NonlinearI(V) characteristics of Luttinger liquids and gated Hall bars

Abstract

Non-linear current voltage characteristics of a disordered Luttinger liquid are calculated using a perturbative formalism. One finds non-universal power law characteristics of the form I(V ) ∼ V 1/(2g−1) which is valid both in the superfluid phase when I is small and also in the insulator phase when I is large. Mesoscopic voltage fluctuations are also calculated. One finds Var(∆V ) ∼ I 4g−3 . Both the I(V ) characteristic and the voltage fluctuations exhibit universal power law behavior at the superfluid insulator transition whereg = 3 2. The possible application of these results to the non-linear transport properties of gated Hall bars is discussed.PACS numbers: 73.20Dx, 73.20Ht, 72.15Rn Recently there has been considerable interest in the transport properties of the Luttinger liquid [1]. There are many reasons for this including applications to quantum wires, quasi-one dimensional organic conductors In their experiments , the point contact tunneling conductance between two ν = 1/3 fractional quantum Hall edge channels is measured as a function of the point contact gate voltage. One observes transmission resonances whose half-widths scale with temperature as T 2/3 . This is in agreement with Kane and Fisher [5]. Off resonance, the conductance scales as T 4 as was also predicted [5].In fact, the experiments of Milliken et al.[4] is only the most recent contribution to the understanding of Luttinger liquid transport. Let us begin our discussion by recounting, in a semi-historical manner, some of the major developments in the field. We will begin by recalling the work by Apel and Rice [6], who used the Kubo formalism to show that the conductance of a clean quantum wire differed from the usual Landauer result that G = e 2 /h [7]. Instead these authors found that G = ge 2 /h, where g ≡ πhv(∂n/∂µ). This result may be obtained as follows: First consider the current injected into the wire from the left reservoir. This is I + giveswhere I B (I) = I[1]− is the backscattering current. Now I B = 0 for a clean wire, so eq. 1 gives G = I/e(µ 1 − µ 2 ) = ge 2 /h.Another major advance in transport theory was the investigation of the superfluid-insulator transition in uniformly disordered Luttinger liquids by Giamarchi and Schultz [8]. These authors found that an infinitesimal amount of disorder will localize a Luttinger liquid with g < 3 2 . Moreover, they find that, if g > 3 2 , disorder is irrelevant and the Luttinger liquid exhibits superfluid behavior.Further progress was made by Kane and Fisher [5] who investigated a Luttinger wire with a single or double barrier. These authors found that V 2kF , the Fourier coefficient of the potential barrier, is highly relevant if g < 1. Because of this, the conductance of a Luttinger liquid with a single impurity vanishes. The existence of conductance resonances may be crudely understood if one views Luttinger liquids as Wigner crystals with quasi-long range positional order [9]. According to this point of view, the coupling of a CDW to a barrier potential may be described by the...

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