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arXiv:1702.08894

Quantum thermalization dynamics with Matrix-Product States

Eyal Leviatan, Frank Pollmann, Jens Hjorleifur Bardarson, David A. Huse, Ehud Altman

Published Feb 28, 2017• 83 citations
arXivHTMLPDF

Abstract

We study the dynamics of thermalization following a quantum quench using tensor-network methods. Contrary to the common belief that the rapid growth of entanglement and the resulting exponential growth of the bond dimension restricts simulations to short times, we demonstrate that the long time limit of local observables can be well captured using the time-dependent variational principle. This allows to extract transport coefficients such as the energy diffusion constant from simulations with rather small bond dimensions. We further study the characteristic of the chaotic wave that precedes the emergence of hydrodynamics, to find a ballistic diffusively-broadening wave-front.

DOI
10.48550/arxiv.1702.08894

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We study the dynamics of thermalization following a quantum quench using tensor-network methods. Contrary to the common belief that the rapid growth of entanglement and the resulting exponential growth of the bond dimension restricts simulations to short times, we demonstrate that the long time limit of local observables can be well captured using the time-dependent variational principle. This allows to extract transport coefficients such as the energy diffusion constant from simulations with rather small bond dimensions. We further study the characteristic of the chaotic wave that precedes the emergence of hydrodynamics, to find a ballistic diffusively-broadening wave-front.
Extracting late-time hydrodynamic transport properties directly from microscopic quantum models, especially in dimensions beyond one, is believed to be classically intractable. This owes, in part, to the extensive growth of entanglement in higher dimensions and often a mismatch between geometric and tensor network locality. Here, we extend the Density Matrix Truncation (DMT) algorithm to accurately and quantitatively predict high-temperature transport coefficients (e.g. diffusion constants) in up to three dimensions. Additionally, we demonstrate the ability to capture the slow crossover dynamics characteristic of weakly broken integrability. The intuition underlying our approach is to preserve the geometrically local information crucial for hydrodynamics throughout the time evolution, despite using a one-dimensional matrix product state ansatz for higher-dimensional calculations. Our work pushes the boundary of classical computability and opens the door to benchmarking the high-temperature, late-time dynamics of generic quantum simulators.
Thermalization of chaotic quantum many-body systems under unitary time evolution is related to the growth in complexity of initially simple Heisenberg operators.Operator growth is a manifestation of information scrambling and can be diagnosed by out-of-time-order correlators (OTOCs).However, the behavior of OTOCs of local operators in generic chaotic local Hamiltonians remains poorly understood, with some semiclassical and large-N models exhibiting exponential growth of OTOCs and a sharp chaos wave front and other random circuit models showing a diffusively broadened wave front.In this paper, we propose a unified physical picture for scrambling in chaotic local Hamiltonians.We construct a random time-dependent Hamiltonian model featuring a large-N limit where the OTOC obeys a Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP) type equation and exhibits exponential growth and a sharp wave front.We show that quantum fluctuations manifest as noise (distinct from the randomness of the couplings in the underlying Hamiltonian) in the FKPP equation and that the noise-averaged OTOC exhibits a crossover to a diffusively broadened wave front.At small N, we demonstrate that operator growth dynamics, averaged over the random couplings, can be efficiently simulated for all time using matrix product state techniques.To show that time-dependent randomness is not essential to our conclusions, we push our previous matrix product operator methods to very large size and show that data for a timeindependent Hamiltonian model are also consistent with a diffusively broadened wave front.
Thermalisation in closed quantum systems occurs through a process of dephasing due to parts of the system outside of the window of observation, gradually revealing the underlying thermal nature of eigenstates. In contrast, closed classical systems thermalize due to dynamical chaos. We demonstrate a deep link between these processes. Projecting quantum dynamics onto variational states using the time-dependent variational principle, results in classical chaotic Hamiltonian dynamics. We study an infinite spin chain in two ways-using the matrix product state ansatz for the wavefunction and for the thermofield purification of the density matrix-and extract the full Lyapunov spectrum of the resulting dynamics. We show that the entanglement growth rate is related to the Kolmogorov-Sinai entropy of dynamics projected onto states with appropriate entanglement, extending previous results about initial entanglement growth to all times. The Lyapunov spectra for thermofield descriptions of thermalizing systems show a remarkable semi-circular distribution.
Out-of-equilibrium dynamics of non-integrable Hamiltonian many-body quantum systems are characterized by highly entangled wave functions. Near-maximal entanglement arises in systems exhibiting thermalization or pre-thermalization, where the system converges to a steady state with a fixed energy density. Classical simulation of the time dependence of such wave functions requires exponential resources. However, typical computations aim to estimate expectation values of local operators and correlation functions to some expected precision. For thermalizing systems at sufficiently high energy densities, such computations can be done without storing the full wave function by instead simulating the evolution of the local operator, which requires significantly fewer resources. Nonetheless, constructing such resource-efficient classical algorithms remains a challenge for intermediate energy densities, where simulating both the wave function and operator evolution is costly. In this paper, we propose a heuristic approach to accelerate the convergence of Matrix Product State (MPS) simulations of expectation values, applicable across a broad range of energy densities. We estimate the desired observables by rescaling the MPS results at low bond dimensions with a factor that depends on the fidelity of the MPS wave function. Using this technique, we simulated the dynamics of the two-dimensional Transverse-Field Ising Model (TFIM) on a 7×87\times87×8 grid with periodic boundary conditions, using a maximum bond dimension of χ=4096χ= 4096χ=4096 on a single A100 GPU, as well as the dynamics of the two-dimensional XY model on grids of size up to 9×99\times99×9. We compare our TFIM results to similar simulations on a digital quantum processor [R. Haghshenas et al., Nature 653, 56 (2026)], demonstrating excellent agreement and confirming the predictive power of our method.
Understanding the emergent system-bath correlations in non-Markovian and non-perturbative open systems is a theoretical challenge that has benefited greatly from the application of Matrix Product State (MPS) methods. Here, we propose an autonmously adapative variant of the one-site Time-Dependent-Variational-Principle (1TDVP) method for many-body MPS wave-functions in which the local bond-dimensions can evolve to capture growing entanglement 'on the fly'. We achieve this by efficiently examining the effect of increasing each MPS bond-dimension in advance of each dynamic timestep, resulting in an MPS that can dynamically and inhomogeneously restructure itself as the complexity of the dynamics grows across time and space. This naturally leads to more efficient simulations, oviates the need for multiple convergence runs, and, as we demonstrate, is ideally suited to the typical, finite-temperature 'impurity' problems that describe open quantum system connected to multiple environments.
We study the thermalization of the transverse field Ising chain with a power law decaying interaction \sim 1/r^{\alpha} ∼ 1 / r α following a global quantum quench of the transverse field in two different dynamical regimes. The thermalization behavior is quantified by comparing the full probability distribution function (PDF) of the evolving states with the corresponding thermal state given by the canonical Gibbs ensemble (CGE). To this end, we used the matrix product state (MPS)-based Time Dependent Variational Principle (TDVP) algorithm to simulate both real time evolution following a global quantum quench and the finite temperature density operator. We observe that thermalization is strongly suppressed in the region with strong confinement for all interaction strengths \alpha α , whereas thermalization occurs in the region with weak confinement.