# Gapless deconfined phase in a $\mathbb{Z}_{N}$ symmetric Hamiltonian created
in a cold-atom setup

Mykhailo V. Rakov  Instytut Fizyki Teoretycznej, Uniwersytet Jagielloński,
ulica Łojasiewicza 11, 30-348 Kraków, Poland  Luca Tagliacozzo  Instituto de
Fisica Fundamental IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain  Maciej
Lewenstein  Institut de Ciencies Fotoniques (ICFO), Av. Karl Friedrich Gauss
3, 08860 Barcelona, Spain  ICREA, Pg. Lluís Companys 23, 08010 Barcelona,
Spain  Jakub Zakrzewski  Instytut Fizyki Teoretycznej, Uniwersytet
Jagielloński, ulica Łojasiewicza 11, 30-348 Kraków, Poland  Mark Kac Complex
Systems Research Center, Uniwersytet Jagielloński,30-348 Kraków, Poland  Titas
Chanda  Department of Physics, Indian Institute of Technology Indore, Khandwa
Road, Simrol, Indore 453552, India

###### Abstract

We investigate a quasi-two-dimensional system consisting of two species of
alkali atoms confined in a specific optical lattice potential [Phys. Rev. A
95, 053608 (2017)]. In the low-energy regime, this system is governed by a
unique $\mathbb{Z}_{N}$ gauge theory, where field theory arguments have
suggested that it may exhibit two exotic gapless deconfined phases, namely a
dipolar liquid phase and a Bose liquid phase, along with two gapped (confined
and deconfined) phases. We address these predictions numerically by using
large-scale density matrix renormalization group simulations. Our findings
provide conclusive evidence for the existence of a gapless Bose liquid phase
for $N\geq 7$. We demonstrate that this gapless phase shares the same critical
properties as one-dimensional critical phases, resembling weakly coupled
chains of Luttinger liquids. In the range of geometries and $N$ considered,
the gapless dipolar phase predicted theoretically is still elusive and its
characterization will probably require a full two-dimensional treatment.

##  I Introduction

Gauge-invariant theories form the bedrock of our current understanding of the
fundamental forces of nature, describing three of the four known forces:
electromagnetic, weak, and strong interactions [1, 2, 3, 4]. They are also
crucial for the low-energy description of topological order in certain
materials within condensed matter physics [5, 6, 7]. While their Lagrangian
formulations on lattices [8, 9, 10, 11, 12, 13, 14] are well-established
through symmetry principles, our understanding of their phase diagrams remains
incomplete, particularly in regimes of strong correlation where traditional
Monte Carlo lattice methods (for a U.S.-centric perspective on the topic, see
[15]) are ineffective. This limitation poses a significant hindrance in the
study of nuclear matter at high fermionic densities, which is crucial for
understanding of the interiors of neutron stars. Besides finite fermionic
densities, other significant challenges include real-time dynamics which are
necessary for understanding the scattering processes in heavy-ion collisions,
and topological terms, which could provide insights into the asymmetry between
matter and antimatter in the universe.

In light of these challenges, the quantum simulation community, following
Feynman’s intuition [16], has begun exploring methods to simulate gauge
theories using ultracold atoms, trapped ions, and superconducting qubits [17,
18, 19, 20, 21, 22, 23, 24]. Recent advancements have refined many of these
initial proposals, and the first experimental realizations of one-dimensional
gauge systems have been achieved [25, 26, 27, 28, 29, 30, 31]. These
experiments include demonstrations of local gauge invariance in large Bose-
Hubbard-like systems. Most of these proposals rely on the microscopic
realization of gauge theories, enforcing local gauge invariance and protecting
it from environmental effects. However, such approaches face significant
challenges, as ultracold atoms are typically dilute and neutral, and their
interactions are limited to two-body interactions. Consequently, implementing
the higher-body interactions necessary for gauge Hamiltonians in two or higher
dimensions is difficult. Even though such higher-order interactions are hard
to obtain, advanced experimental platforms, such as Rydberg atoms, where
tweezers can manipulate atoms sequentially to achieve higher-order
interactions through a series of two-body interactions in the presence of the
so-called Rydberg-blockade can provide access to them [18, 19, 32, 33, 34, 35,
36].

A different approach for simulating gauge theories is based on the observation
that many condensed matter systems can be described by gauge theories at low
energies. This emergent realization of gauge theories contrasts with the
microscopic implementations discussed above. Examples of such systems include
topological spin liquids in highly frustrated magnets and quantum Hall states
[37].

This alternative approach was first explored in Ref. [38], where the authors
proposed an experimental protocol for realizing emergent Abelian lattice gauge
theories using a mixture of two species of ultracold alkali atoms trapped on
an optical lattice. A generalized Bose-Hubbard Hamiltonian can be created by
fast modulation of interspecies interactions, giving rise to exotic,
anisotropic gauge theories at low energies. In two dimensions (2D), such
exotic gauge theories have phase diagrams that differ significantly from their
one-dimensional counterparts.

While Ref. [38] presented the experimental protocol and a qualitative phase
diagram, our work revisits the same systems, providing quantitative results on
some phases using tensor network (TN) [39, 40, 41] simulations. In recent
years, TN methods, free from the limitations associated with traditional Monte
Carlo approaches, have excelled in low-dimensional lattice gauge theory
calculations [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57,
58]. In this work, we consider infinite cylinders with different finite radii,
characterizing the liquid phases with emergent discrete $\mathbb{Z}_{N}$
symmetry and identifying phase boundaries between topological and trivial
phases for various $N$ and cylinder lengths, corroborating the predictions
from Ref. [38]. Specifically, we demonstrate that for $\mathbb{Z}_{N}$
emergent gauge theories with $N\leq 6$, the system exhibits two gapped phases:
the confined and deconfined phases, with the latter possessing non-trivial
topological order. However, for $N\geq 7$, a gapless liquid (deconfined) phase
emerges between these gapped phases. Interestingly, we find that this gapless
phase exhibits critical properties akin to one-dimensional (1D) Bose liquids,
characterized by weakly coupled chains of identical Luttinger liquids.

The paper is organized as follows. In Sec. II we introduce the two-species
cold-atomic setup and describe the emergent $\mathbb{Z}_{N}$ gauge theories
due to inter-species dynamics in the low-energy description. We also present
the global $\mathbb{Z}_{N}$-symmetric model in the dual lattice. In Sec. III,
we examine the phase diagram of the system for different quasi-2D ladder and
cylinder geometries through the lens of entanglement entropy and correlation
length. In Sec. IV, we analyze the gapless deconfined phase by examining
different correlation lengths and the scaling of entanglement entropy.
Finally, we draw our conclusion in Sec. V.

Figure 1: (Color online.) The schematic depiction of the system considered in
this work. (a) The two-species Bose-Hubbard system in a square superlattice
described by Eq. (1). The nearest-neighbor tunneling of the $a$-bosons along
the $y$-direction is modulated by the occupation of the $b$-bosons on the
neighboring sites. (b) The effective Hamiltonian (Eq. (3) or (9)) on the
coarse-grained lattice where two plaquettes along the $x$ direction are
combined into one. Here the relevant degrees of freedom sit on the centers of
the $2\times 1$ square as depicted by the red circles. The blue, green, and
purple arrows depict the two-, three-, and four-plaquette interactions as
described by the second, third, and fourth terms respectively in Eq. (3) or
(9). (c) The effective $\mathbb{Z}_{N}$ gauge theory described by Eq. (6) or
(10) obtained via duality transformation as mentioned in the text. The wiggly
lines describe the gauge or electric-field degrees of freedom located on the
bonds of the coarse-grained lattice. The red arrows denote the magnetic
plaquette term (the first term in Eq. (6) or (10)), while the green and purple
arrows describe the two-body interaction along the $y$-direction given by the
third and fourth terms respectively in the Hamiltonian (6) or (10).

##  II The System

Following the approach presented in Ref. [38], we consider emergent Abelian
lattice gauge theories within the low-energy manifold of two-species cold-atom
dynamics in 2D. The setup involves two bosonic species, referred to as $a$ and
$b$ bosons, on a 2D optical superlattice. A specific configuration of link-
dependent tunneling for the $a$-bosons and the fast modulation of the
interactions between the two species produces the desired effect. Namely, the
$b$-bosons behave as gauge bosons at low energies. The Hamiltonian describing
the combined behavior of the $a$ and $b$ bosons is given by a generalization
of the Bose-Hubbard model:

