Tunneling in Quantum Wires: a Boundary Conformal Field Theory Approach

TL;DR

This work treats tunneling in a one-dimensional interacting electron gas with a localized barrier as a boundary conformal field theory problem. By folding the system into a two-channel model with $U(1)\times U(1)$ symmetry and performing bosonization, the authors construct boundary states (open and periodic) and compute finite-size partition functions to map universal boundary fixed points to conductance regimes. They reproduce and extend Kane-Fisher results for spinless and spin-$\tfrac{1}{2}$ fermions, identify operator contents and ground-state degeneracies, and analyze resonant tunneling via scaling dimensions and Kosterlitz-Thouless–type flows, all consistent with the $g$-theorem. The approach demonstrates a nonperturbative route to exact properties of boundary critical points in irrational $c=4$ theories and clarifies how modular-transformations constrain the spectrum and fixed-point structure in these 1D quantum impurity problems.

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 ``$ε$-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 $\infty$ 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.