# Trimer states with $\mathbb{Z}_{3}$ topological order in Rydberg atom arrays

Giacomo Giudice  Max-Planck-Institute of Quantum Optics, Hans-Kopfermann-
Straße 1, 85748 Garching, Germany  Munich Center for Quantum Science and
Technology (MCQST), Schellingstraße 4, 80799 München, Germany  Federica Maria
Surace  Department of Physics and Institute for Quantum Information and
Matter, California Institute of Technology, Pasadena, CA 91125, USA  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 Giuliano Giudici
Arnold Sommerfeld Center for Theoretical Physics, University of Munich,
Theresienstraße 37, 80333 München, Germany Munich Center for Quantum Science
and Technology (MCQST), Schellingstraße 4, 80799 München, Germany

###### Abstract

Trimers are defined as two adjacent edges on a graph. We study the quantum
states obtained as equal-weight superpositions of all trimer coverings of a
lattice, with the constraint of having a trimer on each vertex: the so-called
trimer resonating-valence-bond (tRVB) states. Exploiting their tensor network
representation, we show that these states can host $\mathbb{Z}_{3}$
topological order or can be gapless liquids with
$\mathrm{U}(1)\times\mathrm{U}(1)$ local symmetry. We prove that this
continuous symmetry emerges whenever the lattice can be tripartite such that
each trimer covers all the three sublattices. In the gapped case, we
demonstrate the stability of topological order against dilution of maximal
trimer coverings, which is relevant for realistic models where the density of
trimers can fluctuate. Furthermore, we clarify the connection between gapped
tRVB states and $\mathbb{Z}_{3}$ lattice gauge theories by smoothly connecting
the former to the $\mathbb{Z}_{3}$ toric code, and discuss the non-local
excitations on top of tRVB states. Finally, we analyze via exact
diagonalization the zero-temperature phase diagram of a diluted trimer model
on the square lattice and demonstrate that the ground state exhibits
topological properties in a narrow region in parameter space. We show that a
similar model can be implemented in Rydberg atom arrays exploiting the
blockade effect. We investigate dynamical preparation schemes in this setup
and provide a viable route for probing experimentally $\mathbb{Z}_{3}$ quantum
spin liquids.

##  I Introduction

When quantum fluctuations meet classical frustration, exotic strongly
correlated states can arise. Paradigmatic examples are resonating valence bond
(RVB) states of hard dimers. They are defined as equal-weight quantum
superpositions of all dimer coverings with one dimer touching each vertex of a
lattice. These many-body states are the ground states of local Hamiltonians
with peculiar properties, such as a ground state degeneracy that depends on
the topology of the system and deconfined excitations that come in pairs [1].
When their correlation length is finite, a stable, topologically ordered,
quantum phase of matter exists with RVB states as its representatives [2].
This kind of topological order is characterized by a local $\mathbb{Z}_{2}$
symmetry closely related to the Gauss’ law in the gauge theory description of
these phases, and it is expected to emerge in dimer models defined on non-
bipartite lattices [1].

((a))

((b))

Figure 1: (a) Maximal trimer configuration with arrows pointing from the
center to the two external vertices of each trimer, on the square and
honeycomb lattice. The net flux for each vertex is $2\,\mathrm{mod}\,3$. The
region enclosed by the dashed line has $N_{s}=3$ vertices and a net flux
$3=2N_{s}\,\mathrm{mod}\,3$. (b) One-parameter family of tRVB states on the
square lattice. The parameter $t=\tan\theta$ weights the presence of straight
trimers, such that for $\theta=0$ ($\pi/2$) the tRVB state is solely made of
bent (straight) trimers.

In this work, we consider RVB states of hard trimers (tRVB). A trimer is an
object made up of two nearest-neighbor edges of a lattice that share a common
vertex [cf. 1(a)]. Hard trimers cannot touch each other, such that each vertex
of the lattice can be covered by at most one trimer, yielding what we refer to
as the trimer constraint. Maximally-packed trimer configurations are obtained
by demanding that exactly one trimer covers each vertex. These configurations
are then promoted to orthogonal quantum states, and the tRVB state is their
equal-weight quantum superposition. On certain lattices, tRVB states are known
to be gapped and to possess a form of topological order with emergent
$\mathbb{Z}_{3}$ gauge symmetry [3, 4, 5], and thus to be good representatives
of a quantum phase with $\mathbb{Z}_{3}$ topological order. However, a general
condition on the lattice geometry for which these states are gapped and have
topological character is still lacking, and physical realizations of this
phase are little known. Here, we identify a necessary condition for having a
gapped tRVB state with $\mathbb{Z}_{3}$ topological order, similar to the
condition of non-bipartite lattices for dimer models. Moreover, we provide a
trimer model where signatures of the above-mentioned topological phase are
identified. This model is physically relevant for experimental platforms based
on Rydberg atom arrays, as we demonstrate by proposing and analyzing a viable
implementation of the trimer constraint in this setup.

Firstly, we focus on the square lattice, where we demonstrate the emergence of
the trimer coverings Hilbert space as a particular limit of a $\mathbb{Z}_{3}$
lattice gauge theory (LGT). We introduce a tensor network (TN) [6, 7, 8]
representation that describes a one-parameter family of tRVB states [Fig.
1(b)]. Utilizing exact and approximate TN methods, we show that all the tRVB
states considered have $\mathbb{Z}_{3}$ topological order except for a fine-
tuned point, where we establish the presence of a
$\mathrm{U}(1)\times\mathrm{U}(1)$ local symmetry that leads to long-range
correlations. We show that topological properties are stable against dilution
of the maximally-packed trimer configurations by studying a tensor-network
perturbation that encodes all hard trimer configurations with _at most_ one
trimer on each vertex. Furthermore, we discuss a general mechanism that
explains the enhancement of the local $\mathbb{Z}_{3}$ conservation law of
Fig. 1(a) to a $\mathrm{U}(1)\times\mathrm{U}(1)$ law, yielding long-range
correlations and ultimately spoiling $\mathbb{Z}_{3}$ topological order. We
support our conclusions with further examples of gapless and gapped tRVB
states, on the triangular and honeycomb lattices.

Secondly, we consider a diluted trimer model on the square lattice with a
single type of trimers, namely the bent blue trimers in Fig. 1(a). The
corresponding tRVB state [Fig. 1(b) for $t=0$] will be shown to be a gapped
$\mathbb{Z}_{3}$ topological liquid, thus motivating the choice of this
geometry. The model Hamiltonian has a control parameter that tunes the density
of trimers. We compute the ground state wavefunction via exact diagonalization
on periodic clusters and show that a $\mathbb{Z}_{3}$ topologically ordered
phase arises at finite density. We show that the blockade effect induced by
van der Waals interactions in Rydberg atom arrays can be used to engineer the
(bent) trimer constraint on the square lattice, by mapping the four possible
trimer orientations on a square into four different excited Rydberg atoms. The
effective Rydberg model is equivalent to the diluted trimer model upon
neglecting some trimer configurations [cf. Fig. 11(a)]. We prove numerically
that removing these configurations from the superposition does not spoil the
$\mathbb{Z}_{3}$ topological nature of the fully-packed tRVB state. We find
that, despite signatures of a topological phase are elusive in the ground
state of the Rydberg Hamiltonian, a semi-adiabatic dynamical preparation
bolsters the topological character of the prepared state.

