# Quantum dimer models with Rydberg gadgets

Zhongda Zeng  Institute for Theoretical Physics, University of Innsbruck,
Innsbruck A-6020, Austria  Institute for Quantum Optics and Quantum
Information, Austrian Academy of Sciences, Innsbruck A-6020, Austria  Giuliano
Giudici  Institute for Theoretical Physics, University of Innsbruck, Innsbruck
A-6020, Austria  Institute for Quantum Optics and Quantum Information,
Austrian Academy of Sciences, Innsbruck A-6020, Austria  planqc GmbH, Garching
D-85748, Germany  Hannes Pichler  Institute for Theoretical Physics,
University of Innsbruck, Innsbruck A-6020, Austria  Institute for Quantum
Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck
A-6020, Austria

###### Abstract

The Rydberg blockade mechanism is an important ingredient in quantum
simulators based on neutral atom arrays. It enables the emergence of a rich
variety of quantum phases of matter, such as topological spin liquids. The
typically isotropic nature of the blockade effect, however, restricts the
range of natively accessible models and quantum states. In this work, we
propose a method to systematically overcome this limitation, by developing
gadgets, i.e., specific arrangements of atoms, that transform the underlying
Rydberg blockade into more general constraints. We apply this technique to
realize dimer models on square and triangular geometries. In these setups, we
study the role of the quantum fluctuations induced by a coherent drive of the
atoms and find signatures of $U(1)$ and $\mathbb{Z}_{2}$ quantum spin liquid
states in the respective ground states. Finally, we show that these states can
be dynamically prepared with high fidelity, paving the way for the quantum
simulation of a broader class of constrained models and topological matter in
experiments with Rydberg atom arrays.

### Introduction. –

Figure 1:  Rydberg gadgets for the dimer constraint. Left and right panels
correspond to the square and triangular lattices, respectively. (a,b) Typical
dimer fully-packed configurations. Red (white) dots represent atoms in the
Rydberg (ground) state (see the inset of (a)). The red crosses indicate the
undesired blockade for the selected blockade radius $R_{b}$. (c,d)
Illustration of the Rydberg gadgets. Gadgets are placed on the vertices of the
lattice and consist of the orange atoms. The green disks indicate the blockade
radius required to realize the dimer constraint. (e,f) Rydberg configurations
in the gadget model corresponding to the dimer coverings in (a,d). Each dimer
consists of one excited atom on the edge and two excited atoms on the gadgets
at the extrema (blue rectangle). The purple-shaded square is the lattice unit
cell.

Preparing and studying strongly correlated many-body quantum states is one of
the central goals of near-term quantum devices [1]. Quantum spin liquids
(QSLs) are a particularly interesting class of such states, that feature many
exotic properties, including long-range entanglement, anyonic excitations, and
topological ground state degeneracy [2, 3, 4, 5]. In addition, QSLs offer
intriguing possibilities for applications in quantum information processing,
where they could form the basis of effective error-correcting schemes for
robust quantum memories [6], or for topological quantum computation [7, 8, 9].
Recently, remarkable progress has been achieved in realizing some of these
topologically ordered states with several quantum simulator platforms, using a
variety of state preparation techniques. In digital approaches, the target
states are prepared by applying circuits consisting of entangling gates to
simple product states [10, 11, 12]. In contrast, in analog approaches,
dynamical preparation protocols with a controllable, time-dependent
Hamiltonian are employed, e.g. using the adiabatic principle [13, 14, 15]. The
latter has the advantage that the prepared states can potentially be
stabilized by a Hamiltonian and dynamical signatures of QLS phases could be
probed.

Experimentally, the analog approach to realizing QSLs in synthetic quantum
matter has been pioneered with Rydberg atom arrays [13]. These platforms
enable the realization of highly programmable spin models, that have been
explored for quantum simulation [16, 17, 18, 19, 20, 13], quantum optimization
[21], and quantum computation [22, 23, 24]. In these setups, atoms are trapped
in optical tweezers and internal states are employed to represent spin degrees
of freedom [25]. Control is enabled by coherent laser drive and the geometric
arrangement of the atoms [26, 27]. A crucial ingredient is the so-called
blockade effect [28, 29, 30], which prevents two atoms from being
simultaneously excited to Rydberg states when their distance is smaller than
the blockade radius. This mechanism, when combined with specific geometries of
the array, can be used to introduce constraints on the allowed spin
configurations, e.g., to mimic the Gauss’s law of lattice gauge theories [31,
32, 33, 34]. Moreover, it is a key ingredient for providing the geometric
frustration that leads to the emergence of QSLs from many-body Rydberg
Hamiltonians [35, 36, 37, 38, 39].

The Rydberg blockade is typically isotropic, strongly restricting the class of
non-trivial constraints that can be implemented in Rydberg atom systems. To
overcome this limitation various approaches have been proposed, ranging from
choosing anisotropic Rydberg states [40] to making use of multi-layer arrays
[38]. Here, we employ a method that avoids these experimental challenges by
generalizing the concept of Rydberg gadgets that have been developed to encode
arbitrary classical constraints, e.g., for quantum optimization problems [41,
42, 43, 44, 45]. A gadget is a particular configuration of Rydberg atoms
designed to provide a specific constraint solely as a result of the isotropic
Rydberg blockade and the lattice geometry. While such constraints allow us to
realize the frustration that can lead to the emergence of a QSL phase [45],
understanding if the accessible quantum fluctuations actually stabilize such a
phase in models constructed from Rydberg gadgets is an open question.

