Conductance renormalization and conductivity of a multi-subband Tomonaga-Luttinger model

We studied the conductance renormalization and conductivity of multisubband Tomonaga-Luttinger models with inter-subband interactions. We found that, as in single-band systems, the conductance of a multi-subband system with an arbitrary number of subbands is not renormalized due to interaction between electrons. We derived a formula for the conductivity in multi-subband models. We applied it to a simplified case and found that inter-subband interaction enhances the conductivity, which is contrary to the intra-subband repulsive interaction, and that the conductivity is further enhanced for a larger number of subbands. 72.10.-d, 72.20.-i, 72.25.-b Typeset using REVTEX 1 Recent studies of low-dimensional systems have brought to light many important properties. For instance, one-dimensional (1D) electron systems, in a low-energy regime, are described not by the Fermi liquid but by a Tomonaga-Luttinger (TL) liquid [1–3]. TomonagaLuttinger liquids that include the effects of the multiple degrees of freedom, such as multichain TL models with the interchain hopping, have been extensively studied. In a bulk system, the interchain hopping between 1D TL chains is relevant, resulting in a strongcoupling regime that includes a spin gap and/or an enhanced superconducting correlation [4–7]. The crossover from TL to Fermi Liquid has also been studied by including the interchain hopping [8]. Regarding the transport properties, for example, a perfect transmission has been suggested in a two-chain system, reflecting the spin gap [9,10]. The interchain conductivity [11] and the Hall effect [12] of a multi-chain system with the interchain hopping have also been discussed. TL liquids have been also studied in mesoscopic quantum wires, especially with respect to the transport properties. The 1D Coulomb drag [13–16] has been studied on 1D two-chain models coupled in a finite region [15] or at a finite point(s) [13,14,16]. In these models, the interchain backward scattering process between electrons, which results in a strong-coupling regime, is essential for the occurrence of a perfect drag [15], a zero-bias anomaly [13], or a power-law temperature dependence of the transconductance [14]. Another TL system with multiple degrees of freedom is a multi-subband TL model with inter-subband forward scattering, where the inter-subband single-particle hopping is forbidden. Although this model is relevant to wide quantum wires with multi-subbands, it has not been well studied for the transport properties, such as conductance and conductivity. In a quantum wire, the long-range Coulomb interaction is not sufficiently screened and the forward scattering processes between electrons with a small momentum transfer play an important role, while the scattering processes with large momentum transfers of the order of the Fermi wave number(s), such as the backward, Umklapp, or inter-subband pair tunneling process, may be neglected. The ground state of the above multi-subband model is in a weak coupling regime without the gapful excitation and is essentially different from 2 the multi-chain model with the interchain hopping or the backward scattering, where the ground state is in a strong coupling regime. For single-band TL models, both the conductance of clean systems [17–24] and the conductivity of dirty systems [25–28] have been studied. The models in refs. [21–24] include the effects of leading wires and show the absence of the conductance renormalization due to the electron-electron interaction, which is consistent with experiments [29]. However, it is not so obvious whether the conductance renormalization of the multi-subband model is absent or not. For example, Liang and co-workers [30] have experimentally found that, in a clean quantum wire, the conductance is smaller than the quantized conductance only in a high in-plane magnetic field, where the two inequivallent spin subbands cross the Fermi level. Hence, the conductance renormalization of a clean multi-subband TL model is also of interest. On the other hand, for a dirty single TL liquid, which can be realized in a long quantum wire where the wire length is longer than the mean-free path, a power-law temperature dependence of the conductivity was observed in experiment [29], which is consistent with the existing theory [25–27]. If we consider a multi-subband system, in a two-subband system, then, as the author and co-workers [28] theoretically found, the inter-subband interaction enhances the conductivity even if the interaction is repulsive, contrary to the intra-subband repulsive interaction. In order to further clarify the multi-subband effect, the conductivity of a TL model with larger number of subbands should be investigated. In this paper, we study the transport properties of the multi-subband TL model with the inter-subband forward scattering, neglecting the large momentum transfer processes, such as backward scatterings. We found that, as in single-band systems, the conductance of a clean multi-subband TL model with an arbitrary number of subbands is not renormalized due to the interaction between electrons. We derived a formula based on the Mori formalism [31,32] for the conductivity of dirty multi-subband TL models. Applying the formula to a multisubband model, we found that the inter-subband interaction enhances the conductivity for an arbitrary number of subbands, and that the conductivity is more enhanced for a larger 3 number of subbands. Conductance of a clean TL model.— Let us start from a N -subband spinless TL model, which includes a spinful model as a special case; i.e., a spinless 2N -subband model is equivalent to a spinful N -subband model. The spinless N -subband TL model can be represented as H = N