Sum rule for transport in a Luttinger liquid with long-range interaction in the presence of an impurity

– We show that the non-linear dc transport in a Luttinger liquid with interaction of finite range in the presence of an impurity is governed by a sum rule which causes the charging energy to vanish. A fundamental problem of electron transport in quantum coherent systems is tunneling through a potential barrier [1]. Since the discovery that interactions play here a crucial role [2], considerable theoretical effort has been devoted to the study of the transport in one-dimensional (1D) interacting electron systems described by a Luttinger model [3], [4] with a potential barrier [5]. It has been shown by renormalization group techniques [6] and by conformal field theory [7] that the current is suppressed at low bias voltage U . The current-voltage characteristic is given by I(U) ∝ U2/g−1 (U → 0), with the interaction constant g (< 1 for repulsive interaction). This implies that even an infinitesimally weak scattering potential is completely insulating. For a tunnel junction connecting two Luttinger systems, it has been shown that a finite range of the interaction induces linear behavior at sufficiently high bias voltage, I(U) = R t (U − Uc) (tunnel resistance Rt) [8], with the “charging energy” Ec ≡ eUc proportional to the interaction potential at zero distance. In this paper, we consider a potential barrier of the height Ub in a Luttinger liquid. The charging energy can be related to the difference of the spectral functions J(ω) and J0(ω) of the elementary excitations with (g < 1) and without (g = 1) interaction, respectively, via the sum rule (h̄ = 1)