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arXiv:2212.14333 · Physical review. B./Physical review. B

Probability transport on the Fock space of a disordered quantum spin chain

Isabel Creed, David E. Logan, Sthitadhi Roy

Published Mar 29, 2023• 14 citations
arXivHTMLPDF

Abstract

Within the broad theme of understanding the dynamics of disordered quantum many-body systems, one of the simplest questions one can ask is, given an initial state, how does it evolve in time on the associated Fock-space graph, in terms of the distribution of probabilities thereon? A detailed quantitative description of the temporal evolution of out-of-equilibrium disordered quantum states and probability transport on the Fock space is our central aim here. We investigate it in the context of a disordered quantum spin chain, which hosts a disorder-driven many-body localization transition. Real-time dynamics/probability transport is shown to exhibit a rich phenomenology, which is markedly different between the ergodic and many-body localized phases. The dynamics is, for example, found to be strongly inhomogeneous at intermediate times in both phases, but while it gives way to homogeneity at long times in the ergodic phase, the dynamics remain inhomogeneous and multifractal in nature for arbitrarily long times in the localized phase. Similarly, we show that an appropriately defined dynamical lengthscale on the Fock-space graph is directly related to the local spin autocorrelation, and as such sheds light on the (anomalous) decay of the autocorrelation in the ergodic phase, and lack of it in the localized phase.

