displaystyle|\psi_{3}\rangle$ $\displaystyle=$
$\displaystyle\cos\phi_{1}|\psi_{3a}\rangle+\sin\phi_{1}|\psi_{3b}\rangle$
$\displaystyle|\psi_{3a}\rangle$ $\displaystyle=$
$\displaystyle\prod_{j=1}^{L/2}|\uparrow_{2j-1,1},\downarrow_{2j-1,2};\downarrow_{2j,1},\uparrow_{2j,2}\rangle,\quad|\psi_{3b}\rangle=|\downarrow_{2j-1,1},\uparrow_{2j-1,2};\uparrow_{2j,1},\downarrow_{2j,2}\rangle,$
(31)

where once again we have chosen $\phi_{1}$ to be independent of $j$. In
contrast, for $\lambda\leq\Delta$, the ground state prefers to have down-spins
on the odd sublattices. Thus one may choose the variational ground state
wavefunction to be
$|\psi_{4}(\phi_{2})\rangle\equiv|\psi_{4}\rangle=|\psi_{2}(\phi=\phi_{2})\rangle$
where $|\psi_{2}\rangle$ is given by Eq. 8. Using these wavefunctions, one can
therefore construct a variational wavefunction given by

$\displaystyle|\Phi_{v}\rangle$ $\displaystyle=$
$\displaystyle\cos\beta|\psi_{3}\rangle+\sin\beta|\psi_{4}\rangle,$ (32)

where $0\leq\beta\leq\pi/2$. The corresponding variational energy can be
computed in a straightforward manner and yields

$\displaystyle E_{1v}$ $\displaystyle=$
$\displaystyle\langle\Phi_{v}|H_{2}|\Phi_{v}\rangle=E_{0}+(\lambda-\Delta)\cos^{2}\beta-w\sin(2\beta)\cos(\phi_{1}-\phi_{2}+\pi).$
(33)

The minimization of $E_{1v}$ leads to

$\displaystyle\phi_{1}$ $\displaystyle=$
$\displaystyle\phi_{2}+\pi,\quad\beta_{0}=\frac{1}{2}\arctan\left[\frac{2w}{\Delta-\lambda}\right].$
(34)

Next, we consider the change in expectation values of the operators $\hat{n}$
and $s_{x}$ across the transition. To this end, we note that using Eqs. 32 and
34, we find

$\displaystyle\langle\hat{n}\rangle_{v}$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{1}{4}\cos^{2}\beta_{0}=\frac{1}{2}-\frac{1}{8}\left(1-\frac{\lambda-\Delta}{\sqrt{(\lambda-\Delta)^{2}+4w^{2}}}\right),$
$\displaystyle\langle s^{\prime}_{x}\rangle_{v}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\langle\sum_{j=1}^{L}\sum_{\ell=1}^{2}\langle\sigma^{x}_{2j-1,\ell}\rangle=\frac{\sin
2\beta_{0}}{4}=\frac{w}{2\sqrt{(\lambda-\Delta)^{2}+4w^{2}}}.$ (35)

These expectations values, can also be computed using the exact ground state
obtained from ED. The two results show excellent match similar to their
counterparts near $\lambda\sim-\Delta$ discussed in the main text. The plot
for $n$ near $\lambda\sim\Delta$, shown in Fig. 2(a), corroborates the above
statement. We note here that the two states $|\psi_{3}\rangle$ and
$|\psi_{4}\rangle$ belong to two different $\mathbb{Z}_{2}$ orders
corresponding to $n=1/4$ and $n=1/2$ respectively. Thus it is natural to
expect a first order transition between them around $\lambda\simeq\Delta$.
However, our numerics from ED seems to indicate a crossover which can be
inferred from $N$ independence of the data. This is most likely a consequence
of the presence of other sector states with fixed $n$. These states do not
feature in this simple minded variational wavefunction computation and their
presence may change this transition to a crossover.

##  Appendix B $\ell_{0}$-leg ladder

Figure 8: (a) Schematic representation of the Rydberg ladder for $\ell_{0}=3$
showing the Rydberg blockade radius (rectangular region) around a Rydberg
excitation (red circle). The blue circles denote sites with blocked Rydberg
excitation. The black circle at $\lambda\sim 0.5$ shows the position of the
quantum phase transition.(b) The plot of Rydberg excitation density $n$ for a
$\ell_{0}=3$ leg ladder as a function of $\lambda$. The red dotted line
corresponds to $w=0$ showing first order transition between the plateaus. The
blue solid lines correspond to $w=0.05$. The black circle shows the position
of the quantum phase transition. All energies are scaled in units of $\Delta$.
See text for details.

In this section, we consider the model $H_{\rm eff}$ for $\ell_{0}$-legged
ladder. For $\ell_{0}>2$, such ladders are difficult to realize in an
experimentally relevant Rydberg atom arrays since periodic boundary conditions
is required along the rung direction. However we study such model Hamiltonians
here since they support phase transitions with non-Ising universality that are
stabilized by order-by-disorder mechanism. In what follows, we shall provide a
general expression for the low-energy effective Hamiltonian
$H^{\prime}_{\ell_{0}}$ for a ladder with $\ell_{0}$ legs. We shall also
provide numerical evidence of the transition for $\ell_{0}=3$ legged ladder.

