# Topological floating phase of dipolar bosons in an optical ladder

Henning Korbmacher  Institut für Theoretische Physik, Leibniz Universität
Hannover, Germany  Gustavo A. Domínguez-Castro  Institut für Theoretische
Physik, Leibniz Universität Hannover, Germany  Mateusz Łącki  Institute of
Theoretical Physics, Jagiellonian University in Krakow, ul. Lojasiewicza 11,
30-348 Kraków, Poland  Jakub Zakrzewski<EMAIL_ADDRESS>Institute
of Theoretical Physics, Jagiellonian University in Krakow, ul. Lojasiewicza
11, 30-348 Kraków, Poland  Mark Kac Complex Systems Research Center,
Jagiellonian University in Krakow, Łojasiewicza 11, 30-348 Kraków, Poland
Luis Santos  Institut für Theoretische Physik, Leibniz Universität Hannover,
Germany

###### Abstract

Ultracold dipolar hard-core bosons in optical ladders provide exciting
possibilities for the quantum simulation of anisotropic XXZ spin ladders. We
show that introducing a tilt along the rungs results in a rich phase diagram
at unit filling. In particular, for a sufficiently strong dipolar strength,
the interplay between the long-range tail of the dipolar interactions and the
tilting leads to the emergence of a quantum floating phase, a critical phase
with incommensurate density-density correlations. Interestingly, the study of
the entanglement spectrum, reveals that the floating phase is topological,
constituting an intermediate gapless stage in the melting of a crystal into a
gapped topological Haldane phase. This novel scenario for topological floating
phases in dipolar XXZ ladders can be investigated in on-going experiments.

Ultracold gases in optical lattices offer excellent possibilities for
simulating spin models relevant to quantum magnetism [1]. For short-range
interacting atoms, super-exchange [2] results in weak nearest-neighbor spin-
spin interactions, typically of Heisenberg type, which for fermions lead to
long-range antiferromagnetic order [3]. Additionally, van der Waals
interacting Rydberg gases naturally realize the quantum Ising model (QIM) [4].
The exquisite control available in these experiments enables the exploration
of various lattice topologies and dimensionalities, including optical ladders.

Figure 1: (a) Scheme of the ladder system. (b) Phase diagram for the nearest-
neighbor model showing the superfluid (SF) phase, the two-site period density
wave (2DW), the polarized (P) phase, and the Haldane (H) phase. (c) Phase
diagram including the dipolar tail (up to the sixth neighbor) showing
additional density modulated phases (3DW and 4DW) and the floating (FL) phase.
The rectangle shows the parameter region studied in Fig. 2. The phase
boundaries are evaluated using DMRG calculations with $L=96$ rungs, and
monitoring the order parameters and correlations discussed in the text.

Spin ladders play a prominent role in quantum magnetism, as an interesting
intermediate case between one- and two-dimensions [5, 6]. The interplay
between leg and rung interactions results in rich physics for two-leg spin
ladders, both in what concerns its highly non-trivial ground-state properties
[7, 8, 9] and its dynamics [10, 11, 12, 13]. These physics have been the focus
of recent research in optical lattice experiments, including the realization
of the Haldane phase using Fermi-Hubbard ladders [14], or the exploration of
ballistic-to-diffusive dynamics in XX ladders using hard-core bosons [15].

Recent experiments with dipolar gases, including magnetic atoms [16, 17, 18,
19], polar molecules [20, 21, 22, 23, 24], and Rydberg gases [25, 26, 27, 28,
29] have realized lattice models with strong inter-site dipolar interactions,
which may go well beyond nearest neighbors [30]. These experiments open
exciting possibilities for the controlled study and simulation of quantum
magnetism. In particular, dipolar particles in optical ladders, formed by
optical lattices or tweezers, provide a well-suited platform for the quantum
simulation of fully anisotropic XXZ ladders.

This Letter illustrates some of these possibilities for the particular case of
a square ladder at unit rung filling. The interplay between hopping, dipolar
interactions, and a tilt of the ladder along the rung direction (readily
possible in experiments) results in rich ground-state physics. Most
interestingly, for finite tilting, the dipolar bosons develop a floating
phase, a critical phase that is intermediate between a crystal and a
disordered system with incommensurate density-density (or equivalently spin-
spin) correlations. The floating phase, originally introduced in the context
of classical commensurate-incommensurate transitions in solids [31, 32], has
more recently been theoretically studied in quantum systems, including the
anisotropic Ising model with competing interactions [33], particular
constrained hard-core boson models [34, 35, 36], and the QIM [37, 38, 39, 40,
41]. Recent experiments on QIM realized with van der Waals interacting Rydberg
atoms have observed the emergence of the floating phase [42].