In this work, we propose simple gadgets that encode the constraint of dimer
models and study the physics associated with the corresponding quantum
fluctuations. Quantum dimer models [46] have recently sparked revived interest
after the theoretical proposal [35] and experimental observation [13] of a
$\mathbb{Z}_{2}$ QSL on a system of Rydberg atoms positioned on the edges of a
kagome lattice, realizing what we call “diluted dimer constraint” (see below).
Here, we focus on geometries where this constraint is not natively provided by
the Rydberg blockade (cf. Fig. 1(a-b)), namely the square and triangular
lattices. We show that in the Rydberg gadget model, i.e., the system in which
gadgets are placed on the vertices of such lattices, the diluted dimer
constraint is recovered in the limit of large density of Rydberg excitations.
We address the effect of quantum fluctuations via analytical perturbation
theory and numerical exact diagonalization (ED) of the Rydberg atom
Hamiltonian. We analyze the ground state phase diagrams of these systems and
demonstrate the onset of $U(1)$ and $\mathbb{Z}_{2}$ QSLs, respectively on the
square and triangular lattice. Moreover, by studying dynamical protocols to
prepare them, we showcase the experimental relevance of our approach.

### Diluted dimer models in Rydberg atom arrays. –

In a Rydberg atom array, neutral atoms are typically trapped into a
configurable lattice geometry and approximated as two-level systems consisting
of the ground state $|g\rangle$ and a highly-excited Rydberg state
$|r\rangle$. Each atom is driven by a coherent laser field with Rabi frequency
$\Omega$ and detuning $\Delta$. The Hamiltonian for this system is [47, 25,
48, 49]

$\hat{H}_{\text{Ryd}}=\Omega\sum_{i}\hat{\sigma}^{x}_{i}-\sum_{i}\Delta_{i}\hat{n}_{i}+\sum_{ij}U_{ij}\hat{n}_{i}\hat{n}_{j},$
(1)

where $\hat{\sigma}^{x}=|r\rangle\langle g|+|g\rangle\langle r|$,
$\hat{n}=|r\rangle\langle r|$, and $i$ labels the position $\bm{x}_{i}$ of
each atom. The interaction is of the van der Waals form
$U_{ij}\sim|\bm{x}_{i}-\bm{x}_{j}|^{-6}$, which is strongly repulsive at short
distances and rapidly decays at large distances. The strong distance
dependence of the interaction strength allows for a blockade approximation,
where interactions within the so-called blockade radius $R_{\textrm{b}}$ are
assumed to be infinite and can be treated as the constraint
$\hat{n}_{i}\hat{n}_{j}=0$ for $|\bm{x}_{i}-\bm{x}_{j}|<R_{\textrm{b}}$, while
interactions at larger distances are neglected.

In the blockade approximation, for certain atom arrangements and a properly
chosen $R_{b}$, the Rydberg atom system can be exactly mapped into a dimer
model subject to the diluted dimer constraint, where _at most_ one dimer
connects to each vertex of a lattice. The mapping consists of interpreting the
atom positions as the midpoint of the edges of a lattice, and identifying a
Rydberg excitation with the presence of a dimer on the corresponding edge.
This correspondence however only holds on certain geometries, such as the ruby
[35] and kagome [15] lattices, which are equivalent to dimer models on the
kagome and honeycomb lattices, respectively.

The effective Hamiltonian induced on the dimer degrees of freedom by Eq. (1)
is given by
$\hat{H}=\Omega\sum_{i}\hat{\sigma}^{x}_{i}-\Delta\sum_{i}\hat{n}_{i}$, where
$i$ labels the edges, and $\hat{\sigma}_{i}^{x}$ adds or removes dimers if
allowed by the diluted dimer constraint. We dub this model diluted dimer model
(DDM) and observe that it is well defined regardless of the mapping into a
Rydberg atom system. In the limit $\Delta/\Omega\rightarrow\infty$, its low-
energy subspace is spanned by the fully-packed dimer configurations with one
dimer per vertex. At finite $\Delta/\Omega$, the off-diagonal term has two,
competing effects on the DDM ground state: it favors coherence between maximal
dimer coverings on the one hand, and lowers the density of dimers, on the
other. Whereas the former typically stabilizes QSL phases, the latter could
spoil it. Therefore, establishing the emergence of QSLs ground states in DDMs
requires a non-perturbative treatment of quantum fluctuations, which we tackle
below with ED methods.

In a generic lattice, the correspondence between Rydberg atom arrays in the
blockade approximation and DDM does not hold. This is illustrated in Fig. 1(a)
and (b) for the square and triangular lattices, respectively. For instance, on
the square lattice (cf. Fig. 1(a)), the distance between atoms in the
horizontal and vertical directions are both equal to the lattice spacing $a$.
The diluted dimer constraint however would correspond to a blockade constraint
that is active in one direction, but not in the other one, which is clearly
incompatible with the isotropic blockade (see Fig. 1(a)). In what follows, we
provide a solution to this issue by inserting gadgets on the lattice vertices
(cf. Fig. 1(c-d)). These gadgets are designed to transmit the blockade of an
excited atom on an edge to all the atoms on adjacent edges, but to no atom on
any other edge. We call the resulting model gadget model. Importantly, while
this gadget model is designed to share similarities with the DDM on the same
lattice, they differ in some key aspects. In particular, the diluted dimer
constraint is enforced on the low-energy spectrum of the gadget model
Hamiltonian only in the large $\Delta/\Omega$ regime. Furthermore, the number
of states in a unit cell is larger in the gadget model, resulting in an
exponentially larger Hilbert space. Nevertheless, as we show below, the gadget
model ground state satisfies the diluted dimer constraint in a wide parameter
range and can even exhibit enhanced QSL features w.r.t. the DDM ground state.

