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We present results of numerical studies of the two dimensional XY model withfour and eight fold symmetry breaking fields. This model has recently beenshown to describe hydrogen induced reconstruction on the W(100) surface. Basedon mean-field and renormalization group arguments,we first show how theinterplay between the anisotropy fields can give rise to different phasetransitions in the model. When the fields are compatible with each other thereis a continuous phase transition when the fourth order field is varied fromnegative to positive values. This transition becomes discontinuous at lowtemperatures. These two regimes are separated by a multicritical point. In thecase of competing four and eight fold fields, the first order transition at lowtemperatures opens up into two Ising transitions. We then use numerical methodsto accurately locate the position of the multicritical point, and to verify thenature of the transitions. The different techniques used include Monte Carlohistogram methods combined with finite size scaling analysis, the real spaceMonte Carlo Renormalization Group method, and the Monte Carlo Transfer Matrixmethod. Our numerical results are in good agreement with the theoreticalarguments.Comment: 29 pages, HU-TFT-94-36, to appear in Phys. Rev. B, Vol 50, November  1, 1994. A LaTeX file with no figure

Numerical studies of the two-dimensionalXYmodel with symmetry-breaking fields