We obtain the complete magnetic phase diagram of the two-dimensional antiferromagnetic Heisenberg model, , within the framework of the O ( N ) nonlinear sigma model. We find two magnetically ordered phases, one with Néel order, for , and another with collinear order, for , separated by a nonmagnetic region, for , where a gapped spin liquid is found. The transition at is of the second order while the one at is of the first order and the spin gaps cross at . Our results are exact at N → ∞ and agree with numerical results from different methods.
The Green's-function Monte Carlo (GFMC) method is used to calculate very accurate ground-state energies of the two-dimensional, spin-1/2 Heisenberg antiferromagnet. The computations are performed on L\ifmmode\times\else\texttimes\fi{}L square lattices up to L=16 with varying uniform magnetization, which allows the extraction of the perpendicular susceptibility (\ensuremath{\chi}) and spin-wave velocity (c). These two quantities are the lattice- or cutoff-dependent parameters that allow one to map the long-wavelength properties of the antiferromagnet onto the nonlinear \ensuremath{\sigma} model and so are of general interest. Systematic errors present in previous GFMC calculations are addressed and corrected to yield results in excellent agreement with other numerical methods. I find, for the ground-state energy per site, -0.669 34(3); the susceptibility renormalization factor, ${\mathit{Z}}_{\mathrm{\ensuremath{\chi}}}$=0.535(5); and the spin-wave velocity renormalization factor,
Zc=1.10(3). Finite-size effects in the extraction of
Zc and ${\mathit{Z}}_{\mathrm{\ensuremath{\chi}}}$ are discussed. The value of ${\mathit{Z}}_{\mathrm{\ensuremath{\chi}}}$ computed here is in agreement with the series-expansion results of Singh and of Zheng, Oitmaa, and Hamer, thereby clearing up a previous inconsistency between the series-expansion and quantum Monte Carlo predictions.
The familiar spin-
21 quantum Heisenberg antiferromanget in a two-dimensional (2D) square lattice is shown, within the nonlinear sigma model approximations, to be another state of matter that has excitations with fractional quantum numbers above a finite energy gap. The 1-skyrmion with an energy $\ensuremath{\approx}2\ensuremath{\pi}J$ is shown to be made of two ``deconfined spinons'' or ``O(3) vortices.'' The many skyrmion operator and the wave functions that we have found are strikingly similar to quantum Hall quasiparticle operators and wave functions. We also predict the presence of finite energy ``spin-
S spinon'' for a general spin S Heisenberg antiferromagnets in 2D. Some consequences are briefly discussed.