Eigenstate Thermalization Hypothesis and Free Probability.

Quantum thermalization is well understood via the eigenstate thermalization hypothesis (ETH). The general form of ETH, describing all the relevant correlations of matrix elements, may be derived on the basis of a "typicality" argument of invariance with respect to local rotations involving nearby energy levels. In this Letter, we uncover the close relation between this perspective on ETH and free probability theory, as applied to a thermal ensemble or an energy shell.

Eigenstate Thermalization and Rotational Invariance in Ergodic Quantum Systems.

Generic rotationally invariant matrix models satisfy a simple relation: the probability distribution of half the difference between any two diagonal elements and the one of off-diagonal elements are the same. In the spirit of the eigenstate thermalization hypothesis, we test the hypothesis that the same relation holds in quantum systems that are nonlocalized, when one considers small energy differences. The relation provides a stringent test of the eigenstate thermalization hypothesis beyond the Gaussian ensemble.

Eigenstate thermalization hypothesis and out of time order correlators.

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.

Measuring effective temperatures in a generalized Gibbs ensemble.

The local physical properties of an isolated quantum statistical system in the stationary state reached long after a quench are generically described by the Gibbs ensemble, which involves only its Hamiltonian and the temperature as a parameter. If the system is instead integrable, additional quantities conserved by the dynamics intervene in the description of the stationary state. The resulting generalized Gibbs ensemble involves a number of temperature-like parameters, the determination of which is practically difficult.

Spatiotemporal Patterns in Ultraslow Domain Wall Creep Dynamics.

In the presence of impurities, ferromagnetic and ferroelectric domain walls slide only above a finite external field. Close to this depinning threshold, they proceed by large and abrupt jumps called avalanches, while, at much smaller fields, these interfaces creep by thermal activation. In this Letter, we develop a novel numerical technique that captures the ultraslow creep regime over huge time scales.

Thermal phase transitions in artificial spin ice.

We use the sixteen-vertex model to describe bidimensional artificial spin ice. We find excellent agreement between vertex densities in 15 differently grown samples and the predictions of the model. Our results demonstrate that the samples are in usual thermal equilibrium away from a critical point separating a disordered and an antiferromagnetic phase in the model.

Solvable model of quantum random optimization problems.

We study the quantum version of a simplified model of optimization problems, where quantum fluctuations are introduced by a transverse field acting on the qubits. We find a complex low-energy spectrum of the quantum Hamiltonian, characterized by an abrupt condensation transition and a continuum of level crossings as a function of the transverse field. We expect this complex structure to have deep consequences on the behavior of quantum algorithms attempting to find solutions to these problems.