The Eigenstate Thermalization Hypothesis and Out of Time Order Correlators

Understand

The Eigenstate Thermalization Hypothesis (ETH) implies a form for the matrix elements of local operators between eigenstates of the Hamiltonian, expected to be valid for chaotic systems.

• Another signal of chaos is a positive Lyapunov exponent, defined on the basis of Loschmidt echo or out-of-time-order correlators.
• For this exponent to be positive, correlations between matrix elements unrelated by symmetry, usually neglected, have to exist.
• The same is true for the peak of the dynamic heterogeneity length, relevant for systems with slow dynamics.