Form factors approach to current correlations in one dimensional systems with impurities

TL;DR

This paper develops an analytical method using massless form factors to compute current correlations in one-dimensional quantum systems with impurities, achieving high accuracy with few terms across all coupling strengths.

Contribution

It introduces a form factor approach for calculating correlations in impurity-laden 1D quantum systems, effective at arbitrary coupling and distances.

Findings

Accurately computes frequency-dependent conductance in Luttinger liquids.

Determines spectral functions in dissipative quantum mechanics.

Analyzes space-dependent susceptibility in the Kondo model.

Abstract

We show how to compute analytically time and space dependent correlations in one dimensional quantum integrable systems with an impurity. Our approach is based on a description of these systems in terms of massless scattering of quasiparticles. Correlators follow then from matrix elements of local operators between multiparticle states, the ``massless form factors''. Although an infinite sum of these form factors has to be considered in principle, we find that for current, spin, and energy operators, only a few (typically two or three) are necessary to obtain an accuracy of more than $1\%$, for {\bf arbitrary coupling strength}, that is all the way from short to large distances. As examples we compute, at zero temperature, the frequency dependent conductance in a Luttinger liquid with impurity, the spectral function in the double well problem of dissipative quantum mechanics and part of the space dependent succeptibility in the Kondo model .