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Tunneling in Quantum Wires: a Boundary Conformal Field Theory Approach

Eugene Wongb, Ian K. Affleck

Published Aug 14, 2016• 73 citations
OpenAlex

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 infinite 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.

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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 infinite 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.
The exact solution of the one-dimensional Hubbard model (for repulsive and attractive U)U)U) is used to calculate the tunneling rates of single electrons and Cooper pairs between two quantum wires. The critical exponents for the tunneling at small frequencies and low temperatures is obtained by means of the mesoscopic energy spectrum and conformal field theory. The situations of crossing wires, tunneling at a boundary, and resonant tunneling (through a bound state) are considered.
Using the principles of the conformal quantum-field theory and the finite size corrections of the energy of the ground and various excited states, we calculate the boundary critical exponents of single- and multicomponent Bethe-Ansatz soluble models. The boundary critical exponents are given in terms of the dressed-charge matrix which has the same form as that of systems with periodic boundary conditions and is uniquely determined by the Bethe-ansatz equations. A Luttinger liquid with open boundaries is the effective low-energy theory of these models. As applications of the theory, the Friedel oscillations due to the boundaries and the tunneling conductance through a barrier are also calculated. The tunneling conductance is determined by a nonuniversal boundary exponent which governs its power law dependence on temperature and frequency. \textcopyright{} 1996 The American Physical Society.
We develop a method to extract the universal conductance of junctions of multiple quantum wires, a property of systems connected to reservoirs, from static ground-state computations in closed finite systems. The method is based on a key relationship, derived within the framework of boundary conformal field theory, between the conductance tensor and certain ground-state correlation functions. Our results provide a systematic way of studying quantum transport in the presence of strong electron-electron interactions using efficient numerical techniques such as the standard time-independent density-matrix renormalization-group method. We give a step-by-step recipe for applying the method and present several tests and benchmarks. As an application of the method, we calculate the conductance of the M fixed point of a Y junction of Luttinger liquids for several values of the Luttinger parameter ggg and conjecture its functional dependence on ggg.