### Diluted dimer models with Rydberg gadgets. –

We now specify the Rydberg gadget models depicted in Fig. 1(c-f). We consider
the square and triangular lattices, with a Rydberg atom on each edge, and a
gadget on each vertex. The gadgets consist of 4 or 6 atoms on the square and
triangular lattices, respectively. Edge and gadget atoms are represented by
blue and orange dots, while the blockade radius is indicated by the green disk
in Fig. 1(c,d). An edge atom blockades all the gadget atoms on adjacent
vertices, except the two gadget atoms that are furthest away (cf. left side of
Fig. 1(c,d)). A gadget atom blockades all atoms in the same gadget and all
edge atoms adjacent to that vertex, excluding the farthest one (cf. right side
of Fig. 1(c,d)). This atom arrangement yields the dimer constraint as follows.
Whenever an atom in a gadget is excited to the Rydberg state, only atoms on
the adjacent edges can be excited to the Rydberg state. This is akin to the
constraint of having only one dimer on any edge connected to a vertex (cf.
Fig. 1(e,f)). Accordingly, we call a gadget _active_ , if it contains a
Rydberg excitation. To favor active gadgets, the detuning for the gadget
atoms, $\Delta_{\rm gad}$, is adjusted relative to the detuning of the edge
atoms, $\Delta$, i.e. $\Delta_{\rm gad}=\alpha\Delta$. In the classical limit
($\Omega=0$) gadgets are active when $\Delta>0$ and $\alpha>3/2$ and
$\alpha>1$ for square and triangular lattices, respectively [50]. In summary,
as shown in Fig. 1(e,f), a single dimer in the gadget model is represented by
an excited atom on an edge, and it is always accompanied by two active gadgets
at the neighboring vertices.

In this setup, the degenerate ground state subspace of the Rydberg Hamiltonian
in the blockade approximation, and for $\Omega=0$ and $\Delta>0$ is the same
as in the associated DDM, and it is composed of the fully-packed dimer
coverings. An important difference between Rydberg gadget models and DDMs is
that the former allows for configurations in which there is more than one
excited atom on edges connecting to the same vertex if the gadget on that
vertex is inactive. However, as we will show below, when $\Delta\gtrsim 0$,
the gadgets activate, thus penalizing these unwanted configurations at low
energy (cf. Fig. 2(e,f)). We will demonstrate that, although the gadget model
is not guaranteed to reproduce the ground state properties of the DDM, it
enables the observation of QSL features in the ground and dynamically prepared
states.

### Ground state phase diagram. –

Figure 2:  Ground state phase diagram. Ground state RVB overlap (a,b),
fidelity susceptibility Eq. (3) (c,d), and dimer-constraint violation
probability (e,f) as a function of $\Delta/\Omega$ and for different values of
$\alpha=\Delta_{\mathrm{gad}}/\Delta$, on the square (a,c,e) and triangular
lattices (b,d,f). The dashed lines indicate the diluted dimer model. Results
are obtained on 10 and 8 unit-cell clusters for the square and triangular
lattices, respectively [50].

We performed ED calculations for the ground state of DDM and gadget models on
finite systems with periodic boundary conditions. We employ a unit cell of 2
and 3 edges for the square and triangular lattices, respectively (cf. Fig
1(a,b)). Since there is one gadget per vertex, the gadget models for these two
lattices have 6 and 9 atoms per unit cell, resulting in a much larger Hilbert
space. Therefore, we focus on periodic clusters of up to 8 and 10 unit cells
for square and triangular lattices [50]. Our figure of merit for the
characterization of a QSL is the overlap with the resonating valence bond
(RVB) state [14, 38, 39], i.e., the equal-weight superposition of fully-packed
dimer coverings $\\{|c\rangle\\}$:

$|\mathrm{RVB}\rangle=\frac{1}{\sqrt{\mathcal{N}}}\sum_{c}|c\rangle,$ (2)

where $\mathcal{N}$ is the total number of these configurations. These states
are the simplest representatives of QSLs and have very different properties on
bipartite and non-bipartite lattices, like the square and triangular lattices.
In the former case they encode the physics of a compact $U(1)$ deconfined
gauge theory and have power-law decaying correlations, while in the latter
they are described by a deconfined $\mathbb{Z}_{2}$ gauge theory and their
correlations are short-range [46, 2, 51, 52, 53, 54, 55].

To identify the boundaries of the gadget model phase diagram, we also consider
the ground state fidelity susceptibility [56, 57]

$\chi=-\frac{2}{\delta\lambda^{2}}\ln\left|\bra{\psi_{0}(\lambda)}\ket{\psi_{0}(\lambda+\delta\lambda)}\right|,$
(3)

where $\lambda=\Delta/\Omega$, $|\psi_{0}(\lambda)\rangle$ is the ground state
wavefunction, and $\delta\lambda$ is a small change in the parameter
$\lambda$. Peaks in $\chi$ serve as a diagnostic tool for the location of
phase transitions.

