Zero-field Partition Function and Free Energy Density of the Two-Dimensional Heisenberg Classical Square Lattice

TL;DR

This paper provides an exact analysis of the zero-field partition function and free energy density for the 2D classical Heisenberg square lattice, revealing phase crossovers and critical behavior at zero temperature.

Contribution

It introduces a rigorous derivation of the zero-field partition function and free energy for the 2D classical Heisenberg model, including phase diagram and critical exponents.

Findings

Identification of three magnetic phases near T=0 K

Exact expression for the free energy density

Confirmation of critical exponent nu=1 at T=0 K

Abstract

We rigorously examine 2d square lattices composed of Ninf{S} classical spins isotropically coupled. If Hsup{ex},inf{i,j} is the local exchange Hamiltonian each operator exp(-beta.Hsup{ex},inf{i,j}) is expanded on the basis of spherical harmonics Yinf{linf{ i,j }, minf{ i,j }}. We derive selection rules for the linf{ i,j}'s and minf{ i,j }'s. For infinite Ninf{S} the value m = 0 is selected. We obtain an exact l-polynomial for the zero-field partition function, valid for any temperature. Its thermal study allows to point out crossovers between the l-eigenvalues. Near Tinf{c} = 0 K we derive a diagram showing three magnetic phases, each one being characterized by the low-temperature behavior of the correlation length. At Tinf{c}= 0 K, we retrieve the critical exponent nu = 1. We identify three regimes: the renormalized classical, the quantum disordered and the quantum critical regimes. We exactly express the free energy density F. For each result we retrieve the corresponding one derived from the renormalization approach.