Abstract
We study the statistics of matrix elements of local operators in the basis of energy eigenstates in a paradigmatic, integrable, many-particle quantum theory, the Lieb-Liniger model of bosons with repulsive delta-function interactions. Using methods of quantum integrability, we determine the scaling of matrix elements with system size. As a consequence of the extensive number of conservation laws, the structure of matrix elements is fundamentally different from, and much more intricate than, the predictions of the eigenstate thermalization hypothesis for generic models. We uncover an interesting connection between this structure for local operators in interacting integrable models and the one for local operators that are not local with respect to the elementary excitations in free theories. We find that typical off-diagonal matrix elements in the same macrostate scale as , where the probability distribution function for is well described by Fréchet distributions and depends only on macrostate information. In contrast, typical off-diagonal matrix elements between two different macrostates scale as , where depends only on macrostate information. Diagonal matrix elements depend only on macrostate information up to finite-size corrections.
- Received 21 January 2024
- Accepted 31 July 2024
DOI:https://doi.org/10.1103/PhysRevX.14.031048
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Popular Summary
To understand how statistical mechanics emerges in many-particle quantum systems initialized in a nonequilibrium quantum state is a long-standing challenge in theoretical physics. In systems where only energy is conserved, key insights into this question are provided by the eigenstate thermalization hypothesis (ETH). However the ETH does not hold in noninteracting theories such as the ideal Bose or Fermi gas—nor in a whole class of “integrable” models—which possess many conserved quantities besides energy. In this study, we address what takes the place of the ETH in such integrable models.
The idea behind the ETH is to express quantum-mechanical operators as matrices in the basis of energy eigenstates, and then to postulate a statistical description of the resulting matrix elements. It follows from this description that many-particle quantum systems initially prepared in far-from-equilibrium states evolve to a state that appears to be in thermal equilibrium.
We find that the statistical structure of matrix elements of local operators in energy eigenstates is fundamentally different in integrable models. In particular, typical matrix elements are much smaller than those in ETH systems and play no role for physical properties. The latter are entirely determined by rare matrix elements, which we characterize.
A key question raised by our findings is how to efficiently sample the “rare” matrix elements and implement novel numerical approaches to obtaining physical properties in integrable models in and out of equilibrium.
