# Transversal effects on the ground-state of hard-core dipolar bosons
in one-dimensional optical lattices

Henning Korbmacher  Institut für Theoretische Physik, Leibniz Universität
Hannover, Appelstr. 2, D-30167 Hanover, Germany  Gustavo A. Domínguez-Castro
Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstr. 2,
D-30167 Hanover, Germany  Wei-Han Li  Max Planck Institute for the Physics of
Complex Systems, Nöthnitzer Str. 38, 01187 Dresden, Germany  Jakub Zakrzewski
Instytut Fizyki Teoretycznej, Uniwersytet Jagielloński, Łojasiewicza 11,
PL-30-348 Kraków, Poland  Mark Kac Complex Systems Research Center,
Uniwersytet Jagielloński, PL-30-348 Kraków, Poland  Luis Santos  Institut für
Theoretische Physik, Leibniz Universität Hannover, Appelstr. 2, D-30167
Hanover, Germany

###### Abstract

Polar lattice gases are usually assumed to have an inter-site interaction that
decays with the inter-particle distance $r$ as $1/r^{3}$. However, a loose-
enough transversal confinement may strongly modify the dipolar decay in one-
dimensional lattices. We show that this modification alters significantly the
ground-state properties of hard-core dipolar bosons. For repulsive inter-site
interactions, the corrected decay alters the conditions for devil’s staircase
insulators, affecting significantly the particle distribution in the presence
of an overall harmonic confinement. For attractive interactions, it results in
a reduction of the critical dipole interaction for the formation of self-bound
clusters, and for a marked enhancement of the region of liquefied lattice
droplets.

###### pacs:

##  I Introduction

Ultracold quantum gases in optical lattices constitute an optimal platform for
studying many-body physics under precisely controlled conditions Bloch  _et
al._ (2008); Gross and Bloch (2017); Langen  _et al._ (2015). In most current
experiments, the interactions between particles are short-range and well-
modeled by a contact pseudopotential. However, seminal experiments on dipolar
systems formed by magnetic atoms de Paz  _et al._ (2013); Baier  _et al._
(2016); Patscheider  _et al._ (2020) and polar molecules Yan  _et al._ (2013);
Li  _et al._ (2023) in optical lattices are starting to explore exciting
physics beyond the short-range scenario. Due to the anisotropic and long-range
character of the dipole-dipole potential, polar gases confined in optical
lattices are characterized not only by on-site interactions, but, crucially,
also by anisotropic inter-site interactions. As a result, dipolar lattice
gases of pinned particles can be employed as quantum simulators for spin
models, whereas itinerant particles realize different forms of the extended-
Hubbard model Lahaye  _et al._ (2009); Baranov  _et al._ (2012). Compared to
their non-dipolar counterparts, dipolar lattice gases present a much richer
ground-state physics, including crystalline phases Burnell  _et al._ (2009)
and supersolids Capogrosso-Sansone  _et al._ (2010), or the Haldane-insulator
phase Dalla Torre  _et al._ (2006).

The spatial decay of the inter-site interactions plays a crucial role in polar
lattice gases. Due to the form of the dipolar interaction in free space, this
decay is typically assumed as $1/r^{3}$, with $r$ the inter-site distance.
However, the interaction decay may be significantly affected by the
confinement transversal to the lattice axis Wall  _et al._ (2013). This
confinement alters the on-site wave functions, introducing a modification of
the inter-site interaction, which may potentially depart very significantly
from the $1/r^{3}$ dependence, and hence alter the equilibrium and out-of-
equilibrium physics of the polar lattice gas Wall  _et al._ (2013); Korbmacher
_et al._ (2023).

In this paper, we show that the modification of the interaction decay in the
presence of a loose-enough transversal confinement results in a significant
modification of the ground-state properties of hard-core dipolar bosons. In
contrast to Ref. Wall  _et al._ (2013), which focused on strong transversal
confinement, we show that the ground state is much more altered in the
experimentally relevant regime of a weak transversal confinement. We first
consider the case of repulsive interactions, showing that the modified decay
results in markedly shifted insulating phases, which translate into a
distorted particle distribution in the presence of an overall harmonic
confinement. For attractive dipoles, we show that the modified decay may
significantly ease the conditions for the realization of self-bound clusters.
Moreover, it results in a much wider parameter region for the observation of
liquefied self-bound droplets Morera  _et al._ (2023) without the need of
super-exchange processes.

Figure 1: (a) Schematic representation of the dipolar gas confined
longitudinally by an optical lattice and transversely by a harmonic trap. (b)
Modified dipole-dipole interaction vs lattice site for different
$\beta_{\text{eff}}$ (see main text), notice the log scale in the vertical
axis.

This manuscript is organized as follows. In Sec. II, we introduce the lattice
model considered, and show how the transversal confinement modifies the inter-
site interaction decay. In Sec. III, we study the effects of the modified
interaction on the phase diagram of repulsive hard-core bosons, whereas Sec.
IV focuses on the attractive case. Finally, in Sec. V, we summarize our
conclusions.

##  II Modified interaction decay

We consider dipolar hard-core bosons of mass $m$ confined longitudinally by a
1D optical lattice, $U_{0}\sin^{2}(\pi z/a)$ with $a$ the lattice constant,
and transversally by an isotropic harmonic potential
$\frac{1}{2}m\omega_{\perp}^{2}(x^{2}+y^{2})$, $\omega_{\perp}$ being the trap
frequency (see Fig. 1(a)). The dipole moments are assumed to be oriented by an
external field on the $xz$ plane forming an angle $\alpha$ with the lattice
axis $z$. Within the tight-binding approximation, the system is well described
by the extended Hubbard model (EHM):

$\hat{H}\\!=\\!\sum_{i}\left[-t\left(\hat{b}^{\dagger}_{i+1}\hat{b}_{i}+\mathrm{H.c.}\right)\\!+\\!\sum_{j>0}V_{j}\hat{n}_{i}\hat{n}_{i+j}-\mu\hat{n}_{i}\right],$
(1)

where $t$ is the tunneling amplitude between nearest neighbors,
$\hat{b}_{i}^{\dagger}$ ($\hat{b}_{i}$) is the creation (annihilation)
operator at site $i$, $\hat{n}_{i}=\hat{b}_{i}^{\dagger}\hat{b}_{i}$ is the
corresponding particle number operator, and $\mu$ is the chemical potential.
The hard-core constraint means that no double occupancy is allowed, i.e.
$(\hat{b}_{i}^{\dagger})^{2}=0$. This restriction can be achieved by means of
strong-enough on-site interactions (which may demand the use of Feshbach
resonances). The inter-site interaction between dipoles separated by $j$ sites
is given by:

$V_{j}=\int d^{3}r\int d^{3}r^{\prime}\
V(\vec{r}-\vec{r}^{\prime})|\phi(\vec{r})|^{2}|\phi(\vec{r}-ja\vec{e}_{z})|^{2},$
(2)

with $V(\vec{r})=\frac{C_{dd}}{4\pi
r^{3}}\left(1-3\frac{(x\sin\alpha+z\cos\alpha)^{2}}{r^{2}}\right)$ being the
dipole-dipole interaction. The coupling constant $C_{dd}$ is $\mu_{0}\mu^{2}$
for particles having a permanent magnetic dipole moment $\mu$ ($\mu_{0}$ is
the permeability of vacuum) and $d^{2}/\varepsilon_{0}$ for particles having a
permanent electric dipole moment $d$ ($\varepsilon_{0}$ is the vacuum
dielectric constant).

The on-site wave function can be written as $\phi(\vec{r})=\psi_{0}(x,y)W(z)$,
where $W(z)$ is the Wannier function associated with the lowest-energy band,
and
$\psi_{0}(x,y)=\frac{e^{-(x^{2}+y^{2})/2\ell_{\perp}}}{\sqrt{\pi}\ell_{\perp}}$
is the ground-state wave function of the transversal confinement, with
$\ell_{\perp}=\sqrt{\hbar/m\omega_{\perp}}$ the harmonic oscillator length.
For deep-enough lattices, $s\gtrsim 5$, where $s=U_{0}/E_{R}$ is the lattice
depth, and $E_{R}=\frac{\pi^{2}\hbar^{2}}{2ma^{2}}$ the recoil energy, the
actual Wannier function can be well approximated (for the purpose of the
calculation of the inter-site interactions) by a Gaussian Bloch  _et al._
(2008), $W(z)\simeq\frac{e^{-z^{2}/2\ell}}{\sqrt{\sqrt{\pi}\ell}}$, with
$\ell=a/(\pi s^{1/4})$ the effective on-site harmonic oscillator length. After
some straightforward algebra, one can show that for $\ell_{\perp}>\ell$
Korbmacher  _et al._ (2023):

$\frac{V_{j}}{E_{R}}=\frac{3B^{3/2}}{2\pi^{2}}(3\cos^{2}\alpha-1)\left(\frac{a_{dd}}{a}\right)f(\sqrt{B}j),$
(3)

where $a_{dd}=mC_{dd}/(12\pi\hbar^{2})$ is the dipolar length,
$B=\frac{\pi^{2}}{2}\frac{\chi}{1-\frac{\chi}{2\sqrt{2}}}$,
$\chi=\hbar\omega_{\perp}/E_{R}$, and

$f(\xi)=2\xi-\sqrt{2\pi}(1+\xi^{2})e^{\xi^{2}/2}\text{erfc}(\xi/\sqrt{2}).$
(4)

By using $V=V_{1}$, the interaction potential in Eq. (3) can be written as
$V_{j}=VG_{j}(B)$, with $G_{j}(B)=f(\sqrt{B}j)/f(\sqrt{B})$. The resulting
interaction decay has hence a universal dependence on the parameter $B$, which
is a function of the confinement parameters only.