Our work shows that dipolar hard-core bosons in ladders constitute a novel
scenario in ultracold gases for the study of quantum floating phases in an
anisotropic XXZ ladder rather than a QIM. In contrast to the QIM, the floating
phase in the XXZ ladder constitutes an intermediate stage in the melting of a
crystalline structure into a disordered yet symmetry-protected gapped
topological Haldane phase. Interestingly, our results based on the degeneracy
of the entanglement spectrum reveal that the resulting floating phase
constitutes an intriguing example of a gapless yet topological phase [43, 44,
45, 46].

## Model.–

We consider dipolar bosons confined in a two-leg ladder system with $L$ rungs
in the $xy$ plane with the legs oriented along $x$ (see Fig. 1 (a)). As in
recent experiments on magnetic atoms [19], we assume strongly repulsive on-
site interactions such that the system is in the hard-core regime, i.e. there
is maximally one particle per site. The dipole moments are oriented by an
external field in a direction $\vec{e}_{d}$ on the $yz$ plane forming an angle
$\theta$ with the $z$-axis. For sufficiently deep lattices, the system is
described by the extended Bose-Hubbard Hamiltonian

$\displaystyle H$ $\displaystyle=$
$\displaystyle-t\sum_{j}\\!\left(\sum_{\alpha}\hat{b}^{\dagger}_{j,\alpha}\hat{b}_{j+1,\alpha}+\hat{b}^{\dagger}_{j,1}\hat{b}_{j,2}+{\mathrm{H}.c.}\\!\right)\\!+\\!\Delta\sum_{j}\hat{n}_{j,2}$
(1) $\displaystyle+$
$\displaystyle\sum_{j,\alpha}\sum_{r>0}V_{r,0}\hat{n}_{j,\alpha}\hat{n}_{j+r,\alpha}+\\!\sum_{j,\alpha\neq\beta}\sum_{r\geq
0}V_{r,1}\hat{n}_{j,\alpha}\hat{n}_{j+r,\beta},$

where $j\in\\{1,\dots,L\\}$ is the rung index, $\alpha,\beta\in\\{1,2\\}$ are
the two legs, $\hat{b}_{j,\alpha}$ ($\hat{b}^{\dagger}_{j,\alpha}$) is the
bosonic annihilation (creation) operator at site $j$ of leg $\alpha$, and
$\hat{n}_{j,\alpha}=\hat{b}^{\dagger}_{j,\alpha}\hat{b}_{j,\alpha}$ is the
corresponding particle number operator. For simplicity, we assume that the
hopping amplitude $t$ is the same along both legs and along the rungs
(although they may be in principle different, offering yet another way of
controlling the properties of the ladder system). We consider leg $2$ to have
a different on-site chemical potential $\Delta>0$ compared to leg $1$. This
energy bias, which may be readily realized experimentally e.g. by employing
tilted ladders, plays a crucial role in the discussion below. The inter-site
dipolar couplings are of the form:

$V_{i_{x},i_{y}}=\frac{V}{\left(i_{x}^{2}+i_{y}^{2}\right)^{3/2}}\left(1-3\frac{i_{y}^{2}}{i_{x}^{2}+i_{y}^{2}}\sin^{2}\theta\right),$
(2)

where $V\equiv V_{1,0}=\frac{C_{dd}}{4\pi\lambda^{3}}$ is the nearest-neighbor
interaction along the leg, with $\lambda$ the lattice spacing,
$C_{dd}=\mu_{0}\mu$ for magnetic dipoles, with $\mu_{0}$ the vacuum
permeability and $\mu$ the magnetic dipole, and $C_{dd}=d^{2}/\varepsilon_{0}$
for electric dipoles, with $\varepsilon_{0}$ the vacuum dielectric constant,
and $d$ the electric dipole moment. We are interested below in the ground-
state phases of the dipolar ladder at unit rung filling, i.e. when
$\sum_{j,\alpha}\langle\hat{n}_{j,\alpha}\rangle=L$.

