GAPS IN THE HEISENBERG-ISING MODEL

TL;DR

This paper investigates the conditions under which energy gaps close in the ground state of the critical Heisenberg-Ising chain at momentum π, revealing special anisotropy values and the scaling behavior of the gap with system size.

Contribution

It provides a detailed analysis of gap closing phenomena at specific anisotropies using Bethe Ansatz, finite-size scaling, and perturbation theory, highlighting new insights into the model's critical behavior.

Findings

Gap closes at anisotropy values $ riangle= ext{cos}(rac{ ext{pi}}{Q})$ for half-filling.

The gap scales as a power of system size with a variable exponent depending on $ riangle$.

For rational fillings, the gap is closed for all $ riangle$, with perturbation theory showing diagram cancellations.

Abstract

We report on the closing of gaps in the ground state of the critical Heisenberg-Ising chain at momentum $\pi$. For half-filling, the gap closes at special values of the anisotropy $\Delta= \cos(\pi/Q)$, $Q$ integer. We explain this behavior with the help of the Bethe Ansatz and show that the gap scales as a power of the system size with variable exponent depending on $\Delta$. We use a finite-size analysis to calculate this exponent in the critical region, supplemented by perturbation theory at $\Delta\sim 0$. For rational $1/r$ fillings, the gap is shown to be closed for {\em all} values of $\Delta$ and the corresponding perturbation expansion in $\Delta$ shows a remarkable cancellation of various diagrams.