For the gadget model on the square lattices, $\chi$ exhibits two peaks,
indicating two qualitative changes in the ground state (cf. Fig. 2(c)). As we
increase $\Delta/\Omega$, the system first undergoes a crossover from a
trivially disordered regime to a “gadget-active” regime [58]. As shown in Fig.
2(e), this process is characterized by a sharp decrease in the ground state
probability $p$ of violating the diluted dimer constraint, given e.g. by
configurations with two dimers connecting to one vertex with an inactive
gadget. Subsequently, the system undergoes a transition to a phase
characterized by a significant overlap with the RVB state, as shown in Fig.
2(a). Since $U(1)$ QSLs are expected to be unstable in 2D [59], we interpret
this RVB phase as an emergence of a finite-size $U(1)$ QSL, which is not
stable in the thermodynamic limit [15].

We note that the RVB overlaps for the DDM and gadget model on the square
lattice converge to the same value in the classical limit
$\Delta/\Omega\rightarrow\infty$, as can be seen from Fig. 2(a). This result
is readily understood through perturbation theory, which yields the same
leading-order effective Hamiltonian for the two models [50]

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} \pgfsys@invoke{\lxSVG@closescope
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}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{11.38092pt}{11.38092pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{}
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@setlinewidth{1.5pt}\pgfsys@invoke{
}{}\pgfsys@moveto{1.4226pt}{0.0pt}\pgfsys@lineto{9.95813pt}{0.0pt}\pgfsys@moveto{1.4226pt}{11.38072pt}\pgfsys@lineto{9.95813pt}{11.38072pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\Big{\rvert}+\textup{h.c.}\Bigr{)},$
(4)

where the thick lines correspond to dimers, and the flipping strength $t$ is
proportional to $\Omega^{4}/\Delta^{3}$ and $\Omega^{12}/\Delta^{11}$ for the
diluted and gadget model, respectively.

In Fig. 2(b,d), we present the numerical results for the triangular lattice,
where the fidelity susceptibility for the gadget model exhibits three peaks.
Similarly to the square lattice, as $\Delta/\Omega$ increases, the system
first undergoes a crossover from the trivially disordered regime to a gadget-
active regime (cf. Fig. 2 (f)). On the triangular lattice, however, a high-RVB
overlap phase is observed only at intermediate values of $\Delta/\Omega$, and
a third peak in $\chi$ indicates a further transition to an ordered valence-
bond solid (VBS) phase. We note that this is not the case for the DDM (dashed
line in Fig. 2(b)), where the RVB overlap monotonically increases with
$\Delta/\Omega$. In contrast to the square lattice, $\mathbb{Z}_{2}$ QSLs can
be stable on the triangular lattice. Therefore, this high-RVB overlap phase
could indicate the emergence of a $\mathbb{Z}_{2}$ QSL phase that survives in
the thermodynamic limit. The fact that RVB overlaps for diluted dimer and
gadget models do not match in the limit $\Delta/\Omega\to\infty$ can again be
understood via perturbation theory. The effective Hamiltonians for the two
models are [50]

$\begin{split}&\hat{H}_{\triangle,\mathrm{diluted}}=-t\sum_{r}\Bigl{(}\Big{\lvert}\leavevmode\hbox
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(5)

The leading-order term for the diluted model is the dimer flipping term from
4th order, with strength $t\propto\Omega^{4}/\Delta^{3}$. For the gadget
model, instead, the leading order is a non-trivial diagonal term with strength
$V\propto\Omega^{6}/\Delta^{5}$ arising at 6th order. In this case, the dimer
flipping term only occurs at 12th order, with a strength proportional to
$\Omega^{12}/\Delta^{11}$. This leads to the difference between the two RVB
overlaps for $\Delta/\Omega\to\infty$.

We stress that the quantitative comparison of ground state RVB overlaps
outlined in Fig. 2 (a) and (b) between DDMs and gadget models is remarkable.
In fact, for fixed $\Delta/\Omega$, RVB overlaps in the gadget model can be
larger than in the DDM despite the much larger Hilbert space dimension of the
former [60]. Moreover, the gadget system possesses the additional parameter
$\alpha=\Delta_{g}/\Delta$, which can be tuned to increase RVB fidelities
further and thus enhance QSL features.

### Dynamical preparation. –

Figure 3:  RVB dynamical preparation infidelity. (a) Dynamical preparation
protocol. (b,c) Infidelities $\mathcal{I}\equiv
1-|\langle\psi(T)|\mathrm{RVB}\rangle|$ obtained from periodic clusters with 6
unit cells as a function of $T/N$ with $T$ the total preparation time and $N$
the number of atoms. The dashed and solid lines are the infidelities obtained
in the diluted dimer model and gadget model, for different values of
$\alpha=\Delta_{\mathrm{gad}}/\Delta$ in the latter. (d,e) Same as (b,c) but
for clusters with 10 and 8 unit cells for the square and triangular lattices,
respectively.

