Free energy asymptotics of the quantum Heisenberg spin chain

TL;DR

This paper analyzes the low-temperature free energy of the 1D ferromagnetic quantum Heisenberg model, showing it matches the ideal Bose gas of magnons, confirming predictions from the spin-wave approximation.

Contribution

It provides rigorous bounds on the free energy, validating the spin-wave approximation for the quantum Heisenberg model in one and two dimensions.

Findings

Free energy asymptotics match the ideal Bose gas of magnons at low temperature.

Upper and lower bounds are established for the free energy.

Results support the spin-wave approximation predictions.

Abstract

We consider the ferromagnetic quantum Heisenberg model in one dimension, for any spin $S\geq 1/2$. We give upper and lower bounds on the free energy, proving that at low temperature it is asymptotically equal to the one of an ideal Bose gas of magnons, as predicted by the spin-wave approximation. The trial state used in the upper bound yields an analogous estimate also in the case of two spatial dimensions, which is believed to be sharp at low temperature.