Tunneling in quantum wires: A boundary conformal field theory approach

Abstract

Tunneling through a localized barrier in a one-dimensional interacting electron gas has been studied recently using Luttinger liquid techniques. Stable phases with zero or unit transmission occur, as well as critical points with universal fractional transmission whose properties have only been calculated approximately, using a type of "ǫ-expansion". It may be possible to calculate the universal properties of these critical points exactly using the recent boundary conformal field theory technique, although difficulties arise from the ∞ number of conformal towers in this c = 4 theory and the absence of any apparent "fusion" principle. Here, we formulate the problem efficiently in this new language, and recover the critical properties of the stable phases.

Junctions of multiple quantum wires with different Luttinger parameters

“…For instance, at low temperature and low bias voltage, a TLL with repulsive interactions (g < 1) is totally disconnected in the presence of an impurity, while one with attractive interactions (g > 1) conducts as in the absence of the impurity. [14][15][16][17][18][19] From a practical viewpoint, junctions of TLL wires serve as important building blocks of quantum circuits, 20,21 and are thus of technological significance. Junctions of three quantum wires, known as Y junctions, also have highly nontrivial transport properties.…”

From integrability to conductance, impurity systems

“…Since this folding idea relies on the vanishing of either the reflection or transmission, our considerations do in general not reduce to that set up, even in the conformal limit. As was already pointed out in [17], and as can be seen directly from (1.1) and (1.2), in that case the conductance is less interesting because it is either zero or perfect for abelian theories.…”

“…It is interesting to note that in the conformal limit, as outlined before equation (2.43), some of the defects, namely D ζ (ψ, ψ) and D λ ± (ψ, ψ), become purely transmitting. Therefore, in contrast to first impression, the folding idea [17] could be employed. We have now enough examples at hand to use them in the following to determine the conductance in a multiparticle system, which we shall do in two alternative ways.…”

“…First we note that the operator (4.12) becomes 16) in the limit R =R = 0 and T =T = 1. The defect should act in this case as the identity operator and, according to (2.2), 17) holds, simply by employing Wick's theorem when carrying out the necessary contractions. For two particles we find,…”

“…There exist various investigations, e.g., [17,13,14,18] for conformal (massless) theories with defect, which exploit the original folding idea of Wong and Affleck [17]. The idea is that a conformal field theory with a purely transmitting or reflecting defect can be mapped into a boundary theory, i.e.…”