We now analyze dynamical schemes typically discussed for preparing QSLs in
experiments. We focus on the preparation protocol illustrated in Fig. 3(a).
The atoms are initialized in their ground state, corresponding to the many-
body ground state of the Hamiltonian Eq. (1) when $\Omega=0$ and $\Delta<0$.
The Rabi frequency is then gradually increased to its maximum value, which we
set to $1$ and fix our time units over the time interval $0<t<T_{1}$.
Following this, the detuning is linearly ramped up from an initial value
$\Delta_{0}<0$ to a positive value $\Delta_{f}$ within the time span
$T_{1}<t<T_{1}+T_{2}$. Finally, the Rabi frequency is gradually decreased to
zero. Here, we set $T_{2}=0.5T$, $T_{1}=T_{3}=0.25T$, $\Delta_{0}=-5$, and
$\Delta_{f}=1.5$ [14].

When the annealing time $T$ goes to infinity, the system evolves adiabatically
and remains in the instantaneous ground state. The final state ends up in the
ground state obtained in the classical limit $\Delta/\Omega\rightarrow\infty$,
implying that the RVB overlap of the final state for $T\to\infty$ has to
approach the RVB overlap of the ground state for $\Delta/\Omega\to\infty$ (cf.
Fig. 2 (a,b)). Conversely, for small $T$, only a few atoms are excited
throughout the dynamics, resulting in a significant number of defects and a
suppressed RVB overlap. Our primary interest lies in the intermediate regime
of total preparation times $T$, where RVB fidelities are expected to be
maximal [14]. In Fig. 3(b-e), we plot the infidelity between the final state
and the RVB state, i.e., $\mathcal{I}\equiv
1-|\langle\psi(T)|\mathrm{RVB}\rangle|$, for the square and triangular
lattices. In both cases, we not only observe an enhanced RVB overlap at
intermediate $T$ w.r.t. the adiabatic case, but also an optimal overlap which
is comparable in the DDMs and gadget models, reaching $\mathcal{I}<10^{-3}$
for the largest systems considered. In Fig. 3(b)-(c), we demonstrate how the
extra control parameter $\alpha$ of the gadget model can be used to further
minimize $\mathcal{I}$. Due to the large Hilbert space dimension, we did not
perform such optimization in the larger periodic clusters in Fig. 3 (d,e).

### Conclusions. –

We proposed a method based on Rydberg gadgets to engineer the constraint of
dimer models in neutral atom arrays in the regime of maximal density of
Rydberg excitations. This method can be realized in experiments simply by
placing the gadgets on the vertices of a lattice. We analyzed the effect of
quantum fluctuations on the gadget model by numerically studying the ground
state phase diagram on the square and triangular lattices, and compared it to
the simpler diluted dimer models, which are not naturally realized with an
isotropic Rydberg blockade in these geometries. Making use of perturbation
theory, we highlighted similarities and differences between DDMs and gadget
models, and showed that the ground state of the latter can exhibit enhanced
QSL features w.r.t. their diluted dimer counterparts. Finally, we demonstrated
that through non-adiabatic dynamical protocols, it is possible to achieve
high-fidelity preparation of $U(1)$ and $\mathbb{Z}_{2}$ spin liquid states on
the square and triangular lattice, respectively.

In this work, we considered the blockade approximation of Rydberg Hamiltonians
and ignored the long-range interaction tails, which is a crude approximation
for real experiments. However, this approximation can be improved by properly
tuning the blockade radius and utilizing different atomic species [50].
Although we only applied Rydberg gadgets to obtain the dimer constraint on
square and triangular lattices, this method can be generalized to a larger
class of models and used to engineer other kinds of constraints which, when
combined with quantum fluctuations, could provide more exotic forms of
topological order [45].

###### Acknowledgements.

### Acknowledgements. –

We are grateful to Giacomo Giudice for providing support on part of the code
used for the numerical simulations, and to Robert Ott for providing feedback
on the manuscript. We thank Tao Shi, Hongzheng Zhao, Robert Ott, and Torsten
Zache for valuable discussions. G.G. acknowledges support from the European
Union’s Horizon Europe program under the Marie Sklodowska Curie Action TOPORYD
(Grant No. 101106005). This work is supported by the European Union’s Horizon
Europe research and innovation program under Grant Agreement No. 101113690
(PASQuanS2.1), the ERC Starting grant QARA (Grant No. 101041435), the EU-
QUANTERA project TNiSQ (N-6001), and by the Austrian Science Fund (FWF)[grant
DOI 10.55776/COE1].

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Supplementary Material

###### Contents

  1. I Atom arrangement and clusters
  2. II Relation between detuning on the edges and on the gadgets
  3. III Derivation of the effective Hamiltonian
  4. IV Rydberg excitation density
  5. V Improving the blockade approximation by multi-species arrays
  6. VI Another application of Rydberg gadgets: the “concentrated” dimer constraint

##  I Atom arrangement and clusters

Figure S1:  Atom arrangement and clusters for ED calculations. (a,d) Depiction
of the atom arrangement for the square (upper) and triangular (bottom)
lattices. (b,e) Clusters with $6$ unit-cells for the square (upper) and
triangular (bottom) lattices. (c) Square lattice cluster with $10$ unit-cells.
(f) Triangular lattice cluster with $8$ unit-cells.