$\displaystyle\hat{H}_{BH}=$
$\displaystyle-J^{b}\sum_{\mathbf{j},\hat{\delta}}\left(\hat{b}^{\dagger}_{\mathbf{j}}\hat{b}_{\mathbf{j}+\hat{\delta}}+\text{h.c.}\right)-J_{1x}^{a}\sum_{j_{x}\in\text{odd}}\left(\hat{a}^{\dagger}_{\mathbf{j}}\hat{a}_{\mathbf{j}+\hat{x}}+\text{h.c.}\right)$
$\displaystyle-J^{a}_{y}\sum_{j_{y}}\left(\hat{a}^{\dagger}_{\mathbf{j}}e^{i\frac{2\pi}{N}(\hat{n}^{b}_{\mathbf{j}+\hat{y}}-\hat{n}^{b}_{\mathbf{j}})}\hat{a}_{\mathbf{j}+\hat{y}}+\text{h.c.}\right)$
$\displaystyle+\frac{U}{2}\sum_{\mathbf{j}}(\hat{a}_{\mathbf{j}})^{2}\hat{a}_{\mathbf{j}},$
(1)

where $N$ is a positive integer. For the schematic depiction of the system,
see Fig. 1(a). The first term in the Hamiltonian describes the nearest-
neighbor tunneling of the $b$-bosons, which is uniform in both the $x$ and $y$
directions with an amplitude $J^{b}$ (here $\hat{\delta}=\hat{x}$ or
$\hat{y}$). The second term represents the nearest-neighbor tunneling of the
$a$-bosons along the $x$-direction with an amplitude $J^{a}_{1x}$ between
sites $\mathbf{j}$ and $\mathbf{j}+\hat{x}$ for odd $j_{x}$. Here, the
tunneling amplitude of $a$-bosons between sites $\mathbf{j}$ and
$\mathbf{j}+\hat{x}$ for even $j_{x}$ is considered vanishingly small compared
to the other terms in the Hamiltonian, and that is why it is neglected. The
third term describes the tunneling of $a$-bosons in the $y$-direction with an
amplitude $J^{a}_{y}$, which is modulated by the occupation of $b$-bosons on
the neighboring sites (see Fig. 1(a)). The final term denotes the onsite
repulsion between $a$-bosons. Here, we consider the limit $U\gg
J^{a}_{1x},J^{a}_{y}$, so that at half-filling, the $a$-bosons effectively
behave as hardcore bosons. Moreover, the onsite repulsion between $b$-bosons
is considered to be vanishingly small, so we have a large occupation of the
$b$-bosons (i.e., $\bar{n}_{b}\gg 1$) at each site.

In the limit $J^{a}_{y}\ll J^{a}_{1x}$, the dynamics of $a$-bosons creates
effective plaquette interactions for the $b$-bosons in a superlattice where
plaquette $\mathbf{p}$ (red shadowed region in Fig. 1(b)) contains the sites
$\mathbf{j}$, $\mathbf{j}+\hat{x}$, $\mathbf{j}+\hat{x}+\hat{y}$,
$\mathbf{j}+\hat{y}$ for odd $j_{x}$ (i.e., every second plaquette of the
original lattice, for details, see [38]). We define a plaquette operator
$\hat{\mathcal{B}}_{\mathbf{p}}$ as

$\hat{\mathcal{B}}_{\mathbf{p}}=\frac{2\pi}{N}\left(\hat{n}^{b}_{\mathbf{j}}-\hat{n}^{b}_{\mathbf{j}+\hat{x}}+\hat{n}^{b}_{\mathbf{j}+\hat{x}+\hat{y}}-\hat{n}^{b}_{\mathbf{j}+\hat{y}}\right),$
(2)

with $\hat{n}^{b}_{\mathbf{j}}$ being the number operator for $b$-bosons, so
that these $\hat{\mathcal{B}}_{\mathbf{p}}$ operators have eigenvalues
$\left\\{\frac{2\pi n}{N}\right\\}$ with
$n\in\left[-\frac{(N-1)}{2},-\frac{(N-2)}{2},\dots,\frac{(N-2)}{2},\frac{(N-1)}{2}\right]$.
In the second order perturbation theory, the effective Hamiltonian of the
$b$-bosons dressed by the dynamics of $a$-bosons reads as (Fig. 1(b)):

$\displaystyle\hat{H}_{P}$
$\displaystyle=-2\sum_{\mathbf{p}}\cos\hat{\mathcal{B}}_{\mathbf{p}}-2g^{2}\sum_{\mathbf{p}}\Bigg{[}\cos\left(\frac{4\pi}{N}\left(\hat{\mathcal{L}}_{\mathbf{p}}-\hat{\mathcal{L}}_{\mathbf{p}-\hat{y}}\right)\right)$
$\displaystyle+2\cos\left(\frac{2\pi}{N}\left(2\hat{\mathcal{L}}_{\mathbf{p}}-\hat{\mathcal{L}}_{\mathbf{p}+\hat{y}}-\hat{\mathcal{L}}_{\mathbf{p}-\hat{y}}\right)\right)$
$\displaystyle+\cos\left(\frac{2\pi}{N}\left(\hat{\mathcal{L}}_{\mathbf{p}}-\hat{\mathcal{L}}_{\mathbf{p}+\hat{x}}-\hat{\mathcal{L}}_{\mathbf{p}+\hat{y}}+\hat{\mathcal{L}}_{\mathbf{p}+\hat{x}+\hat{y}}\right)\right)\Bigg{]},$
(3)

where the coupling constant $g$ depends on $b$-boson filling as well as
tunneling amplitudes of both kinds of bosons as
$g^{2}=\frac{\bar{n}_{b}}{2}\frac{J^{a}_{1x}}{J^{b}}$. Here the plaquette
operator $\hat{\mathcal{L}}_{\mathbf{p}}$ satisfies the following algebra:

$\displaystyle\left[\hat{\mathcal{L}}_{\mathbf{p}},e^{\mp
i\hat{\mathcal{B}}_{\mathbf{p}}}\right]$ $\displaystyle=\pm e^{\mp
i\hat{\mathcal{B}}_{\mathbf{p}}},$
$\displaystyle\left[\hat{\mathcal{B}}_{\mathbf{p}},e^{\pm
i\frac{2\pi}{N}\hat{\mathcal{L}}_{\mathbf{p}}}\right]$
$\displaystyle=\pm\frac{2\pi}{N}e^{\pm
i\frac{2\pi}{N}\hat{\mathcal{L}}_{\mathbf{p}}},$ (4)

i.e., $e^{\pm i\frac{2\pi}{N}\hat{\mathcal{L}}_{\mathbf{p}}}$ fulfills the
role of $\mathbb{Z}_{N}$ ladder operators for the eigenstates of
$\hat{\mathcal{B}}_{\mathbf{p}}$, and conversely $e^{\mp
i\hat{\mathcal{B}}_{\mathbf{p}}}$ for the eigenstates of
$\hat{\mathcal{L}}_{\mathbf{p}}$. It is easy to check that the above
Hamiltonian (3) has a global $\mathbb{Z}_{N}$ symmetry corresponding to
$\left[\prod_{\mathbf{p}}e^{i\hat{\mathcal{B}}_{\mathbf{p}}},\hat{H}_{P}\right]=\left[\prod_{\mathbf{p}}e^{-i\hat{\mathcal{B}}_{\mathbf{p}}},\hat{H}_{P}\right]=0$.
In the limit $N\rightarrow\infty$, we get the global $U(1)$ symmetry.

We note here that the Hamiltonian (3) involves every second plaquette of the
original lattice. To simplify, we can coarse-grain the lattice by grouping two
plaquettes, as illustrated in Fig. 1(b). In this coarse-grained lattice, we
consider the plaquette degrees of freedom to be located at the centers of
these $2\times 1$ squares (red circles in Fig. 1(b)), describing a dual
lattice to the original system.

To arrive at the $\mathbb{Z}_{N}$ gauge theory, we can invert the duality
transformation by defining the electric-field operators on the link
$(\mathbf{j},\hat{x})$ between the sites $\mathbf{j}$ and $\mathbf{j}+\hat{x}$
in the coarse-grained lattice as
$\hat{\mathcal{E}}_{(\mathbf{j},\hat{x})}=\hat{\mathcal{L}}_{\mathbf{p}}-\hat{\mathcal{L}}_{\mathbf{p}-\hat{y}}$
(see Fig. 1(c)). Similarly,
$\hat{\mathcal{E}}_{(\mathbf{j},\hat{y})}=-\hat{\mathcal{L}}_{\mathbf{p}}+\hat{\mathcal{L}}_{\mathbf{p}-\hat{x}}$
on the link $(\mathbf{j},\hat{y})$ between the site $\mathbf{j}$ and
$\mathbf{j}+\hat{y}$. On the other hand, $e^{i\hat{\mathcal{B}}_{\mathbf{p}}}$
can be redefined as
$e^{i\hat{\mathcal{B}}_{\mathbf{p}}}\equiv\hat{U}_{(\mathbf{j},\hat{x})}\hat{U}_{(\mathbf{j}+\hat{x},\hat{y})}\hat{U}_{(\mathbf{j}+\hat{y},\hat{x})}^{\dagger}\hat{U}_{(\mathbf{j},\hat{y})}^{\dagger}$,
where following commutation relations are maintained:

$\displaystyle\left[\hat{\mathcal{E}}_{(\mathbf{j},\hat{\delta})},\hat{U}_{(\mathbf{j},\hat{\delta})}\right]$
$\displaystyle=-\hat{U}_{(\mathbf{j},\hat{\delta})},$
$\displaystyle\left[\hat{\mathcal{E}}_{(\mathbf{j},\hat{\delta})},\hat{U}_{(\mathbf{j},\hat{\delta})}^{\dagger}\right]$
$\displaystyle=\hat{U}_{(\mathbf{j},\hat{\delta})}^{\dagger}.$ (5)

The Hamiltonian for the $\mathbb{Z}_{N}$ gauge theory, derived through the
above duality transformation, is then written as

