The spin-½ Heisenberg antiferromagnet on a square lattice and its application to the cuprous oxides

The spin-½ antiferromagnetic Heisenberg model on a square lattice is used to describe the dynamics of the spin degrees of freedom of undoped copper oxides. Even though the model lacks an exact solution, a solid, accurate, and rather conventional picture emerges from a number of techniques—analytical (spin-wave theory, Schwinger boson mean-field theory, renormalization-group calculations), semianalytical (variational theory, series expansions), and numerical (quantum Monte Carlo, exact diagonalization, etc.). At zero temperature, the effect of the zero-point fluctuations is not strong enough to destroy the antiferromagnetic long-range order, despite the fact that we are dealing with a low-spin low-dimensional system. The corrections to the spin-wave theory, which treats perturbatively the effect of such fluctuations around the classical Néel ground state, appear to be small. At any nonzero temperature the order disappears and the correlation length at low temperature $T(k_B\frac{T}{J}\ll 1$, where $J$ is the antiferromagnetic coupling) follows the singular form $\xi \mathrm{}\left(T\right)=C\exp \left(\alpha \frac{J}{k_B}T\right)$. In the long-wavelength limit and at low $T$, the model has the same behavior as the quantum nonlinear $\sigma$ model in two spatial dimensions and one Euclidean time dimension, which we also study with available analytical and Monte Carlo techniques. The quasiparticles of the theory are bosons; at low $T$ and for wavelengths shorter than the correlation length they are welldefined spin-wave excitations. The spectrum of such excitations and the temperature-dependent correlation length have been determined by neutron and Raman scattering experiments done on ${\mathrm{La}}_2$Cu${\mathrm{O}}_4$. The good agreement of the experimental data with the predictions of this theory suggests that the magnetic state of the undoped materials is the conventional ordered state. We discuss, within a simple mean-field theory, the effect of weak three-dimensional antiferromagnetic coupling and the role of an antisymmetric term, introduced to explain a hidden ferromagnetic behavior of the uniform susceptibility. We find that understanding the copper-oxide antiferromagnetic insulator is only the first essential step towards the development of a theory of the superconductor created upon doping such materials.