DOI
10.1103/physrevb.107.094206

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Within the broad theme of understanding the dynamics of disordered quantum many-body systems, one of the simplest questions one can ask is, given an initial state, how does it evolve in time on the associated Fock-space graph, in terms of the distribution of probabilities thereon? A detailed quantitative description of the temporal evolution of out-of-equilibrium disordered quantum states and probability transport on the Fock space is our central aim here. We investigate it in the context of a disordered quantum spin chain, which hosts a disorder-driven many-body localization transition. Real-time dynamics/probability transport is shown to exhibit a rich phenomenology, which is markedly different between the ergodic and many-body localized phases. The dynamics is, for example, found to be strongly inhomogeneous at intermediate times in both phases, but while it gives way to homogeneity at long times in the ergodic phase, the dynamics remain inhomogeneous and multifractal in nature for arbitrarily long times in the localized phase. Similarly, we show that an appropriately defined dynamical lengthscale on the Fock-space graph is directly related to the local spin autocorrelation, and as such sheds light on the (anomalous) decay of the autocorrelation in the ergodic phase, and lack of it in the localized phase.
The dynamics of interacting quantum many-body systems has two seemingly disparate but fundamental facets. The first is the dynamics of real-space local observables, and if and how they thermalize. The second is to interpret the dynamics of the many-body state as that of a fictitious particle on the underlying Hilbert-space graph. In this work, we derive an explicit relation between these two aspects of the dynamics. We show that the temporal decay of the autocorrelation in a disordered quantum spin chain is explicitly encoded in how the return probability on Hilbert space approaches its late-time saturation. As such, the latter has the same functional form in time as the decay of autocorrelations but with renormalized parameters. Our analytical treatment is rooted in an understanding of the morphology of the time-evolving state on the Hilbert-space graph, and corroborated by exact numerical results.
We investigate the anatomy and complexity of quantum states in Krylov space, in the ergodic and many-body localised (MBL) phases of a disordered, interacting spin chain. The Krylov basis generated by the Hamiltonian from an initial state provides a representation in which the spread of the time-evolving state constitutes a basis-optimised measure of complexity. We show that the long-time Krylov spread complexity sharply distinguishes the two phases. In the ergodic regime, the infinite-time complexity scales linearly with the Fock-space dimension, indicating that the state spreads over a finite fraction of the Krylov chain. By contrast, it grows sublinearly in the MBL regime, implying that the long-time state occupies only a vanishing fraction of the chain. Further, the profile of the infinite-time state along the Krylov chain exhibits a stretched-exponential decay in the MBL regime. This behaviour reflects a broad distribution of decay lengthscales, associated with different eigenstates contributing to the long-time state. Consistently, a large-deviation analysis of the statistics of eigenstate spread complexities shows that while the ergodic regime receives contributions from almost all eigenstates, the complexity in the MBL regime is dominated by a vanishing fraction of eigenstates, which have anomalously large complexity relative to the typical ones.
This article reviews recent progress in understanding the physics of many-body localisation (MBL) in disordered and interacting quantum many-body systems, from the perspective of ergodicity breaking on the associated Fock space. This approach to MBL is underpinned by mapping the dynamics of the many-body system onto that of a fictitious single particle on the high-dimensional, correlated and disordered Fock-space graph; yet, as we elaborate, the problem is fundamentally different from that of conventional Anderson localisation on high-dimensional or hierarchical graphs. We discuss in detail the nature of eigenstate correlations on the Fock space, both static and dynamic, and in the ergodic and many-body localised phases as well as in the vicinity of the MBL transition. The latter in turn sheds light on the nature of the transition, and motivates a scaling theory for it in terms of Fock-space based quantities. We also illustrate how these quantities can be concretely connected to real-space observables. An overview is given of several analytical and numerical techniques which have proven important in developing a comprehensive picture. Finally, we comment on some open questions in the field of MBL where the Fock-space approach is likely to prove insightful.
We explore the Fock-space landscape of eigenstates across the many-body localisation (MBL) transition in a disordered, interacting quantum spin-1/2 chain. Eigenstate expectation values of spatially local observables, which distinguish an MBL phase from an ergodic one, can be represented in terms of eigenstate amplitudes on the Fock space. Motivated by this, we introduce and study spatial correlations on the Fock space. From these, a correlation length emerges, which is found to vary discontinuously across the MBL transition; and is intimately connected to the discontinuous jump in the multifractal exponents characterising the Fock-space wavefunctions. Exploiting the direct connection between the local observables and Fock-space correlations, we show that the discontinuity in the lengthscale also implies discontinuous behaviour of the local observables across the transition. The scaling behaviour of this lengthscale, which is closely connected to that of the inverse participation ratios, shows that the nature of the MBL transition is consistent with it being Kosterlitz-Thouless like, as predicted by recent phenomenological theories. Finally, we also show how correlation functions on the Fock space reveal the inhomogeneities in eigenstate amplitudes on the Fock space in the MBL phase.
We explore the Fock-space structure of eigenstates across the many-body localization (MBL) transition in a disordered, interacting quantum spin-12\frac{1}{2}21​ chain. Eigenstate expectation values of spatially local observables, which distinguish an MBL phase from an ergodic one, can be represented in terms of eigenstate amplitudes on the Fock space. Motivated by this, we introduce and study spatial correlations on the Fock space. From these, a correlation length emerges, which is found to vary discontinuously across the MBL transition, and is intimately connected to the discontinuous jump in the multifractal exponents characterizing the Fock-space wave functions. Exploiting the direct connection between the local observables and Fock-space correlations, we show that the discontinuity in the length scale also implies discontinuous behavior of the local observables across the transition. A scaling theory based on these Fock-space correlations is constructed, which is closely connected to that for the inverse participation ratio. It yields a volume scale in the ergodic phase and a length scale in the MBL phase, whose critical properties suggest a Kosterlitz-Thouless--type scenario for the MBL transition, as is predicted by recent phenomenological theories. Finally, we also show how correlation functions on the Fock space reveal the inhomogeneities in eigenstate amplitudes on the Fock space in the MBL phase.
We examine the standard model of many-body localization (MBL), i.e., the disordered chain of interacting spinless fermions, by representing it as the network in the many-body (MB) basis of noninteracting localized Anderson states. By studying eigenstates of the full Hamiltonian, for strong disorders we find that the dynamics is confined up to very long times to disconnected MB clusters in the Fock space. By keeping only resonant contributions and simplifying the quantum problem to rate equations (REs) for MB states, in analogy with percolation problems, the MBL transition is located via the universal cluster distribution and the emergence of the macroscopic cluster. On the ergodic side, our approximate RE approach to the relaxation processes captures well the diffusion transport, as found for the full quantum model. In a broad transient regime, we find an anomalous, i.e., subdiffusivelike, transport, emerging from weak links between MB states.