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arXiv:cond-mat/9602101

Comment on ``On the Renormalization of Conductance in Tomonaga-Luttinger Liquid''

Satoshi Fujimoto, Norio Kawakami

Published Feb 20, 1996• 0 citations
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Abstract

Recently Kawabata found that the conductance in Tomonaga-Luttinger (TL) liquids should not be renormalized if one incorporates the renormalization of applied fields correctly. This claim is generalized to include the interactions which carry the large momentum transfer. We thus find that Kawabata's conclusion holds generally for the class of TL liquids. A key observation is that such irrelevant interactions merely renormalize the velocity and the TL parameter at the massless TL fixed point.

DOI
10.48550/arxiv.cond-mat/9602101

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Recently Kawabata found that the conductance in Tomonaga-Luttinger (TL) liquids should not be renormalized if one incorporates the renormalization of applied fields correctly. This claim is generalized to include the interactions which carry the large momentum transfer. We thus find that Kawabata's conclusion holds generally for the class of TL liquids. A key observation is that such irrelevant interactions merely renormalize the velocity and the TL parameter at the massless TL fixed point.
In one-dimensional conductors, Coulomb interactions result in correlated electronic systems called Tomonaga-Luttinger liquids (TLL). The TLL physics also applies to other many-body phenomena, providing complementary viewpoints while benefiting from the powerful TLL framework. One such phenomenon is the low-energy conductance suppression of a quantum coherent conductor embedded in a dissipative circuit, an effect called dynamical Coulomb blockade. Here we investigate the basic class of mesoscopic circuits constituted by a short single-channel quantum conductor in series with a resistance R. Remarkably, such circuits can be mapped on a TLL of interaction parameter 1/(1+Re^2/h), with an impurity. From this mapping, generalized to realistic dissipative circuits, a scaling law for the suppressed conductance is derived at R=h/e^2, and small deviations computed for R \neq h/e^2. We find that the scaling law is obeyed by our data for arbitrary quantum channels, emulated by a Ga(Al)As quantum point contact, and by the recent data of Mebrahtu et al. [Nature 488, 61 (2012)] obtained using a carbon nanotube. This demonstrates a highly tunable test-bed for TLL physics, and consolidates a recently proposed phenomenological expression for the conductance of a quantum channel in a linear circuit.
Transport through a spinless Tomonaga-Luttinger (TL) liquid is studied in the presence of a single impurity. A boundary bosonization technique is developed to treat an impurity of arbitrary strength. Our new bosonization formula (i) smoothly connects the ones for the strong and the weak impurity limits, and (ii) reproduces exact scaling equation for the transmission probability in the weak two-body interaction limit. With the use of our formula, TL parameter is determined from the long-distance correlation functions. The result indicates a possible renormalization of TL parameter due to the effect of the single impurity scattering.