Note that due to the hard-core constraint, Eq. (1) is equivalent to the
spin-$1/2$ XXZ ladder Hamiltonian:

$\displaystyle H$ $\displaystyle=$ $\displaystyle
t\sum_{j}\left[\sum_{\alpha}\hat{S}_{j,\alpha}^{+}\hat{S}_{j+1,\alpha}^{-}-\hat{S}_{j,1}^{+}\hat{S}_{j,2}^{-}+\mathrm{H.c.}\right]-\Delta\sum_{j}S_{j,2}^{z}$
(3) $\displaystyle+$
$\displaystyle\sum_{j,\alpha}\sum_{r>0}V_{r,0}S_{j,\alpha}^{z}S_{j+1,\alpha}^{z}+\sum_{j,\alpha\neq\beta}\sum_{r\geq
0}V_{r,1}S_{j,\alpha}^{z}S_{j+r,\beta}^{z},$

where we introduce the spin-$1/2$ operators
${\hat{S}}_{j,\alpha}^{+}=(-1)^{j}{\hat{b}}_{j,\alpha}$ and
${\hat{S}}_{j\alpha}^{z}=1/2-n_{j,\alpha}$, and the tilting $\Delta$ acts as
an effective magnetic field gradient between the two legs.

The results discussed below are obtained from a combination of DMRG and iDMRG,
in order to cross-check the calculations. In the case of finite DMRG, we
usually keep a bond dimension of $\chi=300$ (up to $\chi=450$) and consider
different numbers of rungs $L$, assuming periodic boundary conditions along
the $x$ direction. In the case of iDMRG, we also keep a bond dimension of up
to $\chi=300$ and consider a unit cell of $12$ rungs. Both algorithms were
employed using the TeNPy library [47].

Figure 2: (a) Entanglement entropy $S_{vN}$, (b) $P_{\Lambda}$ and (c) the
maximum of the structure factor $S(k)$ for $k\neq 0$. Results obtained by DMRG
calculations for a ladder with $L=96$ rungs. Figure 3: Structure factor $S(k)$
for $t/V=0.05$ as a function of $\Delta/V$ for $L=12$ (a), $24$ (b), $48$ (c),
and $96$ (d) rungs calculated using DMRG. The results show the emergence
between the density waves of additional intermediate values of the $k$ value
of the maximum of $S(k)$ when $L$ grows, a characteristic feature of the
floating phase. Figure 4: $P_{\Lambda}$ (see text) and central charge $c$
(obtained after a finite size scaling $\sim 1/L$ with errorbars indicated in
light blue) for a cut of $t/V=0.05$. Both results obtained from DMRG with
$\chi=600$ from system sizes $L=96$ and $L\in\\{60,72,84,96\\}$, respectively.

Figure 5:  (a) Phases in the atomic limit ($t=0$) as function of the dipole
orientation $\theta$ and the tilting $\Delta$. The black solid curve indicates
the values employed for the calculation of panel (c). (b) Phase diagram for
$\theta/\pi=0.19$ illustrating that the Haldane phase shrinks, always
surrounding the 2DW, for decreasing $\theta/\pi$. The floating phase shrinks
accordingly. Contours drawn from iDMRG results. (c) Phase diagram along the
black line panel (a), showing how the floating and Haldane phases vanish when
the zig-zag phases start to appear. Contours drawn from DMRG results with
$L=96$ rungs.

## Nearest-neighbor model.–

We consider first dipoles oriented along the $y$ axis ($\theta=\pi/2$). We
neglect at this point the $1/r^{3}$ dipolar tail, and approximate the
Hamiltonian (1) by a nearest neighbor (NN) model, where only $V_{1,0}=V$ and
$V_{0,1}=-2V$ are considered. Under these conditions, with ferromagnetic
(antiferromagnetic) rung (leg) coupling, the XXZ ladder (3) presents for
$\Delta=0$ three possible ground-state phases [8, 9]. Large Ising
interactions, corresponding to low-enough $t/V$ values, result in the so-
called stripe-Néel phase, characterized by staggered magnetization along the
leg direction, which translates into a density wave of period two (2DW) for
the bosons. In the atomic limit ($t=0$), the 2DW is characterized by the two-
rung unit cell $\,\begin{matrix}\vspace*{-0.18cm}\leavevmode\hbox
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() denoting occupied (empty) sites. This spatial order is reflected in a peak
at $k=\pi$ in the structure factor

$\displaystyle
S(k)=\frac{1}{L^{2}}\sum_{i,j}e^{ik(i-j)}\langle\hat{n}_{i}\hat{n}_{j}\rangle,$
(4)

with $\hat{n}_{j}=\sum_{\alpha}\hat{n}_{j,\alpha}$ the occupation of rung $j$.
Dominant exchange (large-enough $t/V$) leads to the XY1 phase, which is a one-
component Luttinger liquid that corresponds to the superfluid (SF) phase of
the dipolar bosons, characterized by polynomically decaying single-particle
correlations $\langle\hat{b}_{i,\alpha}^{\dagger}\hat{b}_{j,\alpha}\rangle$.
Finally, for intermediate $t/V$ values the system is in the Haldane phase, a
gapped disordered phase characterized by a non-zero string order