To this end, we consider an arrangement of $\ell_{0}$ Rydberg chains each
having $2L$ sites so that $N=2L\ell_{0}$ as shown in Fig. 8(a) for
$\ell_{0}=3$. As shown in this figure, we consider a situation where the
presence of a Rydberg excited atom on any site precludes the presence of
another such excitation in all of the $\ell_{0}-1$ sites of the same rung of
the ladder. The model Hamiltonian that we shall consider for these chains is a
generalization for $H_{\rm eff}$ (Eq. 6) and is given by

$\displaystyle H$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{2L}\sum_{\ell=1}^{\ell_{0}}\left(w\tilde{\sigma}_{j,\ell}^{x}-\frac{1}{2}[\Delta(-1)^{j}+\lambda]\sigma_{j,\ell}^{z}\right),\quad\tilde{\sigma}^{x}_{j,\ell}=P_{j-1,\ell}\left(\prod_{\ell^{\prime}\neq\ell}P_{j,\ell^{\prime}}\right)\sigma^{x}_{j,\ell}P_{j+1,\ell}.$
(36)

The phase diagram for such chains for $w=0$ can be obtained in a
straightforward manner. For $\lambda<0$, and $|\lambda|\gg\Delta$, $n=0$.
Around $|\lambda|\simeq\Delta$, it becomes energetically favorable to create
one Rydberg excitation on one of the $\ell_{0}$ sites of an even rung. This
leads to a classical ground state with $n_{0}=1/(2\ell_{0})$; the ground state
is macroscopically degenerate since there are $\ell_{0}^{L}$ Fock states with
$n=1/(2\ell_{0})$. For $\lambda>\Delta$, the ground state corresponds to
maximum possible Rydberg excitations leading to $n=1/\ell_{0}$; it exhibits a
$\ell_{0}$ fold degeneracy. For $w=0$, there are first-order transitions
between these ground states at $\lambda/\Delta\simeq\pm 1$; for finite $w$,
these are related by crossovers. The representative phase diagram, for
$\ell_{0}=3$, is shown in Fig. 8(b). Our numerical analysis also detects, for
$\ell_{0}=3$, a quantum transition within the $n=1/6$ plateau at
$\lambda_{c}/\Delta\simeq 0.52$. A finite-sized scaling analysis near of the
energy gap $\Delta E$, as shown in Fig. 9(a) and 9(b) indicates $z=1$ and
$\nu=5/6\simeq 0.83$ for the transition. Furthermore we define an order
parameter $\hat{O}_{\ell_{0}}=\sum_{j=1}^{L}\sum_{\ell=1}^{\ell_{0}}\exp[2\pi
i\ell/\ell_{0}]\sigma_{2j,\ell}^{z}$ and computes its correlation function for
the $\ell_{0}=3$ chain: $S=\langle\hat{O}_{3}^{\ast}\hat{O}_{3}\rangle/N$. We
find that, using a finite sized scaling analysis, $S\sim N^{2-z-\eta}$ with
$\eta=4/15\simeq 0.267$ and $z=1$ as shown in Figs. 9(c) and 9(d). Thus we
conclude that the transition belongs to the 2D three-state Potts universality
class.

An analytic insight into this transition is provided by a calculation similar
to that outlined in the main text leading to Eq. 17. For an $\ell_{0}$-leg the
effective Hamiltonian receives contribution from processes which are similar
to those shown in Fig. 4 (generalized for $\ell_{0}$ legs) and is given by
[57, 65]

$\displaystyle H_{\rm
eff}^{\ell_{0}}=\frac{-w^{2}}{\Delta-\lambda}\sum_{j=1}^{L}\Big{[}\sum_{\ell=1}^{\ell_{0}}P_{2j,\ell}P_{2j+2,\ell}+\frac{\alpha_{0}}{2}\sum_{\ell,\ell^{\prime}=1,\ell_{0};\ell\neq\ell^{\prime}}P_{2j,\ell^{\prime}}(\sigma^{x}_{2j,\ell}\sigma^{x}_{2j,\ell^{\prime}}+\sigma^{y}_{2j,\ell}\sigma^{y}_{2j,\ell^{\prime}})\Big{]},$
(37)

where $\alpha_{0}=(\Delta-\lambda)/(\Delta+\lambda)$. The emergence of the
phase transition at a critical value of $\lambda/\Delta$ can once again be
understood by noting the competition between the two terms in Eq. 37. The
first term prefers a $\mathbb{Z}_{\ell_{0}}$ symmetry broken ground state with
all Rydberg excitations on any one of the $\ell_{0}$ chains; in contrast, the
second term prefers a linear superposition of all states with one Rydberg
excitation on any site of even rungs of the ladder. This necessitates an
intermediate quantum critical point belonging to $\mathbb{Z}_{\ell_{0}}$
universality class.