Although for sufficiently large distances the potential in Eq. (3) recovers
the standard dipolar tail, i.e. $V_{j\rightarrow\infty}/V\rightarrow 1/j^{3}$,
the modification of the interaction to the neighboring sites may be very
significant (see Fig. 1(b)). The modification of the form of the tail
$V_{j}/V$ is uniquely defined once the ratio $V_{2}/V$ is fixed. Below we
parameterize such a ratio (for any value of $l_{\perp}/l$) by means of the
effective exponent $\beta_{\text{eff}}$, defined as
$V_{2}/V=1/2^{\beta_{\text{eff}}}$ Korbmacher  _et al._ (2023). The ground-
state properties are then fully determined by the value of $V/t$ (which sets
the interaction strength) and $\beta_{\text{eff}}$ (which fixes the form of
the decay $V_{j}/V$). By changing the transversal confinement,
$\beta_{\text{eff}}$ can be tuned within the interval
$0<\beta_{\text{eff}}<3.2$ Korbmacher  _et al._ (2023). When
$\ell_{\perp}<\ell$, $\beta_{eff}>3$, whereas the opposite is true if
$\ell_{\perp}>\ell$. When $\beta_{\text{eff}}=3$, the decay corresponds
exactly to the standard $V_{j}/V=1/j^{3}$, and hence we employ interchangeably
$1/j^{3}$ or $\beta_{\text{eff}}=3$ to denote the standard decay. For
$\ell_{\perp}\ll\ell$, the case discussed in Ref. Wall  _et al._ (2013),
$\beta_{eff}$ is only slightly larger than $3$, and hence the modification to
the $1/j^{3}$ dependence is small. As a result the corrections to the ground-
state properties are minor Wall  _et al._ (2013). In contrast, the deviation
from the standard decay may be large when $\ell_{\perp}>\ell$, leading to a
very significant modification of the ground-state properties, as discussed
below.

##  III Repulsive polar lattice gas

We focus first on the ground-state properties for the case of repulsive inter-
site interactions, $V>0$. In absence of dipolar interaction, the standard
Hubbard model with hard-core bosons may present only two phases, either a
superfluid (SF) phase or a band insulator, with filling factor
$\bar{n}=N/L=1$, with $N$ the number of bosons and $L$ the number of sites.
Note that the latter is equivalent to the vacuum ($\bar{n}=0$) due to
particle/hole symmetry. In the presence of inter-site dipolar interactions,
and depending on the dipole strength $V/t$ and the chemical potential $\mu$
the system may present different insulating phases with commensurate
fractional fillings (devil’s staircase) Burnell  _et al._ (2009). Particularly
relevant are the half-filled density-wave (2DW) ($\bar{n}=1/2$), which for
$t=0$ acquires the form $|\cdots\,\leavevmode\hbox to3.44pt{\vbox
to3.44pt{\pgfpicture\makeatletter\hbox{\hskip 1.72221pt\lower-1.72221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.72221pt}{0.0pt}\pgfsys@curveto{1.72221pt}{0.95116pt}{0.95116pt}{1.72221pt}{0.0pt}{1.72221pt}\pgfsys@curveto{-0.95116pt}{1.72221pt}{-1.72221pt}{0.95116pt}{-1.72221pt}{0.0pt}\pgfsys@curveto{-1.72221pt}{-0.95116pt}{-0.95116pt}{-1.72221pt}{0.0pt}{-1.72221pt}\pgfsys@curveto{0.95116pt}{-1.72221pt}{1.72221pt}{-0.95116pt}{1.72221pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\leavevmode\hbox
to3.84pt{\vbox to3.84pt{\pgfpicture\makeatletter\hbox{\hskip
1.92221pt\lower-1.92221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.72221pt}{0.0pt}\pgfsys@curveto{1.72221pt}{0.95116pt}{0.95116pt}{1.72221pt}{0.0pt}{1.72221pt}\pgfsys@curveto{-0.95116pt}{1.72221pt}{-1.72221pt}{0.95116pt}{-1.72221pt}{0.0pt}\pgfsys@curveto{-1.72221pt}{-0.95116pt}{-0.95116pt}{-1.72221pt}{0.0pt}{-1.72221pt}\pgfsys@curveto{0.95116pt}{-1.72221pt}{1.72221pt}{-0.95116pt}{1.72221pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\leavevmode\hbox
to3.44pt{\vbox to3.44pt{\pgfpicture\makeatletter\hbox{\hskip
1.72221pt\lower-1.72221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.72221pt}{0.0pt}\pgfsys@curveto{1.72221pt}{0.95116pt}{0.95116pt}{1.72221pt}{0.0pt}{1.72221pt}\pgfsys@curveto{-0.95116pt}{1.72221pt}{-1.72221pt}{0.95116pt}{-1.72221pt}{0.0pt}\pgfsys@curveto{-1.72221pt}{-0.95116pt}{-0.95116pt}{-1.72221pt}{0.0pt}{-1.72221pt}\pgfsys@curveto{0.95116pt}{-1.72221pt}{1.72221pt}{-0.95116pt}{1.72221pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\leavevmode\hbox
to3.84pt{\vbox to3.84pt{\pgfpicture\makeatletter\hbox{\hskip
1.92221pt\lower-1.92221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.72221pt}{0.0pt}\pgfsys@curveto{1.72221pt}{0.95116pt}{0.95116pt}{1.72221pt}{0.0pt}{1.72221pt}\pgfsys@curveto{-0.95116pt}{1.72221pt}{-1.72221pt}{0.95116pt}{-1.72221pt}{0.0pt}\pgfsys@curveto{-1.72221pt}{-0.95116pt}{-0.95116pt}{-1.72221pt}{0.0pt}{-1.72221pt}\pgfsys@curveto{0.95116pt}{-1.72221pt}{1.72221pt}{-0.95116pt}{1.72221pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\cdots\rangle$, and
the one-third-filled (3DW) ($\bar{n}=1/3$) $|\cdots\,\leavevmode\hbox
to3.44pt{\vbox to3.44pt{\pgfpicture\makeatletter\hbox{\hskip
1.72221pt\lower-1.72221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.72221pt}{0.0pt}\pgfsys@curveto{1.72221pt}{0.95116pt}{0.95116pt}{1.72221pt}{0.0pt}{1.72221pt}\pgfsys@curveto{-0.95116pt}{1.72221pt}{-1.72221pt}{0.95116pt}{-1.72221pt}{0.0pt}\pgfsys@curveto{-1.72221pt}{-0.95116pt}{-0.95116pt}{-1.72221pt}{0.0pt}{-1.72221pt}\pgfsys@curveto{0.95116pt}{-1.72221pt}{1.72221pt}{-0.95116pt}{1.72221pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\leavevmode\hbox
to3.84pt{\vbox to3.84pt{\pgfpicture\makeatletter\hbox{\hskip
1.92221pt\lower-1.92221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.72221pt}{0.0pt}\pgfsys@curveto{1.72221pt}{0.95116pt}{0.95116pt}{1.72221pt}{0.0pt}{1.72221pt}\pgfsys@curveto{-0.95116pt}{1.72221pt}{-1.72221pt}{0.95116pt}{-1.72221pt}{0.0pt}\pgfsys@curveto{-1.72221pt}{-0.95116pt}{-0.95116pt}{-1.72221pt}{0.0pt}{-1.72221pt}\pgfsys@curveto{0.95116pt}{-1.72221pt}{1.72221pt}{-0.95116pt}{1.72221pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\leavevmode\hbox
to3.84pt{\vbox to3.84pt{\pgfpicture\makeatletter\hbox{\hskip
1.92221pt\lower-1.92221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.72221pt}{0.0pt}\pgfsys@curveto{1.72221pt}{0.95116pt}{0.95116pt}{1.72221pt}{0.0pt}{1.72221pt}\pgfsys@curveto{-0.95116pt}{1.72221pt}{-1.72221pt}{0.95116pt}{-1.72221pt}{0.0pt}\pgfsys@curveto{-1.72221pt}{-0.95116pt}{-0.95116pt}{-1.72221pt}{0.0pt}{-1.72221pt}\pgfsys@curveto{0.95116pt}{-1.72221pt}{1.72221pt}{-0.95116pt}{1.72221pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\leavevmode\hbox
to3.44pt{\vbox to3.44pt{\pgfpicture\makeatletter\hbox{\hskip
1.72221pt\lower-1.72221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.72221pt}{0.0pt}\pgfsys@curveto{1.72221pt}{0.95116pt}{0.95116pt}{1.72221pt}{0.0pt}{1.72221pt}\pgfsys@curveto{-0.95116pt}{1.72221pt}{-1.72221pt}{0.95116pt}{-1.72221pt}{0.0pt}\pgfsys@curveto{-1.72221pt}{-0.95116pt}{-0.95116pt}{-1.72221pt}{0.0pt}{-1.72221pt}\pgfsys@curveto{0.95116pt}{-1.72221pt}{1.72221pt}{-0.95116pt}{1.72221pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\leavevmode\hbox
to3.84pt{\vbox to3.84pt{\pgfpicture\makeatletter\hbox{\hskip
1.92221pt\lower-1.92221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.72221pt}{0.0pt}\pgfsys@curveto{1.72221pt}{0.95116pt}{0.95116pt}{1.72221pt}{0.0pt}{1.72221pt}\pgfsys@curveto{-0.95116pt}{1.72221pt}{-1.72221pt}{0.95116pt}{-1.72221pt}{0.0pt}\pgfsys@curveto{-1.72221pt}{-0.95116pt}{-0.95116pt}{-1.72221pt}{0.0pt}{-1.72221pt}\pgfsys@curveto{0.95116pt}{-1.72221pt}{1.72221pt}{-0.95116pt}{1.72221pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\leavevmode\hbox
to3.84pt{\vbox to3.84pt{\pgfpicture\makeatletter\hbox{\hskip
1.92221pt\lower-1.92221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.72221pt}{0.0pt}\pgfsys@curveto{1.72221pt}{0.95116pt}{0.95116pt}{1.72221pt}{0.0pt}{1.72221pt}\pgfsys@curveto{-0.95116pt}{1.72221pt}{-1.72221pt}{0.95116pt}{-1.72221pt}{0.0pt}\pgfsys@curveto{-1.72221pt}{-0.95116pt}{-0.95116pt}{-1.72221pt}{0.0pt}{-1.72221pt}\pgfsys@curveto{0.95116pt}{-1.72221pt}{1.72221pt}{-0.95116pt}{1.72221pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\cdots\rangle$ (or
equivalently the phase with $\bar{n}=2/3$). Other fractional fillings are
possible but they require significantly larger $V/t$ ratios.

