Two-dimensional quantum Heisenberg antiferromagnet at low temperatures

TLDR

The low‑temperature physics of a two‑dimensional quantum Heisenberg antiferromagnet is captured by a quantum nonlinear σ‑model, which reduces to a classical model with quantum‑renormalized parameters when long‑range order exists at zero temperature, though dynamical predictions remain to be experimentally verified. The authors employ a one‑loop renormalization‑group analysis of the quantum nonlinear σ‑model, supplemented by two‑loop RG and classical lattice simulations, to predict correlation lengths and spin‑correlation functions, and also examine the effect of frustrating next‑nearest‑neighbor couplings. For the spin‑½ Heisenberg model on a square lattice, the predicted correlation length matches experimental data, the La₂CuO₄ measurements rule out models without zero‑temperature long‑range order, and the theory forecasts a few‑meV quasielastic peak at 300 K when kξ≪1.

Abstract

It is argued that the long-wavelength, low-temperature behavior of a two-dimensional quantum Heisenberg antiferromagnet can be described by a quantum nonlinear $\ensuremath{\sigma}$ 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 $\ensuremath{\sigma}$ model are analyzed approximately using the one-loop renormalization-group method. When the model has long-range order at $T=0$, 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 $\ensuremath{\sigma}$ 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 $\frac{T}{\ensuremath{\hbar}}$. Applying this approach to the spin-\textonehalf{} 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 $\frac{\AA{}}{\ensuremath{\hbar}}$. 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 $T=0$. Our theory also predicts a quasielastic peak of a few meV width at 300 K when $k\ensuremath{\xi}\ensuremath{\ll}1$ (where $k$ is wave-vector transfer and $\ensuremath{\xi}$ 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.