The paper is structured as follows. In Sec. II we introduce a one-parameter
family of tRVB states on the square lattice, its TN representation, and a one-
parameter perturbation that lower the density of trimers, preserving the TN
form. We study the state phase diagram of this two-parameters family and
demonstrate the presence of a stable $\mathbb{Z}_{3}$ topologically ordered
phase. In Sec. III we relate gapped tRVB states to the $\mathbb{Z}_{3}$ toric
code, from which we can define string operators and non-local excitations. We
discuss the condition for which a $\mathrm{U}(1)\times\mathrm{U}(1)$ gauge
theory emerges in trimer models, and verify it by considering tRVB states on
various lattice geometries. In Sec. IV.1 we introduce the model of diluted
bent trimers on the square lattice, analyze its ground state properties on
finite periodic systems, and show that a narrow topological phase emerges. We
outline the implementation of this model in Rydberg atom arrays, point out the
differences between the trimer and Rydberg models and discuss the
consequences. Finally, we analyze dynamical preparation protocols to realize
$\mathbb{Z}_{3}$ topologically ordered states in experiments.

##  II tRVB states on the square lattice

Trimers on the square lattice can either be bent or straight [cf. blue and red
trimers in Fig. 1(a)]. Therefore, we can define a one parameter family of tRVB
states on this lattice by weighting each covering $c$ with coefficients that
depend on the number of bent and straight trimers ($N_{\perp}(c)$ and
$N_{\parallel}(c)$ respectively) in the covering. In particular, we introduce
the parameter $\theta\in[0,\pi/2]$ such that the coefficient of a maximally-
packed configuration $c$ is

$\mathcal{W}(c)\leavevmode\nobreak\
=(\cos\theta)^{N_{\perp}(c)}(\sin\theta)^{N_{\parallel}(c)}.$ (1)

The tRVB state then reads

$\Ket{\mathrm{tRVB}(\theta)}=\frac{1}{\mathcal{N}(\theta)}\sum_{\\{c\\}}\mathcal{W}(c)\,\Ket{c},$
(2)

where $\mathcal{N}(\theta)=\sqrt{\sum_{\\{c\\}}[\mathcal{W}(c)]^{2}}$ is a
normalization factor. The angle $\theta$ changes the relative weight of bent
versus straight trimers [see Fig. 1(a)]: in the limit $\theta=0$
($\theta=\pi/2$) only bent (straight) trimers contribute.

In this section, we analyze the topological properties of this state with
tensor network methods. After studying the one-parameter family in Eq. 2, we
add another parameter by considering a diluted TN deformation obtained by
destroying trimers with a certain probability. We map out the state phase
diagram and demonstrate the stability of the topological phase against
dilution.

###  II.1 The tensor network representation of the tRVB model

We turn our attention to the classical statistical-mechanics model whose
partition function is the sum of the squared weights in Eq. 1 of all
maximally-packed trimer configurations on the square lattice. This partition
function can be interpreted as the norm of the quantum state in Eq. 2. Its
tensor-network representation was previously introduced in Ref. [3]: the
partition function $Z$ can be written as the tiling of rank-4 tensors

$Z=\vbox{\hbox{\scalebox{0.5}{\includegraphics{partition}}}}$ (3)

The rank-4 tensor $T$ is constructed in such a way that, once the tensor is
contracted with its neighbors, only the valid trimer configurations survive.
As will be later discussed in Sec. III, it is beneficial to consider an
equivalent representation of this model, where trimer configurations are
mapped to possible ways to position arrow on the links such that there is a
local constraint —or $\mathbb{Z}_{3}$ conservation law— around each edge. As
shown in Fig. 1(a), a given trimer configuration has a one-to-one mapping to a
single arrow configuration obtained by assigning arrows to the links covered
by the trimers, in such a way that each arrow goes from the center of the
trimer to the external vertices. In a fully-packed configuration, each vertex
has either two outgoing arrows or one ingoing arrow. Because the net outgoing
flux for each vertex is $2$ (where the flux is measured $\mathrm{mod}\,3)$, we
obtain that a region of $N_{s}$ vertices has flux $2N_{s}\,\mathrm{mod}\,3$.
This $\mathbb{Z}_{3}$ rule for the flux suggests that the tRVB state can be
described as a $\mathbb{Z}_{3}$ gauge theory and can be a gapped
$\mathbb{Z}_{3}$ quantum spin liquid.

The tensor $T$ is then constructed by labeling each leg with indices
$\\{0,+1,-1\\}$, where $0$ means no arrow, and $+1$ ($-1$) corresponds to an
arrow aligned (anti-aligned) with the direction of the leg. The non-zero
entries are

$\vbox{\hbox{\scalebox{0.5}{\includegraphics{square_tensor}}}}\,.$ (4)

The $\mathbb{Z}_{3}$ symmetry of the underlying gauge theory is reflected in
the tensor. Indeed the symmetry operator $\sigma$, whose matrix representation
111The operator is represented in the basis $\\{0,+1,-1\\}$ defined above. is
later defined in Eq. 12, acts on the tensor as

$\vbox{\hbox{\scalebox{0.5}{\includegraphics{symmetry}}}}\,.$ (5)

We remark that if one wishes to remove the factor $\omega=e^{i2\pi/3}$ from
Eq. 5 it is sufficient to block three consecutive tensors and construct a
$\mathbb{Z}_{3}$-invariant tensor. When interpreting the partition function
encoded in the tensor Eq. 4 as the norm of the tRVB state, each leg of the
tensor is interpreted as the product between the bra and ket virtual layers of
the quantum state.

((a))

((b))

Figure 2: (a) Correlation length $\xi$ as a function of $\theta$, computed
using infinite cylinders of circumference $L$, as well as infinite-size
methods (see main text). (b) Topological entanglement entropy $\gamma\simeq
S_{L}-2S_{L/2}$ extracted from finite-size cylinders.

((a))

((b))

((c))

Figure 3: (a) Link density of the diluted tRVB state $\ket{\Phi(\theta,z)}$,
obtained from CTMRG with an environment bond dimension $D=729$. The grey dots
indicate where its numerical derivative w.r.t. $z$ has an extremum. (b)
Correlation length $\xi$ obtained from the transfer matrix of the diluted tRVB
state computed with CTMRG. Note that $\xi$ diverges at the phase boundary and
at $\theta\to\pi/2$. (c) Details for $\theta=0$: the density displays a non-
analyticity at $z_{\mathrm{\mathrm{c}}}=0.88(6)$ (dashed line), which
corresponds to a divergence of the correlation length (inset).

The properties of the tRVB wavefunction can be extracted by analyzing the row-
wise _transfer operator_

$\mathbb{T}=\vbox{\hbox{\scalebox{0.5}{\includegraphics{transfermatrix}}}}\,.$
(6)

By computing the leading eigenvalues (ordered by magnitude)
$\lambda_{0},\lambda_{1},\dots$ of the transfer operator of length $L$ (with
periodic boundary conditions), we obtain the correlation length $\xi$ as

$\xi=1/\ln\left|\frac{\lambda_{0}}{\lambda_{1}}\right|\,.$ (7)

This quantity bounds all correlation functions in the height direction of the
infinitely-tall cylinder of circumference $L$. The $\mathbb{Z}_{3}$ virtual
symmetry in Eq. 5 can be used to label the eigenvalues of the transfer
operator as $\lambda^{\mathcal{Q}}_{n}$, where $\mathcal{Q}=0,\pm 1$ is the
$\mathbb{Z}_{3}$ charge and $n=0,1\dots$ is eigenvalue index starting from the
largest in magnitude. As expected for a topologically ordered state, this
symmetry is spontaneously broken, yielding an approximate 3-fold degeneracy
for the largest eigenvalues from the three $\mathbb{Z}_{3}$ symmetry sectors:
the gap between the logarithm of these eigenvalues closes exponentially in $L$
[10]. In fact, the spectrum $E=-\log\lambda$ is analogous to the spectrum of a
Hamiltonian with spontaneous symmetry breaking. To compute the correlation
length at finite $L$ we thus consider the two largest eigenvalues in the
$\mathcal{Q}=0$ sector. The results of numerical diagonalizations on finite
cylinders are presented in Fig. 2(a). These finite-size results are compared
to the correlation lengths obtained from the corner-transfer matrix
renormalization group (CTMRG) [11, 12, 13, 14, 15, 16] exploiting the
refletion symmetry along the tensor diagonal [17], as well as from the
variational uniform matrix product state algorithm (VUMPS) [18]. From the
numerical results, we conclude that the correlation length diverges only in
the limit $\theta\to\pi/2$, while it remains finite below that value. As we
will discuss in the next section, long-range correlations concur with the
emergence of a $\mathrm{U}(1)\times\mathrm{U}(1)$ local symmetry for
$\theta=\pi/2$.

