[Paper Review] Bounds on eigenstate thermalization

Shoki Sugimoto, Ryusuke Hamazaki+1 more|arXiv (Cornell University)|Mar 17, 2023

Quantum many-body systems4 citations

TL;DR

This paper establishes rigorous bounds on eigenstate thermalization in quantum many-body systems by analyzing the fluctuations of density matrix deviations from their microcanonical averages. Using Haar measure averaging and concentration of measure techniques, it derives tight upper and lower bounds on the second moments of the 1- and 2-norms of these deviations, showing that the typical deviation scales as $ \mathcal{O}(\sqrt{\log d_{E,\Delta E}/D}) $, confirming the eigenstate thermalization hypothesis in the large-D limit.

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 all few-body operators in a chaotic system; such few-body operators include key quantities in statistical mechanics, such as the total magnetization, the momentum distributions, and their low-order thermal and quantum fluctuations. Here, we formulate a conjecture that for a generic nonintegrable system the ETH holds for all $m$-body operators with $m < α_{\ast} N$ in the thermodynamic limit for some nonzero constant $α_{\ast} > 0$. We first prove this statement for systems with Haar-distributed energy eigenstates to analytically motivate our conjecture. We then verify the conjecture for generic spin, Bose, and Fermi systems with local and few-body interactions by large-scale numerical calculations. Our results imply that generic systems satisfy the ETH for all few-body operators, including their thermal and quantum fluctuations.

Motivation & Objective

• To rigorously quantify the typical size of deviations between eigenstate density matrices and their microcanonical averages in quantum systems.
• To establish bounds on the 1- and 2-norms of these deviations using random matrix theory and Haar measure averaging.
• To confirm the eigenstate thermalization hypothesis (ETH) by showing that fluctuations decay with system size D.
• To derive a concentration inequality for the norm of density matrix deviations under unitary evolution, ensuring typical behavior is close to the average.
• To provide a D-independent upper bound on the Lipschitz constant of the norm function, enabling concentration bounds.

Proposed method

• Uses the Haar measure to compute the expectation of the squared 2-norm of the deviation $ \delta\hat{\rho}_{\alpha\beta} $, yielding $ \mathbb{E}[\|\vec{X}\|_2^2] = \frac{M-1}{D^2-1}\|\hat{X}_0\|_2^2 $.
• Applies fourth moments of the Haar measure to estimate the average squared deviation, leading to bounds on $ \mathbb{E}[(\|\delta\hat{\rho}_{\alpha\beta}\|_2^{(\mathcal{A})})^2] \in \left[\frac{1}{4}\frac{M}{D^2}, \frac{M}{D^2}\right] $.
• Introduces the quantity $ \Lambda_p(\hat{H},\mathcal{A}) \coloneqq \max_{|E_\alpha\rangle,|E_\beta\rangle} \|\delta\hat{\rho}_{\alpha\beta}\|_p^{(\mathcal{A})} $ to characterize the maximum deviation across energy eigenstates.
• Establishes Lipschitz continuity of $ \|\delta\hat{\rho}_{\alpha\beta}\|_p^{(\mathcal{A})} $ as a function of the unitary $ \hat{U} $, with a D-independent bound $ \eta_n $ on the Lipschitz constant.
• Applies a concentration inequality to show that $ \mathbb{E}[(\Lambda_p^{(\hat{H},\mathcal{A})})^n] = \mathbb{E}[(\|\delta\hat{\rho}_{\alpha\beta}\|_p^{(\mathcal{A})})^n] + \mathcal{O}(\sqrt{\log d_{E,\Delta E}/D}) $.
• Uses the bound $ \|\cdot\|_2(A) \leq \|\cdot\|_1(A) \leq D\|\cdot\|_2(A) $ to extend results from the 2-norm to the 1-norm, yielding $ \mathbb{E}[(\|\delta\hat{\rho}_{\alpha\beta}\|_1^{(\mathcal{A})})^2] \in \left[\frac{1}{4}\frac{M}{D^2}, \frac{M}{D}\right] $.

Experimental results

Research questions

• RQ1 What is the typical magnitude of the deviation between an eigenstate density matrix and its microcanonical average?
• RQ2 How do the 1- and 2-norms of this deviation scale with system size D and energy window dimension $ d_{E,\Delta E} $?
• RQ3 Can the eigenstate thermalization hypothesis be rigorously supported by bounding the fluctuations of $ \delta\hat{\rho}_{\alpha\beta} $?
• RQ4 Is the function $ \|\delta\hat{\rho}_{\alpha\beta}\|_p^{(\mathcal{A})} $ Lipschitz continuous in the unitary evolution operator $ \hat{U} $, with a D-independent constant?
• RQ5 What is the concentration behavior of the maximum deviation $ \Lambda_p(\hat{H},\mathcal{A}) $ across energy eigenstates?

Key findings

• The expectation of the squared 2-norm of $ \delta\hat{\rho}_{\alpha\beta} $ is bounded as $ \frac{1}{4}\frac{M}{D^2} \leq \mathbb{E}[(\|\delta\hat{\rho}_{\alpha\beta}\|_2^{(\mathcal{A})})^2] \leq \frac{M}{D^2} $, up to $ \mathcal{O}(1/D^2) $ corrections.
• The 1-norm deviation satisfies $ \frac{1}{4}\frac{M}{D^2} \leq \mathbb{E}[(\|\delta\hat{\rho}_{\alpha\beta}\|_1^{(\mathcal{A})})^2] \leq \frac{M}{D} $, showing a weaker upper bound due to the D-factor in the norm inequality.
• The maximum deviation $ \Lambda_p(\hat{H},\mathcal{A}) $ concentrates around its mean, with the expectation differing by at most $ \mathcal{O}(\sqrt{\log d_{E,\Delta E}/D}) $.
• The Lipschitz constant $ \eta_n $ of the norm function is bounded independently of D, enabling the use of concentration inequalities in high-dimensional systems.
• The derived bounds confirm that eigenstate thermalization holds typically in large systems, as deviations decay with increasing D.
• The result $ \mathbb{E}[(\Lambda_p^{(\hat{H},\mathcal{A})})^n] = \mathbb{E}[(\|\delta\hat{\rho}_{\alpha\beta}\|_p^{(\mathcal{A})})^n] + \mathcal{O}(\sqrt{\log d_{E,\Delta E}/D}) $ establishes that typical deviations are close to the average in high dimensions.