Condensed Matter Theory

Abstract:

We consider the spectral statistics of the Floquet operator for disordered, periodically driven spin chains in their quantum chaotic and many-body localized (MBL) phases. The spectral statistics are characterized by the traces of powers t of the Floquet operator, and our approach hinges on the fact that for integer t in systems with local interactions, these traces can be re-expressed in terms of products of dual transfer matrices, each representing a spatial slice of the system. We focus on properties of the dual transfer matrix products as represented by a spectrum of Lyapunov exponents, which we call spectral Lyapunov exponents. In particular, we examine the features of this spectrum that distinguish chaotic and MBL phases. The transfer matrices can be block diagonalized using time-translation symmetry, and so the spectral Lyapunov exponents are classified according to a momentum in the time direction. For large t we argue that the leading Lyapunov exponents in each momentum sector tend to zero in the chaotic phase, while they remain finite in the MBL phase. These conclusions are based on results from three complementary types of calculation. We find exact results for the chaotic phase by considering a Floquet random quantum circuit with on-site Hilbert space dimension q in the large-q limit. In the MBL phase, we show that the spectral Lyapunov exponents remain finite by systematically analyzing models of noninteracting systems, weakly coupled systems, and local integrals of motion. Numerically, we compute the Lyapunov exponents for a Floquet random quantum circuit and for the kicked Ising model in the two phases. As an additional result, we calculate exactly the higher-point spectral form factors (hpSFFs) in the large-q limit and show that the generalized Thouless time scales logarithmically in system size for all hpSFFs in the large-q chaotic phase.

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.