The leading eigenvector of the transfer operator on a cylinder represents the
diagonal reduced density operator $\rho$ of the infinite half-cylinder, from
which we can as well obtain the entanglement entropy
$S=-\operatorname{Tr}(\rho\ln\rho)$ of this bipartition. The scaling with the
circumference length $L$ of the entanglement entropy obeys

$S_{L}\sim\alpha L-\gamma,$ (8)

where $\gamma$ is a well-known topological correction [19, 20].
$\gamma\simeq\ln 3$ implies that the state is in a gapped $\mathbb{Z}_{3}$
topological phase. In Fig. 2(b) we plot the topological entanglement entropy
obtained from the subtraction $\gamma=S_{L}-2S_{L/2}$, as a function of
$\theta$. While $\gamma$ appears to approach a finite value compatible with
$\ln 3$ for $\theta\neq\pi/2$, a bump occurs in the proximity of this point.
In fact, in the presence of continuous local symmetries such as
$\mathrm{U}(1)\times\mathrm{U}(1)$ the topological correction is expected to
scale logarithmically with $L$ [21].

###  II.2 Stability under dilution of the tRVB state

We now study a deformation of the tRVB state obtained by diluting fully-packed
trimer coverings. This deformation will be relevant for Sec. IV.1, where we
will discuss how to implement trimer models in Rydberg atom arrays. In these
setups, the total occupation can fluctuate, so it is important to consider
imperfect trimer coverings.

Similarly to recent work on dimer models [22], we consider the following
variational ansatz, which depends on two real parameters
$0\leq\theta\leq\pi/2$ and $z\in\mathbb{R}$

$\Ket{\Phi(\theta,z)}\propto\bigotimes_{i,j}\left(\mathbb{1}+z^{2}\Sigma^{-}_{ij}\right)\Ket{\mathrm{tRVB}(\theta)}\,,$
(9)

where $\Sigma^{-}_{ij}$ is the operator that removes a trimer on the edges $i$
and $j$, and $z^{2}$ corresponds to the weight of a removed trimer. In
essence, we add to the fully-packed trimer configurations other trimer
configurations that can be obtained from the former by removing trimers
without moving the remaining ones. Each removed trimer is weighted by $z^{2}$.
In the limit $z\to\infty$, the state is simply the vacuum, while at $z\to 0$
we recover the tRVB state, which we showed to be in a topological phase.
Similarly to $\ket{\mathrm{tRVB}}$, the state $\ket{\Phi}$ has a simple
projected entangled-pair state (PEPS) [8] representation of bond dimension $4$

$\ket{\Phi}=\vbox{\hbox{\scalebox{0.5}{\includegraphics{peps.pdf}}}}\,.$ (10)

Its construction is relegated to Sec. A.1.

In Fig. 3, we study the link density $n$ 222The link density $n$ counts the
ratio of links occupied by a trimer over the total number of links. On the
square lattice the link density of a fully-packed configuration is $1/3$ and
the correlation length of the state as we vary $\theta$ and $z$ using CTMRG.
Remarkably, the topological phase survives up to values of order one of the
dilution strength $z$, and is fairly insensitive to the mixing angle $\theta$.
The transition between the topological and trivial phases appears to be
continuous, as witnessed by a diverging correlation length at the critical
point [Fig. 3(b)]. We do not address the characterization of the universality
class of this phase transition as the critical exponents we could extract from
the available data exhibit strong dependence on the CTMRG environment bond
dimension $D$, even for the largest $D$ we employed. We note that topological
and trivial phases are distinguished by two different degeneracies of the PEPS
cylinder transfer matrix. The spectrum of the transfer matrix is 3-fold
degenerate in the former and 9-fold degenerate in the latter. The 9-fold
degeneracy reflects the full breaking of the
$\mathbb{Z}_{3}\times\mathbb{Z}_{3}$ virtual symmetry of the PEPS double
tensor, which implies the condensation of magnetic and the confinement of
electric excitations [see Sec. III] in the gauge theory picture [10].

##  III tRVB states and lattice gauge theories

To understand the emergence of $\mathbb{Z}_{3}$ topological order in the tRVB
state, it is useful to shed light on its connection with a gauge theory. To
this end, in Sec. III.1 we compare the tRVB state to the ground state of a
$\mathbb{Z}_{3}$ toric code.

((a))

((b))

((c))

((d))

((e))

Figure 4: Definitions of operators (a) $A_{v}$ and (b) $B_{p}$. (c) Mapping
from a trimer configuration to a state in the $\sigma$ basis. (d)
Configurations for a vertex in the trimer model. The 10 total configurations
are obtained from these under rotations. (e) Configurations that satisfy
Gauss’ law but to do not correspond to trimer configurations. The full set of
unallowed configurations (17 in total) is obtained from these under rotations.

((a))

((b))

((c))

Figure 5: (a) String operators in the $\mathbb{Z}_{3}$ toric code: ’t Hooft
lines and ’t Hooft loops (green), Wilson lines and Wilson loops (purple). (b)
Creation of a charge-anticharge pair on a trimer configuration with a string
operator: a monomer is an anticharge, and a charge consists of a pair of
monomers. (c) The correlation length remains finite when interpolating between
the $\mathbb{Z}_{3}$ toric code ($\alpha=0$) and the tRVB state ($\alpha=1$)
at different mixing angles $\theta$. Results obtained by performing CTMRG on
the $\mathbb{Z}_{3}$-charged tensor $T=(1-\alpha)T_{\mathrm{TC}}+\alpha
T_{\mathrm{tRVB}}$.

We argued in Sec. II that the presence of a $\mathbb{Z}_{3}$ local
conservation law of the flux suggests an emergent description as a
$\mathbb{Z}_{3}$ gauge theory, and hence the tRVB state is a good candidate
for being a gapped $\mathbb{Z}_{3}$ quantum spin liquid. However, as we show
below, it may happen that for some trimer models the $\mathbb{Z}_{3}$ local
symmetry is enhanced to a U$(1)\times$U$(1)$ symmetry, in which case the state
is gapless 333As we discuss below, there are exceptions in which, despite the
$\mathrm{U}(1)\times\mathrm{U}(1)$ local, the tRVB state is a gapped symmetry
broken state.. A similar scenario occurs for RVB states of dimer models, that
are known to host gapped $\mathbb{Z}_{2}$ spin liquids only on non-bipartite
lattices; on bipartite lattices, they are described by a U$(1)$ gauge theory,
that does not support a stable topologically ordered phase. In Sec. III.2 we
will formulate a similar criterion for trimer models.

###  III.1 The $\mathbb{Z}_{3}$ toric code

A state that is very similar to the tRVB state and has $\mathbb{Z}_{3}$
topological order is the ground state of the $\mathbb{Z}_{3}$ generalization
of Kitaev’s toric code [25]. We now review this model and show the
similarities and differences between its ground state and the tRVB state.

