The quantized O(1,2)/O(2) x Z2 sigma model has no continuum limit in four dimensions. II. Lattice simulation.

Abstract

A lattice formulation of the $\frac{\mathrm{O}(1,2)}{\mathrm{O}(2)\ifmmode\times\else\texttimes\fi{}{Z}_{2}}$ sigma model is developed, based on the continuum theory presented in the preceding paper. Special attention is given to choosing a lattice action (the "geodesic" action) that is appropriate for fields having noncompact curved configuration spaces. A consistent continuum limit of the model exists only if the renormalized scale constant ${\ensuremath{\beta}}_{R}$ vanishes for some value of the bare scale constant $\ensuremath{\beta}$. The geodesic action has a special form that allows direct access to the small-$\ensuremath{\beta}$ limit. In this limit half of the degrees of freedom can be integrated out exactly. The remaining degrees of freedom are those of a compact model having a $\ensuremath{\beta}$-independent action which is noteworthy in being unbounded from below yet yielding integrable averages. Both the exact action and the $\ensuremath{\beta}$-independent action are used to obtain ${\ensuremath{\beta}}_{R}$ from Monte Carlo computations of field-field averages (two-point functions) and current-current averages. Many consistency cross-checks are performed. It is found that there is no value of $\ensuremath{\beta}$ for which ${\ensuremath{\beta}}_{R}$ vanishes. This means that as the lattice cutoff is removed the theory becomes that of a pair of massless free fields. Because these fields have neither the geometry nor the symmetries of the original model we conclude that the $\frac{\mathrm{O}(1,2)}{\mathrm{O}(2)\ifmmode\times\else\texttimes\fi{}{Z}_{2}}$ model has no continuum limit.