Frequency-temperature crossover in the conductivity of disordered Luttinger liquids

Abstract

The temperature (T ) and frequency (ω) dependent conductivity of weakly disordered Luttinger liquids is calculated in a systematic way both by perturbation theory and from a finite temperature renormalization group (RG) treatment to leading order in the disorder strength. Whereas perturbation theory results in ω/T scaling of the conductivity such scaling is violated in the RG traetment. We also determine the non-linear field dependence of the conductivity, whose power law scaling is different from that of temperature and frequency dependence. Interacting one-dimensional electron systems display a large variety of unusual and interesting phenomena, since not only interactions but also external potentials (periodic or random) and thermal fluctuations have pronounced effects on the behavior of these systems [1,2]. The hallmark of interaction effects in 1d systems is the power law single particle density of states observable in the backscattering from a single impurity in a Luttinger liquid (LL) [3]. While the LL spectral function has been observed experimentally [4], the rich behavior of collective scattering by random impurities has eluded experimental observation so far.In this letter, we focus on the effect of many weak (Gaussian) impurities in a 1d disordered system. For noninteracting electrons, this problem can be solved exactly [5]. Interaction effects are mostly treated perturbatively and described by a dephasing length, which cuts off interference corrections. While this regime of weak interactions has been studied thoroughly [6,7], less is known about the opposite regime of strong interactions. For attractive interactions, an unpinning transition as a function of the interaction strength was found [8,9,10]. In addition, the power law exponent describing the energy dependence of impurity scattering was predicted to flow as a function of energy [9]. Finite temperature effects were partially incorporated by truncating the renormalization group (RG) flow at the de Broglie wave length of plasmon excitations [9]. However, for a complete study of the thermal to quantum crossover, quantum and thermal fluctuations have to be considered on an equal footing [11].We calculate the frequency, temperature, and electric field dependence of the conductivity for spinless 1d fermions to leading order in the disorder strength but for arbitrary short range interactions. We go beyond previous approaches [8,9,10,12] in several respects. From a technical point of view, we i) present a systematic approach to calculate the conductivity from a bosonic self energy in the spirit of [13], both in perturbation theory and from a finite temperature renormalization group [14,15]. This approach can be generalized to higher orders in the impurity strength to obtain weak localization corrections. We ii) explain that due to a symmetry property of the self energy, an imaginary time RG [9,14] has a ballistic density propagator although the retarded density propagator is diffusive. From a physics point of view, we iii) present results for the fr...

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