Two-dimensional quantum Heisenberg antiferromagnet at low temperatures

It is argued that the long-wavelength, low-temperature behavior of a two-dimensional quantum Heisenberg antiferromagnet can be described by a quantum nonlinear $\sigma$ model in two space plus one time dimension, at least in the range of parameters where the model has long-range order at zero temperature. The properties of the quantum nonlinear $\sigma$ model are analyzed approximately using the one-loop renormalization-group method. When the model has long-range order at $T=0\mathrm{}$, the long-wavelength behavior at finite temperatures can be described by a purely classical model, with parameters renormalized by the quantum fluctuations. The low-temperature behavior of the correlation length and the static and dynamic staggered-spin-correlation functions for the quantum antiferromagnet can be predicted, in principle, with no adjustable parameters, from the results of simulations of the classical model on a lattice, combined with a two-loop renormalization-group analysis of the classical nonlinear $\sigma$ model, a calculation of the zero-temperature spin-wave stiffness constant and uniform susceptibility of the quantum antiferromagnet, and a one-loop analysis of the conversion from a lattice cutoff to the wave-vector cutoff introduced by quantum mechanics when the spin-wave frequency exceeds $\frac{T}{\hslash }$. Applying this approach to the spin-½ Heisenberg model on a square lattice, with nearest-neighbor interactions only, we obtain a result for the correlation length which is in good agreement with the data of Endoh et al. on ${\mathrm{La}}_2$Cu${\mathrm{O}}_4$, if the spin-wave velocity is assumed to be 0.67 eV $\frac{\AA }{\hslash }$. We also argue that the data on ${\mathrm{La}}_2$Cu${\mathrm{O}}_4$ cannot be easily explained by any model in which an isolated Cu${\mathrm{O}}_2$ layer would not have long-range antiferromagnetic order at $T=0\mathrm{}$. Our theory also predicts a quasielastic peak of a few meV width at 300 K when $k\xi \ll 1$ (where $k$ is wave-vector transfer and $\xi$ is the correlation length). The extent to which this dynamical prediction agrees with experiments remains to be seen. In an appendix, we discuss the effect of introducing a frustrating second-nearest-neighbor coupling for the antiferromagnet on the square lattice.