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arXiv:1212.3367 · Physical Review A

Equilibration times in clean and noisy systems

Lorenzo Campos Venuti, Sunil Yeshwanth, Stephan Haas

Published Mar 15, 2013• 13 citations
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Abstract

We study the equilibration dynamics of closed finite quantum systems and address the question of the time needed for the system to equilibrate. In particular, we focus on the scaling of the equilibration time ${T}_{\mathrm{eq}}$ with the system size $L$. For clean systems, we give general arguments predicting ${T}_{\mathrm{eq}}=O({L}^{0})$ for clustering initial states, while for small quenches around a critical point we find ${T}_{\mathrm{eq}}=O({L}^{\ensuremath{\zeta}})$ where $\ensuremath{\zeta}$ is the dynamical critical exponent. We then analyze noisy systems where exponentially large time scales are known to exist. Specifically, we consider the tight-binding model with diagonal impurities and give numerical evidence that in this case ${T}_{\mathrm{eq}}\ensuremath{\sim}B{e}^{C{L}^{\ensuremath{\psi}}}$ where $B,C,\phantom{\rule{0.16em}{0ex}}\ensuremath{\psi}$ are observable-dependent constants. Finally, we consider another noisy system whose evolution dynamics is randomly sampled from a circular unitary ensemble. Here, we are able to prove analytically that ${T}_{\mathrm{eq}}=O(1)$, thus showing that noise alone is not sufficient for slow equilibration dynamics.

