By Joseph Maciejko

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The figures illustrate clearly the Coulomb blockade effect. In Fig. 1, we display the Coulomb blockade diamond plot, the differential conductance dJ/dV as function of gate voltage and bias voltage. Note that linear response theory would give only the Vb = 0 line whereas with the nonequilibrium theory, we get the full bias-gate voltage diagram. In Fig. 2, we plot two cuts of Fig. 1 along Vg , for two values of bias voltage: Vb = 0 (equilibrium case for which the differential conductance dJ/dV is just the equilibrium two-terminal conductance G), and Vb = 1 V (out of equilibrium: the equilibrium conductance peaks split due to the applied bias).

32) will give a nonvanishing contribution to the current. We therefore obtain j = e2 d3 p (2π)3 ∂nF dω vp (vp · E) − 2π ∂ω A(p, ω)Λ(p, ω) from which the conductivity tensor is easily extracted, σµν = e2 m2 d3 p (2π)3 dω ∂nF pµ pν − 2π ∂ω A(p, ω)Λ(p, ω) which is equivalent to the Kubo formula result. 3 One-Band Electrons with Spin-Orbit Coupling 51 where λ(p) is odd in p in general due to time-reversal symmetry, but for simplicity we neglect higher-order terms and assume it is only linear in p. Consider for example a two-dimensional electron gas with Rashba and Dresselhaus spin-orbit coupling, HSO = α(py σx − px σy ) + β(px σx − py σy ) where α is the Rashba coupling and β is the linear Dresselhaus coupling.

The only difference is that the action is obtained by integrating the Lagrangian over the Schwinger-Keldysh contour C instead of over the real axis. 10) To calculate the propagator Gn,kα (τ, τ ), it is appropriate to define the Keldysh generating functional Z[¯ η , J], −i Z[¯ η , J] = Tr ρTc e C dτ :H:− n η ¯n dn − kα c†kα Jkα where η¯n and Jkα are Grassmann sources, and the normalization is such that Z[0, 0] = Tr ρ ≡ Z. 11) η¯=0,J=0 The generating functional has a path integral representation, Z[¯ η , J] = ¯ d]ei D[¯ c, c, d, ¯ S[¯ c,c,d,d]+ C dτ (¯ η d+¯ cJ) where we use the shorthand notation η¯d ≡ n η¯n dn and c¯J ≡ kα c¯kα Jkα .

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