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The out-of-time-ordered correlators (OTOCs) have been proposed and widelyused recently as a tool to define and describe many-body quantum chaos. Here,we develop the Keldysh non-linear sigma model technique to calculate thesecorrelators in interacting disordered metals. In particular, we focus on theregularized 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 thethermal density matrix). The calculation of the rate of OTOCs' exponentialgrowth is reminiscent to that of the dephasing rate in interacting metals, buthere it involves two replicas of the system (two "worlds"). The intra-worldcontributions reproduce the dephasing (that would correspond to a decay of thecorrelator), while the inter-world terms provide a term of the opposite signthat exceeds dephasing. Consequently, both regularized and unregularized OTOCsgrow exponentially in time, but surprisingly we find that the correspondingmany-body Lyapunov exponents are different. For the regularized correlator, wereproduce an earlier perturbation theory result for the Lyapunov exponent thatsatisfies the Maldacena-Shenker-Stanford bound. However, the Lyapunov exponentof the unregularized correlator parametrically exceeds the bound. We argue thatthe latter is not a reliable indicator of many body quantum chaos as itcontains additional contributions from elastic scattering events due to virtualprocesses that should not contribute to many-body chaos. These results bring upan important general question of the physical meaning of the OTOCs often usedin calculations and proofs. We briefly discuss possible connections of theOTOCs to observables in quantum interference effects and level statistics via amany-body analogue of the Bohigas-Giannoni-Schmit conjecture.

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