⚛️ quantum physics

Predicting Dynamics from Flows of the Eigenstate Thermalization Hypothesis

This paper introduces the statistical Jacobi approximation (SJA), a procedure that extends the Eigenstate Thermalization Hypothesis (ETH) by generating flow equations for response functions under Hamiltonian perturbations, thereby enabling accurate predictions of far-from-equilibrium quantum dynamics that outperform standard perturbation theory.

Original authors: Dominik Hahn, David M. Long, Marin Bukov, Anushya Chandran

Published 2026-10-19

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CC BY 4.0

Original authors: Dominik Hahn, David M. Long, Marin Bukov, Anushya Chandran

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Isolated quantum systems, such as a collection of atoms trapped in a vacuum, are notoriously difficult to predict once they are disturbed. Even when these systems are known to eventually settle into a calm, thermal state, calculating exactly how they move and change in the moments after a sudden jolt remains a major challenge for physicists. For decades, a guiding principle called the eigenstate thermalization hypothesis has offered a way to describe what these systems look like once they have settled down. This idea suggests that if you look at a single, specific energy state of a chaotic quantum system, the properties you measure there will look exactly like the average properties of a hot, thermal gas. While this hypothesis has been incredibly successful at describing the final state of a system, it has historically failed to predict the journey itself. It acts like a map of the destination without telling you how to get there, leaving scientists unable to forecast the dynamic behavior of a system immediately after it is perturbed.

A team of researchers has now taken a significant step toward turning this static description into a predictive tool for the entire journey. They developed a new method, which they call the statistical Jacobi approximation, to calculate how a quantum system evolves after its underlying energy landscape is slightly altered. The researchers started with a system they already understood well, one that follows the established rules of thermalization. They then introduced a small change to the system's energy structure, a process known as a quench, and asked a difficult question: how do the observable properties of this new, slightly different system behave over time? To answer this, they did not try to solve the complex equations of motion directly. Instead, they treated the mathematical transformation required to switch from the old system to the new one as a series of tiny, random rotations. By assuming that the system remains thermal throughout this transformation, they derived a set of flow equations that track how the system's response functions change as the energy landscape is gradually reshaped.

The core of their approach involves a numerical algorithm known as the Jacobi method, which is typically used to find the energy levels of a matrix by systematically zeroing out the connections between different states. The researchers realized that they could treat the sequence of these zeroing-out steps not as a rigid calculation, but as a statistical process. They imagined the system undergoing a continuous flow where the strongest connections between energy states are smoothed out first, followed by weaker ones. Throughout this process, they assumed the system maintained its thermal character, allowing them to write down equations that describe how the system's ability to respond to disturbances evolves. These equations depend on a single statistical input: a description of how the energy states are connected and how strongly they are linked during the transformation. This input can be measured from small, manageable computer simulations, yet the resulting equations are designed to work for systems of any size.

When the researchers tested their method against exact computer simulations of random quantum systems and one-dimensional chains of spins, the results were striking. The new approach successfully predicted the time evolution of physical quantities, such as the total magnetization of a spin chain or the energy variance of a random system, with high accuracy. Crucially, it outperformed the standard mathematical tools used for such problems, known as time-dependent perturbation theory. While the traditional methods worked well for very short times or extremely weak disturbances, they failed to capture the correct long-term behavior of the system. In contrast, the statistical Jacobi approximation remained accurate across all time scales, correctly predicting both the initial rapid changes and the final steady state the system settles into. The method was particularly effective in cases where the disturbance was strong enough to break the assumptions of traditional theories, yet the system still behaved in a way that allowed for thermalization.

The researchers found that their method works by capturing non-perturbative effects, meaning it accounts for changes that are too large to be described by simply adding up small corrections. By using the statistical distribution of the connections between energy states as a guide, the method effectively bypasses the limitations of standard approximation techniques. In simulations of a mixed-field Ising model, a specific type of magnetic system, the new method accurately reproduced the oscillations and saturation values of the magnetization that other methods missed. The study also showed that the predictions remain stable as the size of the system increases, suggesting that the method can be used to understand large, complex systems based on data gathered from much smaller ones. This stability arises because the most significant changes in the system's behavior are driven by the strongest connections between energy states, which are less affected by the finite size of the computer model.

This work represents a shift from a descriptive to a predictive framework for quantum dynamics. By combining the established idea that individual quantum states are thermal with a statistical treatment of how those states transform, the researchers have created a tool that can forecast the future of a disturbed quantum system. The method does not require solving the full, impossible equations of motion for a large system. Instead, it relies on the statistical properties of the system's internal connections, which can be extracted from small-scale simulations. The results suggest that for systems that are known to thermalize, the path to a new equilibrium can be mapped out with high precision, even when the disturbance is significant. This opens the door to understanding a wide range of dynamic phenomena, from the behavior of materials under sudden stress to the evolution of complex quantum networks, providing a clearer picture of how order emerges from chaos in the quantum world.