To define the $\mathbb{Z}_{3}$ toric code, we introduce clock variables on the
links of our lattice; on each link we define the operators $\sigma$ and
$\tau$, that satisfy the following properties:

$\sigma\tau=\omega\tau\sigma,\hskip 22.76228pt\sigma^{3}=\mathbb{1},\hskip
22.76228pt\tau^{3}=\mathbb{1},$ (11)

where $\omega=e^{2i\pi/3}$. These variables are the $\mathbb{Z}_{3}$
generalizations of the Pauli matrices $\sigma^{z}$ and $\sigma^{x}$, and their
most common matrix representation is

$\sigma=\begin{pmatrix}1&0&0\\\ 0&\omega&0\\\
0&0&\omega^{*}\end{pmatrix},\hskip 22.76228pt\tau=\begin{pmatrix}0&0&1\\\
1&0&0\\\ 0&1&0\end{pmatrix}.$ (12)

We now define the star and plaquette operators as in Figs. 4(a) and 4(b).
Similarly to the case of the $\mathbb{Z}_{2}$ toric code, these operators all
commute: $[A_{v},B_{p}]=0$ for every vertex $v$ and plaquette $p$. We now
define the state $\ket{\psi_{\text{TC}}}$ as the equal-weight superposition of
all the states in the $\sigma$ basis that satisfy the Gauss’ law
$A_{v}=\omega$ for all vertices. Note that this choice differs from the
typical case with $A_{v}=1$ and corresponds to the presence of a background
charge on each vertex of the lattice. Nevertheless, the physical properties
that we are interested in are not altered by this background charge, as a
unitary transformation can be defined to eliminate it. The state defined here
has the property that $B_{p}\ket{\psi_{\text{TC}}}=\ket{\psi_{\text{TC}}}$ for
every plaquette $p$, and is the ground state of the following Hamiltonian

$H_{\text{TC}}=-\sum_{v}(\omega^{*}A_{v}+\omega
A_{v}^{\dagger})-\sum_{p}(B_{p}+B_{p}^{\dagger}).$ (13)

Because star and plaquette operators commute, it is easy to identify the
excitations of the model: we call an excitation with $A_{v}=\omega^{*}$
($A_{v}=1$) a charge (anticharge), while an excitation with $B_{p}=\omega$
($B_{p}=\omega^{*}$) is a vison (antivison). Both “electric”
(charge/anticharge) and “magnetic” (vison/antivison) excitations are gapped.

We now elucidate the connection between the tRVB state and
$\ket{\psi_{\text{TC}}}$. We can map each configuration of fully-packed
trimers to a configuration in the $\sigma$ basis as shown in Fig. 4(c). It is
easily shown that this configuration satisfies Gauss’ law. However, not all
the configurations of the $\mathbb{Z}_{3}$ toric code that satisfy Gauss’ law
correspond to a trimer configuration: as shown in Fig. 4(d), only $10$ of the
$27$ configurations of a vertex correspond to allowed vertex configurations of
the trimer model. Despite this difference, the tRVB state may still have
$\mathbb{Z}_{3}$ topological order like $\ket{\psi_{\text{TC}}}$ if the
missing configurations are recovered under renormalization: for
$0\leq\theta<\pi/2$ the renormalization-group flow from the state
$\ket{\text{tRVB}(\theta)}$ ultimately leads to the toric code ground state,
which is a fixed point under blocking, with correlation length $\xi=0$.
Another approach to establish that the tRVB state and the toric code ground
state describe the same phase consists in showing that no phase transitions
occur when interpolating between the two states. This operation can as well be
interpreted as a smooth interpolation between the two parent Hamiltonians,
since $\mathbb{Z}_{3}$ injectivity is preserved [26, 10]. We interpolate
between the states $\ket{\psi_{\mathrm{TC}}}$ and $\ket{\mathrm{tRVB}}$ by
progressively decreasing the weight of the forbidden configurations in Fig.
4(e). In Fig. 5(c) we plot the correlation length $\xi$ obtained from CTMRG
during the interpolation. The gradual increase of $\xi$ for any
$0\leq\theta<\pi/2$ indicates the absence of phase transitions. We note that
what we observed here differs from what happens in the dimer model of the
kagome lattice, where the RVB state is a fixed point of $\mathbb{Z}_{2}$
topological order, and can be directly mapped into the toric code ground state
[27].

Finally, the connection with the $\mathbb{Z}_{3}$ toric code allows to define
string operators that are useful to detect topological order, namely Wilson
lines and ’t Hooft lines. The latter can be defined as in Fig. 5(a). Because
of Gauss’ law, the value of the ’t Hooft line around a closed loop is equal to
$\omega^{N_{v}+n_{q}-n_{\bar{q}}}$, where $N_{v}$ is the number of vertices,
and $n_{q}$, $n_{\bar{q}}$ are respectively the numbers of charges and
anticharges enclosed by the loop. Similarly, Wilson loops detect the number of
visons/antivisons in a region. Moreover, a ’t Hooft (Wilson) line creates a
vison/antivison (charge/anticharge) pair at the two ends of the line.

We now consider the same string operators on the trimer model. The diagonal
operator (’t Hooft line) is still well defined. Closed ’t Hooft loops count
the number of charges/anticharges in a closed region. Note that, if we
consider a diluted tRVB state, we only allow for the presence of monomers on
vertices, i.e., anticharges having $A_{v}=1$. In this case, a pair of monomers
represents a charge. In contrast with the ’t Hooft line, the off-diagonal
operator (Wilson line) is not well defined on the trimer model, as it can map
a valid trimer configuration to one that contains one of the vertices in Fig.
4(e). However, as shown in Fig. 5(b), on some states it is possible to define
an operator that acts similarly to a Wilson line, and creates a monomer at one
end of the line, and a pair or monomers at the other end. The charges and
anticharges obtained in this way are deconfined if the state has topological
order.
Knowing the operatorial form of Wilson and ’t Hooft lines provides (non-local)
order parameters [28, 29] that can be used to assert the presence of
$\mathbb{Z}_{3}$ topological order, as exploited in [30] for $\mathbb{Z}_{2}$
topological spin liquids in Rydberg atom arrays. Although in Sec. IV.1 we will
not undertake the calculation of these order parameters because of the limited
system sizes, we point out that they might be an effective probe for
experimental realizations of trimer models.

###  III.2 Tripartite trimer models and $\mathrm{U}(1)\times\mathrm{U}(1)$
lattice gauge theories

As shown in Fig. 2, the correlation length of the model diverges for
$\theta=\pi/2$, implying that the tRVB state containing only straight trimers
is gapless. We now explain this result, by proving that for straight trimers
the $\mathbb{Z}_{3}$ symmetry is enhanced to a U$(1)\times$U$(1)$ symmetry. We
define a partition of the square lattice in three sublattices A, B, and C as
in Fig. 6(a). It is easy to check that a straight trimer always covers one and
only one site per type. A similar scenario occurs for dimer models on
bipartite lattices: each dimer covers one site of each type, and the symmetry
is enhanced from $\mathbb{Z}_{2}$ to $\mathrm{U}(1)$. Here, we will show that
the emergent symmetry for straight trimers is
$\mathrm{U}(1)\times\mathrm{U}(1)$. To prove it, we define two electric
fields. The first electric field flows from the A to the B site of each trimer
[Fig. 6(b)], and the second electric field flows from the A to the C site
[Fig. 6(c)]. We then obtain two independent conservation laws, one for each
electric field: consider a region with $N_{A},N_{B},N_{C}$ vertices of the
three types. The net flux going out of the region is $N_{A}-N_{B}$ for the
first electric field and $N_{A}-N_{C}$ for the second electric field.
Therefore, the tRVB state has a local symmetry
$\mathrm{U}(1)\times\mathrm{U}(1)$ and must be gapless, as shown by Polyakov
[31].

\begin{overpic}[width=433.62pt]{u1u1.pdf} \put(0.0,38.0){(a)}
\put(35.0,38.0){(b)} \put(70.0,38.0){(c)} \end{overpic}

((a))

((b))

((c))

Figure 6: (a) Partition of the square lattice in the three sublattices A
(yellow), B (blue) and C (pink). A straight trimer always covers one site of
type A, one of type B and one of type C. (b) First U$(1)$ symmetry: electric
field lines go from the A site to the B site for each trimer. The net flux
going out of the region enclosed by the red line is $-1=N_{A}-N_{B}$. (c)
Second U$(1)$ symmetry: electric field lines go from the A site to the C site
for each trimer. The net flux going out of the region enclosed by the red line
is $-1=N_{A}-N_{C}$.

