We study the spin-1/2 quantum ferromagnetic and antiferromagnetic Heisenberg model using Handscomb’s Monte Carlo (MC) method on square lattices of various sizes. As the temperature is lowered the calculated correlation length in the antiferromagnetic case grows more rapidly than in the ferromagnetic case. We also obtain the correlation length in the leading order of the high-temperature series expansion which, at high temperatures, agrees very well with the MC results. The correlation length obtained from the MC calculation for the ferromagnetic and antiferromagnetic case is compared with existing theories. Taking the average value for the antiferromagnetic coupling between the values suggested by neutron- and Raman-scattering experiments done on ${\mathrm{La}}_2$${\mathrm{CuO}}_4$, we compare our results for the correlation length with those observed by the neutron-scattering experiments. We find that our results for the correlation lengths away from the three-dimensional (3D) Néel temperature $T_N$∼200 K are consistent with the experimental findings. In order to obtain agreement close to the Néel temperature, however, we need to introduce an interlayer coupling between the ${\mathrm{CuO}}_2$ planes. The effect on a 3D coupling is only discussed in the framework of the quantum mechanical nonlinear σ model in three space dimensions. For the case of ${\mathrm{La}}_2$${\mathrm{CuO}}_4$ we find that close to $T_N$ the σ model in 3+1 dimensions reduces to the classical 3D Heisenberg model whose critical properties are known and fit the neutron-scattering data for T∼$T_N$.

• Received 11 April 1988

DOI:https://doi.org/10.1103/PhysRevB.39.575