$\mathcal{O}_{S}=\lim_{|i-j|\rightarrow\infty}\left\langle\delta\hat{n}_{i}e^{i\pi\sum_{i<k<j}\delta\hat{n}_{k}}\delta\hat{n}_{j}\right\rangle,$
(5)

where $\delta\hat{n}_{j}=1-\hat{n}_{j}$. This is a symmetry protected
topological (SPT) phase with a doubly degenerate entanglement spectrum [48],
characterized by $P_{\Lambda}=0$, with $P_{\Lambda}^{2}\equiv\sum_{n\geq
0}|\lambda_{2n}-\lambda_{2n+1}|^{2}$. The entanglement spectrum
$\\{\lambda_{i}\\}$ corresponds to the eigenstates of the reduced density
matrix $\rho_{L/2}$ , obtained after partitioning the system into two halves,
cutting across the legs at the center of the ladder, and tracing the overall
density matrix over one of them.

The above-mentioned phases are adiabatically continued for $\Delta>0$, see
Fig. 1 (b). For growing $\Delta$, the 2DW phase shrinks to lower $t/V$ values
until disappearing at $\Delta/V=4$. For a stronger tilting, the system enters
at low $t/V$ into a polarized (P) phase, in which all particles occupy the
lowest leg. Interestingly, the Haldane phase separates at low $t/V$ the 2DW
and P phases, penetrating at $\Delta/V=4$ all the way down to zero hopping. As
shown below, this is qualitatively very different in the presence of the
dipolar tail.

## Polar lattice gas.–

The phase diagram of the actual polar lattice gas for $\theta=\pi/2$ is shown
in Fig. 1 (c). Although the overall phase diagram, with 2DW, P, Haldane, and
SF phases is qualitatively the same, the non-trivial interplay between the
dipolar tail and the tilting $\Delta$ results in new physics in the
intermediate region in between 2DW and P (see the region within the square in
Fig. 1 (c) and Fig. 2). In the atomic limit ($t=0$) we find two additional
density waves, with periods of $3$ (3DW) and $4$ (4DW) sites, and unit cells
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respectively. The 3DW (4DW) phase is hence characterized by a peak at
$k=2\pi/3$ ($k=\pi/2$) in $S(k)$. These phases survive at finite hopping,
although the 4DW is significantly more robust.

## Topological floating phase.–

Interestingly, in stark contrast to the NN model, the Haldane phase does not
penetrate down to zero hopping through the interstitials in between the
density wave phases. We observe at low hopping $t/V<0.2$ the appearance of a
new, markedly different, gapless phase, characterized by a conformal central
charge $c=1$ (single component Luttinger liquid), see Fig. 2 (b) and Fig. 4,
which we have determined both from our DMRG and iDMRG calculations from the
scaling of the entanglement entropy,
$S_{vN}=-\mathrm{Tr}[\rho_{L/2}\log\rho_{L/2}]$, with the system size $L$ [49]
and the correlation length [50], respectively. The presence of this phase is
marked by a different entanglement entropy (Fig. 2 (a)). More relevantly,
whereas the Haldane phase is a disordered phase with no peak in $S(k\neq 0)$,
the observed phase presents well-defined peaks in $S(k)$ (Fig. 2 (c)). These
peaks, however, do not occur for a restricted set of discrete values linked to
a well-defined density wave, but rather occurs for a value of $k$ that
decreases continuously from $k=\pi$ when $\Delta/V$ increases, as shown in
Fig. 2 (c). This phase hence corresponds to a quantum floating phase, which
results from the fact that the melting of the 2DW crystal occurs by the
proliferation of 3DW unit cells, and then 4DW ones. Due to this, the phase
presents a characteristic marked finite-size dependence, depicted in Fig. 3.

The floating phase, which characterizes the melting of the density waves at
small hopping, fills the whole intermediate region between 2DW and P phases,
surrounding the 3DW and 4DW phases. In contrast to the very recent realization
in QIM ladders using Rydberg atoms [42], in the dipolar XXZ ladder the
floating phase transitions to a disordered but topological Haldane phase.
Moreover, despite of being a gapless phase, the floating phase presents both a
finite string order and a double-degenerate entanglement spectrum, see Fig. 4.
The quantum floating phase constitutes hence an intriguing example of a
gapless topological phase [43, 44, 45, 46].