We are interested in how the modified interaction decay alters the boundaries
of the insulating phases. We employ density-matrix renormalization group
(DMRG) techniques to obtain the ground-state of a system of $L=120$ sites,
assuming periodic boundary conditions. Superfluid (insulating) phases are
characterized by a polynomial (exponential) decay of the single-particle
correlation,
$C_{\text{SF}}(i,j)=\langle\hat{a}_{i}^{\dagger}\hat{a}_{i+j}\rangle$. To
distinguish amongst the different insulating phases, we evaluate the structure
factor
$M(k)=\frac{1}{L}\sum_{j=1}^{L-1}e^{-ikj}\langle\hat{n}_{i}\hat{n}_{i+j}\rangle$,
where $k\in[-\pi,\pi]$ is the quasi-momentum. For an insulating phase with
filling $\bar{n}=\frac{1}{m}$, $M(k)$ presents peaks at $k=\pm\frac{2\pi}{m}$.
Due to particle-hole symmetry, the same is true for the phase with
$\bar{n}=\frac{m-1}{m}$. Note that particle-hole symmetry results in a mirror
symmetry of the phase diagram on the $(t/V,\mu/V)$ plane around the chemical
potential $(\mu/V)_{0}(B)=\sum_{j>0}G_{j}(B)$. In order to compare properly
the results for different $B$ values, we introduce the re-scaled chemical
potential, $\tilde{\mu}=\mu/(2\sum_{j>0}G_{j}(B))$, such that the phase
diagram presents mirror symmetry around $\tilde{\mu}/V=\frac{1}{2}$. With this
shift, the borders of the band insulator, $\bar{n}=1$, and the vacuum, $n=0$,
are given by $\tilde{\mu}/V=0,1$.

Figure 2: (top) Phase diagram in the $(t/V,\tilde{\mu}/V)$ plane for standard
$1/j^{3}$ decay (blue lobes) and the modified dipolar interaction with
$\beta_{\text{eff}}=2$ (red lobes). (bottom) Critical $V/t$ for the 2DW (blue)
and the 3DW (red) for different values of $\beta_{\text{eff}}$, normalized to
the value expected for the $1/j^{3}$ decay. Error bars indicate the exact
value of $(V/t)_{cr}$ and are set by the fidelity of numerical input
parameters.

In Fig. 2 (top), we depict the phase diagram for a $1/r^{3}$ decay (blue), and
for the modified dipolar interaction with $\beta_{\text{eff}}=2$ (red). Due to
the enhanced role of the next-to-NN interaction $V_{2}$ compared to the usual
$1/j^{3}$ decay, the central lobe, which corresponds to the 2DW phase, is
significantly smaller for $\beta_{\text{eff}}=2$. This must be compared to the
results of Ref. Wall  _et al._ (2013), which focused on the case
$\ell_{\perp}\ll\ell$ (and hence $\beta_{\mathrm{eff}}\gtrsim 3$), for which
the modification of the 2DW lobe compared to that expected for the $1/j^{3}$
dependence is very small.

The relative deviation from the results considering the standard $1/j^{3}$
dependence is even more relevant for insulating phases at lower fillings. In
Fig. 2 (top) we observe as well the lobes with filling $\bar{n}=1/3$ and
$\bar{n}=1/4$, and the particle/hole symmetric ones for $\bar{n}=2/3$ and
$\bar{n}=3/4$. Note that these lobes, which are depicted in detail in the
inset, are significantly modified, again due to the enhanced value of $V_{2}$
and $V_{3}$ compared to the standard $1/j^{3}$ decay. In particular, for
$\beta_{\mathrm{eff}}=2$, the critical $V/t$ for the observation of the 3DW is
strongly reduced from a critical $V/t\simeq 47$ to $V/t\simeq 27$. Figure 2
(bottom) shows the critical $(V/t)_{cr}$ for 2DW and 3DW, as a function of
$\beta_{\text{eff}}$, normalized to the value expected for a $1/j^{3}$ decay.
Note that for 2DW (3DW) $(V/t)_{cr}$ increases (decreases) by approximately a
factor of $2$ when reducing $\beta_{\mathrm{eff}}$ down to $1$.

Figure 3: (top) Boundaries of the lobes with $n=1/2$, $1/3$ and $1/4$ for
$t=0$ as a function of the shifted chemical potential $\tilde{\mu}/V$ and
$\beta_{\text{eff}}$. The lobes are evaluated analytically considering a cut-
off of the interactions at $4$ neighbors. (bottom) Spatial density
distribution $\langle n_{i}\rangle$ for $N=39$ bosons, $\Omega/t=0.03$, and
for $1/j^{3}$ decay (blue) and for $\beta_{\text{eff}}=2$ (red) obtained from
DMRG calculations.

The lobes are also changed in their dependence with chemical potential. In
addition to the above-mentioned $(\mu/V)_{0}(B)$ shift, the width (in chemical
potential) of the 2DW (3DW) lobe significantly shrinks (widens) with
decreasing $\beta_{\mathrm{eff}}$. This is illustrated in Fig. 3 (top), where
we consider $t=0$, for which the phase boundaries may be evaluated
analytically.

The modification of the insulating lobes has significant consequences for the
spatial particle distribution in the the presence of an overall harmonic
confinement. The confinement results in an additional term
$\Omega\sum_{i}(i-L/2)^{2}\hat{n}_{i}$ in Eq. (1). For a sufficiently weak
confinement, local-density approximation arguments apply, and the density
profile presents the expected wedding-cake profile. Figure 3 (bottom) shows
the local mean occupation $\langle n_{i}\rangle$ obtained from DMRG
calculations. In the plateaus, which characterize the DW phases, we average
over neighboring sites to flatten the DW oscillations in $\langle
n_{i}\rangle$. In agreement with the phase diagram, the central 2DW plateau
shrinks while the 3DW plateau widens when decreasing $\beta_{\text{eff}}$.

##  IV Attractive polar lattice gas

We analyze at this point the case of $V<0$, focusing first on the simplest
case of just two bosons, and then discussing the formation of self-bound
lattice droplets.

###  IV.1 Dimers

The wave function characterizing a state of two bosons can be separated as
$\Psi(R,r)=e^{iKR}\Phi_{K}(r)$, where $R=(i_{1}+i_{2})/2$ is the center-of-
mass, $r=i_{1}-i_{2}$ is the relative coordinate, and $i_{j=1,2}$ is the
lattice site in which particle $j$ is. The wave function $\Phi_{K}(r)$ depends
on the center-of-mass quasi-momentum $K\in[-\pi,\pi]$, and satisfies the
Schrödinger equation $\hat{H}_{K}|\Phi_{K}\rangle=E_{K}\Phi_{K}$, with

$\displaystyle\hat{H}_{K}$ $\displaystyle=$
$\displaystyle-2t\cos\left(\frac{Ka}{2}\right)\sum_{r\geq
1}(|K,r+1\rangle\langle K,r|+\mathrm{H.c.})$ (5) $\displaystyle+$
$\displaystyle V\sum_{r\geq 1}G_{j}(B)|K,r\rangle\langle K,r|,$

where $|K,r\rangle$ stands for the state with center-of-mass quasi-momentum
$K$, and inter-particle separation $r$. Diagonalizing $\hat{H}_{K}$ for
different $K$ in the Brillouin zone provides the energy spectrum, depicted in
Fig. 4 for $V/t=-4$ and different values of $\beta_{\text{eff}}$. Compared to
the case $1/r^{3}$ for the same $V/t$, the modification of the dipolar tail
results in additional bound eigenstates.

The two-body ground-state is given for all $\beta_{\mathrm{eff}}$ values by a
bound pair with $K=0$, and a spatial distribution peaked at nearest neighbors.
However, the binding becomes stronger, i.e. pairing (and in general the
formation of clusters, as discussed below) demands a smaller $|V|/t$ when
$\beta_{\mathrm{eff}}$ decreases. Furthermore, dimer mobility may be strongly
modified by tuning the transversal confinement. Note that the curvature of the
lowest branch at $K=0$, associated with the effective mass of the ground-state
dimer, is significantly modified as a function of $\beta_{\mathrm{eff}}$. This
is best illustrated with the case of strong $V/t$, for which the lowest bound-
state branch corresponds to bound nearest-neighbor dimers, which move via
second-order hopping with amplitude

$t_{D}=\frac{1}{1-2^{-\beta_{\mathrm{eff}}}}\frac{t^{2}}{V}.$ (6)

This should be compared to the corresponding value $t_{D}=8t^{2}/7V$ for the
case of $1/r^{3}$ decay. Note that e.g. for $\beta_{\text{eff}}=1$,
$t_{D}=2t^{2}/V$, and hence the dimer dynamics is approximately twice faster.

Figure 4: Energy spectrum $E_{K}$ as a function of the center-of-mass quasi-
momentum $K$ for several effective powers of the dipole-dipole interaction. In
all panels the interaction strength is $V/t=-4$.

###  IV.2 Self-bound lattice droplets

The formation of bound dimers in the two-body problem extends in the many-body
case to the formation of self-bound lattice droplets formed by potentially
many particles. These resemble those recently discussed in binary mixtures
Morera  _et al._ (2020, 2021) and cavities Karpov and Piazza (2019, 2022).
Dipolar self-bound lattice droplets have been discussed in the context of out-
of-equilibrium polar lattice gases after a quench of the confinement potential
Li  _et al._ (2020). Very recently, ground-state self-bound dipolar lattice
droplets were studied in Ref. Morera  _et al._ (2023). Although self-bound
lattice droplets present some interesting similarities to quantum droplets in
binary and dipolar Bose-Einstein condensates Böttcher  _et al._ (2020), they
differ from them in the physical mechanism as well as in the fact that lattice
droplets are self-pinned, i.e. they remain for any practical purposes
immobile, due to their large effective mass.

In Ref. Morera  _et al._ (2023), it was shown that droplets can be either a
self-bound Mott insulator (with saturated unit filling) or in a liquefied
state (self-bound but with a filling lower than unity). It was argued that
liquefaction arises due to the interplay between inter-site dipolar attraction
and the super-exchange processes originating from short-range repulsion in
soft-core Bose systems. In the following, we show that self-bound droplets in
hard-core gases (where super-exchange is absent) are generally in either a
saturated or liquid regime, and that the boundaries between the saturated,
liquid, and unbound (gas) phases are strongly dependent on
$\beta_{\mathrm{eff}}$.

