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arXiv:2209.11242 · Physical Review Letters

Prethermalization and the Local Robustness of Gapped Systems

Chao Yin, Andrew Lucas

Published Aug 3, 2023• 19 citations
arXivHTMLPDF

Abstract

We prove that prethermalization is a generic property of gapped local many-body quantum systems, subjected to small perturbations, in any spatial dimension. More precisely, let H_{0} be a Hamiltonian, spatially local in d spatial dimensions, with a gap Δ in the many-body spectrum; let V be a spatially local Hamiltonian consisting of a sum of local terms, each of which is bounded by ε≪Δ. Then, the approximation that quantum dynamics is restricted to the low-energy subspace of H_{0} is accurate, in the correlation functions of local operators, for stretched exponential timescale τ∼exp[(Δ/ε)^{a}] for any a<1/(2d-1). This result does not depend on whether the perturbation closes the gap. It significantly extends previous rigorous results on prethermalization in models where H_{0} was frustration-free. We infer the robustness of quantum simulation in low-energy subspaces, the existence of athermal "scarred" correlation functions in gapped systems subject to generic perturbations, the long lifetime of false vacua in symmetry broken systems, and the robustness of quantum information in non-frustration-free gapped phases with topological order.

DOI
10.1103/physrevlett.131.050402

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We prove that prethermalization is a generic property of gapped local many-body quantum systems, subjected to small perturbations, in any spatial dimension. More precisely, let H_{0} be a Hamiltonian, spatially local in d spatial dimensions, with a gap Δ in the many-body spectrum; let V be a spatially local Hamiltonian consisting of a sum of local terms, each of which is bounded by ε≪Δ. Then, the approximation that quantum dynamics is restricted to the low-energy subspace of H_{0} is accurate, in the correlation functions of local operators, for stretched exponential timescale τ∼exp[(Δ/ε)^{a}] for any a<1/(2d-1). This result does not depend on whether the perturbation closes the gap. It significantly extends previous rigorous results on prethermalization in models where H_{0} was frustration-free. We infer the robustness of quantum simulation in low-energy subspaces, the existence of athermal "scarred" correlation functions in gapped systems subject to generic perturbations, the long lifetime of false vacua in symmetry broken systems, and the robustness of quantum information in non-frustration-free gapped phases with topological order.
In quantum many-body systems subjected to generic perturbations, eigenstates are typically fragile and prone to strong mixing, leading to rapid thermalization in dynamics to an equilibrium state described by statistical mechanics. However, there exist robust counterexamples that defy this typical expectation, exhibiting novel non-equilibrium phenomena and offering potential applications in (quantum) information processing. This dissertation explores such systems and establishes their stability against perturbations with mathematical rigor. First, by generalizing the exponentially slow tunneling effect in single-particle quantum mechanics, we develop a theory of metastability in quantum many-body systems, demonstrating their nonperturbatively long lifetimes. As a canonical application, we provide the first general and tight bounds on the slow decay of false vacua. Second, it is known that ground states in finite spatial dimensions can resist arbitrary local perturbations through topologically ordered phases of matter. We extend this notion of stability to quantum error-correcting codes in infinite dimensions, specifically quantum low-density parity-check (LDPC) codes. These codes are of significant interest due to their promise of lower fault-tolerance overhead compared to finite-dimensional codes. Lastly, we analyze classical counterparts of LDPC codes and prove that their eigenstates remain localized in the many-body Hilbert space under generic perturbations. This finding presents the first unambiguous example of a robust violation of the eigenstate thermalization hypothesis. To achieve these results, we introduce advanced techniques to control operator locality, leveraging and generalizing methods such as Lieb-Robinson bounds, cluster expansions, and Schrieffer-Wolff transformations. These tools not only underpin our specific findings but may also prove valuable for broader challenges in mathematical physics of quantum many-body systems.
We present a mathematical theory of metastable pure states in closed many-body quantum systems with finite-dimensional Hilbert space. Given a Hamiltonian, a pure state is defined to be metastable when all sufficiently local operators either stabilize the state or raise its average energy. We prove that short-range-entangled metastable states are necessarily eigenstates (scars) of a perturbatively close Hamiltonian. Given any metastable eigenstate of a Hamiltonian, in the presence of perturbations, we prove the presence of prethermal behavior: Local correlation functions decay at a rate bounded by a timescale nonperturbatively long in the inverse metastability radius, rather than Fermi’s golden rule. Inspired by this general theory, we prove that the lifetime of the false vacuum in certain d -dimensional quantum models grows at least as fast as exp ( ε − d ) , where ε → 0 is the relative energy density of the false vacuum; this lower bound matches, for the first time, explicit calculations using quantum field theory. We identify metastable states at finite energy density in the PXP model, along with exponentially many metastable states in “helical” spin chains and the two-dimensional Ising model. Our inherently quantum formalism reveals precise connections between many problems, including prethermalization, robust quantum scars, and quantum nucleation theory, and applies to systems without known semiclassical and/or field-theoretic limits.
