Fast Thermalization from the Eigenstate Thermalization Hypothesis

TL;DR

The paper establishes a rigorous link between the Eigenstate Thermalization Hypothesis (ETH) and finite-time thermalization of open quantum systems by deriving a finite-resource Davies-type generator with bath refresh. Under ETH, interactions modeled as i.i.d. random matrices within a near-diagonal band create quantum expanders that enable a fast, classical-like random walk on the energy spectrum, yielding polynomial-time convergence to a Gibbs-like fixed point. It develops two generator frameworks—the rounded (Lbar) and the realistic (L)—and provides explicit resource bounds (bath size, refresh rate, and total run-time) for approximating joint system-bath dynamics at finite times. The approach combines ETH-inspired random-matrix modeling, concentration of measure, and MLSI/approximate tensorization to lift local spectral gaps to global convergence, with detailed analysis of a quasi-free Fermionic bath and a precise accounting of Lamb-shift effects. These results illuminate how chaotic quantum systems thermalize on finite times and suggest quantum Gibbs-sampling avenues with potential algorithmic advantages for thermodynamic tasks.

Abstract

The Eigenstate Thermalization Hypothesis (ETH) has played a major role in understanding thermodynamic phenomena in closed quantum systems. However, its connection to the timescale of thermalization for open system dynamics has remained elusive. This paper establishes a rigorous link between ETH and fast thermalization to the global Gibbs state. Specifically, we demonstrate fast thermalization for a system coupled weakly to a bath of quasi-free Fermions that we refresh periodically. To describe the joint evolution, we derive a finite-time version of Davies' generator with explicit error bounds and resource estimates. Our approach exploits a critical feature of ETH: operators in the energy basis can be modeled by independent random matrices in a near-diagonal band. This gives quantum expanders at nearby eigenstates of the Hamiltonian and reduces the problem to a one-dimensional classical random walk on the energy eigenstates. Our results explain finite-time thermalization in chaotic open quantum systems.

Figures (13)

• Figure 1: The model of thermalization we consider in this work. We weakly couple the system to a quasi-free Fermionic bath that we routinely refresh. The process is markovian and parameterized by the Hamiltonian $\bm{H}_S$, the set of interactions $\bm{A}^a$, the inverse temperature $\beta$, and the bath.
• Figure 2: (Left) The infinite-time limit leads to a classical Markov chain generator (for energy eigenstate inputs). The Gibbs state $\bm{ \sigma}_{\beta}$ is a fixed point. (Middle) The finite-time evolution leads to a "semi-classical" generator that retains coherence within nearby energies ( for block-diagonal inputs in the energy basis). The rounded Gibbs state $\bar{ \bm{ \sigma}}_{\beta}$ is a fixed point. (Right) The operator at rounded frequency $\bm{A}(\bar{\omega})$ is dissected by projectors $\bm{P}_{\bar{\nu}_1}$ and $\bm{P}_{\bar{\nu}_2}$. Coherence remains within each subspace.
• Figure 3: The function $f_{\omega}$ in the ETH ansatz is expected to have most weight below some scale (e.g., $1/n^{1/d}$ for a d-dimensional lattice). The scale $\Delta_{RMT}$ that random matrix behavior kicks in is believed to be smaller and depends on the dynamics of the system dymarsky2018bound. The energy scales in between (the question mark) may partly exhibit random matrix behavior but retain correlation between entries 2020_ETH_small_omega_Richterwang2021eigenstate2021_ETH_OTOC_Brenes.
• Figure 4: The interdependence of the concepts and the parameters in this work. The colored texts and arrows distinguish the notions for the rounded generator and the realistic generator and those in common are in black. This manifests the flexibility of the presented arguments. The main assumption is ETH that the interaction terms are prescribed by i.i.d. random matrices. This gives quantum expanders at local energies and reduces the calculation to a classical random walk on the energy eigenbasis. Interestingly, the decay of correlation in Gibbs state, which featured in the classical martinelli1999lectures and commuting Hamiltonian literature kastoryano2016commutingcapel2021modified, is now a replaceable component. Here, it serves the only purpose that the density of states is Gaussian-like brandao2015equivalence, which, through standard conductance calculation, implies the random walk on the Gibbs distribution mixes rapidly.
• Figure 5: A quantum expander rapidly mixes the inputs. An example is the channel with a few i.i.d. Haar random Kraus operators $\mathcal{N}: = \frac{1}{2\left\vert {a} \right\vert} \sum_a \bm{U}_a [ \cdot ] \bm{U}_a^\dagger + \bm{U}_a^\dagger [ \cdot ] \bm{U}_a$. Intuitively, the i.i.d. Haar random unitaries are "pointing at different directions" so that any input state $\ket{\psi}$ maps to nearly orthogonal states $\{\bm{U}_a\ket{\psi}\}$. Roughly speaking, this connection between random matrices and quantum expander is why ETH leads to thermalization.

Theorems & Definitions (82)

• Theorem 1.1: Implementing the rounded generator, informal
• Theorem 1.2: Implementing the realistic generator, informal
• Theorem 1.3: Convergence of the rounded generator, informal
• Theorem 1.4: Convergence of the realistic generator, informal
• Proposition 2.3.1: The dissipative part is detailed balanced
• proof : Proof of Proposition \ref{["prop:D'_DB"]}
• Proposition 2.3.2: Trace preserving
• Corollary 2.3.1: Gibbs fixed point
• Proposition 2.3.3: Correlators
• proof