$\displaystyle\hat{H}_{G}=$
$\displaystyle-\sum_{\mathbf{j}}\left(\hat{U}_{(\mathbf{j},\hat{x})}\hat{U}_{(\mathbf{j}+\hat{x},\hat{y})}\hat{U}_{(\mathbf{j}+\hat{y},\hat{x})}^{\dagger}\hat{U}_{(\mathbf{j},\hat{y})}^{\dagger}+\text{h.c.}\right)$
$\displaystyle-2g^{2}\sum_{\mathbf{j}}\Bigg{[}\cos\left(\frac{4\pi}{N}\hat{\mathcal{E}}_{(\mathbf{j},\hat{x})}\right)$
$\displaystyle+2\cos\left(\frac{2\pi}{N}\left(\hat{\mathcal{E}}_{(\mathbf{j},\hat{x})}-\hat{\mathcal{E}}_{(\mathbf{j}+\hat{y},\hat{x})}\right)\right)$
$\displaystyle+\cos\left(\frac{2\pi}{N}\left(\hat{\mathcal{E}}_{(\mathbf{j},\hat{y})}-\hat{\mathcal{E}}_{(\mathbf{j}+\hat{y},\hat{y})}\right)\right)\Bigg{]}.$
(6)

Here the first term corresponds to the magnetic plaquette term, while others
are for interactions between the electric fields. The $\mathbb{Z}_{N}$ Gauss
law corresponding to the local $\mathbb{Z}_{N}$ invariance is given by

$\sum_{\hat{\delta}=\hat{x},\hat{y}}\left(\hat{\mathcal{E}}_{(\mathbf{j},\hat{\delta})}-\hat{\mathcal{E}}_{(\mathbf{j}-\hat{\delta},\hat{\delta})}\right)=0.$
(7)

In terms of standard $\mathbb{Z}_{N}$ operators $\hat{P}$ and $\hat{Q}$,
defined as

$\displaystyle\hat{P}=\sum_{n=0}^{N-1}e^{i\frac{2\pi}{N}n}\ket{n}\bra{n},\quad\hat{Q}=\sum_{n=0}^{N-1}\ket{n}\bra{n\text{
mod }N},$ (8)

and satisfying $\hat{Q}\hat{P}=e^{i\frac{2\pi}{N}}\hat{P}\hat{Q}$, the
Hamiltonian (3) in the dual lattice and the $\mathbb{Z}_{N}$ gauge theory
Hamiltonian in Eq. (6) can be rewritten as

$\displaystyle\hat{H}_{P}=-\sum_{\mathbf{p}}\Biggr{[}$
$\displaystyle\hat{P}_{\mathbf{p}}+g^{2}\biggr{(}\hat{Q}^{2}_{\mathbf{p}}\left(\hat{Q}^{\dagger}_{\mathbf{p}-\hat{y}}\right)^{2}+2\,\hat{Q}^{2}_{\mathbf{p}}\hat{Q}^{\dagger}_{\mathbf{p}+\hat{y}}\hat{Q}^{\dagger}_{\mathbf{p}-\hat{y}}$
$\displaystyle+\hat{Q}_{\mathbf{p}}\hat{Q}^{\dagger}_{\mathbf{p}+\hat{x}}\hat{Q}^{\dagger}_{\mathbf{p}+\hat{y}}\hat{Q}_{\mathbf{p}+\hat{x}+\hat{y}}\biggr{)}\Biggr{]}+\text{h.c.},$
(9) $\displaystyle\hat{H}_{G}=-$
$\displaystyle\sum_{\mathbf{j}}\Biggr{[}\hat{Q}_{(\mathbf{j},\hat{x})}\hat{Q}_{(\mathbf{j}+\hat{x},\hat{y})}\hat{Q}_{(\mathbf{j}+\hat{y},\hat{x})}^{\dagger}\hat{Q}_{(\mathbf{j},\hat{y})}^{\dagger}$
$\displaystyle+g^{2}\biggr{(}\hat{P}_{(\mathbf{j},\hat{x})}^{2}+2\,\hat{P}_{(\mathbf{j},\hat{x})}\hat{P}_{(\mathbf{j}+\hat{y},\hat{x})}^{\dagger}+\hat{P}_{(\mathbf{j},\hat{y})}\hat{P}_{(\mathbf{j}+\hat{y},\hat{y})}^{\dagger}\biggr{)}\Biggr{]}$
$\displaystyle+\text{h.c.},$ (10)

respectively. Here, in Eq. (9), we identify $\hat{P}_{\mathbf{p}}\equiv
e^{i\hat{\mathcal{B}}_{\mathbf{p}}}$ and $\hat{Q}_{\mathbf{p}}\equiv
e^{i\frac{2\pi}{N}\hat{\mathcal{L}}_{\mathbf{p}}}$, while in Eq. (10) we use
$\hat{P}_{(\mathbf{j},\hat{\delta})}\equiv
e^{i\frac{2\pi}{N}\hat{\mathcal{E}}_{(\mathbf{j},\hat{\delta})}}$ and
$\hat{Q}_{(\mathbf{j},\hat{\delta})}\equiv\hat{U}_{(\mathbf{j},\hat{\delta})}$.
Unless stated otherwise, in the rest of the paper we will consider Hamiltonian
$\hat{H}_{P}$ (i.e., Eq. (9)) in the dual lattice that possesses global
$\mathbb{Z}_{N}$ symmetry to characterize the properties of the system in the
low-energy manifold. We should note that the duality relation between the
Hamiltonians (9) and (10) is the generalization of well-known Wegner duality
defined between 2D quantum Ising model and $\mathbb{Z}_{2}$ gauge theory [59,
60] (see also [61]).

In case of standard $\mathbb{Z}_{N}$ gauge theories with finite $N$, the phase
diagram consists of two gapped phases – a topologically ordered deconfined
phase at small values of coupling strength $g$ and a confined phase at large
values of $g$ [62, 63, 64], where following Polyakov’s prediction the former
disappears at the $U(1)$ limit of $N\rightarrow\infty$ [65, 66]. In the
present scenario, the confined phase is a symmetry-broken phase due to the
unique nature of the interactions (see [38]). Moreover, using a mean-field
argument [38], it has been conjectured that a gapless Bose liquid phase
appears at an ‘intermediate’ values of coupling strength $g$, where the
population imbalance of the $b$-bosons on plaquettes (given by non-zero
$\hat{\mathcal{B}}_{\mathbf{p}}$ i.e., the magnetic charge) propagates freely
along the $y$-direction. Consequently, the system in this intermediate phase
can be effectively described as a series of one-dimensional gapless phases
stacked together in the $x$-direction. On the other hand, another gapless
phase of dipolar nature has been conjectured to appear for
$N\rightarrow\infty$ and $g\rightarrow 0$ with $gN$ being finite.

In this work, we provide robust numerical validation of these predictions for
finite $N$ by employing density matrix renormalization group (DMRG) algorithm
[67, 68] with matrix-product state (MPS)[39, 40, 41] ansatz. This approach is
applied to quasi-2D ladder and cylinder systems that are infinitely extended
in the $y$-direction but have a finite length or circumference $L_{x}$ in the
$x$-direction. Specifically, we use the infinite-size version of DMRG
(iDMRG)[69], where the underlying infinite MPS (iMPS) extends infinitely in
the $y$-direction and wraps around the cylinder in a zigzag pattern along the
$x$-direction. Moreover, we make use of $\mathbb{Z}_{N}$ symmetry preserving
tensors that conserve the global $\mathbb{Z}_{N}$ symmetry (corresponding to
$\left[\prod_{\mathbf{p}}\hat{P}^{(\dagger)}_{\mathbf{p}},\hat{H}_{P}\right]=0$)
throughout the calculations.

##  III The Phase diagram

In this section, we systematically analyze the phase diagram of Hamiltonian
(9) by examining three different geometries: (1) the 1D case with $L_{x}=1$,
(2) a two-leg ladder geometry with $L_{x}=2$, and (3) cylinder geometries with
$L_{x}>2$ and periodic boundary conditions in the $x$-direction. For this
purpose, we consider two quantities, namely the von Neumann entanglement
entropy of the half-infinite block and the iMPS correlation length for a given
iMPS bond dimension $\chi$. Such technique is called finite-entanglement
scaling and was introduced in a series of publications [70, 71, 72, 73, 74].

The von Neumann entanglement entropy of the half-infinite block for a given
bond dimension $\chi$ is defined as

$S_{\chi}=-\text{Tr}\left[\rho^{L}_{\chi}\ln(\rho^{L}_{\chi})\right]=-\sum_{k}\lambda_{k}^{2}\ln\lambda_{k}^{2},$
(11)

where $\rho^{L}_{\chi}=\text{Tr}_{R}\left[\ket{\psi}\bra{\psi}\right]$
represents the reduced density matrix of the left half of the system obtained
by tracing out the right infinite half of the system, and $\\{\lambda_{k}\\}$
are the corresponding Schmidt coefficients. For an iMPS with bond dimension
$\chi$, the correlation length is given by $\xi_{\chi}=-1/\ln|\epsilon_{2}|$,
where $\epsilon_{2}$ is the second largest eigenvalue of the iMPS transfer
matrix [74]. In gapless systems, where the correlation length diverges,
$\xi_{\chi}$ represents an artificially introduced length scale due to the
finite iMPS bond dimension $\chi$. Typically, $\xi_{\chi}\sim\chi^{\beta}$,
with $\beta$ being a scaling exponent.

