Dr Bruno Bertini

Abstract:

We study the time evolution operator in a family of local quantum circuits with random elds in a xed direction. We argue that the presence of quantum chaos implies that at large times the time evolution operator becomes e ectively a random matrix in the many-body Hilbert space. To quantify this phenomenon we compute analytically the squared magnitude of the trace of the evolution operator the generalised spectral form factor and compare it with the prediction of Random Matrix Theory (RMT). We show that for the systems under consideration the generalised spectral form factor can be expressed in terms of dynamical correlation functions of local observables in the in nite temperature state, linking chaotic and ergodic properties of the systems. This also provides a connection between the many-body Thouless time τth the time at which the generalised spectral form factor starts following the random matrix theory prediction and the conservation laws of the system. Moreover, we explain di erent scalings of τth with the system size, observed for systems with and without the conservation laws.

Abstract:

Over the last decade impressive progress has been made in the theoretical understanding of transport properties of clean, one-dimensional quantum lattice systems. Many physically relevant models in one dimension are Bethe-ansatz integrable, including the anisotropic spin-1/2 Heisenberg (also called the spin-1/2 XXZ chain) and the Fermi-Hubbard model. Nevertheless, practical computations of correlation functions and transport coefficients pose hard problems from both the conceptual and technical points of view. Only because of recent progress in the theory of integrable systems, on the one hand, and the development of numerical methods, on the other hand, has it become possible to compute their finite-temperature and nonequilibrium transport properties quantitatively. Owing to the discovery of a novel class of quasilocal conserved quantities, there is now a qualitative understanding of the origin of ballistic finite-temperature transport, and even diffusive or superdiffusive subleading corrections, in integrable lattice models. The current understanding of transport in one-dimensional lattice models, in particular, in the paradigmatic example of the spin-1/2 XXZ and Fermi-Hubbard models, is reviewed, as well as state-of-the-art theoretical methods, including both analytical and computational approaches. Among other novel techniques, matrix-product-state-based simulation methods, dynamical typicality, and, in particular, generalized hydrodynamics are covered. The close and fruitful connection between theoretical models and recent experiments is discussed, with examples given from the realms of both quantum magnets and ultracold quantum gases in optical lattices.

Abstract:

We study the out-of-equilibrium dynamics of the quantum cellular automaton known as “Rule 54.” For a class of low-entangled initial states, we provide an analytic description of the effect of the global evolution on finite subsystems in terms of simple quantum channels, which gives access to the full thermalization dynamics at the microscopic level. As an example, we provide analytic formulas for the evolution of local observables and Rényi entropies. We show that, in contrast to other known examples of exactly solvable quantum circuits, Rule 54 does not behave as a simple Markovian bath on its own parts, and displays typical nonequilibrium features of interacting integrable many-body quantum systems such as finite relaxation rate and interaction-induced dressing effects. Our study provides a rare example where the full thermalization dynamics can be solved exactly at the microscopic level.