An Introduction to Nonequilibrium Many-Body Theory by Joseph Maciejko

By Joseph Maciejko

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5) where Nα = k c†kα ckα is the number operator for lead α and e > 0 is the electron charge. From the Heisenberg equation of motion i¯hN˙ α = [Nα , H] and Eq. 6) k,n † where we have the mixed lesser Green’s function G< n,kα (t, t ) = i ckα (t )dn (t) . Using the Keldysh technique, we will obtain an expression for the mixed contour-ordered Green’s function Gn,kα (τ, τ ) = −i Tc {dn (τ )c†kα (τ )} and perform analytic continuation to real time to obtain the lesser function G< n,kα (t, t ). For pedagogical reasons, let us first derive the expression from the usual perturbation expansion of the S-matrix and the subsequent application of Wick’s theorem, and then obtain it from the path integral method which is more transparent.

26) which to linear order in the electric field is just the equilibrium Green’s function GR (p, ω). This simplifies drastically the solution of the QBE since all the retarded and advanced quantities can be taken in equilibrium. The QBE is 3 The B = 0 case can also be considered, but then the equilibrium Green’s function has a dependence on the magnetic field which in general breaks translational invariance and makes things more complicated. 2 Linear Response for Steady-State and Homogeneous Systems 45 Fig.

So far we have only rephrased the Kadanoff-Baym equation in Wigner coordinates and made the gradient approximation. 58) is still valid to arbitrary order in the electric field and for both time and space-dependent (but slowly varying) perturbations. Note that this very general equation has in principle to be solved together with the equation for the retarded Green’s function Eq. 59) where the left-hand side is given in Eq. 42) and the anticommutator on the right-hand side is given by Eq. 57). Indeed, the nonequilibrium GR enters the renormalization terms Re G and Re Σ (since Σ R = Σ R [GR ]) in Eq.

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