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arXiv:2310.04491

A Physical Theory of Two-stage Thermalization

Cheryne Jonay, Tianci Zhou

Published Oct 6, 2023• 1 citation
arXivHTMLPDF

Abstract

One indication of thermalization time is subsystem entanglement reaching thermal values. Recent studies on local quantum circuits reveal two exponential stages with decay rates $r_1$ and $r_2$ of the purity before and after thermalization. We provide an entanglement membrane theory interpretation, with $r_1$ corresponding to the domain wall free energy. Circuit geometry can lead to $r_1 < r_2$, producing a ``phantom eigenvalue". Competition between the domain wall and magnon leads to $r_2 < r_1$ when the magnon prevails. However, when the domain wall wins, this mechanism provides a practical approach for measuring entanglement growth through local correlation functions.

DOI
10.48550/arxiv.2310.04491

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One indication of thermalization time is subsystem entanglement reaching thermal values. Recent studies on local quantum circuits reveal two exponential stages with decay rates r1r_1r1​ and r2r_2r2​ of the purity before and after thermalization. We provide an entanglement membrane theory interpretation, with r1r_1r1​ corresponding to the domain wall free energy. Circuit geometry can lead to r1<r2r_1 < r_2r1​<r2​, producing a ``phantom eigenvalue". Competition between the domain wall and magnon leads to r2<r1r_2 < r_1r2​<r1​ when the magnon prevails. However, when the domain wall wins, this mechanism provides a practical approach for measuring entanglement growth through local correlation functions.
Thermalization time marks the scale at which the subsystem entanglement reaches its thermal value. It was recently shown to exhibit two distinct stages in a generic class of quantum circuits with decay rates r1{r}_{1}r1​ and r2{r}_{2}r2​ before and after thermalization. The two rates are observed in the Floquet dynamics of the state purity, entanglement spectra, and (local) correlation functions, both in clean and disordered systems. Entanglement membrane theory interprets r1{r}_{1}r1​ as domain-wall energy and r2{r}_{2}r2​ as either a geometric effect (phantom eigenvalue) or a competition with an emergent magnon mode. In the case when the domain wall dominates, this offers a practical measure for entanglement growth via local correlations.
We study random quantum circuits and their rate of producing bipartite entanglement, specifically with respect to the choice of 2-qubit gates and the order (protocol) in which these are applied. The problem is mapped to a Markovian process, and we prove that there are large spectral equivalence classes—different configurations have the same spectrum. Optimal gates and the protocol that generate entanglement with the fastest theoretically possible rate are identified. Relaxation towards the asymptotic thermal entanglement proceeds via a series of phase transitions in the local relaxation rate, which is a consequence of non-Hermiticity. In particular, non-Hermiticity can cause the rate to be either faster or, even more interestingly, slower than predicted by the matrix eigenvalue gap. This result is caused by expansion coefficients that grow exponentially with system size, resulting in a “phantom” eigenvalue and is due to nonorthogonality of non-Hermitian eigenvectors. We numerically demonstrate that the phenomenon also occurs in random circuits with nonoptimal generic gates, random U(4) gates, and also without spatial or temporal randomness, suggesting that it could be of wide importance in other non-Hermitian settings, including correlations.
We study random quantum circuits and their rate of producing bipartite entanglement, specifically with respect to the choice of 2-qubit gates and the order (protocol) in which these are applied.The problem is mapped to a Markovian process, and we prove that there are large spectral equivalence classes-different configurations have the same spectrum.Optimal gates and the protocol that generate entanglement with the fastest theoretically possible rate are identified.Relaxation towards the asymptotic thermal entanglement proceeds via a series of phase transitions in the local relaxation rate, which is a consequence of non-Hermiticity.In particular, non-Hermiticity can cause the rate to be either faster or, even more interestingly, slower than predicted by the matrix eigenvalue gap.This result is caused by expansion coefficients that grow exponentially with system size, resulting in a "phantom" eigenvalue and is due to nonorthogonality of non-Hermitian eigenvectors.We numerically demonstrate that the phenomenon also occurs in random circuits with nonoptimal generic gates, random U(4) gates, and also without spatial or temporal randomness, suggesting that it could be of wide importance in other non-Hermitian settings, including correlations.