Figure 5: (a) Phase diagram for impenetrable dipolar bosons as a function of
the absolute value of the interaction strength $|V|/t$ and
$\beta_{\text{eff}}$. We found a saturated droplet regime (b), a droplet
superfluid (c), and a unbound phase (gas) (d). The Tonks-Girardeau analysis
corresponds to the color scheme shown, saturated droplet regime (red), droplet
superfluid (blue), and gaseous phase (white). Orange and green markers are
associated with the boundary lines of the saturated droplet and liquid regimes
from DMRG calculations, respectively. The dashed yellow line indicates the
threshold for a two-body bound state (dimer). Lower panels display
characteristic density profiles of the different phases.

In order to analyze self-bound lattice droplets, we develop a variational
approach similar to that used in Ref. Morera  _et al._ (2023). We assume that
the impenetrable lattice gas is well described by a Tonks-Girardeau
$|\Psi_{TG}\rangle=\Pi_{k<k_{F}}\hat{b}_{k}^{\dagger}|0\rangle$ ansatz with
the density $n=k_{F}/\pi$ as a variational parameter, $k_{F}$ being the Fermi
momentum of the fermionized bosons. Although this ansatz is only exact for
non-interacting one-dimensional hard-core bosons, it can perturbatively
capture the main features of dipolar hard-core gases Morera  _et al._ (2023).
Evaluating $\langle\Psi_{TG}|\hat{H}|\Psi_{TG}\rangle$ yields the energy per
particle

$\frac{E[n]}{tN}=-\frac{2\sin
n\pi}{n\pi}+\frac{V}{tn}\sum_{r>0}G_{r}\left[1-\frac{\sin^{2}n\pi
r}{\pi^{2}r^{2}}\right].$ (7)

The first and second terms are associated with the kinetic energy and the
modified dipolar interaction, respectively. Following Ref. Morera  _et al._
(2023), we classify the quantum phases of the dipolar system according to the
value of the density $n_{c}$ at which the energy per particle is minimal. In
the unbound (gas) phase, the dipoles spread uniformly over all available
sites. As a result, the gas phase is characterized by a vanishing density
$n_{c}=0$. In contrast, in a liquid phase, where the droplets are self-bound
and localized at zero pressure Morera  _et al._ (2020, 2021), the energy per
particle takes its minimal value at a finite density $0<n_{c}<1$. Furthermore,
the energy per particle at $n_{c}$ is smaller than the bottom of the
scattering band, i.e $E[n_{c}]/N<-2t$. Lastly, we define the saturated droplet
regime as that in which the energy per particle becomes minimal at $n_{c}=1$.
In such a phase, the droplet is incompressible and hole propagation within the
droplet is inhibited due to the high energy cost of breaking a dipole bond Li
_et al._ (2020). In Fig. 5, we show, in a color scheme, the resulting phase
diagram of impenetrable dipolar bosons as a function of the interaction
strength $|V|/t$ and $\beta_{\text{eff}}$. The saturated droplet regime is
indicated in red, whereas the liquid and gaseous phases in white and blue,
respectively. In stark contrast to the bare dipolar potential $1/r^{3}$ Morera
_et al._ (2023), the modified interaction gives rise to a wide liquefied
region without removing the hard-core constraint.

In addition to the Tonks-Girardeau analysis, we calculate the ground state of
the dipolar system for different values of $|V|/t$ using DMRG simulations with
$N=24$ bosons in $L=80$ sites. In our DMRG simulations, we define the gas-to-
liquid transition at the interaction strength $|V_{c}|/t$ in which the energy
per particle is equal to the bottom of the scattering band $E/N=-2t$.
Meanwhile, we define the saturated droplet regime as that in which the central
density of the ground state reaches unity, i.e. the density distribution
acquires a flat-top profile. Note that the wings of the droplet are not
necessarily saturated, although when $|V|/t$ increases eventually the whole
droplet enters the unit filling regime. Green circles and orange stars in Fig.
5 correspond to our DMRG results for the gas-to-liquid and liquid-to-saturated
boundaries, respectively. A good agreement is found between the Tonks-
Girardeau analysis and the DMRG results. Note as well, that the gas-to-liquid
transition is well-estimated by the threshold of dimer bound-state formation
(dashed line in Fig. 5), obtained from the two-body calculations discussed
above.

##  V Conclusions

A sufficiently loose transversal confinement results in a significant
modification of the inter-site interaction between dipoles in a one-
dimensional optical lattice, which departs from the usually assumed $1/j^{3}$
dependence. We have shown that this modification, which acquires a universal
dependence on the confinement parameters, may significantly modify the ground-
state properties of hard-core bosons. For repulsive dipoles, it leads to a
marked shift of the boundaries of the insulating devil’s staircase phases,
that translates in a significantly modified particle distribution in the
presence of an overall harmonic potential. For attractive dipoles, the
modified interaction decay results in a lower critical dipolar strength for
the formation of self-bound clusters, and to a much wider parameter region for
the observation of liquefied droplets without the need of super-exchange
processes. The discussed effects should play a relevant role in future lattice
experiments on magnetic atoms or polar molecules.

## Acknowledgments

We acknowledge support of the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – Project-ID 274200144 – SFB 1227 DQ-mat within the
project A04, and under Germany’s Excellence Strategy – EXC-2123 Quantum-
Frontiers – 390837967. J.Z. is supported by the National Science Centre
(Poland) under project 2021/43/I/ST3/01142 founded in the OPUS call within the
WEAVE programme.

## References

  * Bloch  _et al._ (2008) I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008).
  * Gross and Bloch (2017) C. Gross and I. Bloch, Science 357, 995 (2017).
  * Langen  _et al._ (2015) T. Langen, R. Geiger, and J. Schmiedmayer, Annual Review of Condensed Matter Physics 6, 201 (2015), https://doi.org/10.1146/annurev-conmatphys-031214-014548 .
  * de Paz  _et al._ (2013) A. de Paz, A. Sharma, A. Chotia, E. Maréchal, J. H. Huckans, P. Pedri, L. Santos, O. Gorceix, L. Vernac, and B. Laburthe-Tolra, Phys. Rev. Lett. 111, 185305 (2013).
  * Baier  _et al._ (2016) S. Baier, M. J. Mark, D. Petter, K. Aikawa, L. Chomaz, Z. Cai, M. Baranov, P. Zoller, and F. Ferlaino, Science 352, 201 (2016).
  * Patscheider  _et al._ (2020) A. Patscheider, B. Zhu, L. Chomaz, D. Petter, S. Baier, A.-M. Rey, F. Ferlaino, and M. J. Mark, Phys. Rev. Res. 2, 023050 (2020).
  * Yan  _et al._ (2013) B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. Hazzard, A. M. Rey, D. S. Jin, and J. Ye, Nature 501, 521 (2013).
  * Li  _et al._ (2023) J.-R. Li, K. Matsuda, C. Miller, A. N. Carroll, W. G. Tobias, J. S. Higgins, and J. Ye, Nature 614, 70 (2023).
  * Lahaye  _et al._ (2009) T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, Reports on Progress in Physics 72, 126401 (2009).
  * Baranov  _et al._ (2012) M. A. Baranov, M. Dalmonte, G. Pupillo, and P. Zoller,  _Chemical Reviews_ , Chemical Reviews 112, 5012 (2012).
  * Burnell  _et al._ (2009) F. J. Burnell, M. M. Parish, N. R. Cooper, and S. L. Sondhi, Physical Review B 80 (2009), 10.1103/physrevb.80.174519.
  * Capogrosso-Sansone  _et al._ (2010) B. Capogrosso-Sansone, C. Trefzger, M. Lewenstein, P. Zoller, and G. Pupillo, Phys. Rev. Lett. 104, 125301 (2010).
  * Dalla Torre  _et al._ (2006) E. G. Dalla Torre, E. Berg, and E. Altman, Phys. Rev. Lett. 97, 260401 (2006).
  * Wall  _et al._ (2013) M. L. Wall, , and L. D. Carr, New Journal of Physics 15, 123005 (2013).
  * Korbmacher  _et al._ (2023) H. Korbmacher, P. Sierant, W. Li, X. Deng, J. Zakrzewski, and L. Santos, Phys. Rev. A 107, 013301 (2023).
  * Morera  _et al._ (2023) I. Morera, R. Ołdziejewski, G. E. Astrakharchik, and B. Juliá-Díaz, Phys. Rev. Lett. 130, 023602 (2023).
  * Morera  _et al._ (2020) I. Morera, G. E. Astrakharchik, A. Polls, and B. Juliá-Díaz, Phys. Rev. Res. 2, 022008 (2020).
  * Morera  _et al._ (2021) I. Morera, G. E. Astrakharchik, A. Polls, and B. Juliá-Díaz, Phys. Rev. Lett. 126, 023001 (2021).
  * Karpov and Piazza (2019) P. Karpov and F. Piazza, Phys. Rev. A 100, 061401 (2019).
  * Karpov and Piazza (2022) P. Karpov and F. Piazza, Phys. Rev. Lett. 128, 103201 (2022).
  * Li  _et al._ (2020) W. Li, A. Dhar, X. Deng, K. Kasamatsu, L. Barbiero, and L. Santos, Phys. Rev. Lett. 124, 010404 (2020).
  * Böttcher  _et al._ (2020) F. Böttcher, J.-N. Schmidt, J. Hertkorn, K. S. H. Ng, S. D. Graham, M. Guo, T. Langen, and T. Pfau, Reports on Progress in Physics 84, 012403 (2020).