In this section, we give more details about the geometric setup used in this
work. In the gadget model, the distance between the gadget atoms and their
corresponding vertices is denoted as $r$. The lattice spacing is denoted as
$a$ (cf. Fig. S1(a,d)). On the square lattice, the gadget method requires: (i)
the edge atom not to be blockaded with the farthest gadget atoms in the two
corresponding vertices ($R_{\mathrm{b}}<r+a/2$); (ii) the edge atom to
blockade the other gadget atoms in the two corresponding vertices
($\sqrt{r^{2}+(a/2)^{2}}<R_{\mathrm{b}}$); and (iii) the atoms in different
gadgets not to blockade each other ($R_{\mathrm{b}}<a-2r$). These conditions
imply

$R_{\min}=a\sqrt{1/4+(r/a)^{2}}<R_{b}<a\min[1-2(r/a),1/2+(r/a)]=R_{\max}.$
(S1)

Since our numerical simulations are in the blockade approximation with no
long-range interaction tails, the choice of $r/a$ and $R_{b}$ is irrelevant,
as long as the exact blockade is imposed on atoms at a distance shorter than
$R_{b}$, with $R_{b}$ satisfying Eq. (S1). However, for this to be a good
approximation of the full Rydberg Hamiltonian we want to maximize the ratio
$R_{\max}/R_{\min}$, such that $U(R_{\min})/U(R_{\max})\gg 1$. For this ratio
to be maximal one must set $r=a/6$, which yields
$U(R_{\min})/U(R_{\max})=\left(4/\sqrt{10}\right)^{6}\simeq 4.1$.

On the triangular lattice similar conditions hold: (i) $R_{\mathrm{b}}<r+a/2$;
(ii) $\sqrt{r^{2}+(a+\sqrt{3}r)^{2}}/2<R_{\mathrm{b}}$; and (iii)
$R_{\mathrm{b}}<a-2r$. Therefore, $R_{b}$ must satisfy

$R_{\min}=\frac{a}{2}\sqrt{\left(\frac{r}{a}\right)^{2}+\left(1+\frac{\sqrt{3}r}{a}\right)^{2}}<R_{b}<a\min\left(1-\frac{2r}{a},\frac{1}{2}+\frac{r}{a}\right)=R_{\max}.$
(S2)

As in the square lattice case, the ratio $R_{\max}/R_{\min}$ is maximal when
$r/a=1/6$, which yields
$U(R_{\min})/U(R_{\max})=\left(4/\sqrt{10+3\sqrt{3}}\right)^{6}\simeq 1.2$.
Differently from the square lattice, the strongest interaction above the
blockade radius is not much smaller than the weakest interaction below the
blockade radius, making the blockade approximation less justified. However, as
we show in Sec. V, this issue can be mitigated by making use of multi-species
arrays.

The exact diagonalization calculations are performed on finite-size systems
with periodic boundary conditions. The clusters we consider are shown in Fig.
S1. Note that the clusters are the same for both the gadget and non-gadget
models. In the gadget models, each unit cell contains 6 atoms on the square
lattice and 9 atoms on the triangular lattice.

##  II Relation between detuning on the edges and on the gadgets

For realizing the dimer model constraint in the gadget model the detuning on
the edges $\Delta$ and on the gadgets $\Delta_{\mathrm{gad}}$ have to satisfy
the relation $\Delta_{\mathrm{gad}}=\alpha\Delta$ with $\alpha>3/2$ and
$\alpha>1$ for the square and triangular lattices, respectively. These minimum
values of $\alpha$ ensure the gadgets’ activation when $\Delta>0$. They can be
inferred by comparing the energy density $\varepsilon_{0}$ of the dimer fully-
packed configurations (i.e., one dimer connects to each vertex) with the
energy density $\varepsilon_{1}$ of the other ground state candidates
configurations, when $\Omega=0$. For both the square and triangular lattices
we have $\varepsilon_{0}=-(1/2+\alpha)\Delta$.

On the square lattice, as $\alpha$ is varied, the dimer fully-packed subspace
competes with the unique gadget fully-inactive configuration shown in Fig.
S2(a), where all edge atoms are in the Rydberg state and all gadget atoms are
in the ground state. The energy density of such configurations is
$\varepsilon_{1}=-2\Delta$. The condition $\varepsilon_{1}>\varepsilon_{0}$
thus implies $\alpha>3/2$.

On the triangular lattice, due to the direct blockade between nearest-neighbor
edge atoms, the states competing with the dimer-fully packed subspace consist
of the “fan” configurations depicted in Fig. S2(b). Their energy is
$\varepsilon_{1}=-3(1+\alpha)\Delta/4$, so that one must set $\alpha>1$ to
have $\varepsilon_{1}>\varepsilon_{0}$.

Figure S2:  Low-energy configurations in the gadget models. (a) Gadget fully-
inactive configuration on the square lattice. (b) “Fan” configurations on the
triangular lattice.

##  III Derivation of the effective Hamiltonian

Figure S3:  Dimer flipping process in the diluted dimer model. (a) Square
lattice. (b) Triangular lattice. These 4th-order processes in perturbation
theory contribute to a dimer-flipping term with strength
$t\propto\Omega^{4}/\Delta^{3}$ to their effective Hamiltonians. Similar
processes also exist in the gadget model, but they require to remove and add
back the gadget atoms, and thus arise at 12th order, with strength
$t\propto\Omega^{12}/\Delta^{11}$.

