We study certain aspects of the nonlinear σ model regularized on the lattice in two space and one Euclidean-time dimensions using the Monte Carlo method. For certain purposes this model is considered as the long-wavelength limit of the quantum spin-S antiferromagnetic Heisenberg model in two space dimensions, where the different spin cases map to different values of the coupling constant g of the σ model. For the value of g that corresponds to the spin-1/2 case on the square lattice, we find that the most probable configurations are characterized by large-ampltiude short-range quantum fluctuations. Such configurations lack smoothness, which, however, can be achieved by means of real-space block-spin transformations. We calculate the Berry phase correlations, namely, the correlation function ${\mathit{C}}_{\mathit{i}\mathit{j}}$==〈exp{iS[Σ(${\mathbf{r}}_{\mathit{i}}$)+(-1${)}^{\mathrm{\Vert }\mathit{i}\mathrm{-}\mathit{j}\mathrm{\Vert }}$Σ (${\mathbf{r}}_{\mathit{j}}$)]}〉, where Σ(${\mathbf{r}}_{\mathit{i}}$) is the area on the unit sphere defined by the path of the spin located at the spatial lattice point ${\mathbf{r}}_{\mathit{i}}$ during its Euclidean-time evolution. We find that the contribution to hole-hole attraction from such correlations is limited to distances of a few lattice spacings. Finally, we examine the possible presence of topological monopole singularities in configurations, and we find only spin-wave excitations.

• Received 18 September 1990

DOI:https://doi.org/10.1103/PhysRevB.43.2615