Non-linear sigma model approach to many-body quantum chaos: regularized and unregularized out-of-time-ordered correlators

In this work, we derive an extended version of Keldysh non-linear sigma model to investigate many-body quantum chaos in a 2D interacting fermion system subject to quenched disorder. This model provides a framework in which the regularized and unregularized out-of-time-ordered correlators with different arrangements of the thermal factors can be studied. The two types of correlators grow exponentially with different growth rates which can be extracted from the mass of the “inter-world” diffuson propagator. This should be compared with its “intra-world” counterpart that gives the dephasing rate corresponding to the phase relaxation of single-particle states. Our result for regularized exponent is consistent with the previous one obtained from the diagrammatic perturbation approach. Furthermore, the regularized (unregularized) exponent obeys (violates) the chaos bound 2πT. We note that the regularized exponent is entirely due to the inelastic collisions between particles, while the unregularized exponent contains an additional contribution from elastic scattering of particles off of the Friedel oscillations of the particle density. Therefore, we believe that the unregularized exponent cannot measure many-body quantum chaos which originates from interactions.