In the blockade approximation, where the blockade constraint is implicitly
assumed, the Hamiltonian reduces to an on-site form, which can be split into a
diagonal term $\hat{H}_{0}=-\sum_{i}\Delta_{i}\hat{n}_{i}$, where
$\Delta_{i}=\Delta$ for edge atoms and $\Delta_{i}=\alpha\Delta$ for gadget
atoms, and an off-diagonal term $\hat{V}=\Omega\sum_{i}\hat{\sigma}_{i}^{x}$.
We take the former as the unperturbed Hamiltonian $\hat{H}_{0}$. Its ground
space is degenerate with energy $E_{0}$, and is made of all the fully-packed
dimer configurations. The off-diagonal term contributes to the creation or
destruction of a dimer. The effective Hamiltonian at leading order $n$ is
written as [61]:

$\hat{H}_{\mathrm{eff}}\sim\hat{P}\hat{V}\left(\hat{Q}\frac{1}{E_{0}-\hat{Q}\hat{H_{0}}\hat{Q}}\hat{Q}\hat{V}\right)^{n-1}\hat{P},$
(S3)

where $\hat{P}$ is the projector onto the ground space of $\hat{H}_{0}$ (the
dimer fully-packed states), and $\hat{Q}=1-\hat{P}$.

For diluted dimer models on both square and triangular lattices, the lowest-
order non-trivial contribution is the flipping of a pair of dimers arising at
order $n=4$, as shown in Fig. S3. The flipping strengths are proportional to
$\Omega^{4}/\Delta^{3}$. In the gadget model, the analogous flipping term
originates at order $n=12$, with strength $\Omega^{12}/\Delta^{11}$. In this
case, the higher order of $\hat{V}$ is needed to enable the dimer to move by
changing the Rydberg state position on the 4 gadgets involved. Since each
gadget requires two flips of the atomic state, there are in total 8 extra
applications of the off-diagonal perturbation w.r.t. the diluted dimer model.
Note that a single-dimer flipping term could exist in finite-size systems with
one narrow direction, such as the clusters in Fig. S1(b,e).

Figure S4:  Example of non-trivial diagonal processes in the triangular gadget
model. In these processes, one edge dimer can be created after removing two
nearby gadget atoms. We focus on the dimer marked with blue dots. The allowed
nearby edges are marked with green dots. For one dimer in the staggered
configuration (a)(left), two such processes (one of which is shown in (a))
contribute to the diagonal matrix element. In the columnar configuration
(b)(left), there are four such processes, one of which is shown in (b).

In the triangular lattice gadget model, the non-trivial diagonal term in Eq.
(6) of the main text results from the direct blockade between atoms on the
edges. In fact, the edge atoms blockade not only the gadget atoms but also the
4 atoms on the nearby edges (cf. Fig. 1(d)). This direct blockade between edge
atoms forbids certain intermediate processes in perturbation theory. For
instance, for the staggered configuration shown in Fig. S4(a), two edge atoms
can be excited after de-exciting two gadget atoms. This process yields back
the initial configuration and thus constitutes a 6th-order diagonal process.
Another example is the columnar configuration, for which more processes are
allowed, as shown in Fig. S4(b). In the more general case, a single dimer (an
excited edge atom) has 10 nearby edges, and 4 of them are directly blockaded
by the edge atom (cf. Fig. 1(d)). We refer to these 4 edges as forbidden
edges. When two dimers form a pair on a plaquette (e.g. ), they share one
forbidden edge. In this case, a dimer can be created on one more nearby edge
w.r.t. local configurations in which the dimers do not form a pair.
Consequently, the pair configurations allow for one more diagonal process at
6th order in perturbation theory, and thus have lower energy compared to
configurations where the plaquette is not flippable. This contributes a non-
trivial diagonal term that counts the number of flippable plaquettes, with
strength $V\propto\Omega^{6}/\Delta^{5}$.

Figure S5:  Ground state Rydberg excitation density. Rydberg excitation
densities (green lines) and their derivatives (red lines) for atoms on the
edges (a,b), on the gadgets (c,d), and in total (e,f), in the ground state of
the gadget model. We set $\alpha=\Delta_{g}/\Delta=5$.

##  IV Rydberg excitation density

The gadget and edge densities, namely the density of Rydberg excitations on
the edges, can serve as another observable to witness the “gadget-active”
regime, which we characterized in Fig. 2 of the main text with a sharp
suppression of the ground state amplitude on atom configurations that violate
the diluted dimer constraint. In Fig. S5, we plot edge, gadget, and total
densities in both gadget models as green solid lines. In the crossover from
the disordered phase to the “gadget-active” regime, the gadget density quickly
increases, while the edge density is suppressed due to the blockade between
edge and gadget atoms. As $\Delta/\Omega$ further increases, other transitions
arise, while the gadgets remain active. Eventually, both edge and gadget
densities saturate to their maximal value. We note that, although the gadget
density is not maximal for intermediate $\Delta/\Omega$, the gadgets
accomplish their function of penalizing configurations that do not satisfy the
diluted dimer constraint. This fact indicates that Rydberg gadgets in this
setup are stable against gadget density fluctuations. In Fig. S5, we also plot
the densities derivatives of $\lambda=\Delta/\Omega$ as red solid lines, and
show that they correctly signal the various phase transitions discussed in the
main text.

Figure S6:  Improvement by using multi-species atoms. The different atom
species are marked by different colors. (a) Square lattice. We set $r=0.16a$,
$R_{\alpha\alpha}=0.45a$, and $R_{\alpha\beta}=0.6a$. (b) Triangular lattice.
We set $r=0.11a$, $R_{\alpha\alpha}=0.45a$, and $R_{\alpha\beta}=0.62a$.

