Nonlinear sigma model approach to many-body quantum chaos: Regularized and unregularized out-of-time-ordered correlators

The out-of-time-ordered correlators (OTOCs) have been proposed and widely used recently as a tool to define and describe many-body quantum chaos. Here, we develop the Keldysh non-linear sigma model technique to calculate these correlators in interacting disordered metals. In particular, we focus on the regularized and unregularized OTOCs, defined as $Tr[\sqrt{\rho} A(t) \sqrt{\rho} A^\dagger(t)]$ and $Tr[\rho A(t)A^\dagger(t)]$ respectively (where $A(t)$ is the anti-commutator of fermion field operators and $\rho$ is the thermal density matrix). The calculation of the rate of OTOCs' exponential growth is reminiscent to that of the dephasing rate in interacting metals, but here it involves two replicas of the system (two "worlds"). The intra-world contributions reproduce the dephasing (that would correspond to a decay of the correlator), while the inter-world terms provide a term of the opposite sign that exceeds dephasing. Consequently, both regularized and unregularized OTOCs grow exponentially in time, but surprisingly we find that the corresponding many-body Lyapunov exponents are different. For the regularized correlator, we reproduce an earlier perturbation theory result for the Lyapunov exponent that satisfies the Maldacena-Shenker-Stanford bound. However, the Lyapunov exponent of the unregularized correlator parametrically exceeds the bound. We argue that the latter is not a reliable indicator of many body quantum chaos as it contains additional contributions from elastic scattering events due to virtual processes that should not contribute to many-body chaos. These results bring up an important general question of the physical meaning of the OTOCs often used in calculations and proofs. We briefly discuss possible connections of the OTOCs to observables in quantum interference effects and level statistics via a many-body analogue of the Bohigas-Giannoni-Schmit conjecture.