Abstract We study the stability with respect to a broad class of perturbations of gapped ground-state phases of quantum spin systems defined by frustration-free Hamiltonians. The core result of this work is a proof using the Bravyi–Hastings–Michalakis (BHM) strategy that under a condition of local topological quantum order (LTQO), the bulk gap is stable under perturbations that decay at long distances faster than a stretched exponential. Compared to previous work, we expand the class of frustration-free quantum spin models that can be handled to include models with more general boundary conditions, and models with discrete symmetry breaking. Detailed estimates allow us to formulate sufficient conditions for the validity of positive lower bounds for the gap that are uniform in the system size and that are explicit to some degree. We provide a survey of the BHM strategy following the approach of Michalakis and Zwolak, with alterations introduced to accommodate more general than just periodic boundary conditions and more general lattices. We express the fundamental condition known as LTQO by means of an indistinguishability radius, which we introduce. Using the uniform finite-volume results, we then proceed to study the thermodynamic limit. We first study the case of a unique limiting ground state and then also consider models with spontaneous breaking of a discrete symmetry. In the latter case, LTQO cannot hold for all local observables. However, for perturbations that preserve the symmetry, we show stability of the gap and the structure of the broken symmetry phases. We prove that the GNS Hamiltonian associated with each pure state has a non-zero spectral gap above the ground state.
Prethermalization refers to the transient phenomenon where a system thermalizes according to a Hamiltonian that is not the generator of its evolution. We provide here a rigorous framework for quantum spin systems where prethermalization is exhibited for very long times. First, we consider quantum spin systems under periodic driving at high frequency ν. We prove that up to a quasi-exponential time τ∗∼ecνlog3ν, the system barely absorbs energy. Instead, there is an effective local Hamiltonian Dˆ that governs the time evolution up to τ∗, and hence this effective Hamiltonian is a conserved quantity up to τ∗. Next, we consider systems without driving, but with a separation of energy scales in the Hamiltonian. A prime example is the Fermi–Hubbard model where the interaction U is much larger than the hopping J. Also here we prove the emergence of an effective conserved quantity, different from the Hamiltonian, up to a time τ∗ that is (almost) exponential in U/J.
An important class of models studied in quantum statistical mechanics consists of frustration-free interactions whose Hamiltonians exhibit a gap in the spectrum above their ground state energies. This paper discusses recent techniques which dramatically extend the class of models to which stability applies when the perturbations exhibit stretched exponential decay at long distances. We consider a class of initial models satisfying an assumption that prevents splitting of the ground state space. This assumption on the structure on the ground state space is referred to as local topological quantum order. A key tool in our analysis is the quasi-local unitary spectral flow automorphism which allows us to decompose the perturbed Hamiltonian into a perturbation model satisfying a relative form bound. We will present a result for the stability of frustration-free, one-dimensional models of this type in the form of an estimate uniform in the system size. We also present additional conditions ensuring the existence of a pure perturbed ground state whose GNS Hamiltonian remains gapped. As an application, we evaluate the stability of the Heisenberg XXZ spin-1/2 chain. Our work shows a new result on the stability of the XXZ model for perturbations respecting a local gauge symmetry and the stability above the translation invariant ground states in the thermodynamic limit. We also discuss challenges to additional conjectured stability results for the XXZ model.
We demonstrate the existence of prethermal discrete time crystals whose subharmonic response is entirely localized to zero-dimensional corner modes. Within the exponentially long prethermal regime, we show that the robustness of these corner modes arises from two related, yet distinct, mechanisms: the presence of a higher-order symmetry-protected topological phase in the effective Hamiltonian, or the emergence of a dynamical constraint that prevents the decay of the corner mode. While the first mechanism ensures the stability of the subharmonic response throughout the entirety of the prethermal regime, it is restricted to initial states in the ground state manifold of the effective Hamiltonian. By contrast, the second mechanism enables the observation of the prethermal time-crystalline order for arbitrary initial states, albeit with a timescale that is not only determined by the frequency of the drive, but also the relative energy scale across the system's sublattices. We characterize these two mechanisms by simulating the dynamics of a periodically driven two-dimensional spin model, and discuss natural extensions of our model to all other dimensions.