##  V Improving the blockade approximation by multi-species arrays

In the main text, we assumed the blockade approximation of the van der Waals
interaction, which implies $U(R_{\min})/U(R_{\max})\gg 1$, where $U(R_{\min})$
($U(R_{\max})$) are the strongest (weakest) interaction above (below) the
blockade radius. As we discussed in Sec. I, this approximation is justified
for the gadget model on the square lattice, where
$U(R_{\min})/U(R_{\max})\simeq 4.1$, but not on the triangular lattice, where
$U(R_{\min})/U(R_{\max})\simeq 1.2$. In this section, we discuss how to
mitigate this issue by employing multi-species atom arrays [62, 63, 64]. A
similar effect can be obtained via multi-layer arrays [65, 43].

On the square lattice, we make use of two species of atoms, denoted as
$\alpha$ and $\beta$. The blockade radii are denoted as
$R_{b}^{\alpha\alpha}$, $R_{b}^{\alpha\beta}$, and $R_{b}^{\beta\beta}$, and
we assume $R_{b}^{\alpha\alpha}=R_{b}^{\beta\beta}$. The atom arrangement and
blockade radii are illustrated in Fig. S6(a), where $\alpha$ and $\beta$ atoms
are represented by orange and purple dots, respectively. The gadget
requirements in terms of the blockade radii are

$\displaystyle
R_{\min}^{\alpha\alpha}=\max\left(2r,\frac{a}{2}-r\right)<R_{b}^{\alpha\alpha}<\frac{a}{2}+r=R_{\max}^{\alpha\alpha},$
(S4) $\displaystyle
R_{\min}^{\alpha\beta}=\sqrt{r^{2}+\left(\frac{a}{2}\right)^{2}}<R_{b}^{\alpha\beta}<\sqrt{(a-r)^{2}+\left(\frac{a}{2}\right)^{2}}=R_{\max}^{\alpha\beta}.$
(S5)

For the blockade approximation to be valid we want
$r_{\alpha\alpha}=U(R_{\mathrm{max}}^{\alpha\alpha})/U(R_{\mathrm{min}}^{\alpha\alpha})\gg
1$ and
$r_{\alpha\beta}=U(R_{\mathrm{max}}^{\alpha\beta})/U(R_{\mathrm{min}}^{\alpha\beta})\gg
1$. We can try to maximize both $r_{\alpha\alpha}$ and $r_{\alpha\beta}$ by
choosing $r/a=0.16$. We get $r_{\alpha\alpha}\simeq 53$ and
$r_{\alpha\beta}\simeq 41$, which are much larger than in the single-species
setup ($r\simeq 4$, cf. Sec. I).

A similar strategy applies to the triangular lattice. However, in this case,
we need to introduce three species of atoms, $\alpha$, $\beta$, and $\gamma$.
Assuming $R_{\alpha\alpha}=R_{\beta\beta}=R_{\gamma\gamma}$ and
$R_{\alpha\beta}=R_{\alpha\gamma}=R_{\beta\gamma}$, the gadget requirements
are

$\displaystyle
R_{\min}^{\alpha\alpha}=\max\left(2r,\frac{a}{2}-r\right)<R_{b}^{\alpha\alpha}<\frac{a}{2}+r=R_{\max}^{\alpha\alpha},$
(S6) $\displaystyle
R_{\min}^{\alpha\beta}=\frac{1}{2}\sqrt{r^{2}+\left(a+\sqrt{3}r\right)^{2}}<R_{b}^{\alpha\beta}<\frac{1}{2}\sqrt{r^{2}+3(a-r)^{2}}=R_{\max}^{\alpha\beta}.$
(S7)

Again, we want to maximize both the intra-species interaction ratio
$r_{\alpha\alpha}$ and the inter-species ratio $r_{\alpha\beta}$. If we set
$r/a=0.11$, we have $r_{\alpha\alpha}\simeq 6.9$ and $r_{\alpha\beta}\simeq
7.5$, which are a notable improvement w.r.t. the single-specie setup ($r\simeq
1.2$, cf. Sec. I).

Figure S7:  Gadget method for constructing the concentrated dimer Hilbert
space.

##  VI Another application of Rydberg gadgets: the “concentrated” dimer
constraint

In this section, we discuss how to apply the gadget method to study another
form of dimer constraint, namely what we call “concentrated” dimer constraint.
This constraint is such that each vertex of the lattice is touched by _at
least_ one dimer. Fig. S7 illustrates the gadgets required to construct this
constraint on square and triangular lattices. The atom arrangement is the same
as with diluted dimer constraint, albeit with a shorter blockade radius.
Specifically, each gadget atom is blockaded with all other gadget atoms on the
same vertex and with the nearest edge atom. At the same time, each edge atom
is blockaded with the two nearest gadget atoms on the corresponding vertices.
In this setup, a dimer is represented by the ground state of an edge atom, and
it is accompanied by the nearest two Rydberg state gadget atoms. The low-
energy subspace comprises minimally packed configurations in the
$\Delta/\Omega\rightarrow\infty$ limit. The system permits the creation of
additional dimers by de-exciting the edge atoms from the Rydberg state to the
ground state. Even though we do not perform large-scale numerical calculations
for this model due to the prohibitive Hilbert space dimension compared with
diluted models, we expect this model can also be used to dynamically prepare
the QSL state in experiments.