Predicting Dynamics from Flows of the Eigenstate Thermalization Hypothesis

TL;DR

The paper tackles the challenge of predicting far‑from‑equilibrium quantum dynamics in well‑thermalizing systems by leveraging the eigenstate thermalization hypothesis (ETH) as a maximal entropy ansatz for matrix elements. It introduces the statistical Jacobi approximation (SJA), which treats Jacobi rotations—used to diagonalize a perturbed Hamiltonian—as a stochastic process under ETH, and derives integrodifferential flow equations for ETH form factors that govern dynamics under H = H0 + J V. By solving these flow equations iteratively, the authors predict quenched dynamics ⟨A(t)⟩H and autocorrelators in the perturbed thermal state, with quantitative agreement to exact numerics in random‑matrix models and one‑dimensional spin chains. The framework is marked by a single statistical input, the decimated‑element distribution ρ_dec, and a thermodynamic‑limit‑friendly structure, offering a scalable path to compute dynamical responses from ETH inputs and suggesting extensions to Floquet systems and OTOCs.

Abstract

Analytical treatments of far-from-equilibrium quantum dynamics are few, even in well-thermalizing systems. The celebrated eigenstate thermalization hypothesis (ETH) provides a post hoc ansatz for the matrix elements of observables in the eigenbasis of a thermalizing Hamiltonian, given various response functions of those observables as input. However, the ETH cannot predict these response functions. We introduce a procedure, dubbed the statistical Jacobi approximation (SJA), to update the ETH ansatz after a perturbation to the Hamiltonian and predict perturbed response functions. The Jacobi algorithm diagonalizes the perturbation through a sequence of two-level rotations. The SJA implements these rotations statistically assuming the ETH throughout the diagonalization procedure, and generates integrodifferential flow equations for various form factors in the ETH ansatz. We approximately solve these flow equations, and predict both quench dynamics and autocorrelators in the thermal state of the perturbed Hamiltonian. The predicted dynamics compare well to exact numerics in both random matrix models and one-dimensional spin chains.

Figures (8)

• Figure 1: The statistical Jacobi approximation (SJA) predicts the dynamics of a well-thermalizing Hamiltonian $H$, given a Hamiltonian $H_0$ satisfying the ETH and observable dynamics generated by the Hamiltonian $H_0$, denoted by $\braket{A(t)}_{H_0}$. Specifically, it takes the form factors appearing in the ETH ansatz for the Hamiltonian $H_0$ [Eq. \ref{['Eq:Spectralfunctions']}] as input, and computes the form factors for the Hamiltonian $H=H_0+JV$. Response functions of the Hamiltonian $H$, e.g., $\braket{A(t)}_{H}$, follow from the form factors.
• Figure 2: Sketch of the Jacobi algorithm: (a) At each iteration step, identify the largest off-diagonal matrix element $w_0$. This is the element to be decimated. (b) Perform the 2-level unitary rotation $R_0$ that sets $H_{a_0b_0}$ to zero. This rotation affects the rows and columns associated with indices $a_0$ and $b_0$. (c) After the rotation, $H_{a_1b_1}=0$. (d) Repeated decimations lead to a full diagonalization of the matrix in $\mathcal{O}(N^2)$ rotations, with $N$ being the matrix size. The grayscale denotes the absolute value of the matrix elements, increasing from white to black.
• Figure 3: Sketch of one update step for the quantity $B^{(n)}_{ij}=\rho^{(n)}_{ij}A^{(n)}_{ji}$. (a) The $(n+1)$th update step affects rows (red) and columns (blue) with indices $a_n$ and $b_n$. The update can be split into three contributions: (b) Rotations between elements $B^{(n)}_{aj}$ and $B^{(n)}_{bj}$ with $j\neq a,b$ (shown in red). (c) Rotations between elements $B^{(n)}_{ia}$ and $B^{(n)}_{ib}$ with $i\neq a,b$ (shown in blue). Contributions (b) and (c) lead to the terms $F_1[B](w,E,\omega)$, $F_2[B](w,E,\omega)$ in the flow equation Eq. \ref{['Eq:resultshort']}. (d) The third contribution, shown in purple, accounts for the update of the elements $B^{(n)}_{ab}$ and $B^{(n)}_{ba}$, leading to the term $G[A,p](w,E,\omega)$ in Eq. \ref{['Eq:resultshort']}. Additionally, there can be correlations between $A^{n}_{ij}$ and $w_n$. These have to be included separately and lead to the term $D[p](w,E,\omega)$ in Eq. \ref{['Eq:resultshort']}.
• Figure 4: Evolution of $\braket{H_0^2}_H$ under a quench Eq. \ref{['eq:Quench1']}, $J=0.5$, $\sigma_\omega=1.5$, $\epsilon=1/3$, $N=2048$. (a) Comparison of exact time evolution (black), with time-dependent perturbation theory (blue), and the statistical Jacobi approximation to different orders (red, orange, and yellow). The data are averaged over 10 random realizations. The agreement with the exact curve improves at second order. (b) Absolute error $|\braket{H_0^2}_{H,\text{approx}}-\braket{H_0^2}_{H,\text{exact}}|$. The iterative solution outperforms time-dependent perturbation theory at intermediate and long times.
• Figure 5: Evolution of $\braket{H_0^2}_H$ under a quench Eq. \ref{['eq:Quench2']}, $\sigma_\omega=0.3$, $\omega_0=0.7$, $N=2048$. Comparison of exact time evolution (black), with time-dependent perturbation theory (TDPT, blue), and the statistical Jacobi approximation to different orders (SJA, red, and orange). (a) $\epsilon=J/\sigma_\omega=2/3$, (b) $\epsilon=J/\sigma_\omega=5/3$. Inset: the absolute error$|\braket{H_0^2}_{H,\text{approx}}-\braket{H_0^2}_{H,\text{exact}}|$. The data are averaged over 10 random realizations. While the oscillations at short times are not accurately captured by any approximate method for strong perturbations, the SJA captures the long-time dynamics in that case.