Title: Response theory of the ergodic many-body delocalized phase: Keldysh Finkel'stein sigma models and the 10-fold way,
Abstract: We derive the finite temperature Keldysh response theory for interacting
fermions in the presence of quenched disorder, as applicable to any of the 10
Altland-Zirnbauer classes in an Anderson delocalized phase with at least a U(1)
continuous symmetry. In this formulation of the interacting Finkel'stein
nonlinear sigma model, the statistics of one-body wave functions are encoded by
the constrained matrix field, while physical correlations follow from the
hydrodynamic density or spin response field, which decouples the interactions.
Integrating out the matrix field first, we obtain weak (anti)localization and
Altshuler-Aronov quantum conductance corrections from the hydrodynamic response
function. This procedure automatically incorporates the correct infrared
physics, and in particular gives the Altshuler-Aronov-Khmelnitsky (AAK)
equations for dephasing of weak (anti)localization due to electron-electron
collisions. We explicate the method by deriving known quantum corrections in
two dimensions for the symplectic metal class AII, as well as the spin-SU(2)
invariant superconductor classes C and CI. We show that conductance corrections
due to the special modes at zero energy in nonstandard classes are
automatically cut off by temperature, as previously expected, while the
Wigner-Dyson class Cooperon modes that persist to all energies are cut by
dephasing. We also show that for short-ranged interactions, the standard
self-consistent solution for the dephasing rate is equivalent to a diagrammatic
summation via the self-consistent Born approximation. This should be compared
to the AAK solution for long-ranged Coulomb interactions, which exploits the
Markovian noise correlations induced by thermal fluctuations of the
electromagnetic field. We discuss prospects for exploring the many-body
localization transition from the ergodic side as a dephasing catastrophe in
short-range interacting models.