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Renyi entropy of chaotic eigenstates
Tsung-Cheng Lu1, Tarun Grover1
1Department of Physics, University of California at San Diego, La Jolla, California 92093, USA.
Physical Review. E
|April 20, 2019
Summary
Researchers derived an analytical expression for Renyi entanglement entropies in chaotic many-body systems. This universal function reveals nonlinear dependencies on subsystem size, challenging previous assumptions about thermalization.
Area of Science:
- Quantum Information Theory
- Statistical Mechanics
- Condensed Matter Physics
Background:
- Entanglement entropy quantifies quantum correlations in many-body systems.
- Ergodicity and the eigenstate thermalization hypothesis (ETH) are key concepts for understanding thermalization in quantum systems.
- Renyi entanglement entropies generalize von Neumann entropy and offer insights into quantum chaos.
Purpose of the Study:
- To derive a universal analytical expression for Renyi entanglement entropies in chaotic many-body systems.
- To investigate the behavior of Renyi entropies in finite-energy density eigenstates, particularly in non-thermal regimes.
- To explore the dependence of Renyi entropies on subsystem size and compare them to thermal states.
Main Methods:
- Derivation of an analytical expression for Renyi entanglement entropies using ergodicity arguments.
- Application of a many-body version of Berry's formula, related to ETH.
- Utilizing an assumption of equal likelihood for complementary subsystem states.
- Exact diagonalization studies on quantum spin-chain Hamiltonians for validation.
Main Results:
- A universal analytical expression for Renyi entanglement entropies is derived for chaotic many-body Hamiltonians.
- The expression is valid for subsystems that are a finite fraction of the total system, where reduced density matrices are not necessarily thermal.
- In the thermodynamic limit, Renyi entropy densities exhibit nonlinear dependence on the subsystem-to-system size ratio (V_A/V), unlike von Neumann entropy density.
- Renyi entropies for n>1 are found to be convex functions of subsystem size, with a volume law coefficient exceeding that of a thermal mixed state.
Conclusions:
- The derived analytical expression provides a novel understanding of entanglement in chaotic quantum systems.
- Renyi entanglement entropies display complex dependencies on subsystem size, deviating from simple thermal behavior.
- The results highlight the importance of considering subsystem size effects in characterizing quantum entanglement and thermalization.
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