Rigorous Bounds on Eigenstate Thermalization

• URL: http://arxiv.org/abs/2303.10069v1
• Date: Fri, 17 Mar 2023 15:52:08 GMT
• Title: Rigorous Bounds on Eigenstate Thermalization
• Authors: Shoki Sugimoto, Ryusuke Hamazaki, Masahito Ueda
• Abstract summary: Eigenstate thermalization hypothesis (ETH) asserts that every eigenstate of a many-body quantum system is indistinguishable from a thermal ensemble. No evidence has been obtained as to whether the ETH holds for $textitany$ few-body operators in a chaotic system.
• Score: 4.511923587827301
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The eigenstate thermalization hypothesis (ETH), which asserts that every eigenstate of a many-body 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 $\textit{any}$ few-body operators in a chaotic system; such few-body operators include crucial quantities in statistical mechanics, e.g., the total magnetization, the momentum distribution, and their low-order 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 particle-number density for Bose and Fermi systems. Thermalization of $\textit{typical}$ systems for $\textit{any}$ few-body operators is thus rigorously proved.

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