###  III.1 One-dimensional scenario

Figure 2: (Color online.) (a) The entanglement entropy $S_{\chi}$ and (b) the
correlation length $\xi_{\chi}$ for the ground state of the
$\mathbb{Z}_{N}$-symmetric 1D system (Eq. (12)) as a function of $g$,
calculated for $N=5,6,7,8$ with iMPS bond dimension $\chi=256$. Only two
gapped phases are observed for $N\leq 6$, while a gapless phase emerges at
$N=7$ around $g=0.5$ and becomes more robust at $N=8$. Vertical dashed lines
indicate the gapless phases for $N=7$ and $8$.

In 1D geometry, the last term in the Hamiltonian (9), i.e., the four-plaquette
term, is absent, and the system is then described by the Hamiltonian:

$\displaystyle\hat{H}_{P}^{1D}=$
$\displaystyle-\sum_{\mathbf{p}}\left[\hat{P}_{\mathbf{p}}+g^{2}\left(\hat{Q}^{2}_{\mathbf{p}}\left(\hat{Q}^{\dagger}_{\mathbf{p}-\hat{y}}\right)^{2}+2\,\hat{Q}^{2}_{\mathbf{p}}\hat{Q}^{\dagger}_{\mathbf{p}+\hat{y}}\hat{Q}^{\dagger}_{\mathbf{p}-\hat{y}}\right)\right]$
$\displaystyle+\text{h.c.}$ (12)

In terms of gauge theory, such a 1D system of plaquettes in the dual lattice
corresponds to a local $\mathbb{Z}_{N}$ invariant system in a two-leg ladder
geometry described by Hamiltonian (10) where the last term, i.e., the
$\hat{P}_{(\mathbf{j},\hat{y})}\hat{P}_{(\mathbf{j}+\hat{y},\hat{y})}^{\dagger}$
term, is absent.

First, we observe that in absence of the three-plaquette term (i.e., the last
term) in Eq.(12), the odd and even $\mathbb{Z}_{N}$ sectors decouple for even
$N$ due to the squared $\hat{Q}^{2}_{\mathbf{p}}$ operators present in the
second term. This decoupling maps the system into two copies of a
$\mathbb{Z}_{N/2}$ symmetric $N/2$-state quantum Clock model [75, 76]. It has
become clear that for the $q$-state quantum Clock model, while there are two
gapped phases for $q\leq 4$, an intermediate gapless phase appears for $q\geq
5~{}$[77, 78]. In our scenario, we first confirm (results not shown) that the
system described by Eq. (12), without the last term, exhibits an intermediate
gapless phase for $N\geq 10$, regardless of whether $N$ is even or odd.

The presence of the three-plaquette term in Eq. (12) introduces coupling
between the odd and even sectors, preventing the system from being mapped into
two independent copies of the $N/2$-state quantum clock model. This coupling
fundamentally alters the structure and behavior of the system, raising a
critical question: does the gapless intermediate phase, observed in the
absence of the three-plaquette term, persist when this term is included? If
so, at what value of $N$ does this phase begin to appear? To address this, we
carefully examine the phase diagram of the system in the presence of the
three-plaquette interaction as elaborated below.

Figure 2 shows the entanglement entropy and the correlation length for the
ground state of Hamiltonian (12) for different values of $N$ calculated for
iMPS bond dimension $\chi=256$. Both the entanglement entropy and the
correlation length approach to zero in the limit $g\rightarrow 0$ for any
values of $N\leq 8$. Therefore, we detect a (trivial) gapped phase in the
small $g$ limit. In the dual picture of $\mathbb{Z}_{N}$ gauge theory, this
gapped phase corresponds to the deconfined phase with non-trivial topological
order [79, 80, 81]. The non-trivial topological character of this deconfined
phase is verified by checking the degeneracies in the entanglement spectrum
[82] of the ground state of Hamiltonian (10) in the small $g$ limit.

The correlation length approaches to zero also in the high $g$ limit
indicating the presence of another gapped phase. This gapped phase at strong
coupling corresponds to a spontaneous symmetry broken (SSB) phase where global
$\mathbb{Z}_{N}$ symmetry of Hamiltonian (12) gets broken. That is why the
entanglement entropy saturates to $S\rightarrow\ln N$ for $g\rightarrow\infty$
as in our calculation iMPS ansatz preserves the global $\mathbb{Z}_{N}$
symmetry. In case of symmetry-broken states, the order parameter
$\sum_{\mathbf{p}}\braket{\hat{Q}_{\mathbf{p}}+\hat{Q}^{\dagger}_{\mathbf{p}}}$
becomes non-zero in this large-$g$ phase. Within our $\mathbb{Z}_{N}$-symmetry
preserving iMPS calculations, this symmetry-breaking can be verified by
checking
$|\braket{\hat{Q}_{\mathbf{p}}\hat{Q}^{\dagger}_{\mathbf{p}+R\hat{y}}}|\approx
1$ for large $R$. In the dual gauge-invariant system, this strong coupling
phase corresponds to an SSB confined phase with
$\sum_{\mathbf{j},\hat{\delta}}\braket{\hat{P}_{(\mathbf{j},\hat{\delta})}+\hat{P}_{(\mathbf{j},\hat{\delta})}^{\dagger}}$
being the order parameter. Moreover, we notice that there is only one cusp
(maxima) in the profile of entropy (correlation length) for $N\leq 6$,
indicating a direct phase transition between these gapped phases.

Starting from $N=7$, the phase diagram undergoes a notable change (Fig. 2)
with the emergence of an additional phase at intermediate values of $g$111It
should be noted that we observe a robust gapless phase starting from $N=7$.
Determining whether the gapless phase also exists for $N=6$ or even $N=5$
would require a very fine-grained grid along the $g$-axis, significantly
increasing computational complexity. Therefore, we refrain from conducting
such an analysis.. The high correlation lengths observed in this intermediate
phase suggest its critical nature. Therefore, the inclusion of the three-
plaquette term causes the gapless phase to appear at a lower value of $N$
compared to its absence. Since gapless phases in 1D are essentially Luttinger
liquids [83, 84], we anticipate the same in this system. In the following Sec.
IV, we validate this expectation by examining the scaling of entanglement
entropy. In the dual $\mathbb{Z}_{N}$ gauge theory in two-leg ladder geometry,
this gapless phase corresponds to a gapless deconfined phase. For larger
values of $N$, we find that the region of this gapless phase increases, which
indicates the robustness of this phase with respect to increasing $N$. We
should note that the origin of this gapless phase is fundamentally different
from the Coulomb phase recently observed in $\mathbb{Z}_{N}$ lattice gauge
theories in the presence of Higgs matter in two-leg ladder geometry [85], as
such Coulomb phase disappears for pure gauge theory, i.e., in absence of the
Higgs matter.

Figure 3:  (Color online.) The entanglement entropy $S_{\chi}$ and the
correlation length $\xi_{\chi}$ of the ground state of the system (9) in a
two-leg ladder geometry as a function of $g$, calculated for $N=6,7$ and $8$.
The data indicates the presence of two gapped phases for $N=6$. At $N=7$ the
gapless phase emerges in the area $0.43\lesssim g\lesssim 0.53$, while at
$N=8$ it encompasses a wider region $0.39\lesssim g\lesssim 0.58$. The black
vertical lines demarcate the gapless region.

###  III.2 Two-leg ladder system

In this subsection, we consider the simplest extension beyond 1D: a two-leg
ladder geometry (i.e., $L_{x}=2$), which includes the four-plaquette term (the
last term in Eq.(9)). In this setup, the system maps to a three-leg ladder
system with open boundary conditions along the $x$-direction in the dual
$\mathbb{Z}_{N}$ gauge theory (Eq.(10)). The key question we aim to address
here is whether the conclusions drawn in the previous subsection still hold as
we make the transition from a 1D to a quasi-2D system. While we expect the
gapped phases to persist, the fate of the gapless phase remains uncertain. The
four-plaquette term may act as a relevant perturbation, potentially causing a
gap to open.

Figure 4:  (Color online.) The entanglement entropy $S_{\chi}$ of the ground
state of the $\mathbb{Z}_{8}$-symmetric system (9) embedded in cylinders with
transverse dimensions $L_{x}=3$ and $L_{x}=4$ as a function of $g$. Similar to
the situations for $L_{x}=1,2$, we observe two gapped phases, confined and
deconfined for large and small values of $g$ respectively, and a gapless phase
sandwiched between these two gapped phases at intermediate values of $g$. The
gapless phase persists in the range $0.35\lesssim g\lesssim 0.53$ for
$L_{x}=3$ and $0.35\lesssim g\lesssim 0.52$ for $L_{x}=4$.

In Fig. 3, we show the entanglement entropy $S_{\chi}$ and the correlation
length $\xi_{\chi}$ for the ladder system for different values of $N$ and iMPS
bond dimension $\chi$ as depicted in the figure. Both the entropy and the
correlation length show similar features as in the 1D scenario. While for
$N\leq 6$ the system exhibits only two gapped phases (the deconfined and
confined phases in the language of $\mathbb{Z}_{N}$ gauge theory)222We stress
again that there is a possibility of having a minuscule gapless region
sandwiched between the confined and deconfined regions for $N=6,5$. However,
distinguishing this phase from a direct critical phase transition between the
confined and deconfined phases is very challenging with iDMRG simulations with
finite bond dimension and finite grid-spacing in $g$., we observe extended
gapless region for $N\geq 7$ in the intermediate values of $g$. By monitoring
large values of correlation length that increases with increasing iMPS bond
dimension, we identify the gapless phase to be within the range $0.43\lesssim
g\lesssim 0.53$ for $N=7$ and $0.39\lesssim g\lesssim 0.58$ for $N=8$. We
should note that nature of this gapless phase can be described by coupled
pillars of Luttinger liquids, as the four-plaquette term now couples the two
chains (along $y$ direction) of former Luttinger liquids along the
$x$-direction. We leave this discussion for Sec. IV.

