False signatures of non-ergodic behavior in disordered quantum many-body systems

TL;DR

This work analyzes how disorder averaging in disordered quantum many-body systems can produce non-Gaussian signatures in local observables that masquerade as ETH breakdown. By decomposing observables into components parallel and perpendicular to the Hamiltonian moments, the authors derive a microcanonical framework showing that the leading observable behavior $s_l^z(E)$ is linearly related to the local disorder $h_l$ and that the orthogonal (non-commuting) part scales as $O(1/\sqrt{D})$ in ergodic systems. They demonstrate that the observed disorder-induced structures in $\langle n|\hat{s}_l^z|n\rangle$ arise from the energy-window choice; removing the microcanonical contribution $s_l^z(E)$ yields Gaussian, ETH-consistent distributions in both eigenstates and long-time dynamics. The results offer a practical prescription for correctly diagnosing ETH in disordered systems and connect equilibrium eigenstate properties to non-equilibrium quench outcomes, with implications for interpreting experiments in quasiperiodic and random disorder regimes.

Abstract

Ergodic isolated quantum many-body systems satisfy the eigenstate thermalization hypothesis (ETH), i.e., the expectation values of local observables in the system's eigenstates approach the predictions of the microcanonical ensemble. However, the ETH does not specify what happens to expectation values of local observables within an energy window when the average over disorder realizations is taken. As a result, the expectation values of local observables can be distributed over a relatively wide interval and may exhibit nontrivial structure, as shown in [Phys. Rev. B \textbf{104}, 214201 (2021)] for a quasiperiodic disordered system for site-resolved magnetization. We argue that the non-Gaussian form of this distribution may \textit{falsely} suggest non-ergodicity and a breakdown of ETH. By considering various types of disorder, we find that the functional forms of the distributions of matrix elements of the site-resolved magnetization operator mirror the distribution of the onsite disorder. We argue that this distribution is a direct consequence of the local observable having a finite overlap with moments of the Hamiltonian. We then demonstrate how to adjust the energy window when analyzing expectation values of local observables in disordered quantum many-body systems to correctly assess the system's adherence to ETH, and provide a link between the distribution of expectation values in eigenstates and the outcomes of quench experiments.

Figures (12)

• Figure 1: The probability distribution of site-resolved magnetization, $s_l^z$, for disorder-averaged GOE matrices (a) and the HSC-QP Hamiltonian at $W=0.5$ (b) at different system sizes, $L$. For both (a) and (b) panels, $100$ eigenstates are considered at the rescaled energy $\epsilon = 0.5$. The number of disorder realizations is varied, being at least $7500$ for $L\leq 16$ and $60$ for $L=18$ for panel (a), while for HSC-QP case (b) it is $10^4$ disorder realizations for $L\leq 16$ and $2500$ for $L=18$. The insets show the standard deviations of respective distributions, decreasing exponentially with $L$ for GOE (a) and saturating for HSC-QP model.
• Figure 2: The distribution of onsite disorder is depicted for various types of disorders, including random (a), cosine (b), binary (c), tertiary (d), and triangle-wave (e). Additionally, the corresponding distributions of site-resolved magnetization are shown in (f-j), considering $100$ states near $\epsilon =0.5$ averaged over at least $200$ disorder realizations for a system of size $L=18$.
• Figure 3: The distribution of the difference of adjacent onsite disorder, $\delta h_l$, is depicted for various types of disorders, including random (a), cosine (b), binary (c), tertiary (d), and triangle-wave (e). Additionally, the corresponding distributions of differences of adjacent site-resolved magnetization, $\delta s_l^z$, (f-j), considering $100$ states near $\epsilon =0.5$, averaged over at least $200$ disorder realizations for a system of size $L=18$.
• Figure 4: The site-resolved magnetization, $s_l^z$, for a single disorder realization of the HSC QP model at different rescaled energies $\epsilon$ for $L=18$ and $W=0.5$. The dark blue and red scatter plots correspond to sites $l=9$ and $l=10$. The dashed, vertical lines indicate the rescaled energy values that give the maximum value of $\mathrm{DoS}$ and the average rescaled energy $\braket{\epsilon}$. The bold blue lines illustrate the analytical curves obtained by expanding the microcanonical expectation value, $\mathcal{O}(\epsilon)$, to the first order in the Hamiltonian's moment.
• Figure 5: (a,b) The probability distribution for the site-resolved magnetization, $s_l^z$, and (c,d) the site-resolved magnetization after removing the approximate microcanonical expectation, $\tilde{s}_l^z$. Results are shown, respectively, for the HSC model with Random (a,c) and QP (b,d) disorder. For all panels $L=18$, $W=0.5$, with $100$ eigenstates in intervals around $\braket{\epsilon}$ and $\braket{\epsilon}\pm \varepsilon$ for at least $1300$ disorder realizations.