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arXiv:2104.00699 · Physical Review Research

Constraint-induced breaking and restoration of ergodicity in spin-1 PXP models

Bhaskar Mukherjee, Zi Cai, W. Vincent Liu

Published Aug 31, 2021• 28 citations
arXivHTMLPDF

Abstract

Eigenstate thermalization hypothesis (ETH) has played a pivotal role in understanding ergodicity and its breaking in isolated quantum many-body systems. The recent experiment on 51-atom Rydberg quantum simulator by Bernien et al. [Nature (London) 551, 579 (2017)] and subsequent theoretical analysis have shown that hardcore kinetic constraint can lead to weak ergodicity breaking. In this work, we demonstrate, using $1d$ spin-1 PXP models, that miscellaneous type of ergodicity can be realized by adjusting the hardcore constraints between different components of nearest-neighbor spins. This includes ETH violation due to emergent shattering of Hilbert space into exponentially many subsectors of various sizes, a novel form of nonintegrability with an extensive number of local conserved quantities and strong ergodicity. We analyze these different forms of ergodicity and study their impact on the nonequilibrium dynamics of a ${\mathbb{Z}}_{2}$ initial state. We use forward-scattering approximation to understand the amount of ${\mathbb{Z}}_{2}$ oscillation present in these models. Our work shows that not only ergodicity breaking but an appropriate choice of constraints can lead to restoration of ergodicity as well.

