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arXiv:2305.08123 · Physical review. B./Physical review. B

Quantum scar states in coupled random graph models

Bhilahari Jeevanesan

Published Aug 14, 2023• 2 citations
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Abstract

We analyze the Hilbert space connectivity of the $L$ site PXP model by constructing the Hamiltonian matrices via a Gray code numbering of basis states. The matrices are all formed out of a single Hamiltonian-path backbone and entries on skew diagonals. Starting from this observation, we construct an ensemble of related Hamiltonians based on random graphs with tunable constraint degree and variable network topology. We study the entanglement structure of their energy eigenstates and find two classes of weakly entangled midspectrum states. The first class contains scars that are approximate products of eigenstates of the subsystems. Their origin can be traced to the near orthogonality of random vectors in high-dimensional spaces. The second class of scars has $log2$ entanglement entropy and is tied to the occurrence of special types of subgraphs. The latter states have some resemblance to the Lin-Motrunich $\sqrt{2}$ scars.

DOI
10.1103/physrevb.108.075131

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We analyze the Hilbert space connectivity of the LLL site PXP model by constructing the Hamiltonian matrices via a Gray code numbering of basis states. The matrices are all formed out of a single Hamiltonian-path backbone and entries on skew diagonals. Starting from this observation, we construct an ensemble of related Hamiltonians based on random graphs with tunable constraint degree and variable network topology. We study the entanglement structure of their energy eigenstates and find two classes of weakly entangled midspectrum states. The first class contains scars that are approximate products of eigenstates of the subsystems. Their origin can be traced to the near orthogonality of random vectors in high-dimensional spaces. The second class of scars has log2log2log2 entanglement entropy and is tied to the occurrence of special types of subgraphs. The latter states have some resemblance to the Lin-Motrunich 2\sqrt{2}2​ scars.
We investigate the emergence of quantum scars in a general ensemble of random Hamiltonians (of which the PXP is a particular realization), that we refer to as quantum local random networks. We find a class of scars, that we call “statistical”, and we identify specific signatures of the localized nature of these eigenstates by analyzing a combination of indicators of quantum ergodicity and properties related to the network structure of the model. Within this parallelism, we associate the emergence of statistical scars to the presence of “motifs” in the network, that reflects how these are associated to links with anomalously small connectivity. Most remarkably, statistical scars appear at well-defined values of energy, predicted solely on the base of network theory. We study the scaling of the number of statistical scars with system size: by continuously changing the connectivity of the system we find that there is a transition from a regime where the constraints are too weak for scars to exist for large systems to a regime where constraints are stronger and the number of statistical scars increases with system size. This allows to define the concept of “statistical robustness” of quantum scars.
Quantum many-body scars are atypical, highly nonthermal eigenstates embedded in a sea of thermal eigenstates that have been observed in, for example, kinetically constrained quantum many-body models. These special eigenstates are characterized by a bipartite entanglement entropy that scales as most logarithmically with the subsystem size. We use numerical optimization techniques to investigate if quantum many-body scars of the experimentally relevant PXP model can be well approximated by Gaussian states. Gaussian states are described by a number of parameters that scales quadratically with system size, thereby having a much lower complexity than generic quantum many-body states, for which this number scales exponentially. We find that while quantum many-body scars can typically be well approximated by (symmetrized) Gaussian states, this is not the case for ergodic (thermal) eigenstates. This observation suggests that the non-ergodic part of the PXP Hamiltonian is related to certain quadratic parent Hamiltonians, thereby hinting on the origin of the quantum many-body scars.
Following the recent observation of wave function revivals in large Rydberg atom quantum simulators, much effort has focused on understanding the emergence of many-body scars in nonintegrable quantum systems. Here we explore the origin of scarred wave function revivals in a family of models obtained by deforming the graph adjacency matrix of the PXP model---the effective model of Rydberg atoms in the strong Rydberg blockade regime. We consider deformations that either enhance the Rydberg constraint, ultimately resulting in an effective tight-binding model of two hypercubes joined at a single vertex, or relax the constraint until reaching the free spin-1/2 model. In the former case, we argue that the model of two joined hypercubes captures the essential features of many-body scarring present in the PXP model. On the other hand, relaxing the constraint leads to a sequence of new scarred models, some with more robust scarring signatures than the PXP model, as can be understood from the graph-theoretic viewpoint. Our results shed light on the nature of scarring in the PXP model by identifying its simple parent model, while also highlighting its distinction from the free-spin precession.
Quantum many-body scar is a recently discovered phenomenon weakly violating eigenstate thermalization hypothesis, and it has been extensively studied across various models. However, experimental realizations are mainly based on constrained models such as the PXP model. Inspired by recent experimental observations on the superconducting platform in the works by Zhang et al. [Nat. Phys. 19, 120 (2023)] and Yao et al. [Nat. Phys. (2023)], we study a distinct class of quantum many-body scars based on a half-filling hard-core Bose-Hubbard model, which is generic to describe in many experimental platforms. It is the so-called Hilbert-space quantum scar as it originates from a subspace with a hypercube geometry weakly connecting to other thermalization regions in Hilbert space. Within the hypercube, a pair of collective Fock states do not directly connect to the thermalization region, resulting in slow thermalization dynamics with remarkable fidelity revivals with distinct differences from dynamics of other initial states. This mechanism is generic in various real-space lattice configurations, including one-dimensional Su-Schrieffer-Heeger chain, comb lattice, and even random dimer clusters consisting of dimers. In addition, we develop a toy model based on Hilbert hypercube decay approximation, to explain the spectrum overlap between the collective states and all eigenstates. Furthermore, we explore the Hilbert-space quantum scar in two- and three-dimensional Su-Schrieffer-Heeger many-body systems, consisting of tetramers or octamers, respectively. This study makes quantum many-body scar state more realistic in applications such as quantum sensing and quantum metrology.
Quantum many-body scars enable persistent non-ergodic dynamics in otherwise thermalizing systems, yet their stabilization typically relies on fine-tuned initial states or engineered Hamiltonian perturbations. Here we show that lattice geometry alone can serve as a powerful and experimentally accessible control knob for inducing and enhancing scarring. By transforming a one-dimensional chain into a quasi-one-dimensional triangle-decorated lattice, we find that the fully polarized state -- normally thermalizing in the PXP model -- exhibits pronounced fidelity revivals, slow entanglement growth, and strong overlap with a tower of weakly entangled eigenstates. We trace this behavior to a geometry-induced restructuring of the constrained Hilbert space, whereby the adjacency graph decomposes into hypercube subgraphs that enforce coherent population transfer and stabilize an emergent approximate su(2)\mathrm{su}(2)su(2) algebra. We propose a direct implementation in programmable arrays of tweezer-trapped Rydberg atoms, where the triangle-decorated geometry can be realized using spatial light modulators and the resulting scarring dynamics probed via time-resolved measurements of excitation density. Our results establish lattice connectivity as a design principle for engineering non-ergodic dynamics in constrained quantum systems.
Recently a class of quantum systems exhibiting weak ergodicity breaking has attracted much attention. These systems feature in the energy spectrum a special band of eigenstates called quantum many-body scar states. One prototype model that is known to host scar states is the so-called PXP model. In this work we study the fate of quantum many-body scar states in a 2D PXP model against random disorders. We show that in both the square lattice and the honeycomb lattice the scar states can persist up to a finite disorder strength, before eventually being erased by the disorder. We further study the localization properties of the system in the presence of even stronger disorders and show that whether a full localization transition occurs depends on the type of disorder we introduce. Our study thus reveals the fascinating interplay between disorder and quantum many-body scarring in a 2D system.