Refinements of the Eigenstate Thermalization Hypothesis under Local Rotational Invariance via Free Probability

TL;DR

This work refines the Eigenstate Thermalization Hypothesis (ETH) by embedding matrix-element statistics into free-probability and distinguishing global versus local rotational invariance. It develops a quantitative framework using free cumulants and Weingarten calculus to obtain leading and first subleading contributions to multi-point ETH correlations, with explicit energy dependence via local free cumulants. The results connect ensemble averages over random-basis transformations to empirical energy-window averages, and are validated in non-integrable Floquet models, showing robust agreement with predicted scaling. The approach provides a principled route to understand subleading corrections to ETH and their potential imprint on long-time thermalization and chaotic dynamics, with extensions to operator-valued settings and banded/unitary structures.

Abstract

The Eigenstate Thermalization Hypothesis (ETH) was developed as a framework for understanding how the principles of statistical mechanics emerge in the long-time limit of isolated quantum many-body systems. Since then, ETH has shifted the attention towards the study of matrix elements of physical observables in the energy eigenbasis. In this work, we revisit recent developments leading to the formulation of full ETH, a generalization of the original ETH ansatz that accounts for multi-point correlation functions. Using tools from free probability, we explore the implications of local rotational invariance, a property that emerges from the statistical invariance of observables under random basis transformations induced by small perturbations of the Hamiltonian. This approach allows us to make quantitative predictions and derive an analytical characterization of subleading corrections to matrix-element correlations, thereby refining the ETH ansatz. Moreover, our analysis links the statistical properties of matrix elements under random basis changes to the empirical averages over energy windows that are usually considered when dealing with a single instance of the ensemble. We validate our analytical predictions through comparison with numerical simulations in non-integrable Floquet systems.

Figures (7)

• Figure 1: Illustrative examples of the random matrix $\mathcal{U}$ in Eq. \ref{['ASS']}, employed to rotate the observable $A$. Different refined toy models for deriving ETH are shown. In (a) the matrix $\mathcal{U}$ is a global Haar unitary matrix, so that $A$ exhibits consequently global rotational invariant. Local rotational invariance is achieved by incorporating a local structure, as in (b) and (c): in (b) the matrix $\mathcal{U}$ is composed of disjointed blocks of independent Haar unitaries, each acting only on a local interval of states. The last more refined toy model in (c) is a smoothed version of the previous one, see Sec. \ref{['sec_change']} for discussion.
• Figure 2: (a) Graphical representation of some partitions of four elements (in blue) together with their respective Kreweras complements (in red). The unique crossing partition for $n=4$ is shown in the lower right corner, and has no Kreweras complement. (b) $NC(4)$ (sub)lattice of non-crossing partitions; lines between rows are connecting partitions at a unit distance. At the bottom, the discrete partition (finest), at the top the trivial one (coarsest). Correspondingly, $NC(4)$ can be seen as the geodesic interval $[e,\gamma_4]$, subset of the Cayley graph, for which permutations from one row to the other are related via a single transposition. Each set of permutations following a path from $e$ to $\gamma_4$ defines a single geodesic in the interval; an explicit example is highlighted in green, and on the right.
• Figure 3: Numerical results for the Floquet system in \ref{['Floquet_operator']}, and for the observable $A=\sum_i Z_i/L$. We numerically reproduce Eqs. \ref{['4p_1_ene_Floquet']}--\ref{['4p_2_ene_Floquet']}, for $\omega_1=\omega_2=\omega$ chosen in $[-2\pi,2\pi]$. The error bars are given by averaging each product, for each $\omega$, over different values of the smoothing parameter $\Delta$, as explained in the text. (a),(b),(c) refer to \ref{['4p_1_ene_Floquet']}, while (d),(e),(f) to \ref{['4p_2_ene_Floquet']}. In (a),(d) we compare the right-hand side (purple curve) with the respective leading order (pink curve) for fixed $L=16$; for (a) the leading order is zero, while in (d) the data nicely reproduce the expected leading order factorization. In (b),(e) we compare the difference between the right-hand side and the leading contribution (brown curve) with the respective subleading correction (orange curve) for fixed $L=16$. The data verify the subleading correction to the factorization. In (c),(f) we represent the scalings of all the curves as a function of $D$. The scalings were studied for values $L\in[8,10,12,14,16]$, and for a fixed value of $\omega^*\approx-2.13$. While all the curves in (c) scale as $1/L^2D^2$, since the leading order is zero, in (f) we observe the different scalings between the leading (purple and pink curves), as $1/L^2D^2$ and the subleading (brown and orange curves), as $1/L^2D^3$.
• Figure 4: Numerical comparison between the Floquet systems with and without time reversal symmetry, that is, belonging to the orthogonal or the unitary symmetry class, with and without disorder, presented in App. \ref{['app_num_GOE']}. We numerically reproduce equation Eq. \ref{['GOE_eq']}. The error bar is given by averaging $\overline{A_{ij}A_{ji}}(0)$ over different values of the smoothing parameter $\Delta$, as explained in the text. For the first column $L\in [8,10,12,14,16]$, while for the second $L \in [6,8,10,12]$. We compare $|\overline{A_{ii}^2}-\overline{A_{ii}}^2|$ (brown curve) with the respective subleading correction $|\overline{A_{ij}A_{ji}}(0)|$ (orange curve), as functions of $D$, and we calculate the ratio $r=|\overline{A_{ii}^2}-\overline{A_{ii}}^2|/|\overline{A_{ij}A_{ji}}(0)|$ for the biggest system size for each figure. All the curves well reproduce the expected scaling of $1/LD$ for the system without disorder, and $1/D$ for the system with disorder. In (a) and (b) we notice that the two curves almost coincide, reproducing the result for the unitary symmetry class \ref{['ex_ii_LRI']}. In (c) and (d), instead, $|\overline{A_{ii}^2}-\overline{A_{ii}}^2|$ is approximately twice as large as the subleading one, as expected for the orthogonal symmetry class \ref{['GOE_eq2']}.
• Figure 5: Numerical results for the Floquet system with disorder presented in App. \ref{['app_obs']}, and for the observable $A=X_{L/2}$. Similarly to Fig. \ref{['fig_numerics']}, we numerically reproduce equations Eqs. \ref{['4p_1_ene_Floquet']}--\ref{['4p_2_ene_Floquet']}, for $\omega_1=\omega_2=\omega$ chosen in $[-2\pi,2\pi]$. The error bar is given by averaging each product, for each $\omega$, over different values of the smoothing parameter $\Delta$, as explained in the text. (a),(b),(c) refer to \ref{['4p_1_ene_Floquet']}, while (d),(e),(f) to \ref{['4p_2_ene_Floquet']}. In (a),(d) we compare the right-hand side (purple curve) with the respective leading order (pink curve) for fixed $L=12$; for (a) the leading order is zero, while in (d) the data very well reproduce the expected leading order factorization. In (b),(e) we compare the difference between the right-hand side and the leading contribution (brown curve) with the respective subleading correction (orange curve) for fixed $L=12$. The data verify the subleading correction to the factorization. In (c),(f) we represent the scalings of all the curves as a function of $D$. The scalings were studied for values $L\in[6,8,10,12]$, and for a fixed value of $\omega^*\approx-2.13$. While all the curves in (c) scale as $1/D^2$, since the leading order is zero, in (f) we observe the different scalings between the leading (purple and pink curves), as $1/D^2$ and the subleading (brown and orange curves), as $1/D^3$.