Abstract
Using numerical exact diagonalization, we study matrix elements of a local spin operator in the eigenbasis of two different nonintegrable quantum spin chains. Our emphasis is on the question to what extent local operators can be represented as random matrices and, in particular, to what extent matrix elements can be considered as uncorrelated. As a main result, we show that the eigenvalue distribution of band submatrices at a fixed energy density is a sensitive probe of the correlations between matrix elements. We find that, on the scales where the matrix elements are in a good agreement with all standard indicators of the eigenstate thermalization hypothesis, the eigenvalue distribution still exhibits clear signatures of the original operator, implying correlations between matrix elements. Moreover, we demonstrate that at much smaller energy scales, the eigenvalue distribution approximately assumes the universal semicircle shape, indicating transition to the random-matrix behavior, and in particular that matrix elements become uncorrelated.
Highlights
Questions of equilibration and thermalization in isolated quantum many-body systems have experienced an upsurge of interest both from the theoretical and the experimental side over the last decades [1,2,3]
The eigenstate thermalization hypothesis (ETH) has been established as a key concept to explain the emergence of thermodynamic behavior, by assuming a certain matrix structure of physical operators O in the eigenbasis of generic Hamiltonians H [4,5,6]
The ETH is believed to hold for nonintegrable models and physical observables
Summary
Questions of equilibration and thermalization in isolated quantum many-body systems have experienced an upsurge of interest both from the theoretical and the experimental side over the last decades [1,2,3]. [29,30,31,32] for related studies of the ETH and the emergence of quantum chaos in these models.) Going beyond the “standard” indicators of the ETH, we investigate the existence of the scale ERMT below which random matrix theory (RMT) prevails To this end, we establish the eigenvalue spectrum of O as a sensitive probe of the correlations between the Omn. While the spectrum of the full spin operator includes only two eigenvalues ±1/2, we focus on the spectrum of band submatrices at a fixed energy density Ewhere the Omn are restricted to a narrow band |En − Em| ωc. IV, where we put our findings into context with previous studies of the ETH and outline future directions of research
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