Eigenstate thermalization hypothesis and out of time order correlators.

Area of Science:

• Quantum chaos
• Statistical mechanics
• Condensed matter theory

Background:

• The eigenstate thermalization hypothesis (ETH) describes matrix elements of local operators in chaotic quantum systems.
• Chaos in quantum systems is often characterized by a positive Lyapunov exponent, observable via Loschmidt echoes or out-of-time-order correlators.
• Slow dynamics in certain systems are associated with dynamic heterogeneity length peaks.

Purpose of the Study:

• To explore the role of correlations between matrix elements in chaotic systems.
• To extend the eigenstate thermalization hypothesis to include these previously neglected correlations.
• To connect these correlations to phenomena like positive Lyapunov exponents and dynamic heterogeneity.

Main Methods:

• Analysis of matrix elements of local operators between Hamiltonian eigenstates.
• Investigation of correlations between matrix elements unrelated by symmetry.
• Examination of correlations between matrix elements of different operators.

Main Results:

• Identified the necessity of correlations between matrix elements for a positive Lyapunov exponent.
• Showed that these correlations are also relevant for the dynamic heterogeneity length.
• Formulated a generalized ETH that encompasses these correlations.

Conclusions:

• A generalized form of the eigenstate thermalization hypothesis is proposed.
• This generalized ETH accounts for correlations between matrix elements, crucial for understanding quantum chaos and slow dynamics.
• The findings provide a more comprehensive framework for chaotic quantum systems.