Thermodynamics of the spin square

TL;DR

This paper derives explicit analytical expressions for thermodynamic quantities of a four-spin antiferromagnetic Heisenberg ring, comparing quantum and classical results and analyzing the classical limit.

Contribution

It provides the first explicit analytical formulas for thermodynamic properties of the classical four-spin ring and compares quantum and classical behaviors across different spin quantum numbers.

Findings

Analytical expressions for density of states, partition function, and thermodynamic quantities.

Good agreement between quantum and classical results at high spin quantum numbers.

Low-temperature discrepancies diminish as spin quantum number increases.

Abstract

Small spin systems at the interface between analytical studies and experimental application have been intensively studied in recent decades. The spin ring consisting of four spins with uniform antiferromagnetic Heisenberg interaction is an example of a completely integrable system, in the double sense: quantum mechanical and classical. However, this does not automatically imply that the thermodynamic quantities of the classical system can also be calculated explicitly. In this work, we derive analytical expressions for the density of states, the partition function, specific heat, entropy, and susceptibility. These theoretical results are confirmed by numerical tests. This allows us to compare the quantum mechanical quantities for increasing spin quantum numbers $s$ with their classical counterparts in the classical limit $s\to \infty$. As expected, a good agreement is obtained, except for the low temperature region. However, this region shrinks with increasing $s$, so that the classical state variables emerge as envelopes of the quantum mechanical ones.