Coulomb Gas on the Keldysh Contour: Anderson-Yuval-Hamann representation of the Nonequilibrium Two Level System

The nonequilibrium tunnelling center model of a localized electronic level coupled to a fluctuating two-state system and to two electronic reservoirs, is solved via an Anderson-Yuval-Hamann mapping onto a plasma of alternating positive and negative charges time-ordered along the two "Keldysh" contours needed to describe nonequilibrium physics. The interaction between charges depends both on whether their time separation is small or large compared to a dephasing scale defined in terms of the chemical potential difference between the electronic reservoirs and on whether their time separation is larger or smaller than a decoherence scale defined in terms of the current flowing from one reservoir to another. A renormalization group transformation appropriate to the nonequilibrium problem is defined. An important feature is the presence in the model of a new coupling, essentially the decoherence rate, which acquires an additive renormalization similar to that of the energy in equilibrium problems. The method is used to study interplay between the dephasing-induced formation of independent resonances tied to the two chemical potentials and the decoherence which cuts off the scaling and leads to effectively classical long-time behavior. We determine the effect of departures from equilibrium on the localization-delocalization phase transition.