###  III.3 Three- and four-legged cylinder system

Next, we analyze the fate of the intermediate gapless phase by increasing the
transverse dimension $L_{x}$. Specifically, we consider the Hamiltonian (9) in
three- and four-legged cylinder geometries, i.e., with periodic boundary
condition along $x$ axis with $L_{x}=3$ and $L_{x}=4$. In the dual lattice,
such systems translate to $\mathbb{Z}_{N}$ gauge theory (i.e., Eq. (10))
embedded in cylinders with transverse dimension $L_{x}=3$ and $L_{x}=4$
respectively.

The numerical result for the entanglement entropy $S_{\chi}$ for the ground
state of $\mathbb{Z}_{8}$-symmetric Hamiltonian (9) for different iMPS bond
dimension $\chi$ is given in Fig. 4. The qualitative scenario remains
effectively unchanged from the previous cases. From the cusps in the entropy
profile, we detect the gapless region in the range $0.35\lesssim g\lesssim
0.53$ for $N=7$ and $0.35\lesssim g\lesssim 0.52$ for $N=8$. It is therefore
evident that the gapless region remains robust at larger transverse dimension
$L_{x}$, and we expect it to remain true even for fully extended 2D case.

System | At small $g$ | At intermediate $g$ ($N\geq 7$) | At large $g$
---|---|---|---
$\hat{H}_{P}$ (Eq. (9)) | Trivial gapped phase | Gapless 1D liquids | SSB phase
(Global $\mathbb{Z}_{N}$ symmetry) |  | (Weekly coupled Luttinger liquids) | $\sum_{\mathbf{p}}\braket{\hat{Q}_{\mathbf{p}}+\hat{Q}^{\dagger}_{\mathbf{p}}}\neq 0$
$\hat{H}_{G}$ (Eq. (10)) | Deconfined phase | Gapless deconfined phase | SSB confined phase
(Local $\mathbb{Z}_{N}$ symmetry) | (Topologically ordered) |  | $\sum_{\mathbf{j},\hat{\delta}}\braket{\hat{P}_{(\mathbf{j},\hat{\delta})}+\hat{P}_{(\mathbf{j},\hat{\delta})}^{\dagger}}\neq 0$
Table 1: Different phases found in our study for cylinder/ladder systems with
width $L_{x}\leq 4$.

The summary of the phase diagram found in our calculation is presented in the
Table 1. It should be noted that contrary to the predictions of [38], we do
not find a gapless at $g\rightarrow 0$ predicted for large $N$. This is,
probably, because we have limited our analysis for finite $N$ 333We note that
we have verified this by calculating the ground state of the system till
$N=15$ for a few small values of $g$.. Increasing $N$ to very large numbers
would increase the computational cost due to the increment in the local
Hilbert space dimension. The absence of this gapless phase at low coupling may
also be attributed to the limited cylinder width which we consider with MPS
ansatz.

##  IV Characterization of the gapless deconfined phases

In this section, we first characterize the gapless deconfined phases appearing
for $N\geq 7$ by scrutinizing different correlation functions. Next, we
determine the central charge of the underlying conformal field theory (CFT) in
these critical regions from the scaling of entanglement entropy.

Figure 5: (Color online.) (a)-(b) The power-law decay of the correlation
functions $\mathcal{C}_{P}$ and $\mathcal{C}_{Q}$ inside the gapless phase for
$N=8$ and $L_{x}=1,2$. We set iMPS bond dimension $\chi$ to 512 and 1280
respectively for $L_{x}=1$ and $L_{x}=2$. The deviation from power-law decay
seen for $L_{x}=2$ at large $R$ is due the finite correlation length
$\xi_{\chi}$ imposed by finite $\chi$. Both $x$ and $y$ axes are in log-scale.
(c)-(d) The power-law exponents $b_{p}$ and $b_{q}$, extracted by fitting the
$\mathcal{C}_{P}$ and $\mathcal{C}_{Q}$ data respectively, in the entire
gapless region for $N=8$ and $L_{x}=1,2$. The shaded regions indicate the
error bars and the vertical dashed lines depict the extent of the gapless
regions. Here, we set iMPS bond dimension $\chi$ to 512 and 640 respectively
for $L_{x}=1$ and $L_{x}=2$.

###  IV.1 Correlation functions

We consider the following two-point connected correlation functions along the
$y$-direction:

$\displaystyle\mathcal{C}_{P}(R):=\braket{\hat{P}^{\dagger}_{\mathbf{p}}\hat{P}_{\mathbf{p}+R\hat{y}}}-\braket{\hat{P}^{\dagger}_{\mathbf{p}}}\braket{\hat{P}_{\mathbf{p}+R\hat{y}}},$
$\displaystyle\mathcal{C}_{Q}(R):=\braket{\hat{Q}^{\dagger}_{\mathbf{p}}\hat{Q}_{\mathbf{p}+R\hat{y}}}-\braket{\hat{Q}^{\dagger}_{\mathbf{p}}}\braket{\hat{Q}_{\mathbf{p}+R\hat{y}}}.$
(13)

In our calculations, since we are employing $\mathbb{Z}_{N}$ symmetry
preserving iMPS ansatz, the local expectation values
$\braket{\hat{Q}^{\dagger}_{\mathbf{p}}}$ or $\braket{\hat{Q}_{\mathbf{p}}}$
necessarily vanishes.

At the small-$g$ gapped phase, both the correlations decay exponentially with
$R$ as it is expected for a trivial gapped phase. On the other hand, at the
strong-coupling gapped phase, $\mathcal{C}_{P}$ decays to zero exponentially,
while $|\mathcal{C}_{Q}|$ saturates to one for large enough $R$ due to SSB.
For the gapless phase at intermediate $g$, we expect both the correlations to
decay algebraically to zero: $|\mathcal{C}_{P}(R)|\sim R^{-b_{p}}$ and
$|\mathcal{C}_{Q}(R)|\sim R^{-b_{q}}$ as long as $R\lesssim\xi_{\chi}$. Figure
5(a) and (b) display the power-law behavior of these correlators inside the
gapless region for $N=8$ and $L_{x}=1$ and $2$. By extracting the exponent
$b_{p}$ for the correlator $\mathcal{C}_{P}$ (Fig. 5(c)) we find that
$b_{p}\approx 2$, i.e., $|\mathcal{C}_{P}(R)|\sim 1/R^{2}$, irrespective of
the values of $g$. In the case of $\mathcal{C}_{Q}$ the decay is much slower
where $b_{q}$ varies with varying $g$ (Fig. 5). This slow decay of
$\mathcal{C}_{Q}$ is expected as this gapless phase is adjacent to the SSB
phase where $b_{q}\rightarrow 0$.

###  IV.2 Scaling of entanglement entropy

To understand the nature of the gapless phase, we consider the scaling of
entanglement entropy with respect to the correlation length. For 1D (or
quasi-1D, as in the present scenario) critical systems, the entanglement
entropy increases logarithmically with the correlation length following the
well-known relation [86, 87, 88]:

$S_{\chi}=\frac{c}{6}\log\xi_{\chi}+b^{\prime},$ (14)

where $c$ is the central charge of the underlying CFT and $b^{\prime}$ is a
non-universal constant.

Figure 6: (Color online.) The scaling of entanglement entropy $S_{\chi}$ as a
function of the correlation length $\xi_{\chi}$ according to Eq. (14) for
different system-width $L_{x}$ and $N$. The extracted values of the central
charge, along with other details, are given in Table 2.

By fitting Eq.(14) to the entropic data for various iMPS bond dimensions
within the gapless phase (see Fig.6), we find the central charge $c=1$ for
$L_{x}=1$ in both $\mathbb{Z}_{7}$ and $\mathbb{Z}_{8}$ symmetric systems.
This result aligns with expectations for 1D gapless phases described by
Luttinger liquid theory. For $L_{x}\geq 2$, the four-plaquette term in Eq.(9)
couples $L_{x}$ identical copies of such Luttinger liquids. In principle, this
coupling could be relevant and potentially open a (partial) gap in the system,
resulting in $c\leq L_{x}$. However, the scaling analysis presented in Fig.6
shows $c=L_{x}$ up to small numerical errors. This indicates that the four-
plaquette interaction is an irrelevant perturbation, and following $c$-theorem
[89], the system essentially consists of $L_{x}$ decoupled Luttinger liquids
in the long-wavelength (low-energy, infrared) limit. Therefore, the gapless
phase at intermediate $g$ effectively remains a set of pure 1D critical
states, which we expect to hold even for large $L_{x}$ and in a fully extended
2D setting.

The extracted numerical values of $c$ along with other system parameters,
considered for the plot of Fig. 6, are presented in the Table 2.