## Dependence on the dipole orientation.–

Up to this point, we have focused on the case $\theta=\pi/2$. Figure 5 (a)
shows the results for the atomic limit ($t=0$) as a function of $\theta$ and
$\Delta/V$. For $\theta/\pi<0.17$, a value close but not equal to the magic
angle $\theta_{M}=\arcsin(1/\sqrt{3})$ at which the rung interaction $V_{0,1}$
vanishes, the 2DW transforms into a zigzag wave (ZZ2) [44, 51], that
corresponds to the Néel phase of XXZ spin ladders, which presents also period
2, but has a unit cell $\,\begin{matrix}\vspace*{-0.18cm}\leavevmode\hbox
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1.72221pt\lower-1.72221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{1.72221pt}{0.0pt}\pgfsys@curveto{1.72221pt}{0.95116pt}{0.95116pt}{1.72221pt}{0.0pt}{1.72221pt}\pgfsys@curveto{-0.95116pt}{1.72221pt}{-1.72221pt}{0.95116pt}{-1.72221pt}{0.0pt}\pgfsys@curveto{-1.72221pt}{-0.95116pt}{-0.95116pt}{-1.72221pt}{0.0pt}{-1.72221pt}\pgfsys@curveto{0.95116pt}{-1.72221pt}{1.72221pt}{-0.95116pt}{1.72221pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{matrix}$.
Additional density waves occur, with period 3 (ZZ3) and 4 (ZZ4) with unit cell
$\,\begin{matrix}\vspace*{-0.18cm}\leavevmode\hbox to3.44pt{\vbox
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} \pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\\\
\vspace*{0.05cm}\leavevmode\hbox to3.84pt{\vbox
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} \pgfsys@invoke{\lxSVG@closescope
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to3.44pt{\vbox to3.44pt{\pgfpicture\makeatletter\hbox{\hskip
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} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\end{matrix}$ and
$\,\begin{matrix}\vspace*{-0.18cm}\leavevmode\hbox to3.44pt{\vbox
to3.44pt{\pgfpicture\makeatletter\hbox{\hskip 1.72221pt\lower-1.72221pt\hbox
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} \pgfsys@invoke{\lxSVG@closescope
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} \pgfsys@invoke{\lxSVG@closescope
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to3.84pt{\vbox to3.84pt{\pgfpicture\makeatletter\hbox{\hskip
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} \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
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} \pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\\\
\vspace*{0.05cm}\leavevmode\hbox to3.84pt{\vbox
to3.84pt{\pgfpicture\makeatletter\hbox{\hskip 1.92221pt\lower-1.92221pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\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{
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to3.44pt{\vbox to3.44pt{\pgfpicture\makeatletter\hbox{\hskip
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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} \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{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{{{}}{}{}{}{}{}{}{}{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{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
to3.44pt{\vbox to3.44pt{\pgfpicture\makeatletter\hbox{\hskip
1.72221pt\lower-1.72221pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\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}}\end{matrix}$,
respectively. When $\theta$ decreases the Haldane phase shrinks in favor of
the SF (see Fig. 3 (b) for $\theta/\pi=0.19$). The appearance of the zigzag
phases correlates with the disappearance of the Haldane and the floating
phases (see Fig. 3 (c)).

## Conclusions.–

We have shown that dipolar particles in optical ladders constitute an
interesting novel scenario for the realization of a quantum floating phase,
which interestingly is gapless but topological, intermediate between a crystal
and a gapped topological disordered Haldane phase. The floating phase results
from the interplay between the $1/r^{3}$ tail of the dipolar interaction and
the tilting of the ladder along the rungs. The observation of the floating
phase demands $V/t\gtrsim 5$. Since on-going experiments with erbium atoms
have already achieved $V/t=6$ [19], the realization of the topological
floating phase in dipolar XXZ ladders should be within reach of current state-
of-the-art experiments. In the future, it would be interesting to investigate
the nature of the phase transitions associated to crystal melting, which may
present a highly non-trivial nature [52, 34, 35, 36, 41], to study the non-
equilibrium dynamics of the dipolar XXZ ladder, and to extend the analysis to
multi-leg ladders.

###### Acknowledgements.

H.K., G.A.D.-C. and L.S acknowledge the 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). M.Ł.
acknowledges support from the National Science Centre (Poland) via Opus Grant
No. 2019/35/B/ST2/00838. M.Ł. gratefully acknowledges Polish high-performance
computing infrastructure PLGrid (HPC Centers: ACK Cyfronet AGH) for providing
computer facilities and support within computational grant no.
PLG/2024/017242. J.Z. acknowledges support by the National Science Centre
(Poland) under the OPUS call within the WEAVE program 2021/43/I/ST3/01142. The
research has also been supported by the Priority Research Area (DigiWorld)
under the Strategic Programme Excellence Initiative at Jagiellonian University
(M.Ł., J.Z.).

