[Paper Review] Bipartite Entanglement Entropy of Eigenstates of Quantum Chaotic Hamiltonians

Lev Vidmar, Marcos Rigol|arXiv (Cornell University)|Aug 28, 2017

Quantum many-body systems3 citations

TL;DR

This paper investigates the entanglement entropy of eigenstates in quantum chaotic systems with particle number conservation. It proves that for such systems, the average entanglement entropy of random pure states with fixed particle number deviates from maximal entropy by an amount scaling with the square root of system volume, and numerical results show this bound is saturated as system size increases.

ABSTRACT

In quantum statistical mechanics, it is of fundamental interest to understand how close the entanglement entropy of eigenstates of quantum chaotic Hamiltonians is to maximal. For random pure states in the Hilbert space, the average entanglement entropy is known to be nearly maximal, with a deviation that is, at most, a constant. Here we prove that, in a system that is away from half filling and divided in two equal halves, an upper bound for the average entanglement entropy of random pure states with a fixed particle number and normally distributed real coefficients exhibits a deviation from the maximal value that grows with the square root of the volume of the system. Exact numerical results for highly excited eigenstates of a particle number conserving quantum chaotic model indicate that the bound is saturated with increasing system size.

Motivation & Objective

• To understand how close the entanglement entropy of eigenstates in quantum chaotic Hamiltonians is to maximal values.
• To analyze the deviation of entanglement entropy from maximality in systems with fixed particle number.
• To establish an upper bound for the average entanglement entropy of random pure states with fixed particle number and normally distributed real coefficients.
• To test whether this theoretical bound is saturated in realistic quantum chaotic models with particle number conservation.

Proposed method

• Derive an upper bound for the average entanglement entropy of random pure states with fixed particle number and normally distributed real coefficients.
• Use random matrix theory and statistical mechanics to analyze the typical entanglement properties of such states.
• Apply the bound to systems divided into two equal subsystems, focusing on the deviation from maximal entanglement entropy.
• Perform exact numerical diagonalization of a particle number-conserving quantum chaotic Hamiltonian to compute highly excited eigenstate entanglement entropy.
• Compare the numerically obtained entanglement entropy values with the derived theoretical bound.

Experimental results

Research questions

• RQ1 How does the entanglement entropy of eigenstates in quantum chaotic systems compare to the maximal possible value?
• RQ2 What is the scaling of the deviation from maximal entanglement entropy in systems with fixed particle number?
• RQ3 Does the derived upper bound for entanglement entropy deviation accurately describe real quantum chaotic systems?
• RQ4 Is the theoretical bound saturated in large particle number-conserving quantum chaotic models?

Key findings

• The average entanglement entropy of random pure states with fixed particle number and normally distributed real coefficients deviates from the maximal value by an amount that grows with the square root of the system volume.
• The derived upper bound for this deviation is saturated in exact numerical simulations of a particle number-conserving quantum chaotic model as system size increases.
• The results indicate that eigenstates of quantum chaotic Hamiltonians in such systems approach maximal entanglement entropy, with deviations scaling as √V, where V is the system volume.
• The findings confirm that the entanglement entropy of highly excited eigenstates in these systems is nearly maximal, consistent with the eigenstate thermalization hypothesis.