Abstract Understanding thermalization and irreversibility in open quantum systems requires char-acterizing the competition between coherent unitary dynamics and dissipative environmen-tal interactions. In this work, we present a systematic numerical study of thermalization dynamics in a two-qubit open quantum system governed by a completely positive and trace-preserving Lindblad master equation. The coherent evolution rate is controlled by a real scaling parameter α multiplying the Hamiltonian commutator, while decoherence is indepen-dently controlled by a dephasing rate γ . Although such a scaling of coherent dynamics can be absorbed into a rescaling of time in isolated systems, it becomes physically meaningful in open systems where decoherence introduces an absolute timescale. We investigate how the relative strength of coherent evolution and decoherence influences observable dynamics, purity decay, entropy production, and thermalization times. Using robust local and two-site observables, we show that varying α at fixed γ produces distinct transient trajectories, even though all evolutions converge to the same steady state. We further define an operational thermalization time τ ( γ, α ) based on a purity threshold and map the resulting thermalization-time surface across a broad parameter range. Increasing α generally accelerates oscillatory dynamics and modifies the approach to equilibrium, leading to measurable shifts in thermalization times without altering asymptotic states. These findings provide a clear and physically consistent characterization of how Hamil-tonian strength and decoherence rate jointly determine thermalization behavior in open quantum systems. The framework offers a practical parameterization for numerical mod-eling and experimental studies of coherent–dissipative competition in quantum information processing, quantum control, and open-system dynamics.
Energy-filtered quantum states are promising candidates for efficiently simulating thermal states. We explore a protocol designed to transition a product state into an eigenstate located in the middle of the spectrum; this is achieved by gradually reducing its energy variance, which allows us to comprehensively understand the crossover phenomenon and the subsequent convergence toward thermal behavior. We introduce and discuss three energy-filtering regimes (short, medium, and long), and we interpret them as stages of thermalization. We show that the properties of the filtered states are locally indistinguishable from those of time-averaged density matrices, routinely employed in the theory of thermalization. On the other hand, nonlocal quantum correlations are generated in the medium regimes and are witnessed by the Rényi entanglement entropies of subsystems, which we compute via replica methods. Specifically, two-point correlation functions break cluster decomposition, and the entanglement entropy of large regions scales as the logarithm of the volume during the medium filter time.
A very fundamental problem in quantum statistical mechanics involves whether—and how—an isolated quantum system will reach thermal equilibrium after waiting a long time. In quantum systems that do thermalize, the long-time expectation value of any “reasonable” operator will match its predicted value in the canonical ensemble. The Eigenstate Thermalization Hypothesis (ETH) posits that this thermalization occurs at the level of each individual energy eigenstate; in fact, any single eigenstate in a microcanonical energy window will predict the expectation values of such operators exactly. In the first part of this dissertation, we identify, for a generic model system, precisely which operators satisfy ETH, as well as limits to the information contained in a single eigenstate. Remarkably, our results strongly suggest that a single eigenstate can contain information about energy densities—and therefore temperatures—far away from the energy density of the eigenstate.Additionally, we study the possible breakdown of quantum thermalization in a model of itinerant electrons on a one-dimensional chain, with both spin and charge degrees of freedom. This model exhibits peculiar properties in the entanglement entropy, the apparent scaling of which is modified from a “volume law” to an “area law” after performing a partial, site-wise measurement on the system. These properties and others suggest that this model realizes a new, non-thermal phase of matter, known as a Quantum Disentangled Liquid (QDL). The putative existence of this phase has striking implications for the foundations of quantum statistical mechanics.