Coulomb gas on the Keldysh contour: Anderson-Yuval-Hamann representation of the nonequilibrium two-level system

The nonequilibrium tunneling 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 acquired by 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 effectively to classical long-time behavior. We determine the effect of departures from equilibrium on the localization-delocalization phase transition.

Figure 1

Sketch of energy levels and cutoffs; band energy cutoffs chosen for each lead to be symmetrical about Fermi energy in that lead. (A) (Upper panel) chemical potential difference $\Delta \mu ={\mu }_L-{\mu }_R=0$. (B) (Middle panel) Chemical potential difference $\Delta \mu \ne 0$ but less than cutoff $2{\xi }^{-1}$. (C) (Lower panel) Chemical potential difference $\Delta \mu$ greater than cutoff $2{\xi }^{-1}$; no dissipative processes possible.

Figure 2

Examples of spin-flip events in the Keldysh two axis representation (left side) and in the single axis representation employing quantum and classical fields (right side). The charges $n_i$ are indicated on the figure. In the top and bottom examples, the fields $q=+1$ for all instantons. In the middle example, $q=+1$ for the left hand pair of instantons and $=-1$ for the right hand pair. Although the first and the third examples have the same quantum field configuration, the different positions of the times on the Keldysh contour lead to different phase factors.

Figure 3

Scheme for integrating out close pairs.

Figure 4

Close pairs that are nearest neighbors to charge at time $t_i$.

Figure 5

(Color online) Crossover function $g_{\mathit{neq}}$ (in units of $\frac{\pi }{2}{Q}_M\Delta \mu$) describing additive renormalization of decay constant ${\Gamma }_{\mathit{neq}}$, for hard (solid line) and soft (dashed line) cutoffs.