Figure 9:  Exact numerics confirming the presence of the three-state Potts
transition for $\ell_{0}=3$. (a): Plot of $(\Delta E)N^{z}$ for a three-leg
ladder as a function of $\lambda$ for several $N$ showing a crossing at
$\lambda=\lambda_{c}\simeq 0.501$ for $z=1$. (b) Plot of $(\Delta E)N^{z}$ as
a function of $N^{1/\nu}(\lambda-\lambda_{c})$ showing perfect scaling
collapse for $z=1$ and $\nu\simeq 1/1.2=0.83$.(c): Plot of $SN^{2-z-\eta}$ as
a function of $\lambda$ showing a crossing at $\lambda_{c}\simeq 0.502$ for
$\eta\simeq 0.27$. (d) Plot of $SN^{2-z-\eta}$ as a function of
$\lambda-\lambda_{c})N^{1/\nu}$ showing a scaling collapse for $\nu=1/1.2$ and
$\eta=0.27$. All energies are scaled in units of $\Delta$. See text for
details.

To estimate the position of this critical point, we once again consider an
excited state over the $\mathbb{Z}_{\ell_{0}}$ symmetry broken ground state
$|\psi^{\prime}_{G}\rangle$ given by

$\displaystyle|\psi^{\prime}_{G}\rangle$ $\displaystyle=$
$\displaystyle|\downarrow_{1,1}\downarrow_{1,2}..\downarrow_{1,\ell_{0}};\uparrow_{2,1}\downarrow_{2,2}..\downarrow_{2,\ell_{0}};....\uparrow_{2j,1}\downarrow_{2j,2}..\downarrow_{2j,\ell_{0}}...;\uparrow_{2L,1}\downarrow_{2L,2}..\downarrow_{2\ell_{0}}\rangle.$
(38)

Such an excited state can be created by constructing a linear superposition of
states with a Rydberg excitation on any one of the sites of one of the even
rungs of the ladder for which $j^{\prime}=2j$; for all $j^{\prime}\neq 2j$,
the Rydberg excitation reside at $\ell_{0}=1$. Such an excited state can be
represented as

$\displaystyle|\psi^{\prime}_{\rm ex}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\ell_{0}}}\left(|\downarrow_{1,1}\downarrow_{1,2}..\downarrow_{1,\ell_{0}};\uparrow_{2,1}\downarrow_{2,2}..\downarrow_{2,\ell_{0}};....\uparrow_{2j,1}\downarrow_{2j,2}..\downarrow_{2j,\ell_{0}}...;\uparrow_{2L,1}\downarrow_{2L,2}..\downarrow_{2\ell_{0}}\rangle\right.$
$\displaystyle\left.+|\downarrow_{1,1}\downarrow_{1,2}..\downarrow_{1,\ell_{0}};\uparrow_{2,1}\downarrow_{2,2}..\downarrow_{2,\ell_{0}};....\downarrow_{2j,1}\uparrow_{2j,2}..\downarrow_{2j,\ell_{0}}...;\uparrow_{2L,1}\downarrow_{2L,2}..\downarrow_{2\ell_{0}}\rangle\right.$
$\displaystyle\left.+....|\downarrow_{1,1}\downarrow_{1,2}..\downarrow_{1,\ell_{0}};\uparrow_{2,1}\downarrow_{2,2}..\downarrow_{2,\ell_{0}};....\downarrow_{2j,1}\downarrow_{2j,2}..\uparrow_{2j,\ell_{0}}...;\uparrow_{2L,1}\downarrow_{2L,2}..\downarrow_{2\ell_{0}}\rangle\right),$

The energy cost of creating such an excited state can be easily computed
following the method outlined in the main text and leads to $\delta E_{\rm
ex}^{\ell_{0}}=\delta E_{1}^{\ell_{0}}+\delta E_{2}^{\ell_{0}}$ where

$\displaystyle\delta E_{1}^{\ell_{0}}$ $\displaystyle=$
$\displaystyle\frac{w^{2}(1-1/\ell_{0})}{\Delta-\lambda},\quad\delta
E_{2}^{\ell_{0}}=-\frac{w^{2}(\ell_{0}-1)}{\Delta+\lambda}.$ (40)

The energy gap vanishes when $\delta E^{\ell_{0}}_{\rm ex}=0$ which leads to
an estimate of the critical point given by

$\displaystyle\lambda_{c}=\Delta(\ell_{0}-1)/(\ell_{0}+1).$ (41)

For $\ell_{0}=2$, this reproduces Eq. 20 of the main text; for $\ell_{0}=3$,
it provide an estimate of the three-state Potts critical point to be
$\lambda_{c}=\Delta/2$ which matches quite well with exact numerical results
shown Fig. 9.

Before ending this section, we note that we expect that for $\ell_{0}>4$, the
transition shall be first order; however we have not been able to carry out ED
studies on sufficiently large system size to ascertain this due to increased
dimension of the constrained Hilbert space.

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