## References

  * Bloch  _et al._ [2008] I. Bloch, J. Dalibard, and W. Zwerger, Many-body physics with ultracold gases, Rev. Mod. Phys. 80, 885 (2008).
  * Trotzky  _et al._ [2008] S. Trotzky, P. Cheinet, S. Fölling, M. Feld, U. Schnorrberger, A. M. Rey, A. Polkovnikov, E. A. Demler, M. D. Lukin, and I. Bloch, Time-resolved observation and control of superexchange interactions with ultracold atoms in optical lattices, Science 319, 295 (2008).
  * Mazurenko  _et al._ [2017] A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, A cold-atom fermi–hubbard antiferromagnet, Nature 545, 462 (2017).
  * Browaeys and Lahaye [2020] A. Browaeys and T. Lahaye, Many-body physics with individually controlled rydberg atoms, Nature Physics 16, 132 (2020).
  * Dagotto and Rice [1996] E. Dagotto and T. M. Rice, Surprises on the way from one- to two-dimensional quantum magnets: The ladder materials, Science 271, 618 (1996).
  * Dagotto [1999] E. Dagotto, Experiments on ladders reveal a complex interplay between a spin-gapped normal state and superconductivity, Reports on Progress in Physics 62, 1525 (1999).
  * Mikeska and Kolezhuk [2004] H.-J. Mikeska and A. K. Kolezhuk, One-dimensional magnetism, in  _Quantum Magnetism_, edited by U. Schollwöck, J. Richter, D. J. J. Farnell, and R. F. Bishop (Springer Berlin Heidelberg, Berlin, Heidelberg, 2004) pp. 1–83.
  * Hijii  _et al._ [2005] K. Hijii, A. Kitazawa, and K. Nomura, Phase diagram of $\mathrm{S}=\frac{1}{2}$ two-leg $xxz$ spin-ladder systems, Phys. Rev. B 72, 014449 (2005).
  * Li  _et al._ [2017] S.-H. Li, Q.-Q. Shi, M. T. Batchelor, and H.-Q. Zhou, Groundstate fidelity phase diagram of the fully anisotropic two-leg spin-1/2 xxz ladder, New Journal of Physics 19, 113027 (2017).
  * Steinigeweg  _et al._ [2014] R. Steinigeweg, F. Heidrich-Meisner, J. Gemmer, K. Michielsen, and H. De Raedt, Scaling of diffusion constants in the spin-$\frac{1}{2}$ xx ladder, Phys. Rev. B 90, 094417 (2014).
  * Schubert  _et al._ [2021] D. Schubert, J. Richter, F. Jin, K. Michielsen, H. De Raedt, and R. Steinigeweg, Quantum versus classical dynamics in spin models: Chains, ladders, and square lattices, Phys. Rev. B 104, 054415 (2021).
  * Rakovszky  _et al._ [2022] T. Rakovszky, C. W. von Keyserlingk, and F. Pollmann, Dissipation-assisted operator evolution method for capturing hydrodynamic transport, Phys. Rev. B 105, 075131 (2022).
  * Domınguez-Castro  _et al._ [2023] G. Domınguez-Castro, T. Bilitewski, D. Wellnitz, A. M. Rey, and L. Santos, Relaxation in dipolar spin ladders: from pair production to false-vacuum decay (2023), arXiv:2311.18091 [cond-mat.quant-gas] .
  * Sompet  _et al._ [2022] P. Sompet, S. Hirthe, D. Bourgund, T. Chalopin, J. Bibo, J. Koepsell, P. Bojović, R. Verresen, F. Pollmann, G. Salomon, C. Gross, T. A. Hilker, and I. Bloch, Realizing the symmetry-protected haldane phase in fermi–hubbard ladders, Nature 606, 484 (2022).
  * Wienand  _et al._ [2023] J. F. Wienand, S. Karch, A. Impertro, C. Schweizer, E. McCulloch, R. Vasseur, S. Gopalakrishnan, M. Aidelsburger, and I. Bloch, Emergence of fluctuating hydrodynamics in chaotic quantum systems (2023), arXiv:2306.11457 [cond-mat.quant-gas] .
  * 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, Nonequilibrium quantum magnetism in a dipolar lattice gas, 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, Extended bose-hubbard models with ultracold magnetic atoms, Science 352, 201 (2016).
  * Fersterer  _et al._ [2019] P. Fersterer, A. Safavi-Naini, B. Zhu, L. Gabardos, S. Lepoutre, L. Vernac, B. Laburthe-Tolra, P. B. Blakie, and A. M. Rey, Dynamics of an itinerant spin-3 atomic dipolar gas in an optical lattice, Phys. Rev. A 100, 033609 (2019).
