Signatures of diffusion and ballistic transport in the stiffness, dynamical correlation functions, and statistics of one-dimensional systems

Abstract

Integrable and nonintegrable systems have very different transport properties. In this paper, we highlight these differences for specific one-dimensional models of interacting lattice fermions. The technique used is a finite temperature numerical calculation of the adiabatic stiffness (also called the Drude weight or charge stiffness) and isothermal stiffness (also called the "Meissner" stiffness) in electrical and energy transport, and the momentum dependent dynamical conductivities σ(q,ω) and κ(q,ω). We apply a flux twist to break the Kramers degeneracy; thus, allowing us to focus on the effect of the dynamical degeneracies in the integrable system. In this situation, we show that the isothermal stiffness goes to zero rapidly with the system size for both types of systems even at high temperatures; while the adiabatic stiffness appears to go to zero in the nonintegrable system and to a finite value in the integrable one. We analyze this difference in terms of the statistics of the current matrix elements and the degeneracies of the systems, and show that in the integrable system, despite the presence of degeneracies, the dominant contribution to the adiabatic stiffness comes from large-current-carrying nondegenerate states. We also show that energy transport at nonzero ω and q occurs within a banded continuum in the integrable system indicative of ballistic transport while the nonintegrable system shows diffusion but with the existence of overdamped excitations at large values of momentum.