Technical Summary: Predicting Dynamics from Flows of the Eigenstate Thermalization Hypothesis

Problem Statement
Analytical treatments of far-from-equilibrium quantum dynamics remain scarce, even for systems that are well-thermalizing. While the Eigenstate Thermalization Hypothesis (ETH) provides a successful post hoc description of observables in thermalizing systems—ansatzing matrix elements in the energy eigenbasis given specific response functions—it lacks predictive power regarding those response functions themselves. Specifically, the standard ETH cannot predict how the form factors (smooth functions governing matrix elements) evolve when the underlying Hamiltonian is perturbed. Consequently, there is no established analytical framework to predict the dynamics of an observable $A$ under a perturbed Hamiltonian $H = H_0 + JV$ , given only the dynamics under the unperturbed Hamiltonian $H_0$ and the statistical properties of the perturbation.

Methodology: The Statistical Jacobi Approximation (SJA)
The authors introduce the Statistical Jacobi Approximation (SJA), a procedure that updates the ETH ansatz following a perturbation to the Hamiltonian. The core of the method relies on the Jacobi diagonalization algorithm, which transforms the eigenbasis of $H_0$ into the eigenbasis of $H$ through a sequence of two-level unitary rotations.

Statistical Flow: Instead of performing exact diagonalization, the SJA treats the Jacobi algorithm statistically. It assumes that the ETH holds throughout the diagonalization process. The algorithm proceeds by iteratively zeroing out the largest off-diagonal matrix elements ( $w$ ) of the perturbation. The authors parametrize the progress of this diagonalization not by the iteration count, but by the magnitude of the decimated element $w$ , which flows from an initial value $w_0$ (the largest off-diagonal element of $JV$ in the $H_0$ basis) to zero.
Form Factor Evolution: The method tracks the evolution of "form factors" (e.g., $B(w, E, \omega)$ , $A(w, E)$ , $p(w, E)$ ) which encode the matrix elements of the observable and the density matrix in the rotating basis.
Flow Equations: By averaging over the statistical distribution of Jacobi rotations (specifically the joint number density of decimated elements and the matrix elements of the observable between the rotated states), the authors derive integrodifferential flow equations for these form factors. These equations describe how the form factors change as $w$ decreases.
Iterative Solution: Since the flow equations are difficult to solve exactly, the authors solve them iteratively as a series expansion in the perturbation strength $J$ . The leading order of this expansion recovers second-order time-dependent perturbation theory (TDPT), but higher-order terms incorporate non-perturbative corrections derived from the statistical input of the Jacobi algorithm.

Key Contributions

Derivation of Flow Equations: The paper derives a set of linear integrodifferential equations (Eqs. 38, 45, 46, 47) governing the evolution of ETH form factors under Jacobi rotations. These equations depend on the initial form factors (from $H_0$ ) and a statistical input: the joint number density of decimated elements $\rho_{dec}$ and the joint density involving the observable $\rho_{dec, A}$ .
Non-Perturbative Predictions: While the solution is expanded in $J$ , the method is non-perturbative in nature because the statistical input (the Jacobi spectral function $|f_{Jac}|^2$ ) is obtained from the exact numerical implementation of the Jacobi algorithm on small systems. This captures effects that standard perturbation theory misses, particularly when the spectral function of the perturbation differs significantly from the Jacobi spectral function.
Unified Framework for Dynamics: The SJA provides a unified approach to predict both quench dynamics (evolution from a stationary state of $H_0$ under $H$ ) and thermal autocorrelators (evolution under $H$ starting from a thermal state of $H$ ).

Results
The authors validate the SJA against numerically exact diagonalization in two distinct settings:

Random Matrix Models: In models where matrix elements are uncorrelated, the SJA predictions for the time evolution of observables (e.g., $\langle H_0^2(t) \rangle$ ) agree quantitatively with exact numerics. The iterative SJA solutions outperform second-order TDPT, particularly in capturing the correct long-time steady-state values. The method remains accurate even for perturbations where the spectral functions of the perturbation and the Jacobi flow diverge significantly.
One-Dimensional Spin Chains: The SJA is tested on a mixed-field Ising model (a structured, thermalizing system). Here, the SJA again outperforms linear response and second-order TDPT. Specifically, higher-order SJA approximations correctly capture the amplitude of oscillations and the long-time saturation values of the total magnetization, whereas lower-order perturbative methods fail to reproduce the steady state accurately.
Finite-Size Scaling: The authors demonstrate that the statistical inputs (joint number densities) and the resulting SJA predictions exhibit weak dependence on system size for well-thermalizing systems. This suggests that the method can predict the thermodynamic limit dynamics using statistical data computed from relatively small system sizes.

Significance and Claims
The paper claims to take the "first steps" toward transforming the ETH from a descriptive hypothesis into a predictive theory for quantum dynamics. The significance of the SJA lies in its ability to:

Bridge the gap between the static ETH ansatz and dynamic response functions under perturbations.
Provide a systematic way to go beyond standard perturbation theory without requiring full diagonalization of large systems.
Offer a framework where the "path to generalization" is clear: one can apply the same flow equation derivation to generalized ETH ansatze (e.g., for out-of-time-ordered correlators) to predict more complex dynamical quantities.

The authors emphasize that while the Jacobi algorithm itself is not competitive with modern tensor-network methods for exact diagonalization at fixed large system sizes, the SJA is less plagued by finite-size effects. This is because the flow equations are derived in the thermodynamic limit, and the statistical distribution of rotations stabilizes with increasing system size, allowing for accurate predictions of large-system dynamics based on small-system statistics.

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