The argument reported above can be generalized to any trimer model on a two-
dimensional lattice. We posit that a trimer model is tripartite if three
sublattices can be defined, such that a trimer always covers one site for each
sublattice. Note that this definition depends both on the lattice and on the
class of trimers considered. If a trimer model is tripartite, the tRVB state
has a local U$(1)\times$U$(1)$ symmetry. In the absence of lattice symmetry
breaking, the emergence of this continuous local symmetry leads to a gapless
spin liquid state akin to RVB states in dimer models on bipartite lattices. We
can thus conclude that a necessary condition for having a gapped
$\mathbb{Z}_{3}$ spin liquid from a tRVB state is that the trimer model is
_not_ tripartite. We remark that this condition is not sufficient, as
demonstrated by the examples that we provide below.

Let us first consider the tRVB state on the honeycomb lattice [Fig. 7(a)].
This trimer model is not tripartite, so this state on the honeycomb lattice
can be a gapped state with $\mathbb{Z}_{3}$ topological order. The numerics in
Fig. 8 confirm that this is the case. In Fig. 8(b) we show that the
correlation length of the tRVB state on a finite cylinder converges to a
finite value as the circumference increases. In Fig. 8(b) we show that the
entanglement entropy of a half-infinite cylinder exhibits a $-\ln 3$
correction to its area law scaling. Finally, the blue hexagons in Fig. 8(c)
demonstrate that the logarithmic gap between the leading eigenvalues of the
neutral and charged sectors closes exponentially, pointing to the spontaneous
breaking of $\mathbb{Z}_{3}$ virtual symmetry.

((a))

((b))

((c))

((d))

((e))

Figure 7: (a) Trimer model on the honeycomb: the lattice is not tripartite.
(b, c) Two possible tripartitions of the trimer model on the kagome lattice.
We define trimers of type I (straight), II (bent, with angle $60^{\circ}$),
and III (with angle $120^{\circ}$). (b) The lattice is tripartite if no type
III trimers are included. (c) Similarly, the lattice is tripartite if no type
I trimers are included. (d) Trimer model on the triangular lattice: the
lattice is tripartite for triangular trimers. (e) The triangular lattice is
tripartite for trimers of type I and II, not tripartite for trimers of type
III.

Let us now turn our attention to the kagome lattice. In this case, various
types of trimers can be defined. If we consider the tripartition of the
lattice shown in Fig. 7(b), we note that some types of trimers (I and II)
cover sites of different types, while trimers of type III do not. Therefore,
we deduce that type III trimers are needed to have a $\mathbb{Z}_{3}$ spin
liquid phase. Similarly, from the tripartition in Fig. 7(c), we find that type
II trimers are also needed. This result is in agreement with Ref. [5], where
it was shown that a gapped tRVB state with topological order is found only
when all types of trimers are included. This lattice exhibits a counterexample
that shows how our “non-tripartibility” condition is not sufficient for gapped
$\mathbb{Z}_{3}$ topological order. In fact, in Ref. [5] it was proven that
the tRVB state with trimers of type I and III possesses a $\mathrm{U}(1)$
local symmetry that spoils $\mathbb{Z}_{3}$ topological order although the
trimer model cannot be tripartite. Trimer models on this geometry also provide
an example of $\mathrm{U}(1)\times\mathrm{U}(1)$ symmetric tRVB state that is
not gapless but has symmetry breaking: the tRVB state with trimers of type II
only is tripartite, but all trimer coverings break the two-fold rotation of
the lattice that maps upper into lower triangles.

((a))

((b))

((c))

((d))

Figure 8: (a) Scaling of the entanglement entropy of the tRVB state on the
honeycomb lattice as a function of the number of tensors $N$ around the
cylinder. We extract a topological correction $\gamma\simeq\ln 3$. (b)
Correlation length of the tRVB state on the honeycomb lattice. (c) Logarithmic
gap between the leading eigenvalues of the $\mathcal{Q}=0$ and
$\mathcal{Q}=\pm 1$ sectors for the tRVB state on the honeycomb (blue
hexagons) and triangular (orange triangles). On the triangular lattice, only
triangular trimers are considered. Transfer matrix eigenvalues are labeled as
$\lambda^{\mathcal{Q}}_{n}$ where $\mathcal{Q}$ is the $\mathbb{Z}_{3}$
symmetry sector and $n=0,1\dots$ is the position in the spectrum starting from
the largest in magnitude. (d) Correlation length of the tRVB on the triangular
lattice. Similar to dimer models on bipartite lattices, it grows linearly with
the number of tensors $N$ along the circumference. The TN description of these
models can be found in App. A.

Finally, let us consider the triangular lattice. As can be inferred from Fig.
7(d), the tRVB state of triangular trimers 444The triangular trimers
considered here do not consist of two edges sharing a common vertex, but
rather of triangular plaquettes with hard constraints. Nevertheless, the
discussion on the general conditions for $\mathbb{Z}_{3}$ or
U$(1)\times$U$(1)$ gauge symmetry is applicable here in the same way. (grey
triangles in the figure) is $\mathrm{U}(1)\times\mathrm{U}(1)$ symmetric
because the model is tripartite: this finding agrees with Ref. [33], where a
U$(1)\times$U$(1)$ conservation law (for “left-” and “right-movers”) was found
in the classical configurations. From numerical diagonalization of the
transfer matrix on finite-size cylinders, we deduce that this tRVB state is
indeed gapless, as demonstrated in Fig. 8(d), where we show that for the sizes
accessible with our numerics, the correlation length scales linearly with the
circumference of the cylinder. We refer to Sec. A.3 for the explicit TN
representation of this tRVB state. Using the same definitions of trimers as
for the kagome lattice, we have that trimers of type I (straight) on the
triangular lattice are also tripartite and expected to spoil gapped
topological order, while trimers of type III do not respect the tripartition
[Fig. 7(e)]. This implies that tRVB states on this lattice can have
topological order only if these trimers are included. We let a complete
analysis of this family of tRVB states for future work.

Figure 9: (a) Periodic clusters employed for the exact diagonalization of the
Hamiltonian in Eq. 14. $36$, $48$, and $60$ stand for the number of edges
inside the cluster. The turquoise regions are the subsystems used to compute
the entanglement entropy. We measure their perimeter $L$ in units of an edge
of the square lattice.

##  IV Diluted trimer models and Rydberg atoms

In the previous sections we have shown that RVB states of trimers can be
gapped and have topological character. When these conditions are met they are
good candidates for representing a stable phase with $\mathbb{Z}_{3}$
topological order. It is thus natural to ask if simple Hamiltonians exist that
have tRVB-like phases at zero temperature. As trimer states are TN states with
finite bond dimension, they are exact ground states of local Hamiltonians with
finite range. However, it is known that such Hamiltonians can be rather
complex and include fairly unphysical operators [26, 27, 34]. In particular,
parent Hamiltonians of tRVB states on certain lattices are discussed in Refs.
[3, 5]. Here, instead, we introduce a simple trimer model on the square
lattice and study its ground state phase diagram via exact methods, providing
evidence of a tRVB-like phase with $\mathbb{Z}_{3}$ topological order.
Moreover, we show that a similar model can be implemented in Rydberg atom
arrays and that hallmarks of $\mathbb{Z}_{3}$ topological order can be
observed employing semi-adiabatic dynamical preparation schemes.