# Metahouse: noise-insulating chamber based on periodic structures

Mariia Krasikova<EMAIL_ADDRESS>School of Physics and
Engineering, ITMO University, St. Petersburg 197101, Russia  Sergey Krasikov
School of Physics and Engineering, ITMO University, St. Petersburg 197101,
Russia  Anton Melnikov<EMAIL_ADDRESS>Chair of Vibroacoustics of
Vehicles and Machines, Technical University of Munich, Garching b. München
85748, Germany  Yuri Baloshin  School of Physics and Engineering, ITMO
University, St. Petersburg 197101, Russia  Steffen Marburg  Chair of
Vibroacoustics of Vehicles and Machines, Technical University of Munich,
Garching b. München 85748, Germany  David Powell  School of Engineering and
Information Technology, University of New South Wales, Northcott Drive,
Canberra, Australian Capital Territory 2600, Australia  Andrey Bogdanov
School of Physics and Engineering, ITMO University, St. Petersburg 197101,
Russia

###### Abstract

Noise pollution remains a challenging problem requiring the development of
novel systems for noise insulation. Extensive work in the field of acoustic
metamaterials has led to occurrence of various ventilated structures which,
however, are usually demonstrated for rather narrow regions of the audible
spectrum. In this work, we further extend the idea of metamaterial-based
systems developing a concept of a metahouse chamber representing a ventilated
structure for broadband noise insulation. Broad stop bands originate from
strong coupling between pairs of Helmholtz resonators constituting the
structure. We demonstrate numerically and experimentally the averaged
transmission $-43$ dB within the spectral range from $1500$ to $16500$ Hz. The
sparseness of the structure together with the possibility to use optically
transparent materials suggest that the chamber may be also characterized by
partial optical transparency depending on the mutual position of structural
elements. The obtained results are promising for development of novel noise-
insulating structures advancing urban science.

##  I Introduction

Acoustic metamaterials and phononic crystals have been subjects of keen
scientific interest for a couple of decades as they provide novel routes
towards efficient control of acoustic field [1, 2, 3, 4]. In addition to the
realization of negative refractive index [5], superlenses [6, 7], holograms
[8], and acoustic cloaks [9], recent advances include the development of non-
reciprocal systems [10], topological insulators [11, 12], nonlinear [13],
tunable [14], coding [15], and programmable metasurfaces [16]. Acoustic
metasurfaces were also explored as potential platforms for analog computing
[17] and, vice versa, advances in computer science and artificial intelligence
boost design procedures to achieve desired properties of metamaterials and
metasurfaces [18, 19, 20, 21]. Metamaterials can be also used as a platform to
explore analogies of the quantum concepts, such as Hall effect [22, 23], spin
properties [24, 25, 26, 27], skyrmions [28], and twistronics [29].

One of the developing branches in the field of acoustic metamaterials is
dedicated to realization of novel noise-insulating systems. Increasing noise
pollution in urban areas is one of the endangering trends affecting global
health and ecological environment [30, 31, 32, 33, 34]. Solution of this
problem requires the development of new methods and materials allowing
broadband passive noise insulation. Conventionally used systems are typically
represented by bulky structures imposing harsh engineering restrictions on
buildings and constructions [35]. Frequency range of noise mitigation has to
be compromised with mass and volume of the used materials. Moreover, some
crucial properties like ventilation or optical transparency are usually
incompatible with such systems.

Figure 1: Metahouse concept. The Helmholtz resonators are shown with gray only
for visual clearance and can be made of transparent materials. The inner space
of the developed chamber is protected from acoustic noises within a broad
audible spectral range. At the same time, the structure itself is ventilated
and optically transparent. Two former parameters depend solely on the distance
between the structural elements and their size, which also determine the
spectral range of blocked noises.

Instead of the conventional mass-density law, reflection and attenuation of
sound in metamaterials relies mostly on periodicity and shape of the
structural elements rather than their material properties. One of the
important options enabled by metamaterials is the possibility to implement
structures allowing air flow [36, 37, 38, 39]. A variety of designs includes
perforated membranes [40, 41], space coiling structures [42, 43, 44, 45, 46],
and metacages [47, 48] have been presented. Nevertheless, despite the plethora
of achievable physical effects, acoustic metamaterials rarely find real-life
applications. These structures usually have complicated designs and a narrow
operating range.

In this paper, we propose a noise-insulating ventilated metahouse chamber
allowing access of light into the inner area. The chamber is designed to be
simple for manufacturing and assembling. At the same time, the requirements
for materials are easily met and the structure can be made of ecologically
friendly and optically transparent materials. The chamber is based on coupled
Helmholtz resonators with C-shaped cross-section and exploits band-gap
formation as the main mechanism for sound insulation. The artistic
illustration of the concept is presented in Fig. 1. It is proposed that such
structures may be incorporated into urban parks to create noise-free
relaxation zones for visitors.

We start with designing a 2D metamaterial-based periodic structure and tuning
its parameters for maximization of the integral width of band-gaps in the
audible spectral range. Then, we perform an experimental demonstration of
stop-bands corresponding to band-gaps of the infinite system. Finally, we
provide an experimental demonstration of a finite-size prototype of the
metahouse chamber. All experimental results are verified by numerical
simulations.

##  II Results

###  II.1 Design of a unit cell

The chosen unit cell consists of two coupled Helmholtz resonators with
C-shaped cross-section, as shown in Fig. 2(a). Such resonators can be 3D
printed or easily manufactured via simple mechanical processing of
commercially available polymer pipes. Similar geometries were already
considered for noise mitigation purposes [49, 48, 50] and we have just
utilized the fact that local coupling between the resonators may increase the
integral width of band gaps [51]. In this paper we consider only $\Gamma X$
band-gaps since the implemented experimental realization of the semi-infinite
structure implies that the incident plane wave has only $k_{x}$ component.
Performing the tuning of geometrical parameters, we have optimised the system
to further increase the integral width of band-gaps $f_{\Gamma X}$ defined as
the sum of the widths of individual band-gaps [see Figs. 2(b) and 2(c)]:

$f_{\Gamma X}=\sum f_{\Gamma X}^{(i)}$ (1)

where $i$ stays for the number of a band-gap and index $\Gamma X$ indicates
that the considered band-gaps lie within $\Gamma X$ interval of the Brillouin
zone [see Fig. 2(b) and 2(c)]. Here it should be noted that summation is done
over the band-gaps within the considered frequency ranges. For simplicity, we
have considered a 2D geometry assuming that the resonators are infinite in the
direction perpendicular to the surface of the cross-section.

Figure 2: Design of the unit cell. (a) Illustration of the considered unit
cell consisting of two locally coupled Helmholtz resonators. Width of the unit
cell is $a_{x}$ and the height is $a_{y}$. The resonators are characterized by
the inner radius $r$ and the outer radius $R$. Slit width $w$ and distance
between the resonators $d$ are tuning parameters. The resonators are rotated
around their out-of-plane axes by the angles $\theta_{1}$ and $\theta_{2}$.
Dashed red lines indicate reflection symmetry axes of the cell. (b) The
corresponding reciprocal unit cell. (c) Arbitrary band diagram representing
labeling of individual $\Gamma X$ band-gaps used for calculation of the
integral width of $\Gamma X$ band-gaps $f_{\Gamma X}$.

Since the system is scalable, for the tuning procedure we have normalized all
geometric parameters to the outer radius of the resonators $R$. The size of
the unit cell was fixed at values of $a_{x}/R=4.5$ and $a_{y}/R=2.25$. The
inner radius of the resonators $r$ is fixed as $r/R=0.905$. The distance
between the centers of resonators $d$ as well as the width of slits $w$ were
varied as parameters directly affecting the local coupling. The frequencies
are also normalized by $R/c_{\mathrm{air}}$, where
$c_{\mathrm{air}}=343\,\mathrm{m/s}$ is the speed of sound in air.

Figure 3: Band structure and optimization. (a) Integral width of band-gaps as
a function of slit width $w$ and distance between the resonators $d$. Points A
and B indicate two local maxima of the function $f_{\Gamma X}$, occurring near
the points with $w/R=0.75$, $d/R=2.25$ and $w/R=1.55$, $d/R=2.4$,
respectively. The corresponding unit cells are shown in panels (b) and (c).
Note that the map is rescaled using a power-law relationship to provide better
visualization. Band diagrams for the structures with such configurations are
demonstrated in panels (d) and (e). Shaded green areas indicate the $\Gamma X$
band-gaps.

The colormap shown in Fig. 3(a) demonstrates the integral width of $\Gamma X$
band-gaps $f_{\Gamma X}$ as a function of $w$ and $d$. Clearly, the presence
of finite-size slits results in the increase of $f_{\Gamma X}$. At the same
time, the dependence is nonlinear and there are two local maxima corresponding
to the points with $d/R=2.25$, $w/R=0.75$ [point A, see Fig. 3(b)] and
$d/R=2.4$, $w/R=1.55$ [point B, see Fig. 3(c)]. The corresponding band
structures demonstrating large $\Gamma X$ band-gaps are shown in Fig. 3(d) and
3(e), respectively. It is also important to note that within the considered
frequency range the structure may possess the features of a metamaterial as
well as a phononic crystal. So there may be several mechanisms of band-gap
formation involved.

Following the adjustment of slit widths and distances between the resonators,
we analyze the effect of resonators’ rotation on the integral width of band-
gaps. Figure 4 shows the integral width of band-gaps calculated for the range
of rotational angles $\theta_{1}$ and $\theta_{2}$ [see Fig. 4(a)]. Rotation
of the resonators results in a pronounced decrease of $f_{\Gamma X}$ in the
vicinity of the point $\theta_{1}=0^{\circ}$, $\theta_{2}=180^{\circ}$,
corresponding to the case of CC-like geometry. Such a reduction of the
integral width of band-gaps is associated with the decrease of local coupling
strength [51] which in the case of CC-like geometry is weak. In the
Supplementary Information we demonstrate that the proposed geometry is indeed
characterized by much stronger local coupling allowing increase of integral
width of band gaps by nearly 30 % comparing to the CC-like geometry.
Considering the point A, for CC-like geometry $f_{\Gamma X}$ reduces by more
than 20 % in comparison to the optimal case with
$\theta_{1}=\theta_{2}=0^{\circ}$ [see Fig. 4(b)]. Similarly, for point B the
corresponding reduction is about 15 % [see Fig. 4(b)], however, there are
configurations which are less beneficial. Figure 4 also demonstrates the
robustness of the considered system against rotations of the resonators.
Indeed, the calculations indicate that rotation within approximately
$15^{\circ}$ almost does not perturb values of $f_{\Gamma X}$. This fact is
important for practical realization of the proposed structure since it allows
to reduce requirements for mounting tolerances.