  * Su  _et al._ [2023] L. Su, A. Douglas, M. Szurek, R. Groth, S. F. Ozturk, A. Krahn, A. H. Hébert, G. A. Phelps, S. Ebadi, S. Dickerson, F. Ferlaino, O. Marković, and M. Greiner, Dipolar quantum solids emerging in a hubbard quantum simulator, Nature 622, 724 (2023).
  * Yan  _et al._ [2013] B. Yan, S. A. Moses, B. Gadway, J. P. Covey, K. R. A. Hazzard, A. M. Rey, D. S. Jin, and J. Ye, Observation of dipolar spin-exchange interactions with lattice-confined polar molecules, Nature 501, 521 (2013).
  * Christakis  _et al._ [2023] L. Christakis, J. S. Rosenberg, R. Raj, S. Chi, A. Morningstar, D. A. Huse, Z. Z. Yan, and W. S. Bakr, Probing site-resolved correlations in a spin system of ultracold molecules, Nature 614, 64 (2023).
  * Holland  _et al._ [2023] C. M. Holland, Y. Lu, and L. W. Cheuk, On-demand entanglement of molecules in a reconfigurable optical tweezer array, Science 382, 1143 (2023).
  * Bao  _et al._ [2023] Y. Bao, S. S. Yu, L. Anderegg, E. Chae, W. Ketterle, K.-K. Ni, and J. M. Doyle, Dipolar spin-exchange and entanglement between molecules in an optical tweezer array, Science 382, 1138 (2023).
  * Carroll  _et al._ [2024] A. N. Carroll, H. Hirzler, C. Miller, D. Wellnitz, S. R. Muleady, J. Lin, K. P. Zamarski, R. R. W. Wang, J. L. Bohn, A. M. Rey, and J. Ye, Observation of generalized t-j spin dynamics with tunable dipolar interactions (2024), arXiv:2404.18916 [cond-mat.quant-gas] .
  * de Léséleuc  _et al._ [2019] S. de Léséleuc, V. Lienhard, P. Scholl, D. Barredo, S. Weber, N. Lang, H. P. Büchler, T. Lahaye, and A. Browaeys, Observation of a symmetry-protected topological phase of interacting bosons with rydberg atoms, Science 365, 775 (2019).
  * Lienhard  _et al._ [2020] V. Lienhard, P. Scholl, S. Weber, D. Barredo, S. de Léséleuc, R. Bai, N. Lang, M. Fleischhauer, H. P. Büchler, T. Lahaye, and A. Browaeys, Realization of a density-dependent peierls phase in a synthetic, spin-orbit coupled rydberg system, Phys. Rev. X 10, 021031 (2020).
  * Signoles  _et al._ [2021] A. Signoles, T. Franz, R. Ferracini Alves, M. Gärttner, S. Whitlock, G. Zürn, and M. Weidemüller, Glassy dynamics in a disordered heisenberg quantum spin system, Phys. Rev. X 11, 011011 (2021).
  * Scholl  _et al._ [2022] P. Scholl, H. J. Williams, G. Bornet, F. Wallner, D. Barredo, L. Henriet, A. Signoles, C. Hainaut, T. Franz, S. Geier, A. Tebben, A. Salzinger, G. Zürn, T. Lahaye, M. Weidemüller, and A. Browaeys, Microwave engineering of programmable $xxz$ hamiltonians in arrays of rydberg atoms, PRX Quantum 3, 020303 (2022).
  * Bornet  _et al._ [2023] G. Bornet, G. Emperauger, C. Chen, B. Ye, M. Block, M. Bintz, J. A. Boyd, D. Barredo, T. Comparin, F. Mezzacapo, T. Roscilde, T. Lahaye, N. Y. Yao, and A. Browaeys, Scalable spin squeezing in a dipolar rydberg atom array, Nature 621, 728 (2023).
  * Chanda  _et al._ [2024] T. Chanda, L. Barbiero, M. Lewenstein, M. J. Mark, and J. Zakrzewski, Recent progress on quantum simulations of non-standard bose-hubbard models (2024), arXiv:2405.07775 .
  * Pokrovsky and Talapov [1979] V. L. Pokrovsky and A. L. Talapov, Ground state, spectrum, and phase diagram of two-dimensional incommensurate crystals, Phys. Rev. Lett. 42, 65 (1979).
  * Bak [1982] P. Bak, Commensurate phases, incommensurate phases and the devil’s staircase, Reports on Progress in Physics 45, 587 (1982).
  * Rieger and Uimin [1996] H. Rieger and G. Uimin, The one-dimensional annni model in a transverse field: analytic and numerical study of effective hamiltonians, Zeitschrift für Physik B Condensed Matter 101, 597 (1996).