((a))

((b))

((c))

((d))

((e))

Figure 10: (a) Overlap between the ground state of the Hamiltonian Eq. 14 and
the tRVB state (solid line), and the diluted tRVB state Eq. 9 optimized over
$z$ (dashed line). The shaded red region set approximate boundaries for the
topologically ordered phase, where the fidelity with the tRVB state is
maximized. (b) Ground state fidelity susceptibility per link
$\mathcal{F}=(1-|\braket{\mathrm{GS}(\lambda)}{\mathrm{GS}(\lambda+d\lambda)}|)/Nd\lambda^{2}$,
with $\lambda=\Omega/\Delta$. Two peaks appear for the larger clusters,
pointing at the presence of an intermediate phase for
$0.95\lesssim\Delta/\Omega\lesssim 1.15$. (c) Optimal value of $z$ that
maximizes the overlap between the ground state and the diluted tRVB state Eq.
9. $|z|\lesssim 0.4$ in the intermediate phase, a value that lies deep in the
topological phase in the state phase diagram plotted in Fig. 3. (d) Ground
state topological entanglement entropy computed by subtracting the entropies
of the square-shaped regions in Fig. 9 with $L=4$ and $L=8$. (e) Scaling of
the entanglement entropy of the ground state for $L=4,6,8,10$ and
$\Omega/\Delta=1$.

###  IV.1 An effective trimer models on the square lattice

We consider the Hilbert space spanned by all diluted trimer configurations of
bent trimers on the square lattice, i.e. with at most one trimer per vertex of
the square lattice, and take the following model Hamiltonian

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(14)

$R_{\frac{\pi}{2}}$ denotes the terms that can be obtained by 90-degree
rotations from those given in Eq. (14). The fist term coherently creates and
destroys bent trimers (subject to the hard trimer constraint), whereas the
second, diagonal term acts like a chemical potential for trimers. The ratio
$\Delta/\Omega$ controls the density of trimers in the ground state. For large
and negative $\Delta/\Omega$ the ground state is trivial and adiabatically
connected to the vacuum. For $\Delta/\Omega=+\infty$ the classical ground
space is exponentially degenerate and consists of all maximally-packed trimer
coverings, corresponding to a link density $\braket{n}=1/3$. By treating
pertubatively the off-diagonal diagonal term, it is easy to see that the first
non trivial process in this subspace occurs at fourth order and produces
resonances between pairs of trimer coverings differing only on two nearby
squares. Therefore, at large $\Delta/\Omega$ a valence bond solid (VBS) ground
state is expected to emerge, with a maximal density of resonating
“plaquettes”, i.e., resonating pairs of nearby squares 555We note that the
number of plaquette coverings is not finite when the system is infinite, and a
quantum order by disorder scenario is likely to occur.. At finite
$\Delta/\Omega$, quantum fluctuations act in two ways: they create defects in
the trimer coverings by lowering the density and build coherent superpositions
of high density components. As we showed in Sec. II, topological order can
survive at finite dilution, implying that a diluted tRVB state might also
arise from this setup.

To understand the character of the ground state at intermediate
$\Delta/\Omega$ we performed exact diagonalization calculations on periodic
clusters of up to $60$ edges of the square lattice [Fig. 9]. In Fig. 10(a) we
plot the overlap between the ground state and the pure tRVB state (solid
line), and the ground state fidelity with the diluted tRVB state Eq. 9 for
$\theta=0$ optimized over $z$ (dashed line). The optimal values of $z$ as a
function of $\Delta/\Omega$ are shown in Fig. 10(c). Remarkably the overlap
displays a maximum at $\Delta/\Omega\simeq 1$, pointing at the presence of an
intermediate tRVB-like phase. The maximum fidelity is greatly improved when
optimized with the diluted tRVB state. We note that the optimal value of $z$
near the maximum is perfectly consistent with the topologically ordered phase
in the state phase diagram in Fig. 3. The occurrence of an intermediate phase
is also witnessed by the presence of two peaks in the ground state fidelity
susceptibility per link
$\mathcal{F}=(1-|\braket{\mathrm{GS}(\lambda)}{\mathrm{GS}(\lambda+d\lambda)}|)/Nd\lambda^{2}$,
where $\lambda=\Omega/\Delta$ 666The parameter $\lambda$ is chosen to make
more distinguishable the two peaks in $\mathcal{F}$. for the $48$\- and
$60$-links clusters, as depicted in the Fig. 10(b). To confirm the nature of
the intermediate phase in an unbiased way, in Fig. 10(d) we show the
topological entanglement entropy extracted from $\gamma\simeq S_{2L}-2S_{L}$
as functions of $\Delta/\Omega$. Here $L$ is the length of the contour of the
subsystem, in units of one edge of the square lattice, and the subsystems
employed for the computation are depicted in Fig. 9. The value of $\gamma$
obtained near the tRVB fidelity maximum is remarkably close to the value $\ln
3$, hinting at the emergence of $\mathbb{Z}_{3}$ topological order.

###  IV.2 The trimer constraint with Rydberg atoms

We now turn to a discussion of potential realizations of trimer models and
tRVB states with experiments based on Rydberg atom arrays. In these systems,
neutral atoms are individually trapped and arranged in a desired lattice
configuration using optical tweezers [37, 38]. Spin models can then be
realized by manipulating the internal degrees of freedom of each atom with an
external laser field [39, 40, 41, 42, 43, 44]. Specifically, we consider a
situation where a laser induces a coherent coupling from the atomic ground
state $\ket{g}$ to a highly excited Rydberg state $\Ket{r}$. The frequency
mismatch between the laser frequency and the transition frequency between
those two states, i.e. the laser detuning, is denoted by $\Delta$. The
coupling strength for this transition, i.e. the Rabi frequency, is denoted by
$\Omega$, and is proportional to the laser amplitude. Importantly, two atoms
that are both in the Rydberg state interact via a Van der Waals process, whose
strength decays with the sixth power of the atomic separation. As a result,
the Hamiltonian governing the dynamics of this system is given by [45]

$H_{\mathrm{Ryd}}=\frac{\Omega}{2}\sum_{i}\sigma_{i}^{x}-\Delta\sum_{i}n_{i}+C\sum_{i>j}\frac{n_{i}n_{j}}{|\vec{x}_{i}-\vec{x}_{j}|^{6}}\quad,$
(15)

where, $\vec{x}_{i}$ is the position of atom $i$, and we defined
$\sigma_{i}^{x}=\Ket{g}_{i}\\!\Bra{r}+\Ket{r}_{i}\\!\Bra{g}$ and
$n_{i}=\Ket{r}_{i}\\!\Bra{r}$. The parameter $C$ depends on the Rydberg state.
The interplay between the laser parameters and the geometry of the atom
arrangement gives rise to a variety of phenomena [46, 47, 48, 49, 50, 51].
Most of them are based on the Rydberg blockade effect, that prohibits the
simultaneous excitation of two atoms located at a distance
$r<R_{b}=(V/\Omega)^{1/6}$. Below we show that this effect can be used to
implement trimer constraints in Rydberg atom arrays. For example, it is easy
to prove that the hard trimer constraint for triangular trimers on the
triangular lattice in Fig. 7(d) is equivalent to a Rydberg blockade constraint
on a honeycomb lattice: the atoms sit on the centers of the original
triangular lattice and a Rydberg excitation represents a triangular plaquette;
the blockade radius $R_{b}$ is chosen such that two atoms cannot be
simultaneously excited if and only if they belong to the same hexagon. We now
show that the Rydberg blockade effect also allows to realize a bent trimer
model on the square lattice very similar to the one outlined above. A sketch
of the implementation is depicted in Fig. 11. Rydberg atoms are placed on the
corners of a square lattice such that an excited atom is mapped to a bent
trimer. The basic idea is to exploit the blockade radius to mimic the hard
trimer constraint. However, while the latter is anisotropic, the blockade
effect is not, as long as the Rydberg state is rotational invariant.
Nevertheless, we can avoid the use of anisotropic Rydberg states by dividing
the atoms into two groups, according to the sublattice of the square to which
they are closer. The two groups are then arranged onto two planes at a
distance $h$. Consequently, atoms between different planes will be blockaded
if their planar distance is less than $\sqrt{R_{b}^{2}-h^{2}}$, where $R_{b}$
is the blockade radius. By properly choosing $h$ and the atoms positions
inside the plaquettes of the square lattice, it is possible to realize a
trimer constraint, as demonstrated by Fig. 11(b). This trimer constraint is
such that some trimer configurations are locally prohibited. The latter are
trimer coverings that include two trimers with the same orientation that are
“wedged” diagonally as in Fig. 11(b).