DOI
10.1103/physrevresearch.3.033201

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Eigenstate thermalization hypothesis (ETH) has played a pivotal role in understanding ergodicity and its breaking in isolated quantum many-body systems. The recent experiment on 51-atom Rydberg quantum simulator by Bernien et al. [Nature (London) 551, 579 (2017)] and subsequent theoretical analysis have shown that hardcore kinetic constraint can lead to weak ergodicity breaking. In this work, we demonstrate, using 1d1d1d spin-1 PXP models, that miscellaneous type of ergodicity can be realized by adjusting the hardcore constraints between different components of nearest-neighbor spins. This includes ETH violation due to emergent shattering of Hilbert space into exponentially many subsectors of various sizes, a novel form of nonintegrability with an extensive number of local conserved quantities and strong ergodicity. We analyze these different forms of ergodicity and study their impact on the nonequilibrium dynamics of a Z2{\mathbb{Z}}_{2}Z2​ initial state. We use forward-scattering approximation to understand the amount of Z2{\mathbb{Z}}_{2}Z2​ oscillation present in these models. Our work shows that not only ergodicity breaking but an appropriate choice of constraints can lead to restoration of ergodicity as well.
Experimental observation of coherent oscillations in a Rydberg atom chain [H. Bernien et al., Nature (London) 551, 579 (2017)] has led to the discovery of quantum many-body scars (QMBSs) which is a new paradigm for ergodicity breaking. The experimental findings in the Rydberg chain can be well captured by a kinetically constrained model called the ``PXP'' model, which has been shown to host the eigenstate thermalization hypothesis (ETH)-violating scar states in the middle of the spectrum. Much effort has been put into identifying similar kinetically restricted systems that show a violation of ETH. In this paper, we study the QMBSs that can arise in one such model, namely the spin-1 Kitaev chain, where owing to some conserved quantities, the Hilbert space gets fragmented into unequal disconnected subspaces. Recently, You et al. [Phys. Rev. Res. 4, 013103 (2022)] showed that the ground state sector of this chain can be mapped exactly onto the prototypical PXP model and thus hosts QMBSs. Here, we demonstrate that the phenomenon of scarring is also present in other sectors, and in particular, we identify a sector that exhibits substantially more scarring than the ground state one. We propose an initial state and numerically demonstrate that its fidelity revivals are pronounced and longer lived than those in the PXP model.
The eigenstate thermalization hypothesis (ETH) provides the prevailing framework for understanding quantum thermalization and ergodicity in isolated many-body systems. Yet, no general theory describes the continuous onset of ETH violation between the conventional ETH and its complete breakdown. Here, we introduce a two-parameter ansatz for the ETH violation that unifies and distinguishes two mechanisms: fading ergodicity and trapped ergodicity. While fading ergodicity captures the established route to ergodicity breaking, trapped ergodicity describes a distinct scenario in which ETH is violated in finite systems but restored in the thermodynamic limit. We test this framework in the spin-1/2 J1J_1J1​-J2J_2J2​ chains with on-site disorder and linear potential. In both cases, we find that the observed ETH violation is consistent with trapped ergodicity.
In classical systems, chaos is clearly defined via the behavior of trajectories. In quantum systems with a classical analogue one finds that the transition from regular to chaotic dynamics is signified by a change in the spectral statistics. This has been found to remain true for quantum systems with no classical analogue, including many-body systems. Furthermore, quantum chaotic systems explore all the allowed configurations in the Hilbert space, i.e. they are ergodic, while integrable systems, and systems in the many-body localized phase, are restricted to a certain subspace of the available phase space, and hence strongly break ergodicity. In this dissertation, we study the intermediate behavior between ergodicity and localization, i.e. the weak breaking of ergodicity. The model examined is the PXP spin chain model, where spins are allowed to flip only under certain kinetic constraints. We start by reproducing some already established results. First, we explore the eigenstate thermalization hypothesis (ETH) for this model and demonstrate the existence of a small number of states, throughout the PXP spectrum, that violate the ETH. Then we study the level-spacing statistics of the model, a well-known quantum chaos diagnostic, which turns out to be close to semi-Poisson and approach Wigner--Dyson statistics for large system sizes. Moreover, we examine various aspects of the model that have not been studied before. For example, the eigenvector component statistics, another quantum chaos diagnostic, for the PXP model turn out to be non-Gaussian. Finally, we perform a quench, in order to study how the energy spreads throughout the system, and observe ballistic fronts.
Thermalization in closed quantum systems and its breakdown are central to understanding the emergence of statistical behavior in isolated many-body systems. Kinetic constraints can hinder thermalization by restricting the allowed dynamics and giving rise to non-ergodic behavior. While various aspects of ergodicity breaking in kinetically constrained models have been extensively studied in Hermitian systems, their behavior in non-Hermitian kinetically constrained models remains less explored. In particular, the inequivalence of left and right eigenstate ensembles in non-Hermitian systems raises the question of whether they exhibit qualitatively different dynamical and eigenstate properties. In this work, we study non-Hermitian generalizations of the paradigmatic PXP and PPXPP models, focusing on their Krylov complexity, eigenstate structure, and dynamics. By exploiting the Arnoldi's algorithm, we find distinct behavior of Krylov complexity in the left and right eigenstate ensembles as non-Hermiticity is increased: in the right ensemble, it exhibits persistent oscillations with an increasing period, while in the left ensemble, it gradually saturates. The characteristic arch-shaped profile of the Arnoldi coefficients is also modified differently in the two ensembles, with a Hermitian-like arch re-emerging in the left ensemble at large non Hermiticity. Additionally, in the non-Hermitian PXP model, the right ensemble retains the characteristic scar structure, while the left ensemble develops additional non-scar eigenstates with anomalously low entanglement and large overlap with the vacuum state. These findings reveal that non-Hermiticity can affect Krylov dynamics and eigenstate structure differently in the left and right ensembles, demonstrating the need to treat them separately when characterizing non-Hermitian kinetically constrained systems.
Kinetically constrained spin systems play an important role in understanding key properties of the dynamics of slowly relaxing materials, such as glasses. Recent experimental studies have revealed that manifest kinetic constraints govern the evolution of strongly interacting gases of highly excited atoms in a noisy environment. Motivated by this development we explore which types of kinetically constrained dynamics can generally emerge in quantum spin systems subject to strong noise and show how, in this framework, constraints are accompanied by conservation laws. We discuss an experimentally realizable case of a lattice gas, where the interplay between those and the geometry of the lattice leads to collective behavior and time-scale separation even at infinite temperature. This is in contrast to models of glass-forming substances which typically rely on low temperatures and the consequent suppression of thermal activation.