Rigorous Bounds on Eigenstate Thermalization
Abstract
The eigenstate thermalization hypothesis (ETH), which asserts that every eigenstate of a manybody quantum system is indistinguishable from a thermal ensemble, plays a pivotal role in understanding thermalization of isolated quantum systems. Yet, no evidence has been obtained as to whether the ETH holds for $\textit{any}$ fewbody operators in a chaotic system; such fewbody operators include crucial quantities in statistical mechanics, e.g., the total magnetization, the momentum distribution, and their loworder thermal and quantum fluctuations. Here, we identify rigorous upper and lower bounds on $m_{\ast}$ such that $\textit{all}$ $m$body operators with $m < m_{\ast}$ satisfy the ETH in fully chaotic systems. For arbitrary dimensional $N$particle systems subject to the Haar measure, we prove that there exist $N$independent positive constants ${\alpha}_L$ and ${\alpha}_U$ such that ${\alpha}_L \leq m_{\ast} / N \leq {\alpha}_U$ holds. The bounds ${\alpha}_L$ and ${\alpha}_U$ depend only on the spin quantum number for spin systems and the particlenumber density for Bose and Fermi systems. Thermalization of $\textit{typical}$ systems for $\textit{any}$ fewbody operators is thus rigorously proved.
 Publication:

arXiv eprints
 Pub Date:
 March 2023
 DOI:
 10.48550/arXiv.2303.10069
 arXiv:
 arXiv:2303.10069
 Bibcode:
 2023arXiv230310069S
 Keywords:

 Condensed Matter  Statistical Mechanics;
 Quantum Physics
 EPrint:
 8 pages, 1 figure (Supplemental Material: 20 pages)