((a))

((b))

Figure 11: (a) Mapping between Rydberg atoms and bent trimers on the square
lattice. Atoms are placed at each corner of the square lattice such that an
excited atom is mapped to a bent trimer on that corner. (b) Atoms are split
into two groups (green and red) depending on the sublattice of the square
lattice bipartition to which the corresponding corner belongs. The groups are
arranged onto two planes at distance $h$ such that the 2D blockade radius
between atoms of different colors is $\sqrt{R_{b}^{2}-h^{2}}$, where $R_{b}$
is the 3D blockade radius. Tuning $h$ and the distance of the atoms from the
vertexes allows realizing a constraint equivalent to the trimer constraint
where the “wedged" trimer configurations on the bottom left are blockaded (and
its 90-degree rotations).

Before addressing the Rydberg model, we study the effect of removing these
coverings from the fully-packed tRVB state. In Figs. 12(c) and 12(d) we plot
the lowest logarithmic gaps in the spectrum of the cylinder transfer matrix of
the corresponding TN state as functions of the cylinder circumference $L$. The
TN representation is outlined in Sec. A.1. Despite a level crossing occurs at
finite $L$, the gap between the neutral and charged sectors eventually closes
exponentially (green circles), whereas the neutral gap (blue circles) appears
to be increasing for the available $L$s. From infinite-size calculations we
can infer that this gap converges to $\simeq 0.6$, yielding a correlation
length $\xi\simeq 1.7$. We note that this value is larger than the correlation
length of the unrestricted trimer state $\xi\simeq 1.1$ [cf. Fig. 2(a)]. This
fact is expected, as removing these configurations pushes away the tRVB state
from the $\mathbb{Z}_{3}$ toric code fixed point, for which $\xi=0$. These
results demonstrate that $\mathbb{Z}_{3}$ topological order is preserved.
Although we did not study TN perturbations that lower the density of trimers,
we expect a diluted version of this tRVB state to host a topologically ordered
phase.

We now focus on the Rydberg model arising from the implementation explained
above. For simplicity, we neglect interactions beyond the blockade, so that
the effective Hamiltonian is the same as Eq. 14, with the caveat that all the
states containing wedged trimers as in Fig. 11(b) are not included in the
Hilbert space of diluted trimer coverings. The exact diagonalization of this
restricted trimer model displays no evidence of an intermediate topological
phase in the ground state, rather a single phase transition between a
disordered phases and a plaquette phase can be identified. Therefore, we
conclude that if such a phase exists it is extremely narrow. In fact, the
topological entanglement entropy extracted from the finite size cluster $48$
in Fig. 9 exhibits a peak approaching $\gamma=\ln 3$ that is much sharper than
in the unrestricted model, as we show in Fig. 12(b). the black lines are the
ground state curves for $\gamma$ in the restricted (solid line) and
unrestricted (dashed line) trimer models. However, below we provide numerical
evidence that this witness of $\mathbb{Z}_{3}$ topological order is stabilized
by a dynamical preparation protocol regularly used in experiments [48].

((a))

((b))

((c))

((d))

Figure 12: (a) The protocol employed for the semi-adiabatic dynamical
preparation. The vacuum $\Ket{0}$ is evolved with a time-dependent Hamiltonian
with $\Delta(t)$ and $\Omega(t)$ as depicted in the figure. First, $\Omega$ is
switched on from $0$ to $1$ at constant $\Delta=\Delta_{0}$ with a smoothed
linear ramp. $\Delta$ is then increased linearly from $\Delta_{0}=-1.5$ to
$\Delta_{1}=+3$ of constant $\Omega$. The total sweep duration is $T$. (b)
Topological entanglement entropy of the dynamically prepared state, for
different preparation times. The black line corresponds to the ground state of
the Hamiltonian Eq. 14 ($T=\infty$). Solid and dashed lines are obtained in
the effective Rydberg model where the “wedged” trimer configuration of Fig.
11(b) are not included in the Hilbert space and the unrestricted diluted
trimer model, respectively. Results are obtained on the periodic cluster $48$
of Fig. 9. (c) $E=-\log\lambda$, where $\lambda$ are the eigenvalues of the
cylinder transfer matrix of the tensor network representation of the
restricted tRVB state of bent trimers on the square lattice. $N$ is the number
of tensors along the circumference, $\mathcal{Q}$ is the $\mathbb{Z}_{3}$
virtual charge ($\mathcal{Q}=\pm 1$ sectors are exactly degenerate), $n$ is
the eigenvalue index in the sector with charge $\mathcal{Q}$. (d) Exponential
scaling of the gap between the smallest $E$s in the neutral and charged
sectors, signaling spontaneous symmetry breaking of the $\mathbb{Z}_{3}$
virtual symmetry of the tensor.

The initial state is the vacuum, subsequently evolved with the time-dependent
Hamiltonian $H(t)=H(\Omega(t),\Delta(t))$. To prepare the ground state of
$H(t)$ the variation of the time-dependent couplings has to be perfectly
adiabatic. In real experiments, this is very hard in practice, due to limited
coherence time. Thus it is often preferable to consider non-adiabatic state
preparation schemes. In fact, as demonstrated in Ref. [48, 22], non-adiabatic
effects can even enhance topological order in the prepared state w.r.t. the
ground state. In the following, we show that a similar result is observed
here. Specifically, we study the dynamical preparation process depicted in
Fig. 12(a). The vacuum state $\Ket{0}$ is evolved with the time-dependent
Hamiltonian $H(\Omega(t),\Delta(t))$, where $\Delta(0)=-2.5$, $\Omega(t)=0$,
such that $\Ket{0}$ is the ground state at $t=0$. A first (smoothed) linear
ramp turns on the effective Rabi frequency until the final value $\Omega=1$ is
reached. The latter sets our units of energy and time. A second ramp is used
to drive the detuning from $\Delta(0)=-1.5$ to $\Delta(T)=3$, where $T$ is the
total sweep time, and the final value of $\Delta$ is chosen to be well beyond
the peaks in tRVB overlap and topological entanglement entropy of the ground
state. The slopes of the two ramps decrease with increasing $T$ and are fixed
by requiring that the switching on of $\Omega$ ($\Delta$) takes $T/3$
($2T/3$). In both the restricted and unrestricted models a phase transition is
crossed during the second ramp.
In Fig. 12 we plot the topological entanglement entropy of the state during
the preparation sweep, for different total sweep times $T$, for the $48$
cluster in Fig. 9, that corresponds to $96$ atoms in the mapping of Fig.
11(a). The result indicates that topological properties are stabilized in the
prepared state when the preparation is not adiabatic, i.e. for short and
intermediate $T$. Remarkably, the peaks in the topological entropy correction
disappear in this regime, and the latter witnesses a topological state when
$\Delta/\Omega\gtrsim 1$. For the largest $T$s the ground state curve is
recovered ($T=\infty$). We note that this phenomenon occurs in both the
restricted and unrestricted models [cf. dash and solid lines in Fig. 12(b)].
We remark that it might be possible to engineer other implementations of the
Hamiltonian in Eq. 14 that do not require a restriction of the diluted trimer
Hilbert space.