DOI
10.1103/physreva.87.032108

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We study the equilibration dynamics of closed finite quantum systems and address the question of the time needed for the system to equilibrate. In particular, we focus on the scaling of the equilibration time Teq{T}_{\mathrm{eq}}Teq​ with the system size LLL. For clean systems, we give general arguments predicting Teq=O(L0){T}_{\mathrm{eq}}=O({L}^{0})Teq​=O(L0) for clustering initial states, while for small quenches around a critical point we find Teq=O(L\ensuremathζ){T}_{\mathrm{eq}}=O({L}^{\ensuremath{\zeta}})Teq​=O(L\ensuremathζ) where \ensuremathζ\ensuremath{\zeta}\ensuremathζ is the dynamical critical exponent. We then analyze noisy systems where exponentially large time scales are known to exist. Specifically, we consider the tight-binding model with diagonal impurities and give numerical evidence that in this case Teq\ensuremath∼BeCL\ensuremathψ{T}_{\mathrm{eq}}\ensuremath{\sim}B{e}^{C{L}^{\ensuremath{\psi}}}Teq​\ensuremath∼BeCL\ensuremathψ where B,C,\ensuremathψB,C,\phantom{\rule{0.16em}{0ex}}\ensuremath{\psi}B,C,\ensuremathψ are observable-dependent constants. Finally, we consider another noisy system whose evolution dynamics is randomly sampled from a circular unitary ensemble. Here, we are able to prove analytically that Teq=O(1){T}_{\mathrm{eq}}=O(1)Teq​=O(1), thus showing that noise alone is not sufficient for slow equilibration dynamics.
One of the main questions of research on quantum many-body systems following unitary out of equilibrium dynamics is to find out how local expectation values equilibrate in time. For non-interacting models, this question is rather well understood. However, the best known bounds for general quantum systems are vastly crude, scaling unfavorable with the system size. Nevertheless, empirical and numerical evidence suggests that for generic interacting many-body systems, generic local observables, and sufficiently well-behaved states, the equilibration time does not depend strongly on the system size, but only the precision with which this occurs does. In this discussion paper, we aim at giving very simple and plausible arguments for why this happens. While our discussion does not yield rigorous results about equilibration time scales, we believe that it helps to clarify the essential underlying mechanisms, the intuition and important figure of merits behind equilibration. We then connect our arguments to common assumptions and numerical results in the field of equilibration and thermalization of closed quantum systems, such as the eigenstate thermalization hypothesis as well as rigorous results on interacting quantum many-body systems. Finally, we complement our discussions with numerical results - both in the case of examples and counter-examples of equilibrating systems.
Isolated many-body quantum systems quenched far from equilibrium can eventually equilibrate, but it is not yet clear how long they take to do so. To answer this question, we use exact numerical methods and analyze the entire evolution, from perturbation to equilibration, of a paradigmatic disordered many-body quantum system in the chaotic regime. We investigate how the equilibration time depends on the system size and observables. We show that if dynamical manifestations of spectral correlations in the form of the correlation hole (``ramp'') are taken into account, the time for equilibration scales exponentially with system size, while if they are neglected, the scaling is better described by a power law with system size, although with an exponent larger than what is expected for diffusive transport.
Markovian open-system dynamics have widespread applications throughout quantum information science, including algorithmic state preparation. Their convergence is commonly quantified using the worst case global trace distance between the evolving and stationary states. However, this criterion can be unnecessarily stringent when only physically relevant observables are of interest. Here we introduce and study observable-specific mixing times. We prove that, for quasi-local, rapidly mixing Lindbladians, sums of geometrically local observables equilibrate in a time independent of system size, in contrast to the logarithmic dependence of global state mixing. This separation reduces the runtime of dissipative quantum algorithms, including quantum Gibbs samplers, for estimating quantities such as the Gibbs state energy and local order parameters, yielding an overall scaling that is linear in system size. Complementing this quantum result, we develop a quantum-inspired classical algorithm for estimating the same quantities. Its runtime is likewise linear in system size, but scaling exponentially in O(log⁡(1/ε)D)\mathcal{O}\big(\log(1/ε)^D\big)O(log(1/ε)D), where DDD denotes the spatial dimension of the lattice. We further analyse non-interacting Lindbladians over qudits, fermions, and bosons, demonstrating that locality of observables is not always necessary for a qualitatively faster mixing. Small-scale simulations of quantum Gibbs samplers reveal no large hidden constants in our asymptotic analysis and show that the theoretical predictions closely capture the finite-size dynamics.
A finite quantum system evolving unitarily equilibrates in a probabilistic fashion. In the general many-body setting the time fluctuations of an observable A are typically exponentially small in the system size. We consider here quasifree Fermi systems where the Hamiltonian and observables are quadratic in the Fermi operators. We first prove a bound on the temporal fluctuations ΔA(2) and then map the equilibration dynamics to a generalized classical XY model in the infinite temperature limit. Using this insight, we conjecture that, in most cases, a central limit theorem can be formulated, leading to what we call Gaussian equilibration: observables display a Gaussian distribution with relative error ΔA/A[over ¯]=O(L(-1/2)), where L is the dimension of the single-particle space. The conjecture, corroborated by numerical evidence, is proven analytically under mild assumptions for the magnetization in the quantum XY model and for a class of observables in a tight-binding model. We also show that the variance is discontinuous at the transition between a quasifree model and a nonintegrable one.
Understanding equilibration times in closed quantum systems is essential for characterising their approach to equilibrium. Chaotic many-body systems are paradigmatic in this context: they are expected to thermalise according to the eigenstate thermalisation hypothesis and exhibit spectral properties well described by random matrix theory (RMT). While RMT successfully captures spectral correlations, its ability to provide quantitative predictions for equilibration timescales has remained largely unexplored. Here, we study equilibration within RMT using the framework of equilibration as dephasing, focusing on closed systems whose Hamiltonians are drawn from the Gaussian unitary ensemble (GUE). We derive an analytical expression that approximates the average equilibration time of the GUE and show that it is independent of both the initial state and the choice of observable, a consequence of the rotational invariance of the GUE. Numerical simulations confirm our analytical expression and demonstrate that our approximation is in close agreement with the true average equilibration time of the GUE. We find that the equilibration time decreases with system size and vanishes in the thermodynamic limit. This unphysical result indicates that the true equilibration timescale of realistic chaotic many-body systems must be dominated by physical features not captured by random matrix ensembles -- the GUE in particular.
In the first part of this dissertation, we study the dynamics of isolated and clean quantum systems out of equilibrium. We initially address the Kibble-Zurek (KZ) problem of determining the dynamical evolution of a system close to its critical point under slow changes of a control parameter. We formulate a scaling limit in which the nonequilibrium behavior is universal and discuss the universal content. We then report computations of some scaling functions in model Gaussian and large-N problems. Next, we apply KZ scaling to topologically ordered systems with no local order parameter. In the examples of the Ising gauge theory and the SU(2)k phases of the Levin-Wen models, we observe a slow, coarsening dynamics for the string-net that underlies the physics of the topological phase at late times for ramps across transitions that reduce topological order. We conclude by studying quenches in the quantum O(N) model in the infinite N limit in varying spatial dimensions. Despite the failure to equilibrate owing to an infinite number of emergent conservation laws, the qualitative features of late time states following quenches is predicted by the equilibrium phase diagram. In the second part of this dissertation, we explore the relationship between entanglement and topological order in fractional quantum Hall (FQH) phases. In 2008, Li and Haldane conjectured that the entanglement spectrum (ES), a presentation of the Schmidt values of a real space cut, reflects the energy spectrum of the FQH chiral edge. Specifically, both spectra should have the same quasi-degeneracy of eigenvalues everywhere in the phase. We offer an analytic, microscopic proof of this conjecture in the Read-Rezayi sequence of model states. We further identify a different ES that reflects the bulk quasihole spectrum and prove a bulk-edge correspondence in the ES. Finally, we show that the finite-size corrections of the ES of the Laughlin states reveal the fractionalization of the underlying quasiparticles.