Figure 4: Rotation of the resonators. The panel (a) illustrate the definition
of the rotational angles $\theta_{1}$ and $\theta_{2}$. The strongly coupled
geometry is characterized by $\theta_{1}=\theta_{2}=0^{\circ}$ while the case
of the CC-like geometry corresponds to $\theta_{1}=0^{\circ}$,
$\theta_{2}=180^{\circ}$. The colormaps (b) and (c) show the integral width of
band-gaps as a function of the rotational angles such that the resonators are
characterized by (b) $w/R=0.75$, $d/R=2.25$ and (c) $w/R=1.55$, $d/R=2.4$.
These parameters correspond to the points labeled in Fig. 3 as A and B,
respectively.

###  II.2 Experimental verification of stop-bands

To experimentally verify the presence of wide band-gaps, we consider a
configuration of the structure corresponding to the point A in Fig. 3. We
imitate the semi-infinite structure, such that it is finite along one axis and
infinite along the two other ones (see Methods section for details). The
samples fabricated via 3D printing are shown in Fig. 5(a) while the schematic
view and photograph of the setup are demonstrated in Figs. 5(b) and 5(c),
respectively. Note that of instead the complete resonators we consider their
halves utilizing the symmetry of the field distribution within the unit cell,
as discussed in Methods section. The measurements were provided for the unit
cell characterized by $a_{x}=240\,\mathrm{mm}$ and $a_{y}=120\,\mathrm{mm}$
such that the outer radius of the resonators is $R=53\,\mathrm{mm}$ and the
inner radius is $r=48\,\mathrm{mm}$. Slit width is then $w=40\,\mathrm{mm}$
and the distance between the resonators is $d=120\,\mathrm{mm}$. The
considered frequency range is 200 – 2060 Hz.

The obtained transmission spectra are demonstrated in Figs. 5(e) – 5(g), where
the blue lines correspond to numerical simulations (for a 2D system) and the
red lines represent the experimental measurements. The green shaded areas
indicate the $\Gamma X$ band-gaps of the equivalent infinite 2D system shown
in Fig. 5(a). For convenience, two frequency axes are shown, the normalized
(top) and the un-normalized (bottom). The measurements are provided for $1$,
$2$, and $3$ pairs of resonators in order to explicitly demonstrate the
evolution of stop-bands with respect to a change in the structure thickness.
In accordance with a simple intuition, increasing the number of pairs results
in broadening of the stop-bands. It should be noted that, unlike the noise
barrier consisting of a single Helmholtz resonator [48], a wide stop-band can
be observed even for the one-pair thickness. Practically, for the case of $3$
pairs of coupled resonators, the stop-bands cover more than 80 % of the
considered spectral range. But even for one pair the stop-bands are quite
large and pronounced, mostly going below $-20$ dB.

Figure 5: Transmission of semi-infinite structure. (a) Photograph of the
samples characterized by $R=53$ mm, $r=48$ mm, $w=40$ mm, and height $h=60$
mm. (b) Schematic picture and (c) photo of the experimental setup. (d) Band
diagram for the system with the unit cell with $a_{x}=240$ mm, $a_{y}=120$ mm
and $d=120$ mm. For convenience, two frequency axes are shown, the normalized
one and the one in conventional units. Measured and simulated (for a 2D
system) transmission spectra for (e) one, (f) two, and (g) three pairs of
coupled Helmholtz resonators. Experimentally obtained values are shown by red
lines, while the numerical results are demonstrated by the blue lines. Green
shaded areas indicate $\Gamma X$ band-gaps of the corresponding infinite 2D
structure.

###  II.3 Noise-insulating chamber

The finite-size prototype of the metahouse chamber is shown in Fig. 6(a). It
is an axially symmetric structure consisting of $30$ radially oriented arrays
with $3$ pairs of resonators each. The outer radius of the chamber is $300$ mm
and the inner one is $120$ mm. Height of the structure is $60$ mm. The
resonators are characterized by $R=6.7$ mm, $r=6$ mm, and $w=5$ mm. Along the
arrays, the distance between centers of the resonators (in $xy$ plane) is
$d=15$ mm. The transmission spectrum is shown in Fig. 6(c) with experimental
values indicated by red line and the numerical simulations (for a 2D system)
by the blue one. Shaded green areas indicate $\Gamma X$ band-gaps of the
corresponding infinite 2D system considered in the previous Section.

The measurements were performed in a box with the walls covered by foam rubber
pyramids [see Fig. 6(b)]. However, it has to be noted that the height of the
box is $60$ mm while the wavelength of generated waves is smaller than this
value for almost whole range of spectra (see Methods section). Since all of
the structures are uniform in the vertical direction, it is unlikely that the
higher order modes would be excited. Still, assuming that the system is not
ideal it may be not a perfect equivalent of the 2D system which explains some
differences between the numerical and experimental results.

Figure 6: Transmission of the metahouse chamber. (a) The manufactured
prototype of the metahouse chamber. There are $30$ arrays forming a toroid
with the outer radius $300$ mm and the inner one $120$ mm. Each array consists
of $3$ pairs of resonators with $R=6.7$ mm, $r=5.5$ mm, and $w=5$ mm. The
distance between centers of the resonators along the axis of an array is
$d=15$ mm. (b) Photograph of the experimental setup. The microphone is located
inside the chamber and the loudspeaker is placed at one of the walls. (c)
Measured transmission spectrum (red line) and the corresponding numerical
values obtained for the equivalent 2D chamber (blue line). Shaded green areas
correspond to the equivalent infinite 2D periodic structure [see Fig. 3].

##  III Discussion

The developed metahouse chamber demonstrates good insulating properties with
the average transmission of $-43$ dB in the broad spectral range from $1500$
to $16500$ Hz [Fig. 6(c)]. The average value was calculated as the arithmetic
mean within the spectral range. Ventilation properties as well as partial
optical transparency are defined by geometric properties of the structure.
Analysis of the air flow is presented in the Supplementary Information. While
the concept was demonstrated for the case when the resonators are located
quite close to each other, the structure may be optimized to increase the
sparseness. For example, numerical calculations done for the finite-size
metacage with the number of arrays reduced by a half also demonstrate
reasonable noise-insulation properties with the average transmission -16 dB
(see Supplementary Information). Tuning of parameters and optimization of the
structure may enhance the results allowing proper air flow and optical
transparency implying compromise between sparseness and sound transmission. In
addition, the structure can be made of optically transparent materials, such
as the commercially available PLA filament for 3D printers [52]. The
possibility to achieve simultaneous air ventilation with optical transparency
is a unique feature of the developed structure resulting from the proposed
design. In Table 1 we also provide values of acoustic impedance for several
materials which may be used for construction of the metahouse chamber. It is
important to highlight that the range of possible materials is huge and hence
the conventional widespread materials can be used.

Table 1: Acoustic impedance $Z$ of some materials which can be used for construction of the metahouse chamber. Material |  $Z$ (MRayl) | Reference
---|---|---
Polylactic acid (PLA) | 2.56 – 2.77 | [53]
Polypropylene | 2.3 – 2.4 | [54]
Polymethyl methacrylat | 3.14 – 3.69 | [55]

##  IV Methods

###  IV.1 Numerical simulations

All numerical simulations are performed for 2D systems using finite element
method (FEM) via COMSOL Multiphysics, Pressure Acoustics module. Throughout
the calculations, the boundaries of resonators are assumed to be sound-hard
walls. Such an approximation is justified by high impedance contrast between
the material of resonators and air. For example, the acoustic impedance of
polypropylene is about $2.4\,\mathrm{MRayl}$ [54] which is several orders
higher than for air. Similar contrasts of impedances is also observed for
other materials which can be used for construction of the chamber, see Table
1. It is also should be mention that any thermo-viscous boundary effects are
neglected, since there are no narrow channels in the considered structures.

The semi-infinite structure is modelled as an array made of $N$ halved
resonators placed into a 2D waveguide with sound hard walls. The distance
between the walls is equal to $a_{y}/2$. The ends of the waveguide are
modelled as perfectly matched layers (PML). The incident plane wave with
amplitude $p_{0}$ propagates along the direction of periodicity ($x$-axis,
parallel to the walls of the waveguide). Pressure values are calculated at the
sample point located at the distance $x_{t}=15$ mm from the rightmost
resonator. Parameters of the resonators correspond to those of the fabricated
samples.

In the model for the finite-size case the chamber is placed into the 2D box in
which all walls are PML such that the distance between the walls is $1210$ mm
and $510$ mm along $x$ and $y$ axis, respectively. The structure is located at
the center of the waveguide and the sample point is placed at the center of
the structure. Incident waves with amplitude $p_{0}$ are generated by a point
source located near one of the walls. Geometric size of the 2D chamber
correspond to the parameters of the manufactured structure.

###  IV.2 Experimental setup

The experimental measurements are provided inside a waveguide with rectangular
cross-section. The walls and the bottom of the waveguide are made of aluminum
with thickness of $15$ mm. The lid is made of $6$ mm thick transparent
plexiglass. The height of the inner cross-section of the waveguide is fixed
and equal to $60$ mm. However, the width is adjustable and can be varied from
$0$ to $650$ mm via displacement of one of the walls. The loudspeaker (Visaton
BF 45/4) is located at one end of the waveguide and the microphone (BOYA BY-
PVM1000) is placed at the other one. While the full length of the waveguide is
135 cm, the distance between the microphone and the loudspeaker is 107 cm. The
rest of the waveguide’s length is occupied by two pieces of foam rubber, each
14 cm long. The loudspeaker and the microphone are embedded into these
inserts.

The microphone is directly connected to a USB audio interface Roland Rubix22.
To connect the loudspeaker to the same sound card we use an amplifier based on
the Yamaha YDA138-E microchip. The sound card itself is connected to a
personal computer via USB interface. Control of signal generation and
recording is realized via specially developed software based on the Python
programming language (in particular, generation and recording of signals
exploit the sound device module [56]).

