Low-temperature asymptotics for transverse autocorrelator of the magnetically polarized Ising chain studied by ordinary and nested Dyson equations

TL;DR

This paper develops low-temperature asymptotics for the transverse autocorrelator in the magnetically polarized Ising chain using Dyson equations, distinguishing between ordinary and nested approaches and their respective contributions.

Contribution

It introduces two versions of the Plakida-Tserkovnikov algorithm for low-temperature analysis and derives corresponding Dyson equations, clarifying their roles in capturing magnon processes.

Findings

Ordinary Dyson equation correctly accounts for magnon creation.

Nested Dyson equation captures magnon to bound two-magnon transitions.

Results can be extended to the polarized XXZ-chain.

Abstract

Suggesting two versions for the Plakida-Tserkovnikov algorithm breakdown in the low-temperature regime, we derive ordinary and nested Dyson equations for the transverse autocorrelator of the magnetically polarized Ising chain. Using them we get the corresponding low-temperature asymptotics for the autocorrelator. We show that the ordinary Dyson equation results in a correct $o({\rm e}^{-\beta E_{gap}})$ account of the magnon creation process, while the nested Dyson equation additionally gives the correct $o({\rm e}^{-\beta E_{gap}})$ contribution associated with transitions from magnons to bound two-magnon states. The obtained result may be useful for the extension of the suggested approach on the polarized $XXZ$-chain.