On singularities of dynamic response functions in the massless regime of the XXZ spin-1/2 chain

TL;DR

This paper provides an exact analysis of the singular behavior of dynamical response functions near excitation thresholds in the massless regime of the XXZ spin-1/2 chain, confirming and extending non-linear Luttinger liquid predictions.

Contribution

It introduces a first-principles method to analyze response function singularities in the XXZ chain, avoiding reliance on phenomenological models or field theory correspondence.

Findings

Confirmed power-law behavior near thresholds as predicted by non-linear Luttinger theory

Identified new edge exponents due to non-convex dispersion relations and slow velocity branches

Derived explicit amplitudes for response function singularities

Abstract

This work extracts, by means of an exact analysis, the singular behaviour of the dynamical response functions -- the Fourier transforms of dynamical two-point functions -- in the vicinity of the various excitation thresholds in the massless regime of the XXZ spin-1/2 chain. The analysis yields the edge exponents and associated amplitudes which describe the local behaviour of the response function near a threshold. The singular behaviour is derived starting from first principle considerations: the method of analysis \textit{does not rely, at any stage}, on some hypothetical correspondence with a field theory or other phenomenological approaches. The analysis builds on the massless form factor expansion for the response functions of the XXZ chain obtained recently by the author. It confirms the non-linear Luttinger based predictions relative to the power-law behaviour and of the associated edge exponents which arise in the vicinity of the dispersion relation of one massive excitation (hole, particle or bound state). In addition, the present analysis shows that, due to the lack of strict convexity of the particles dispersion relation and due to the presence of slow velocity branches of the bound states, there exist excitation thresholds with a different structure of edge exponents. These origin from multi-particle/hole/bound state excitations maximising the energy at fixed momentum.