Eigenstate thermalization for observables that break Hamiltonian symmetries and its counterpart in interacting integrable systems

A new study explores how quantum systems behave when their inherent symmetries are broken, revealing patterns that could inform future technologies. Researchers from Penn State University investigated the matrix elements of observables in spin-chain models, comparing chaotic and integrable systems. This work helps bridge the gap between theoretical predictions and experimental observations in quantum mechanics.

The key finding is that in quantum-chaotic systems, the off-diagonal matrix elements follow a Gaussian distribution, as predicted by the eigenstate thermalization hypothesis (ETH). For example, in the chaotic model with parameters like Δ=0.55 and λ=1, the variance of these elements scales inversely with the Hilbert space dimension, specifically as 1/D. In contrast, integrable systems, such as the XXZ chain with λ=0, show skewed log-normal-like distributions, indicating deviations from ETH but still with variances scaling as 1/D.

ology involved using full exact diagonalization on spin-1/2 chains with up to 22 lattice sites, focusing on observables that break translational symmetry, like nearest-neighbor interactions. The researchers analyzed matrix elements connecting energy eigenstates from different quasimomentum sectors, ensuring all symmetries were resolved. This approach builds on s described in the paper, such as computing variances and scaled variances to study low-frequency behavior.

analysis, based on figures from the paper, shows that in chaotic systems, the scaled variance |f_O(0,ω)|² exhibits a plateau at low frequencies, consistent with diffusive scaling where the Thouless energy scales as 1/L². For instance, Figure 6 illustrates this for observables like Û_n and Û_nn, with data collapsing when plotted against ωL². In integrable systems, the scaled variance V_O(0,ω) shows ballistic scaling at low frequencies, as seen in Figure 12, where V_O(0,ω)/L plotted against ωL in system-size-independent curves. The data indicates that for integrability-breaking observables, V_O(0,ω) approaches a non-zero value as ω→0, unlike integrability-preserving ones where it may vanish.

This research matters because it enhances our understanding of quantum thermalization and equilibration, which are crucial for developing quantum computers and simulating complex materials. By showing how symmetries affect matrix elements, the study provides insights into controlling quantum systems for practical applications, such as improving energy efficiency in quantum devices.

Limitations include the use of finite-size systems, as noted in the paper's discussion of finite-size effects that disrupt scaling in larger chains. For example, deviations from Gaussianity in distributions, as seen in Figure 2, are attributed to these effects, and the exact behavior in the thermodynamic limit remains partially unknown. The study also does not address all types of observables or real-world implementations, leaving room for future research on broader system classes.