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We derive the finite temperature Keldysh response theory for interactingfermions in the presence of quenched disorder, as applicable to any of the 10Altland-Zirnbauer classes in an Anderson delocalized phase with at least a U(1)continuous symmetry. In this formulation of the interacting Finkel'steinnonlinear sigma model, the statistics of one-body wave functions are encoded bythe constrained matrix field, while physical correlations follow from thehydrodynamic density or spin response field, which decouples the interactions.Integrating out the matrix field first, we obtain weak (anti)localization andAltshuler-Aronov quantum conductance corrections from the hydrodynamic responsefunction. This procedure automatically incorporates the correct infraredphysics, and in particular gives the Altshuler-Aronov-Khmelnitsky (AAK)equations for dephasing of weak (anti)localization due to electron-electroncollisions. We explicate the method by deriving known quantum corrections intwo dimensions for the symplectic metal class AII, as well as the spin-SU(2)invariant superconductor classes C and CI. We show that conductance correctionsdue to the special modes at zero energy in nonstandard classes areautomatically cut off by temperature, as previously expected, while theWigner-Dyson class Cooperon modes that persist to all energies are cut bydephasing. We also show that for short-ranged interactions, the standardself-consistent solution for the dephasing rate is equivalent to a diagrammaticsummation via the self-consistent Born approximation. This should be comparedto the AAK solution for long-ranged Coulomb interactions, which exploits theMarkovian noise correlations induced by thermal fluctuations of theelectromagnetic field. We discuss prospects for exploring the many-bodylocalization transition from the ergodic side as a dephasing catastrophe inshort-range interacting models.

Response theory of the ergodic many-body delocalized phase: Keldysh Finkel'stein sigma models and the 10-fold way