$N$ |  $L_{x}$ |  $g$ |  $\chi$-range considered | $c$
---|---|---|---|---
7 | 1 | 0.55 | [64, 512] | 1.00(2)
8 | 1 | 0.55 | [64, 512] | 0.98(2)
7 | 2 | 0.5 | [256, 1280] | 1.98(2)
8 | 2 | 0.5 | [256, 1280] | 2.02(3)
8 | 3 | 0.45 | [256, 2048] | 3.03(4)
8 | 4 | 0.45 | [256, 2048] | 4.0(1)
Table 2: The numerically extracted values of the central charge $c$ by fitting
the entropic data to Eq. (14) for different system (and iMPS) parameters. The
values of $g$ are chosen in such a way that the system remains within the
gapless region.

##  V Conclusion

Ultracold atoms in particular optical lattice potentials allow for the
creation of models with local symmetries. In the current work, we have
considered a specific optical lattice setup involving two species of ultracold
bosons. In the low-energy manifold, inter-species dynamics give rise to unique
Abelian $\mathbb{Z}_{N}$ gauge theories. By using infinite-system density
matrix renormalization group simulations, we have shown that such quantum
systems exhibit particularly interesting phases of matter. While for $N\leq 6$
only two gapped phases exist as in standard $\mathbb{Z}_{N}$ gauge theories,
we confirm the presence of a gapless (deconfined) phase for $N\geq 7$ as well
as its robustness with respect to increasing $N$. This gapless phase exhibits
the same conformal critical properties as the one-dimensional Bose liquid
phase, in line with field theoretical predictions [38], where its description
is same as the collection of weakly coupled chains of identical Luttinger
liquids.

On the other hand, we cannot confirm the existence of another gapless
(presumably dipolar liquid) phase. This is probably a consequence of finite
$N$ considered here or the physical geometry of the system we can explore, in
which one of the two dimensions is bound to be very short. As a result, even
if we cannot observe it, we cannot exclude the possibility of its existence in
the thermodynamic limit of the U(1) symmetric theory where the dipolar liquid
phase is conjectured to exist, i.e., in the limit ($L_{x}\to\infty,g\to 0$,
$N\to\infty$, $gN={\rm const}$). Numerical calculations at large values of $N$
are beyond the scope of the current manuscript.

###### Acknowledgements.

The work of M.V.R. was funded by the National Science Centre, Poland, under
the OPUS call within the WEAVE program 2021/43/I/ST3/01142. L.T. acknowledges
support from the Proyecto Sinérgico CAM Y2020/TCS-6545 NanoQuCo-CM, the CSIC
Research Platform on Quantum Technologies PTI-001, and from the Grant
TED2021-130552B-C22 funded by MCIN/AEI/10.13039/501100011033 and by the
“European Union NextGenerationEU/PRTR”, and Grant PID2021-127968NB-I00 funded
by MCIN/AEI/10.13039/501100011033. M.L. acknowledges support from: European
Research Council AdG NOQIA; MCIN/AEI (PGC2018-0910.13039/501100011033,
CEX2019-000910-S/10.13039/501100011033, Plan National FIDEUA
PID2019-106901GB-I00, Plan National STAMEENA PID2022-139099NB, I00, project
funded by MCIN/AEI/10.13039/501100011033 and by the “European Union
NextGenerationEU/PRTR” (PRTR-C17.I1), FPI); QUANTERA DYNAMITE PCI2022-132919,
QuantERA II Programme co-funded by European Union’s Horizon 2020 program under
Grant Agreement No 101017733); Ministry for Digital Transformation and of
Civil Service of the Spanish Government through the QUANTUM ENIA project call
- Quantum Spain project, and by the European Union through the Recovery,
Transformation and Resilience Plan - NextGenerationEU within the framework of
the Digital Spain 2026 Agenda; Fundació Cellex; Fundació Mir-Puig; Generalitat
de Catalunya (European Social Fund FEDER and CERCA program, AGAUR Grant No.
2021 SGR 01452, QuantumCAT U16-011424, co-funded by ERDF Operational Program
of Catalonia 2014-2020); Barcelona Supercomputing Center MareNostrum
(FI-2023-3-0024), HORIZON-CL4-2022-QUANTUM-02-SGA PASQuanS2.1, 101113690, EU
Horizon 2020 FET-OPEN OPTOlogic, Grant No 899794, EU Horizon Europe Program
(grant agreement No 101080086 NeQSTGrant Agreement 101080086 — NeQST); ICFO
Internal “QuantumGaudi” project; European Union’s Horizon 2020 program under
the Marie Sklodowska-Curie grant agreement No 847648; “La Caixa” Junior
Leaders fellowships, La Caixa” Foundation (ID 100010434): CF/BQ/PR23/11980043.
The work of J.Z. was funded by the National Science Centre, Poland, project
2021/03/Y/ST2/00186 within the QuantERA II Programme that has received funding
from the European Union Horizon 2020 research and innovation programme under
Grant agreement No 101017733. A support by the Strategic Programme Excellence
Initiative (DIGIWorkd) at Jagiellonian University is acknowledged. We
gratefully acknowledge Polish high-performance computing infrastructure PLGrid
(HPC Center: ACK Cyfronet AGH) for providing computer facilities and support
within computational grant no. PLG/2024/017289. The iDMRG simulations have
been performed using the TeNPy library [90]. Views and opinions expressed in
this work are, however, those of the authors only and do not necessarily
reflect those of the European Union, European Commission, European Climate,
Infrastructure and Environment Executive Agency (CINEA), or any other granting
authority. Neither the European Union nor any granting authority can be held
responsible for them.

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# Gravitationally sensitive structured x-ray optics using nuclear resonances

Shin-Yu Lee  Department of Physics, National Central University, Taoyuan City
32001, Taiwan  Sven Ahrens  Shanghai Normal University, Shanghai 200234, China
Wen-Te Liao<EMAIL_ADDRESS>Department of Physics, National Central
University, Taoyuan City 32001, Taiwan  Physics Division, National Center for
Theoretical Sciences, Taipei 10617, Taiwan  Center for Quantum Technology,
Hsinchu 30013, Taiwan

###### Abstract

Einstein’s general theory of relativity not only revolutionized human
understanding of the universe, but also brought many gravitational
applications in large scale, such as gravitational-wave astronomy [1],
gravitational lensing [2], and the operation of the global positioning system
[3]. However, it still remains a challenge to implement applications for
gravitational effects at small spacial extensions on Earth. Here, we
investigate a structured waveguide system that allows for the control of an
x-ray profile at altitude separations of millimeters and even shorter using
the nuclear resonant scattering of x rays [4, 5]. Our present results suggest
a potential compact scheme for turning the Earth’s gravity into a practical
application of x-ray optics.

quantum optics,interference effect

The Pound–Rebka experiment [6] has demonstrated a unique system for probing
the gravitational red-shift effect by exploiting an extremely narrow nuclear
linewidth in combination of a high x-ray energy in the Mössbauer effect [7].
Moreover, advances in modern x-ray light sources and optics have raised the
field of x-ray-nuclei interactions to a new level of accuracy where coherent
quantum control comes into play [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21]. A combination of nuclear quantum coherence and its sensitivity to
gravity will potentially lead to a new type of x-ray optics whose performance
depends on the gravitational red-shift in addition to the typical Zeeman shift
[8, 12] and Doppler shift [14, 17, 18, 19]. In this context, we investigate a
system that emulates the Schrödinger equation [22] and is sensitive to
gravity. The present scheme allows for a systematic generation of structured x
rays [23, 24, 25, 26] via changing the altitude, the x-ray photon energy, or
the external magnetic field of a our system for a given waveguide structure.

Table 1:  Nuclear and waveguide material parameters. For each isotope $X$ we present the nuclear transition energy $E_{t}$ and the radiative decay rate $\Gamma$. The last six columns list the x-ray index of refraction $n_{e}=1-\delta+i\beta$ for SXWG materials [5, 27, 28].  $X$ |  $E_{t}$ (keV) |  $\Gamma$ (MHz) | $\delta_{X}(10^{-6})$ | $\delta_{\mathrm{C}}(10^{-6})$ | $\delta_{\mathrm{Pt}}(10^{-5})$ | $\beta_{X}(10^{-9})$ | $\beta_{\mathrm{C}}(10^{-9})$ | $\beta_{\mathrm{Pt}}(10^{-6})$
---|---|---|---|---|---|---|---|---
45Sc | 12.4 | 2.18$\times 10^{-6}$ | 3.84 | 2.97 | 2.091 | 131.9 | 1.78 | 2.737
57Fe | 14.413 | 7.05 | 7.43 | 2.20 | 1.607 | 338.9 | 0.93 | 2.49
73Ge | 13.275 | 0.24 | 5.41 | 2.59 | 1.622 | 508.9 | 1.32 | 2.947
181Ta | 6.238 | 0.11 | 67.74 | 11.77 | 8.731 | 7987.3 | 32.76 | 12.616
182Ta | 16.273 | 2.45$\times 10^{-6}$ | 10.41 | 1.72 | 1.304 | 1062.2 | 0.55 | 1.642

