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arXiv:2309.00482 · SciPost Physics

Quantum Monte Carlo simulations in the restricted Hilbert space of Rydberg atom arrays

Pranay Patil

Published Jan 27, 2026• 1 citation
arXivHTMLPDF

Abstract

Rydberg atom arrays have emerged as a powerful platform to simulate a number of exotic quantum ground states and phase transitions. To verify these capabilities numerically, we develop a versatile quantum Monte Carlo sampling technique which operates in the reduced Hilbert space generated by enforcing the constraint of a Rydberg blockade. We use the framework of stochastic series expansion and show that in the restricted space, the configuration space of operator strings can be understood as a hard rod gas in d+1 d + 1 dimensions. We use this mapping to develop cluster algorithms which can be visualized as various non-local movements of rods. We study the efficiency of each of our updates individually and collectively. To elucidate the utility of the algorithm, we show that it can efficiently generate the phase diagram of a Rydberg atom array, to temperatures much smaller than all energy scales involved, on a kagomé link lattice. This is of broad interest as the presence of a Z_2 Z 2 spin liquid has been hypothesized recently.

DOI
10.21468/scipostphys.20.1.022

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Rydberg atom arrays have emerged as a powerful platform to simulate a number of exotic quantum ground states and phase transitions. To verify these capabilities numerically, we develop a versatile quantum Monte Carlo sampling technique which operates in the reduced Hilbert space generated by enforcing the constraint of a Rydberg blockade. We use the framework of stochastic series expansion and show that in the restricted space, the configuration space of operator strings can be understood as a hard rod gas in d+1 d + 1 dimensions. We use this mapping to develop cluster algorithms which can be visualized as various non-local movements of rods. We study the efficiency of each of our updates individually and collectively. To elucidate the utility of the algorithm, we show that it can efficiently generate the phase diagram of a Rydberg atom array, to temperatures much smaller than all energy scales involved, on a kagomé link lattice. This is of broad interest as the presence of a Z_2 Z 2 spin liquid has been hypothesized recently.
Arrays of Rydberg atoms are a powerful platform to realize strongly-interacting quantum many-body systems. A common Rydberg Hamiltonian is free of the sign problem, meaning that its equilibrium properties are amenable to efficient simulation by quantum Monte Carlo (QMC). In this paper, we develop a Stochastic Series Expansion QMC algorithm for Rydberg atoms interacting on arbitrary lattices. We describe a cluster update that allows for the efficient sampling and calculation of physical observables for typical experimental parameters, and show that the algorithm can reproduce experimental results on large Rydberg arrays in one and two dimensions.
The quantum dimer and loop models attract great attention, partially because the fundamental importance in the phases and phase transitions emerging in these prototypical constrained systems, and partially due to their intimate relevance toward the on-going experiments on Rydberg atom arrays in which the blockade mechanism naturally enforces the local constraint. Here we show, by means of the sweeping cluster quantum Monte Carlo method, the complete ground state phase diagram of the fully packed quantum loop model on the square lattice. We find between the lattice nematic (LN) phase with strong dimer attraction and the staggered phase (SP) with strong dimer repulsion, there emerges a resonating plaquette (RP) phase with off-diagonal translational symmetry breaking. Such a quantum phase is separated from the LN via a first order transition and from the SP by the famous Rokhsar-Kivelson point. Our renormalization group analysis reveals the different flow directions, fully consistent with the order parameter histogram in Monte Carlo simulations. The realization and implication of our phase diagram in Rydberg experiments are proposed.
This thesis presents computational studies of three different models of many-body physics with direct or indirect constraints. The presence of constraints in complex many-body systems calls for non-trivial numerical algorithms to study them. The first two models which have a direct form of local constraint are the Rokhsar-Kivelson Quantum Dimer model (QDM) and a classical statistical mechanics model of non-intersecting loops with attractive interactions, both on the square lattice. The investigations of such constrained models have found a recent resurgence with their direct realizations on Rydberg atom arrays quantum simulators. The study of the classical model uses a Monte Carlo directed loop algorithm while the QDM calls for a novel Quantum Monte Carlo scheme based on the framework of Stochastic Series Expansions called the Sweeping Cluster Algorithm (SCA). We present a modification of the SCA in order to render simulations fully ergodic at finite temperature. For both models, our numerical studies show the existence of a critical phase separated by a phase transition at finite temperature to an ordered phase of dimers or loops which spontaneously breaks certain lattice symmetries. We show that for the case where the interaction is attractive this phase transition is of Kosterlitz-Thouless type and can be understood by constructing a coarse-grained field theory through a height mapping. The finite temperature phase diagram of the QDM presents an unusual re-entrance behavior in the critical phase. The final part of this thesis deals with the role of non-abelian symmetries in the thermalization process of quantum many-body systems. We study the high-energy eigenstates of a SU(3) symmetric spin chain in presence of disorder. While the model does not directly have constraints, we perform exact diagonalization in a constrained basis of Young tableau making use of the full SU(3) symmetry of the model. By looking at the commonly used probes for thermalization (spectral statistics, distribution of local observables and scaling of entanglement entropy), we show that the model exhibits a non-ergodic regime over a broad range of system sizes for strong enough disorder, contrasting with the rapid thermalization observed at weak disorder.
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 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.
Quantum spin liquids are exotic phases of matter whose low-energy physics is described as the deconfined phase of an emergent gauge theory. With recent theory proposals and an experiment showing preliminary signs of Z 2 topological order [G. Semeghini , ], Rydberg atom arrays have emerged as a promising platform to realize a quantum spin liquid. In this work, we propose a way to realize a U(1) quantum spin liquid in three spatial dimensions, described by the deconfined phase of U(1) gauge theory in a pyrochlore lattice Rydberg atom array. We study the ground state phase diagram of the proposed Rydberg system as a function of experimentally relevant parameters. Within our calculation, we find that by tuning the Rabi frequency, one can access both the confinement-deconfinement transition driven by a proliferation of “magnetic” monopoles and the Higgs transition driven by a proliferation of “electric” charges of the emergent gauge theory. We suggest experimental probes for distinguishing the deconfined phase from ordered phases. This work serves as a proposal to access a confinement-deconfinement transition in three spatial dimensions on a Rydberg-based quantum simulator.
Leveraging the rapid development of quantum simulators, the intriguing phenomena of quantum string are observed across various quantum simulation platforms. However, the complex interplay between the quantum strings cannot be well analyzed due to the limited system size in real quantum simulators. Here, with the help of a newly developed quantum Monte Carlo method, we can simulate a larger-scale Kagome Rydberg atom array, providing an ideal playground for studying quantum strings. By introducing a novel edge pinning method, the ends of a quantum string can be attached to edges so that the flexible manipulation of the quantum string becomes possible. Due to the geometric constraint, the quantum strings are unidirectional, which strongly complicates their interplay. To quantitatively describe the quantum string, we built a one-dimensional effective model. With both analytic and numerical methods, rich physics can be found, including ``geometric breaking", heart-like superposition state of quantum strings, and the attractive inter-string interactions. This work can benefit the comprehension of quantum strings and may also shed light on the simulation of high-energy physics.