To imitate semi-infinite structure in the experiment we consider an array
consisting of $N$ pairs of coupled resonators. Proper boundary conditions —
namely, sound hard walls at geometrical symmetry axes — allow to imitate
periodicity along the two other axes ($y$ and $z$ axes). The numerical 2D
model corresponds to a structure infinite in the $z$ direction, while in the
experiment it is mimicked by keeping the waveguide width and height smaller
than $\lambda/2$ for the considered spectral range. The operational range of
the setup for the case of the semi-infinite structure is $200$ – $2100$ Hz. In
order to imitate an infinite structure, all walls have to be totally
reflective. At the same time, symmetry of the unit cell and of field
distributions within it have to be preserved. Therefore, we select the size of
the cross-section such that it’s width and height are smaller than the
smallest wavelength in the spectrum of a generated signal. Otherwise, the
walls have to be perfectly symmetric and elements within the waveguide have to
be centered ideally, which is hardly achievable. In order to increase the
reliable frequency range in the waveguide below $\lambda/2$, we therefore only
use half instead of full resonators. Such modification is valid since it does
not violate symmetry of the structure. The resonators were manufactured by 3D
printing (PLA filament). Note, that to simplify placement of resonators inside
the waveguide, there are small place-holders in the slits of the resonators.

All measurements were conducted for generated chirped signal with the
frequency changing from $200$ to $2100$ Hz. For each measurement the signal
was generated several times and then the measured spectra were averaged in
Fourier space. Sampling frequency was always fixed at 44100 Hz. Following the
conventional definition, the transmission was defined as

$T=20\log\left(p_{\mathrm{tr}}/p_{\mathrm{ref}}\right)$ (2)

where $p_{\mathrm{ref}}$ is a pressure amplitude of the wave in the absence of
a structure (reference signal) and $p_{\mathrm{tr}}$ is a pressure amplitude
of the transmitted wave.

The measurement of transmission for the finite-size chamber were conducted in
the same waveguide, as for the case of semi-infinite structure. However, the
distance between the walls was increased to $650$ mm. The walls and the ends
were covered by foam rubber pyramids (with height $50$ mm and square base $50$
by $50$ mm). The chamber was located at the center of the waveguide such that
microphone (BOYA BY-M1) was placed inside the structure (at the center). All
other elements and connections between them were the same as for the semi-
infinite system. The procedure of signal generation and averaging also
remained the same, however the frequency range was changed to be from $1500$
to $16500$ Hz.

## References

  * [1] Cummer, S. A., Christensen, J. & Alù, A.  Controlling sound with acoustic metamaterials.  _Nature Reviews Materials_ 1, 1–13 (2016).
  * [2] Zangeneh-Nejad, F. & Fleury, R.  Active times for acoustic metamaterials.  _Reviews in Physics_ 4, 100031 (2019).
  * [3] Liu, J., Guo, H. & Wang, T.  A Review of Acoustic Metamaterials and Phononic Crystals.  _Crystals_ 10, 305 (2020).
  * [4] Liao, G. _et al._ Acoustic Metamaterials: A Review of Theories, Structures, Fabrication Approaches, and Applications.  _Advanced Materials Technologies_ 6, 2000787 (2021).
  * [5] Kaina, N., Lemoult, F., Fink, M. & Lerosey, G.  Negative refractive index and acoustic superlens from multiple scattering in single negative metamaterials.  _Nature_ 525, 77–81 (2015).
  * [6] de Rosny, J. & Fink, M.  Overcoming the Diffraction Limit in Wave Physics Using a Time-Reversal Mirror and a Novel Acoustic Sink.  _Physical Review Letters_ 89, 124301 (2002).
  * [7] Li, J., Fok, L., Yin, X., Bartal, G. & Zhang, X.  Experimental demonstration of an acoustic magnifying hyperlens.  _Nature Materials_ 8, 931–934 (2009).
  * [8] Zhang, J., Tian, Y., Cheng, Y. & Liu, X.  Acoustic holography using composite metasurfaces.  _Applied Physics Letters_ 116, 030501 (2020).
  * [9] Chen, H. & Chan, C. T.  Acoustic cloaking in three dimensions using acoustic metamaterials.  _Applied Physics Letters_ 91, 183518 (2007).
  * [10] Guo, X., Ding, Y., Duan, Y. & Ni, X.  Nonreciprocal metasurface with space–time phase modulation.  _Light: Science & Applications_ 8, 123 (2019).
  * [11] Peng, Y.-G. _et al._ Experimental demonstration of anomalous Floquet topological insulator for sound.  _Nature Communications_ 7, 13368 (2016).
  * [12] Xue, H. _et al._ Observation of an acoustic octupole topological insulator.  _Nature Communications_ 11, 2442 (2020).
  * [13] Fang, X., Wen, J., Bonello, B., Yin, J. & Yu, D.  Ultra-low and ultra-broad-band nonlinear acoustic metamaterials.  _Nature Communications_ 8, 1288 (2017).
  * [14] Chen, S. _et al._ A Review of Tunable Acoustic Metamaterials.  _Applied Sciences_ 8, 1480 (2018).
  * [15] Xie, B. _et al._ Coding Acoustic Metasurfaces.  _Advanced Materials_ 29, 1603507 (2017).
  * [16] Tian, Z. _et al._ Programmable Acoustic Metasurfaces.  _Advanced Functional Materials_ 29, 1808489 (2019).
  * [17] Zuo, S., Wei, Q., Tian, Y., Cheng, Y. & Liu, X.  Acoustic analog computing system based on labyrinthine metasurfaces.  _Scientific Reports_ 8, 10103 (2018).
  * [18] Donda, K. _et al._ Ultrathin Acoustic Absorbing Metasurface Based on Deep Learning Approach.  _Smart Materials and Structures_ (2021).
  * [19] Xu, Z.-X., Zheng, B., Yang, J., Liang, B. & Cheng, J.-C.  Machine-Learning-Assisted Acoustic Consecutive Fano Resonances: Application to a Tunable Broadband Low-Frequency Metasilencer.  _Physical Review Applied_ 16, 044020 (2021).
  * [20] Gao, N., Wang, M., Cheng, B. & Hou, H.  Inverse design and experimental verification of an acoustic sink based on machine learning.  _Applied Acoustics_ 180, 108153 (2021).
  * [21] Ding, H. _et al._ Deep Learning Enables Accurate Sound Redistribution via Nonlocal Metasurfaces.  _Physical Review Applied_ 16, 064035 (2021).
  * [22] Zhou, W., Su, Y., Muhammad, Chen, W. & Lim, C. W.  Voltage-controlled quantum valley Hall effect in dielectric membrane-type acoustic metamaterials.  _International Journal of Mechanical Sciences_ 172, 105368 (2020).
  * [23] Fan, X.-D. & Zhang, L.  Acoustic orbital angular momentum Hall effect and realization using a metasurface.  _Physical Review Research_ 3, 013251 (2021).
  * [24] Shi, C. _et al._ Observation of acoustic spin.  _National Science Review_ 6, 707–712 (2019).
  * [25] Long, Y. _et al._ Realization of acoustic spin transport in metasurface waveguides.  _Nature Communications_ 11, 4716 (2020).
  * [26] Yang, C., Tan, Y.-T., Chen, H. & Ren, J.  Real spin angular momentum and acoustic spin torque in a topological phononic crystal.  _Journal of Applied Physics_ 129, 135106 (2021).
  * [27] Wang, S. _et al._ Spin-orbit interactions of transverse sound.  _Nature Communications_ 12, 6125 (2021).  eprint 2105.13558.
  * [28] Ge, H. _et al._ Observation of Acoustic Skyrmions.  _Physical Review Letters_ 127, 144502 (2021).
  * [29] Gardezi, S. M., Pirie, H., Carr, S., Dorrell, W. & Hoffman, J. E.  Simulating twistronics in acoustic metamaterials.  _2D Materials_ 8, 031002 (2021).
  * [30] Stansfeld, S. A. & Matheson, M. P.  Noise pollution: Non-auditory effects on health.  _British Medical Bulletin_ 68, 243–257 (2003).
  * [31] Singh, N. & Davar, S. C.  Noise Pollution-Sources, Effects and Control.  _Journal of Human Ecology_ 16, 181–187 (2004).
  * [32] Goines, L. & Hagler, L.  Noise pollution: A modern plague.  _Southern Medical Journal_ 100, 287–294 (2007).
  * [33] Kunc, H. P. & Schmidt, R.  The effects of anthropogenic noise on animals: A meta-analysis.  _Biology Letters_ 15, 20190649 (2019).
  * [34] Münzel, T., Sørensen, M. & Daiber, A.  Transportation noise pollution and cardiovascular disease.  _Nature Reviews Cardiology_ 18, 619–636 (2021).
  * [35] Hopkins, C.  _Sound Insulation_ (Routledge, London, 2007).
  * [36] Xu, Z.-X. _et al._ Topology-Optimized Omnidirectional Broadband Acoustic Ventilation Barrier.  _Physical Review Applied_ 14, 054016 (2020).
  * [37] Sun, M., Fang, X., Mao, D., Wang, X. & Li, Y.  Broadband Acoustic Ventilation Barriers.  _Physical Review Applied_ 13, 044028 (2020).
  * [38] Kumar, S., Xiang, T. B. & Lee, H. P.  Ventilated acoustic metamaterial window panels for simultaneous noise shielding and air circulation.  _Applied Acoustics_ 159, 107088 (2020).
  * [39] Dong, R., Mao, D., Wang, X. & Li, Y.  Ultrabroadband Acoustic Ventilation Barriers via Hybrid-Functional Metasurfaces.  _Physical Review Applied_ 15, 024044 (2021).
  * [40] Ma, G., Yang, M., Yang, Z. & Sheng, P.  Low-frequency narrow-band acoustic filter with large orifice.  _Applied Physics Letters_ 103, 011903 (2013).
  * [41] Langfeldt, F., Kemsies, H., Gleine, W. & von Estorff, O.  Perforated membrane-type acoustic metamaterials.  _Physics Letters A_ 381, 1457–1462 (2017).
  * [42] Cheng, Y. _et al._ Ultra-sparse metasurface for high reflection of low-frequency sound based on artificial Mie resonances.  _Nature Materials_ 14, 1013–1019 (2015).
  * [43] Chen, Z. _et al._ An open-structure sound insulator against low-frequency and wide-band acoustic waves.  _Applied Physics Express_ 8, 107301 (2015).
  * [44] Zhang, H.-L. _et al._ Omnidirectional ventilated acoustic barrier.  _Applied Physics Letters_ 111, 203502 (2017).
  * [45] Ghaffarivardavagh, R., Nikolajczyk, J., Anderson, S. & Zhang, X.  Ultra-open acoustic metamaterial silencer based on Fano-like interference.  _Physical Review B_ 99, 024302 (2019).
  * [46] Yu, X. _et al._ Sound transmission through a periodic acoustic metamaterial grating.  _Journal of Sound and Vibration_ 449, 140–156 (2019).
  * [47] Shen, C., Xie, Y., Li, J., Cummer, S. A. & Jing, Y.  Acoustic metacages for sound shielding with steady air flow.  _Journal of Applied Physics_ 123, 124501 (2018).
  * [48] Melnikov, A. _et al._ Acoustic metamaterial capsule for reduction of stage machinery noise.  _The Journal of the Acoustical Society of America_ 147, 1491–1503 (2020).
  * [49] Melnikov, A. _et al._ Acoustic meta-atom with experimentally verified maximum Willis coupling.  _Nature Communications_ 10, 3148 (2019).
  * [50] Elford, D. P., Chalmers, L., Kusmartsev, F. V. & Swallowe, G. M.  Matryoshka locally resonant sonic crystal.  _The Journal of the Acoustical Society of America_ 130, 2746–2755 (2011).
  * [51] Hu, G., Tang, L., Das, R., Gao, S. & Liu, H.  Acoustic metamaterials with coupled local resonators for broadband vibration suppression.  _AIP Advances_ 7, 025211 (2017).
  * [52] Matos, B. D. M. _et al._ Evaluation of commercially available polylactic acid (PLA) filaments for 3D printing applications.  _Journal of Thermal Analysis and Calorimetry_ 137, 555–562 (2019).
  * [53] Parker, N. G., Mather, M. L., Morgan, S. P. & Povey, M. J. W.  Longitudinal acoustic properties of poly(lactic acid) and poly(lactic- co -glycolic acid).  _Biomedical Materials_ 5, 055004 (2010).
  * [54] Selfridge, A. R.  Approximate Material Properties in Isotropic Materials.  _IEEE Transactions on Sonics and Ultrasonics_ 32, 381–394 (1985).
  * [55] Carlson, J. E., van Deventer, J., Scolan, A. & Carlander, C.  Frequency and temperature dependence of acoustic properties of polymers used in pulse-echo systems.  In _IEEE Symposium on Ultrasonics, 2003_ , vol. 1, 885–888 Vol.1 (2003).
  * [56] Play and Record Sound with Python.  spatialaudio.net (2021).

