Correlations in one dimensional quantum impurity problems with an external field or a temperature

TL;DR

This paper advances the theoretical understanding of low energy excitations and response functions in one-dimensional quantum impurity problems, extending previous results to include temperature effects and higher-order frequency responses.

Contribution

It provides new exact expressions for response function derivatives at zero frequency, incorporating external fields and temperature effects, using renormalized form-factors and linear response theory.

Findings

Exact expressions for derivatives of response functions at zero frequency.

Extension of results to finite temperature and no external field.

Recovery of known Hall conductance results within the linear response framework.

Abstract

We discuss in more details the theory of low energy excitations in quantum impurity problems with an external field at vanishing temperature, giving further support to results of the previous paper. We then extend these results to the next order at low frequency, obtaining in particular the exact expression, as a function of the bias, of the first two derivatives of the response function $\chi''(\omega)$ at $\omega=0$ in the double well problem of dissipative quantum mechanics. We also extend our approach to the case of non vanishing temperature and no external field. Fendley et al. had obtained in that case an expression for the Hall conductance with a single impurity using a Landauer-B\"uttiker type approach. We recover their result in the framework of linear response theory using renormalized form-factors. We also obtain, as a function of the temperature, the first derivative of the response function $\chi''(\omega)$ at $\omega=0$ in the double well problem of dissipative quantum mechanics.