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arXiv:2211.01381

Quantum Spin Puddles and Lakes: NISQ-Era Spin Liquids from Non-Equilibrium Dynamics

Sahay, Rahul, Ashvin Vishwanath, Ruben Verresen

Published Nov 2, 2022• 0 citations
arXivHTMLPDF

Abstract

While many-body systems can host long-ranged entangled quantum spin liquids (QSLs), the ingredients for realizing these as ground states can be prohibitively difficult. In many circumstances, one requires (i) a constrained Hilbert space and (ii) an extensive quantum superposition. The paradigmatic example is the toric code, or $\mathbb{Z}_2$ spin liquid, which is a superposition of closed loop states. We show how non-equilibrium Hamiltonian dynamics can provide a streamlined route toward creating such QSLs. Rather than cooling into the ground state of a Hamiltonian, we show how a simple parameter sweep can dynamically project a family of initial product states into the constrained space, giving rise to a QSL. For the toric code, this is achieved in systems with a separation in energy scales between the $e$- and $m$-anyons, where one can sweep in a way that is adiabatic (sudden) with respect to the former (latter). Although this separation of scales does not extend to the thermodynamic limit, we analytically and numerically show that this method efficiently prepares a spin liquid in finite-sized regions, which we brand ``quantum spin lakes.'' This mechanism elucidates recent experimental and numerical observations of the dynamical state preparation of the ruby lattice spin liquid in Rydberg atom arrays. In fact, the slow dynamics of $m$-anyons suggest that we can capture spin lake preparation by simulating the dynamics on tree lattices, which we confirm with tensor network simulations. Finally, we use this mechanism to propose new experiments, e.g., for preparing a finite-sized $U(1)$ spin liquid as a honeycomb Rokhsar-Kivelson dimer model using Rydberg atoms -- which is remarkable given its equilibrium counterpart is unstable in $2 + 1$D. Our work opens up a new avenue in the study of non-equilibrium physics, as well as the exploration of exotic states of finite extent in NISQ devices.