##  V Acknowledgements

The authors thank Aleksandr Kalganov for the help with design and assembly of
the experimental setup and Mikhail Kuzmin for sample fabrication. The authors
also thank Nikita Olekhno for fruitful discussions and suggestions. This
research was supported by Priority 2030 Federal Academic Leadership Program.

##  VI Author contributions

M.K. designed the structure, experimental samples, and the experimental setup,
and performed experimental measurements. S.K. and M.K. provided numerical
calculations. S.K. participated in the experimental setup assembly and
measurements. A.M., Y.B., S.M., and D.P. provided guidance on all aspects of
the work. A.B. suggested the idea and supervised the project. All authors
contributed to writing and editing of the manuscript.

##  VII Competing interests

The authors declare no competing interests.

Supplementary Information

##  VIII Coupled Helmholtz resonators

Since it is claimed that the broadband noise-insulation is achieved due to the
strong local coupling of the Helmholtz resonators we would like to provide a
proof that the coupling is indeed strong. For that we consider a pair of the
resonators in a free space and calculate pressure inside one of them, assuming
that the incident wave with the amplitude $p_{0}=1\,\mathrm{Pa}$ propagates
along the $x$-axis [see Figs. 7(a) and 7(d)]. The resulting spectra are shown
in Figs. 7(b) and 7 (e). The splitting of the Helmholtz resonance decreasing
with the increase of the distance between the resonators is the manifestation
of the coupling between the resonators. For the case of the considered
geometry [Fig. 7(d)], the splitting is much larger than for the CC-like one
[Fig. 7(a)]. In addition, the field distribution in the proposed geometry is
fundamentally different from the CC-like geometry since it has additional
mirror symmetry with respect to $y$-axis. Hence, the split Helmholtz resonance
occurs as in-phase and out-of-phase excitation of the resonators characterized
by symmetric and anti-symmetric field distributions, respectively [see Figs.
7(c) and 7(f)].

Figure 7: Coupling between two resonators. The system consists of a pair of
the Helmholtz resonators located in a free space, such that the incident
pressure field propagates along the $x$-axis. Two cases are considered, namely
(a) the CC-like geometry and (d) the strongly coupled resonators. Panels (b)
and (e) shows corresponding frequency spectra calculated at the sampling point
located at the center of the rightmost resonator (the red dot). The spectral
distances between the split resonances for the case of $d/R=2.25$ are
indicated by $\Delta$. Distributions of the scattered pressure field for the
case of $d/R=2.25$ are shown in panels (c) and (f); the frequencies correspond
to the resonances indicated by the vertical dashed lines on the spectra.

In the case of an infinite arrays, the local coupling results in the
pronounced increase of the integral width of band-gaps [51]. Figure 8
demonstrates the difference between the band structures for the infinite
arrays of resonators with zero slits (which are just circular pipes), CC-like
geometry and strongly coupled resonators. Indeed, the introduction of slits to
the pipes results in a formation of an additional band [see Fig. 8(b)] as well
as the additional band-gap, as consequence. Further, an enhancement of the
local coupling leads to the formation of two more $\Gamma$-$X$ band-gaps. In
this case, the out-of-phase excitation of the Helmholtz resonance corresponds
to the flat band indicating strong localization of the field inside the
resonators as well as the absence of coupling between the neighboring couples.
In turn, the other mode, which is flat only within the $\Gamma X$ interval, is
associated with the in-phase excitation of the Helmholtz resonance.

Figure 8: Deformation of band structure. Band diagrams for the case of (a)
pipes, (b) CC-like geometry (c) strongly coupled resonators. Shaded green
areas indicate the $\Gamma$-$X$ band-gaps. The following parameters of the
structures are used: (b) $w/R=0.75$; (c) $w/R=0.75$, $d/R=2.25$; for each
panel $r/R=0.905$. Field distributions shown in panel (d) correspond to the
labeled blue points in the band diagrams.

To trace the deformation of band structure caused by introduction of slit and
the consequent enhancement of the local coupling, we consider field
distributions within the unit cells (at $\Gamma$-point). For the structure
made of pipes there are two modes with zero frequency. One of them, labelled
as a1-1 in Fig. 8(d), is a structural resonance, typical for periodic
structures. This mode is present in both other band diagrams [b1 in Fig. 8(b)
and c1 in Fig. 8(c)]. Another mode, labelled as a1-2, is the flat band with
zero frequency actually corresponding to the eigenmode of a Helmholtz
resonator with zero slit. As soon as the slit is introduced, this band deforms
and shifts to the area of larger frequencies. Hence, the mode a1-2 of pipe
arrays can be associated with the mode b2, which is a Helmholtz resonance of a
CC-like structure. When the coupling is enhanced, this mode splits on two [c2
and c3 in Fig. 8(c)]. In this case, the field distribution patterns within a
single unit cell of the periodic structure are similar to those observed for a
single pair of resonators presented in Fig. 7(f). The anti-symmetric mode,
labelled as c3 is perfectly localized within the resonators meaning that there
is no coupling between neighbouring pairs, which is the reason why the
corresponding band is flat. Also, there is another additional mode c4, which
may be considered as the additional structural resonance resulting from the
deformation of the unit cell.

Another type of modes are the modes of a circular pipe, which also correspond
to flat bands in each of the diagrams. For the case of pipes, these are two
degenerate modes a2-1 and a2-2. When the slit is introduced, the degeneracy
lifts due to the break of symmetry, but the mode with similar field
distribution remains in the system. Then, the structure with the strong
coupling is characterized by two similar nearly-degenerate modes (c6-1 and
c6-2). Finally, the mode c5 is the structural resonance, such that similar
resonances are present in all of the considered structures, but at higher
frequencies.

##  IX Ventilation properties of the structures

In order to demonstrate the ventilation properties of the proposed structures,
we calculate the laminar air flow with the help of COMSOL Multiphysics
software package. For all cases, the normal inflow velocity was taken to be
$U_{0}=10^{-4}\,\mathrm{m/s}$ and the no-slip wall boundary condition was
used. The results shown in Fig. 9(a) demonstrate that the semi-infinite
structure does not violate the air flow, suggesting that the structure might
be suitable for noise insulation of ventilated ducts.

Figure 9: Air flow through the designed structures. (a) Semi-infinite
structure in the duct, (b) metahouse chamber and (c) sparse metahouse chamber
considered in the main text. For each panel left boundary is considered to be
an inlet port with normal inflow velocity $U_{0}=10^{-4}\,\mathrm{m/s}$. The
direction of the flow is shown by black arrows while the color indicates its
velocity $U$.

Switching to the case of metahouse chamber, Fig. 9(b) shows that the laminar
flow inside the structure is quite weak. Flow velocity at the center of the
chamber is about $10^{-5}\,\mathrm{m/s}$, which is one order smaller than the
inflow velocity. Hence, the initial structure for which the experimental
measurements were done is poorly ventilated. At the same time, considering the
use of such structures in the real-world conditions of urban parks, it can be
hypothesised that air will definitely penetrate inside the chamber which at
the same time may provide some protection from wind. Increasing the sparseness
enhances the air flow inside the structure, as is seen in Fig. 9(c). However,
in this case the flow velocity inside the chamber is higher than the inlet
one. The reason for this is the cone-like narrowing of the channels in-between
the arrays of the resonators resulting in the flow acceleration. While the
sparsed structure has a half of the initial number of the resonators, its
noise-insulating properties are still quite reasonable. In this case the
numerically calculated average transmission is about $-16\,\mathrm{dB}$ as the
Fig. 10 shows. The conclusion which can be made from the Figs. 9 and 10 is
that it is possible to find a compromise between the ventilating and noise-
insulating properties of the proposed metahouse structure by varying its
sparseness.

Figure 10: Transmission of the sparse metahouse chamber. Schematic
representation of the (a) initial [see Fig. 6 from the main text] and (b)
sparse structures. (c) Simulated transmission spectra for both cases. Average
transmission for the sparse structure is $-16$ dB and for the initial
structure is $-57$ dB.