  * Fendley  _et al._ [2004] P. Fendley, K. Sengupta, and S. Sachdev, Competing density-wave orders in a one-dimensional hard-boson model, Phys. Rev. B 69, 075106 (2004).
  * Chepiga and Mila [2019] N. Chepiga and F. Mila, Floating phase versus chiral transition in a 1d hard-boson model, Phys. Rev. Lett. 122, 017205 (2019).
  * Giudici  _et al._ [2019] G. Giudici, A. Angelone, G. Magnifico, Z. Zeng, G. Giudice, T. Mendes-Santos, and M. Dalmonte, Diagnosing potts criticality and two-stage melting in one-dimensional hard-core boson models, Phys. Rev. B 99, 094434 (2019).
  * Weimer and Büchler [2010] H. Weimer and H. P. Büchler, Two-stage melting in systems of strongly interacting rydberg atoms, Phys. Rev. Lett. 105, 230403 (2010).
  * Bermudez  _et al._ [2012] A. Bermudez, J. Almeida, K. Ott, H. Kaufmann, S. Ulm, U. Poschinger, F. Schmidt-Kaler, A. Retzker, and M. B. Plenio, Quantum magnetism of spin-ladder compounds with trapped-ion crystals, New Journal of Physics 14, 093042 (2012).
  * Rader and Läuchli [2019] M. Rader and A. M. Läuchli, Floating phases in one-dimensional rydberg ising chains (2019), arXiv:1908.02068 [cond-mat.quant-gas] .
  * Chepiga and Mila [2021] N. Chepiga and F. Mila, Kibble-zurek exponent and chiral transition of the period-4 phase of rydberg chains, Nature Communications 12, 414 (2021).
  * Maceira  _et al._ [2022] I. A. Maceira, N. Chepiga, and F. Mila, Conformal and chiral phase transitions in rydberg chains, Phys. Rev. Res. 4, 043102 (2022).
  * Zhang  _et al._ [2024] J. Zhang, S. H. Cantú, F. Liu, A. Bylinskii, B. Braverman, F. Huber, J. Amato-Grill, A. Lukin, N. Gemelke, A. Keesling, S.-T. Wang, Y. Meurice, and S. W. Tsai, Probing quantum floating phases in rydberg atom arrays (2024), arXiv:2401.08087 [quant-ph] .
  * Kestner  _et al._ [2011] J. P. Kestner, B. Wang, J. D. Sau, and S. Das Sarma, Prediction of a gapless topological haldane liquid phase in a one-dimensional cold polar molecular lattice, Phys. Rev. B 83, 174409 (2011).
  * Ruhman  _et al._ [2012] J. Ruhman, E. G. Dalla Torre, S. D. Huber, and E. Altman, Nonlocal order in elongated dipolar gases, Phys. Rev. B 85, 125121 (2012).
  * Thorngren  _et al._ [2021] R. Thorngren, A. Vishwanath, and R. Verresen, Intrinsically gapless topological phases, Phys. Rev. B 104, 075132 (2021).
  * Fraxanet  _et al._ [2022] J. Fraxanet, D. González-Cuadra, T. Pfau, M. Lewenstein, T. Langen, and L. Barbiero, Topological quantum critical points in the extended bose-hubbard model, Phys. Rev. Lett. 128, 043402 (2022).
  * Hauschild and Pollmann [2018] J. Hauschild and F. Pollmann, Efficient numerical simulations with Tensor Networks: Tensor Network Python (TeNPy), SciPost Phys. Lect. Notes , 5 (2018).
  * Pollmann  _et al._ [2010] F. Pollmann, A. M. Turner, E. Berg, and M. Oshikawa, Entanglement spectrum of a topological phase in one dimension, Phys. Rev. B 81, 064439 (2010).
  * Ejima  _et al._ [2014] S. Ejima, F. Lange, and H. Fehske, Spectral and entanglement properties of the bosonic haldane insulator, Phys. Rev. Lett. 113, 020401 (2014).
  * Calabrese and Lefevre [2008] P. Calabrese and A. Lefevre, Entanglement spectrum in one-dimensional systems, Phys. Rev. A 78, 032329 (2008).
  * Gammelmark and Zinner [2013] S. Gammelmark and N. T. Zinner, Dipoles on a two-leg ladder, Phys. Rev. B 88, 245135 (2013).
  * Huse and Fisher [1982] D. A. Huse and M. E. Fisher, Domain walls and the melting of commensurate surface phases, Phys. Rev. Lett. 49, 793 (1982).