DOI
10.48550/arxiv.2211.01381

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Quantum spin liquids are elusive long-range entangled states. Motivated by experiments in Rydberg quantum simulators, recent excitement has centered on the possibility of dynamically preparing a state with quantum spin liquid correlation even when the ground state phase diagram does not exhibit such a topological phase. Understanding the microscopic nature of such quantum spin "lake" states and their relationship to equilibrium spin liquid order remains an essential question. Here, we extend the use of approximately symmetric neural quantum states for real-time evolution and directly simulate the dynamical preparation in systems of up to N=384N=384N=384 atoms. We analyze a variety of spin liquid diagnostics as a function of the preparation protocol and optimize the extent of the quantum spin lake thus obtained. In the optimal case, the prepared state shows spin-liquid properties extending over half the system size, with a topological entanglement entropy plateauing close to γ=ln⁡2γ= \ln 2γ=ln2. We extract two physical length scales λλλ and ξξξ which constrain the extent of the quantum spin lake ℓ\ellℓ from above and below.
The exploration of quantum spin liquids (QSLs) has been guided by different approaches including the resonating valence bond (RVB) picture, deconfined lattice gauge theories, and the Kitaev model.More recently, a spin-liquid ground state was numerically established on the ruby lattice, inspired by the Rydberg blockade mechanism.Here, we unify these varied approaches in a single parent Hamiltonian, in which local fluctuations of anyons stabilize deconfinement.The parent Hamiltonian is defined on kagomé triangleseach hosting four RVB-like states-and includes only Ising interactions and single-site transverse fields.In the weak-field limit, the ruby spin liquid and exactly soluble kagomé dimer models are recovered, while the strong-field limit reduces to the Kitaev honeycomb model, thereby unifying three seemingly different approaches to QSLs.We similarly obtain the chiral Yao-Kivelson model, the honeycomb toric code, and a new spin-1 quadrupolar Kitaev model.The last is shown to be in a QSL phase by a nonlocal mapping to the kagomé Ising antiferromagnet.We demonstrate various applications of our framework, including (a) an adiabatic deformation of the ruby lattice model to the exactly soluble kagomé dimer model, conclusively establishing the QSL phase in the former and (b) demystifying the dynamical protocol for measuring offdiagonal strings in the Rydberg implementation of the ruby lattice spin liquid.More generally, we find an intimate connection between Kitaev couplings and the repulsive interactions used for emergent dimer models.For instance, we show how a spin-1=2 XXZ model on the ruby lattice encodes a Kitaev honeycomb model, providing a new route toward realizing the latter in cold-atom or solid-state systems.
Locally constrained gauge theories underpin our understanding of fundamental interactions in particle physics and the emergent behaviour of quantum materials. In strongly correlated systems, they can give rise to quantum spin liquids that lack conventional order and are defined by coherent superpositions of an extensive number of many-body configurations. Realising and probing such exotic states experimentally is an outstanding challenge both in solid-state and synthetic quantum systems, not least due to the difficulty of detecting the fragile coherences between many-body states. Here, we report a large-scale (>3,000 sites) realisation of a two-dimensional U(1) lattice gauge theory with ultracold atoms in a square optical superlattice and demonstrate non-equilibrium preparation of extended regions of U(1) quantum spin liquids. We demonstrate Gauss's law validity in a quench experiment, enabled by a new microscopy technique for detecting doubly occupied sites. We observe characteristic real-space correlations and momentum-space pinch points, hallmarks of the emergent U(1) gauge structure. Using round-trip interferometric protocols, we directly observe large-scale coherence between many-body configurations, providing strong evidence for quantum spin liquid regions extending over ~100 lattice sites. Our results establish non-equilibrium quantum simulation protocols as a powerful route for accessing and probing exotic, highly-entangled states beyond those hosted by the engineered Hamiltonian in thermal equilibrium.
Quantum phases with topological order, such as quantum spin liquids, have been the focus of explorations for several decades. Such phases feature a number of remarkable properties including long-range quantum entanglement. Moreover, they can be potentially exploited for the realization of robust quantum computation, as exemplified by the paradigmatic toric code model. While some indications that such phases may be present in frustrated condensed matter systems have been previously reported, so far quantum spin liquids have eluded direct experimental detection. In this talk, I will show how a programmable quantum simulator based on Rydberg atom arrays can be used to realize and probe quantum spin liquid states. In our approach, atoms are placed on the links of a kagome lattice and coherent evolution under Rydberg blockade enables the transition into frustrated quantum states with no local order. We detect the onset of a quantum spin liquid phase of the toric code type by measuring topological string operators in two complementary bases. The properties of this state are further revealed using a lattice with non-trivial topology, representing a step towards the realization of a topological qubit. Our observations open the door to the controlled experimental exploration of topological quantum matter, and could enable the investigation of new methods for topologically protected quantum information processing.
Arrays of Rydberg atoms constitute a highly tunable, strongly interacting venue for the pursuit of exotic states of matter. We develop a strategy for accessing a family of fractionalized phases known as quantum spin liquids in two-dimensional Rydberg arrays. We specifically use effective field theory methods to study arrays assembled from Rydberg chains tuned to an Ising phase transition that famously hosts emergent fermions propagating within each chain. This highly entangled starting point allows us to naturally access spin liquids familiar from Kitaev's honeycomb model---albeit from an entirely different framework. In particular, we argue that finite-range repulsive Rydberg interactions, which frustrate nearby symmetry-breaking orders, can enable coherent propagation of emergent fermions between the chains in which they were born. Delocalization of emergent fermions across the full two-dimensional Rydberg array yields a gapless Z2{\mathbb{Z}}_{2}Z2​ spin liquid with a single massless Dirac cone. Here, the Rydberg occupation numbers exhibit universal power-law correlations that provide a straightforward experimental diagnostic of this phase. We further show that explicitly breaking symmetries perturbs the gapless spin liquid into gapped, topologically ordered descendants: Breaking lattice symmetries generates toric-code topological order, whereas introducing Floquet-mediated chirality generates non-Abelian Ising topological order. In the toric-code phase, we analytically construct microscopic incarnations of non-Abelian defects, which can be created and transported by dynamically controlling the atom positions in the array. Our work suggests that appropriately tuned Rydberg arrays provide a cold-atoms counterpart of solid-state ``Kitaev materials'' and, more generally, it spotlights a different angle for pursuing experimental platforms for Abelian and non-Abelian fractionalization.
The presence of strong-interactions and entanglement in a quantum many-body system can lead to several exotic states of matter. Quantum Spin liquids are one such set of exotic states that have attracted significant interest due to their interesting properties and possible technological applications. The rapid progress in cold atom experiment setups, particularly the Rydberg atom simulators, has opened up new avenues to study these spin liquids in one and two-dimensions. This calls for advanced theoretical techniques to investigate the non-equilibrium properties of isolated pure quantum states. This thesis aims to develop numerical algorithms to study the non-equilibrium properties of spin liquids on a square lattice following a quantum quench using the framework of tensor networks. A variational form of Projected Entangled Pair State (PEPS) ansatz is used to describe the spin liquids. A new symmetric form of the Corner Transfer Matrix Renormalization group (CTMRG), the algorithm necessary to construct the reduced density matrix of the wavefunction in the thermodynamic limit for a complex-valued PEPS wavefunction, is described. To study the quench dynamics, two algorithms called simple-update and full update, along with the necessary modifications to adapt them to study spin liquid wavefunctions, are explained. The tools are then used to study the non-equilibrium dynamics of the resonating valence bond (RVB) state with nearest-neighbor (NN) valence bond configurations, a spin-liquid state that is known to exhibit critical dimer-dimer correlations, within the PEPS manifold.