Figure 1:  (a) the incident x ray (red upward arrow) drives a nuclear
transition $|g\rangle\rightarrow|e\rangle$ with detuning $\Delta+\Delta_{G}$
(gray vertical double arrows). $\Delta$ is the x-ray detuning, and
$\Delta_{G}$ is the x-ray gravitational red shift. (b) a hard x ray of
transverse mode $\psi_{1}$ (red arrow) propagates through a structured
platinum cladding waveguide with a periodic nuclear distribution whose spatial
concentration is indicated by the horizontal green-yellow legend. An output x
ray of mode $\psi_{2}$ is measured by a downstream position-sensitive detector
(blue thick disc). The top black curves and pictures in gray level represent
the intra-waveguide x-ray intensity. (c) the SXWG altitude-dependent nuclear
coherence $\rho_{eg}$ between the nuclear ground state $|g\rangle$ and the
excited state $|e\rangle$ of the isotope 45Sc. Red-dashed (blue-solid) line
depicts the imaginary (real) part of the nuclear $\rho_{eg}$. Black horizontal
double arrow indicates the full altitude width $\Delta Z_{G}$ at the half
maximum $|Re\left[\rho_{eg}\right]|$. (d) the transversely gradient (along x)
and the longitudinally periodic (along y) electronic refractive index of an
SXWG. Between the platinum claddings the intra-waveguide structure, made of
carbon and an isotope $\mathrm{X}$, drives the transition from the x-ray
ground state $\psi_{1}$ (red-solid line) to the first excited state $\psi_{2}$
(blue-solid line).  Figure 2:  A full cycle of the x-ray Rabi oscillation
between $\psi_{1}$ and $\psi_{2}$ is illustrated by (a) the fidelity $F_{1}$
(red-dashed line) and $F_{2}$ (green-solid line) and (b) the normalized intra-
waveguide x-ray intensity distribution. The input x-ray in the state
$\psi_{1}$ splits and reaches the maximum transverse double-hump separation of
the state $\psi_{2}$ at $y=2$mm where the maximum $F_{2}$ also occurs. For
$y>2$mm the x-ray confluence reflects the second half Rabi cycle, and the
transverse x-ray pattern returns toward the state $\psi_{1}$.

Figure 3:  (a) Earth’s gravity changes the x-ray propagation in the waveguide
composed of 45Sc nuclei. Three cases at different altitudes $z=2.32$cm,
$z=2$cm, and $z=1.72$cm where x rays propagate with a detuning $\Delta=19.36$
and gravitational red shifts $\Delta_{G}=-21.88$, $\Delta_{G}=-18.86$, and
$\Delta_{G}=-16.22$, respectively. (b, d, and f) the fidelity $F_{1}$ (red-
dashed line) and $F_{2}$ (green-solid line) for $z=2.32$cm, $z=2$cm, and
$z=1.72$cm, respectively. (c, e, and g) the normalized intra-waveguide x-ray
intensity distribution $|E\left(x,y\right)|^{2}$ at altitude $z=2.32$cm,
$z=2$cm, and $z=1.72$cm from the top to the bottom.

Figure 4:  (a) altitude-dependent x-ray fidelity $F_{2}$ through an SXWG
composed of 45Sc nuclei. (b) the FWHM $\Delta Z_{G}$ on Earth is dependent on
the quality factor $Q$ of nuclear resonances for different nuclear species.

The system is depicted in Figure 1 with the description as follows. An x ray
drives a nuclear transition from the ground state $|g\rangle$ to the excited
state $|e\rangle$ with the total detuning
$\Delta_{t}\Gamma=\left(\Delta_{G}+\Delta\right)\Gamma$ in Fig. 1(a). We
emphasize that the gravitational red shift $\Delta_{G}\simeq-
zE_{t}GM_{E}/\left(\hbar\Gamma c^{2}R_{E}^{2}\right)$ has to be taken into
account when the system is located at different altitude $z$ relative to where
x rays are emitted. Here $E_{t}$ is the nuclear transition energy, $G$ is the
gravitational constant, $M_{E}$ is the mass of the Earth, and $R_{E}$ is the
average radius of the Earth. $\Delta\Gamma$ is the x-ray detuning, and
$\Gamma$ the spontaneous decay rate of the excited state $|e\rangle$. Our
system of nuclear resonant scattering of x rays can be described by the
optical-Bloch equation [9, 21]:

$\displaystyle\partial_{t}{\rho_{eg}}$ $\displaystyle=$
$\displaystyle-\Gamma\left[\frac{1}{2}-i\left(\Delta+\Delta_{G}\right)\right]\rho_{eg}+\frac{i}{2}\frac{P}{\hbar}E,$
(1) $\displaystyle\frac{1}{c}\partial_{t}E+\partial_{y}E$ $\displaystyle=$
$\displaystyle\frac{-1}{2ik}\nabla_{\perp}^{2}E-\frac{k}{2i}(n_{e}^{2}-1)E+i\eta\rho_{eg},$
(2)

where $\rho_{eg}$ is the coherence of a nuclear two-level system, $E$ is the
x-ray electric field strength, $\hbar$ is the reduced Planck constant, $k$ is
the x-ray wavenumber, and $n_{e}$ is the index of refraction from electrons.
Further, we denote the transverse Laplacian
$\nabla_{\perp}^{2}=\partial^{2}_{x}+\partial^{2}_{z}$, and the coupling
constant $\eta=2\hbar\Gamma\xi/\left(PL\right)$, where $L$ the length of the
waveguide, $P$ is the nuclear transition dipole moment, and $\xi$ is the
nuclear resonant thickness. The steady state of Eqs. (1-2) leads to the
analytic solution
$\rho_{eg}=PE\left(i-2\Delta_{t}\right)/\left[\hbar\Gamma\left(1+4\Delta_{t}^{2}\right)\right]$
and the optical Schrödinger equation [22] (see supplemental information)

$i\hbar c\partial_{y}E=-\frac{\hbar^{2}}{2m_{e}}\nabla_{\perp}^{2}E+\hbar
c\left[\frac{k(1-n_{e}^{2})}{2}+\frac{4\xi\Delta_{t}}{L\left(1+4\Delta_{t}^{2}\right)}\right]E,$
(3)

with the effective mass $m_{e}=\hbar k/c$. Equation (3) suggests a structured
x-ray waveguide (SXWG) system composed of resonant nuclei, as depicted in Fig.
1(b), with a high degree of freedom for simulating different quantum systems
via spatial engineering of $n_{e}$ and $\xi$. Fig. 1(c) displays the altitude
dependent real part $Re\left[\rho_{eg}\right]$ (blue-solid) and imaginary part
$Im\left[\rho_{eg}\right]$ (red-dashed line) of $\rho_{eg}$ with $\Delta=0$
for the isotope 45Sc. $Re\left[\rho_{eg}\right]$ describes the refractive
index from nuclei and results in the gravitational effects in our system as
revealed by the last term of Eq. (3). In cantrast, $Im\left[\rho_{eg}\right]$
of Lorentzian line shape represents the nuclear absorption of x rays and leads
to the Pound–Rebka experiment [6].

In the following, we use the SXWG to simulate Rabi oscillations of x ray in a
finite square well of width $L_{x}$. As illustrated in Fig. 1(d), we introduce
a platinum cladding $n_{e}=1-\delta_{\mathrm{Pt}}+i\beta_{\mathrm{Pt}}$ for
$|x|>L_{x}/2$ to constitute a finite square well potential, which provides
with a transverse confinement to the x ray propagation direction [21]. The
cladding material leads to the energy eigenfunctions of the $E$ field in Eq.
(3) which read as (see supplemental information)

$\psi_{n}\left(x\right)=\sqrt{\frac{2}{L_{x}}}\sin\left[\frac{n\pi}{L_{x}}\left(x+\frac{L_{x}}{2}\right)\right],$
(4)

with the eigen angular frequencies
$\omega_{n}=n^{2}\pi^{2}c/\left(2kL_{x}^{2}\right).$ Inside the square well
$|x|\leq L_{x}/2$, we perturb the system by a periodic (gradient) particle
distribution of isotope $X$ and carbon along the y direction (x direction),
where
$\xi\left(x,y\right)=\left(\varrho/2\right)\left[1+\left(2x/L_{x}\right)\sin\left(k_{d}y\right)\right]$
and
$n_{e}\left(x,y\right)=1-\delta_{\mathrm{C}}\left(x,y\right)+i\beta_{\mathrm{C}}\left(x,y\right)-\delta_{\mathrm{X}}\left(x,y\right)+i\beta_{\mathrm{X}}\left(x,y\right)$
(see supplemental information for the form of $n_{e}$). In Fig. 1(d)
subscripts Pt, C, and X represent platinum, carbon and resonant nucleus,
respectively. We list other relevant material parameters in Table 1. With the
above density modulation, the last term in Eq. (3) effectively becomes the
electric dipole Hamiltonian in an oscillating field which perturbes the square
well potential. This plays the key role to drive the x-ray Rabi oscillation
with gravitational sensitivity. When the resonant condition
$ck_{d}=\omega_{n+1}-\omega_{n}$ is fulfilled, the periodic structure of the
refractive index drives the dipole transition $\psi_{n}\rightarrow\psi_{n+1}$
with the effective Rabi frequency (see supplemental information)

$\Omega_{n}=\frac{16\varrho
n\left(n+1\right)c}{\pi^{2}\left(2n+1\right)^{2}L}\left[\frac{k\left(\delta_{\mathrm{C}}-\delta_{\mathrm{Sc}}\right)}{2N\sigma_{0}}+\frac{2\Delta_{t}}{1+4\Delta_{t}^{2}}\right].$
(5)

A propagating x ray experiences a constant SXWG-induced Rabi frequency, and
the condition for having a $m\pi$ pulse is $|\Omega_{n}|L/c=m\pi$. We use
$m=2$ to demonstrate the Rabi oscillation between the x-ray ground state
$\psi_{1}$ and the first excited state $\psi_{2}$ by numerically solving Eq.
(3).

