Thermalization and quantum chaos

Numerically computed spectral form factor K(t) for different system sizes L of a kicked spinless fermion chain (a), (b) with and (c) without particle-number conservation. Adapted from Phys. Rev. E 102, 060202(R) (2020)

The motion in classical mechanics can be regular for integrable systems, and irregular or chaotic for nonintegrable systems. The quantum chaos is a branch of physics that tries to identify and understand the chaotic motion of nonintegrable systems when the quantum effects are significant. One key goal of quantum chaos is to establish a relationship between the universal spectral fluctuations of chaotic quantum systems and the random matrix theory (RMT). It took significant research efforts spanning over twenty years to obtain such a goal for single-particle systems whose corresponding classical dynamics are fully chaotic. Only recently, a series of studies can make further progress in establishing such a relationship for low dimensional and locally interacting, nonintegrable many-body systems where local degrees of freedom, e.g., spin-1/2’s, fermions, qubits, have no classical limit. These studies have analytically computed the spectral form factor (SFF) characterizing spectral fluctuations at high energies, and the derived SFF shows a good agreement with the RMT form, which is solely determined by the symmetry of underlying dynamical systems. The study of quantum chaos and its connection to RMT is essential in describing ergodicity and thermalization in closed quantum systems. 

We extend the recent effort to identify microscopic mechanisms of quantum chaos to strongly interacting fermions, bosons and their mixtures with nearest-neighbor hopping processes and long-range pairwise interactions in the presence or absence of conserved particle number. A new dynamical chaos mechanism has been found which maps the SFF to an average recurrence probability of a classical Markov chain with transition probabilities given as square-moduli of hopping amplitudes. We discover universal non-abelian symmetries of the Markov matrices whose subleading eigenvalues determine the system-size scaling of Thouless time to reach universal RMT form for spectral form factor in these systems. These symmetries lead to identical quantum chaotic features in the studied bosonic and fermionic models in the presence of particle number conservation. In the absence of particle number conservation, the bosonic and fermionic models display different system-size scaling of the Thouless time. Our results provide a useful tool to investigate the ergodic phase of long-range interacting systems with the disorder which have been investigated in recent years in the context of many-body localization transition.

R. Singh, R. Moessner, and D. Roy, Phys. Rev. B 95, 094205 (2017)

D. Roy and T. Prosen, Phys. Rev. E 102, 060202(R) (2020)

M. Ljubotina, D. Roy, and T. Prosen, Phys. Rev. B 106, 054314 (2022)

D. Roy, D. Mishra, and T. Prosen, Phys. Rev. E 106, 024208 (2022)

V. Kumar and D. Roy, Phys. Rev. E 109, L032201 (2024)