##  V Conclusions

We showed that maximally-packed trimer states can be simple representatives of
quantum spin liquids with $\mathbb{Z}_{3}$ topological order. By mapping
trimer configurations into the Hilbert space of a lattice gauge theory, we
identified a condition on the lattice geometry and trimer model that leads to
the emergence of a $\mathrm{U}(1)\times\mathrm{U}(1)$ and a tRVB state with
infinite correlation length. We verified this condition by performing
numerical checks on several trimer models with TN methods. We demonstrated
that when tRVB states are gapped, $\mathbb{Z}_{3}$ topological order is stable
against fluctuations in the number of trimers. We did so by studying a TN
perturbation that represents a diluted tRVB state on the square lattice and
showing that it hosts a wide topologically ordered phase in the state phase
diagram. Finally, we considered a simple model Hamiltonian on the square
lattice that exhibits signatures of a tRVB-like phase, where the ground state
is well approximated by the $\mathbb{Z}_{3}$ topologically ordered diluted TN
perturbation previously studied. We provided an implementation of a very
similar model by exploiting the blockade effect in Rydberg atom arrays, and
show that hallmarks of a $\mathbb{Z}_{3}$ quantum spin liquid can be observed
in non-adiabatic dynamical preparation schemes.

Our findings open new future directions for the quantum simulation of
topological phases of matter. The necessary condition for having
$\mathbb{Z}_{3}$ topological order that we formulated depends solely on the
geometry of the model and can therefore guide the search for quantum spin
liquids in various experimental implementations, including—but not limited
to—Rydberg atom arrays. In this respect, it would be interesting to study more
extensively the realization of tRVB states both as ground states of realistic
Hamiltonians and as dynamically-prepared non-equilibrium states. In addition,
our approach can be naturally extended from trimer- to polymer-RVB states,
which can support the emergence of $\mathbb{Z}_{n}$ topological order. A
systematic study of such states can similarly be performed efficiently with
tensor network methods and is left for future work. Finally, an interesting
direction is the related problem of quantum spin liquid phases in SU(3) and
SU(N) symmetric models. In certain models, trimers (polymers) can be
interpreted as simplified versions of SU(3) (SU(N)) spin singlets; it remains
an open question to what extent this interpretation can be used to infer the
properties of RVB states of singlets.

###### Acknowledgements.

We acknowledge inspiring discussions with Norbert Schuch, Juraj Hasik, and
Andrej Gendiar. TN calculations were performed by resolving symmetries at the
tensor level with the TensorKit.jl package [52]. This work has been supported
by the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme through the ERC-CoG SEQUAM (No. 863476), the
ERC-CoG QSIMCORR (No. 771891), and ERC-StG QARA (No. 101041435), as well as
the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy (EXC-2111 – 390814868).

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##  Appendix A Details on the tensor-network representations

###  A.1 tRVB model on the square lattice

####  A.1.1 Symmetries

The tensor defined in Eq. 4 enjoys a reflection symmetry along the diagonal,
which can be exploited in the CTMRG algorithm. The corresponding transfer
operator is not self-adjoint, which requires one to compute both fixed points
for the VUMPS algorithm. However, the fixed point in one direction can be
readily converted into the the fixed point into the other direction, since

$\vbox{\hbox{\scalebox{0.5}{\includegraphics{permutation}}}}\,,$ (16)

where $P$ is the operator that permutes the $\pm 1$ indices.

####  A.1.2 Diluted tRVB PEPS

The PEPS representation for the diluted tRVB in Eq. 9 is constructed by
introducing two species of trimers; one that appear on the physical layer and
ones that do not. Graphically, we have the following construction:

$\ket{\Phi}=\vbox{\hbox{\scalebox{0.5}{\includegraphics{peps_projected}}}}.$
(17)

We label the indices of the tensor $\tilde{T}$ with
$\\{0,+1,-1,+\tilde{1},-\tilde{1}\\}$. We now have two pairs of uncoupled
charges, each of which have the similar diagrams as Eq. 4. In order to
preserve the correspondence with the partition function in Eq. 3, we must also
take the square root of each weight. The black dot on each link represents a
projector, which signals the presence of one type of trimer on the physical
layer, and gives a fugacity contribution $z$ to the second type

$\vbox{\hbox{\scalebox{0.5}{\includegraphics{projector}}}}\,.$ (18)

The corresponding PEPS tensor, formed by contracting one $\tilde{T}$ and two
projectors, has bond dimension $D=5$. It should be noted that the double-layer
tensor—coming from the contraction of the norm $\braket{\Phi}{\Phi}$—can be
reduced to dimension 9 (instead of the naïvely expected $5^{2}$). We observe
this by performing Gaussian elimination on the tensor, which is exact up to
numerical precision.

####  A.1.3 Restricted tRVB model

Figure 13: Allowed diagrams at each vertex for the restricted tRVB model, up
to rotations of $\pi/2$.

The mapping to Rydberg atoms in Sec. IV.1 suggests the ansatz state in which
we restrict the trimer configurations to not have any “wedged” configurations,
as shown in Fig. 11. We start from the arrow representation in Fig. 4(d), but
we introduce new color labels

$\includegraphics[width=140.92517pt]{restricted_indices.pdf}\,.$ (19)

Referring to Fig. 13, we start by labelling the two outgoing arrows of a bent
trimer with different colors (blue and orange). To enforce the additional
constraint, we introduce a dashed arrow which continues anticlockwise from the
blue arrow. Notice that the arrow does not convey any charge information, as
all dashed lines correspond to a $\mathbb{Z}_{3}$-charge 0. A vertex with the
middle of the trimer can not couple to it directly since it requires two gray
lines. The other diagrams then account for the possible ways of closing the
dashed lines. Accounting for rotations, in total there are 28 distinct
diagrams.

The construction of the tensor network from Fig. 13 is straightforward, and is
similar to Eq. 4. We must however remember that the mapping from the arrow
representation to the index of the tensor is different if a leg of the tensor
is ingoing or outgoing. Each leg of the double-layer tensor is then eight-
dimensional.

###  A.2 tRVB model on the honeycomb lattice

On the honeycomb lattice there is only one type of trimer. On each vertex of
the lattice we can place the tensor

$\left.\vbox{\hbox{\scalebox{0.5}{\includegraphics{honeycomb_tensor}}}}\
\right\\}\mbox{$\forall$ rotations}$ (20)

to obtain the corresponding partition function. To convert the problem into a
TN on the square lattice, we define the tensor

$\vbox{\hbox{\scalebox{0.5}{\includegraphics{honeycomb_contraction}}}}\,.$
(21)

Similarly to the case of square lattice in Eq. 5, this tensor obeys a similar
transformation:
$(\sigma^{\dagger}\otimes\sigma^{\dagger}\otimes\sigma\otimes\sigma)T=\omega^{2}T$.

###  A.3 tRVB model on the triangular lattice

On the triangular lattice an efficient TN representation can be found in the
dual-lattice picture, where a site is defined on each triangular face,
similarly to how one would propose a Rydberg implementation. In this picture,
only one site around each original vertex of the triangular lattice can be
occupied. Correspondingly, in the TN picture we define a $\delta$-tensor on
each face, representing the site

$\vbox{\hbox{\scalebox{0.5}{\includegraphics{triangular_delta}}}}\,,$ (22)

which is connected to three rank-6 constraint tensors

$\left.\vbox{\hbox{\scalebox{0.5}{\includegraphics{triangular_constraint}}}}\
\right\\}\mbox{$\forall$ rotations}\,.$ (23)

We can then bring the problem back to a TN contraction on the square lattice
by first defining a decomposition of the constraint tensor

$\vbox{\hbox{\scalebox{0.5}{\includegraphics{triangular_decomposition}}}}\,,$
(24)

and performing the contraction

$\vbox{\hbox{\scalebox{0.5}{\includegraphics{triangular_contraction}}}}\,.$
(25)