The solution of Eq. (3) in Fig. 2 represents an x ray which propagates through
an SXWG with with $X=^{45}$Sc, $\varrho=26.25$, natural scandium particle
density $N=3.99\times 10^{28}$m-3, nuclear resonance absorption cross section
$\sigma_{0}=12.6$(kbarn), $L_{x}=100$nm, $L=4$mm, $k_{d}=22.778\times
10^{3}$rad/m, $\Delta=4\Gamma$, and $\Delta_{G}=0$. The process is visualized
in terms of the fidelity
$F_{n}\left(y\right)=|\int_{-\infty}^{\infty}\psi_{n}^{\ast}\left(x\right)E\left(x,y\right)dx|^{2}/\int_{-\infty}^{\infty}|E\left(x,y\right)|^{2}dx$
in Fig. 2(a), and the normalized intra-waveguide x-ray intensity distribution
$|E\left(x,y\right)|^{2}/\int_{-\infty}^{\infty}|E\left(x,y\right)|^{2}dx$ in
Fig. 2(b). The alternation of $F_{1}$ and $F_{2}$ in Fig. 2(a) clearly
demonstrates that the ground-state x ray enters the SXWG at $y=0$, and is then
coherently promoted to the first excited state when approaching $y=2$mm. After
that, the x ray returns to the ground state and finishes a full Rabi cycle at
$y=4$mm. One can also observe the same phenomenon at the intra-waveguide x-ray
intensity which evolves back and forth between states $\psi_{1}$ and
$\psi_{2}$ in Fig.2(b).

The above x-ray Rabi oscillation suggests a systematic way to generate the
$\psi_{n}$ mode with a sequence of SXWGs driving $\Delta n=1$ transitions,
where one can raise the quantum number one by one. Specifically, one can
connect two different SXWGs to accomplish a $\psi_{1}\rightarrow\psi_{3}$
transition, where the upstream SXWG drives a $\psi_{1}\rightarrow\psi_{2}$
transition, and the downstream SXWG achieves a $\psi_{2}\rightarrow\psi_{3}$
promotion. Moreover, one can even design multiple SXWG modules for $\Delta
n=2$ or any dipole forbidden transitions. Thus, all combinations of SXWG
modules open the capability to generate high order x-ray modes starting from
the ground state $\psi_{1}$. It is worth mentioning another possible
application using dual SXWGs with a gap in between as an x-ray interferometer.
While the upstream SXWG causes the $\psi_{1}\rightarrow\psi_{2}$ transition as
a beam splitter, the downstream SXWG leads to the return of
$\psi_{2}\rightarrow\psi_{1}$ as a beam combiner. Furthermore, in the gap
between two SXWGs one can introduce a phase modulator to impose a phase shift
at one branch of the split state $\psi_{2}$, e.g., $x>0$ at $y=2$mm in Fig. 2.
A controllable interference due to the phase modulation is expected to happen
at the end of the downstream SXWG.

We are ready to demonstrate the Earth’s gravitational effect on our SXWG
system. Given that the gravitational redshift $\Delta_{G}$ significantly
changes the nuclear coherence $\rho_{eg}$ and Rabi frequency $\Omega_{n}$ in
Eq. (5) in two millimeter on Earth as demonstrated in Fig. 1(c), this
sensitivity potentially allows for turning gravity into a practical use, e.g.,
gravitationally sensitive x-ray optics. For illustrating the effect, we
numerically solve Eq. (3) and use the isotope 45Sc in an SXWG with parameters
$\varrho=8.38$, $L_{x}=100$nm, $L=2$mm, $k_{d}=23.778\times 10^{3}$rad/m, and
$\Delta=19.36$ to show the gravitational effect. Fig. 4(a) illustrates three
cases where the above discussed SXWG is located at $z=2.32$cm, $z=2$cm, and
$z=1.72$cm from the top down. An incident x ray with the transverse mode
$\psi_{1}$ is deflected upward and experiences a gravitational redshift
(vertical upward arrow with color gradient). The $\Delta_{G}$ will change when
the x ray illuminates the SXWG at different altitudes. The total detuning for
each case is specified at the level-scheme plot, namely,
$\Delta+\Delta_{G}=-2.52$, $\Delta+\Delta_{G}=0.5$, and
$\Delta+\Delta_{G}=3.14$ from the top down. We emphasize that the periodic
particle density modulation effectively plays the role of a resonant field,
and it always resonantly drives a transition between the x-ray modes in a
cladding waveguide for three cases. However, various $\Delta_{G}$ change the
effective coupling strength $\Omega_{n}$ and result in different outputs. The
scattered/split x rays reflect the output mode and can be measured by a
downstream position-sensitive detector. The $x$-dependent photon number counts
show the output $|E\left(x,L\right)|^{2}$ and reveal the Earth’s gravitational
effect. We depict the normalized $|E\left(x,y\right)|^{2}$ for $z=2.32$cm,
$z=2$cm, and $z=1.72$cm in Fig. 4(c, e, and g), respectively. The intra-
waveguide intensity shows that the x ray significantly gets split in Fig. 4(e)
under a half Rabi cycle $\psi_{1}\rightarrow\psi_{2}$ in the SXWG at $z=2$cm.
In contrast, Fig. 4(c and g) depicts only a transverse broadening of the x ray
due to a small $|\Omega_{1}|$. Fig. 4(b,d, and f) illustrate
$F_{1}\left(y\right)$ (red-dashed line) and $F_{2}\left(y\right)$ (green-solid
line) for $z=2.32$cm, $z=2$cm, and $z=1.72$cm, respectively. One can clearly
see that the x ray experiences Rabi flopping and becomes $\psi_{2}$ at the
resonant altitude $z=2$cm as also pointed out by Fig. 4(d and e). Given that
$|\Omega_{1}|$ decreases when the SXWG leaves the resonant altitude, the x ray
mostly remains in the initial mode $\psi_{1}$, namely,
$F_{1}\left(y\right)>F_{2}\left(y\right)$ at $z=2.32$cm and $z=1.72$cm. We
depict the output altitude-dependent $F_{2}$ at $y=2$mm in Fig. 4(a). As a
result, different x-ray splitting is expected to occur when lifting an SXWG
composed of 45Sc at only a millimeter altitude change.

We quantify the gravitational sensitivity of the SXWG by the full altitude
width $\Delta Z_{G}$ at the half maximum $Re\left[\rho_{eg}\right]$

$\Delta
Z_{G}=\sqrt{3}\left(\frac{\hbar\Gamma}{E_{t}}\right)\frac{c^{2}R_{E}^{2}}{GM_{E}},$
(6)

as indicated by the black-horizontal double arrow in Fig. 1(c). The introduced
$\Delta Z_{G}$ is a measure for the sensitivity of the x-ray-nucleus coupling
to the change of the SXWG vertical location. With the definition of the
quality factor of a nuclear resonance $Q=E_{t}/\left(\hbar\Gamma\right)$, we
can see that $\Delta Z_{G}$ is proportional to $1/Q$. Fig. 4(b) exemplifies
the implication of Eq. (6) for our system on Earth in a double-logarithmic
plot, where we mark the isotopes 45Sc, 57Fe, 67Zn, 73Ge, 103Rh, 107Ag, 109Ag,
181Ta, 182Ta, 229Th, and 249Bk, according their $Q$ factor. Some of the
nuclear parameters are listed in Table 1. Remarkably, the advantage of a very
high $Q\sim 10^{19}$ of 45Sc or 182Ta nuclear resonance endues an SXWG with a
gravitational sensitivity to only millimeter altitude change. Notably, it is
also possible to get sub-millimeter $\Delta Z_{G}$ using 229Th whose $Q\sim
10^{20}$ [29], and micron $\Delta Z_{G}$ with 107Ag and 109Ag whose $Q\sim
10^{22}$. It deserves to mention that 103Rh whose $Q>10^{23}$ [27] even
results in a nanometer $\Delta Z_{G}$, and it may lead to gravitational
application in mesoscopic scale.

In conclusion we have put forward a controllable SXWG system that potentially
turns gravity into an application of x-ray optics. A periodic intra-waveguide
structure, e.g., nuclear optical lattice 57Fe/56Fe bilayers in Ref.[30], can
drive a transition between x-ray modes. The x-ray transverse mode experiences
Rabi oscillation when propagating in an SXWG. Our scheme allows for
applications like a systematic production of structured x rays and an x-ray
interferometer without any beam splitter. Remarkably, a significant change of
a gravitionally induced splitting of x rays could happen by lifting our SXWG
made of, e.g., 45Sc or 182Ta, at only a millimeter scale.

S.-Y. L. and W.-T. L. are supported by the National Science and Technology
Council of Taiwan (Grant No. 110-2112-M-008-027-MY3, 110-2639-M-007 -001-ASP,
111-2923-M-008-004-MY3 & 111-2639-M-007-001-ASP). S. A. is supported by
National Science Foundation of China (Grant No. 11975155).

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