We analyze exactly in the limit n -+ oo, the n-component continuous-spin model with a cubic field term Ai N~g~'. Full "four-field" tricritical behavior is exhibited for dimensionalities d & 3. However, orthodox tricritical scaling is shown to be impossible for 3 & d &4: to obtain the correct nonclassical spherical-model exponents on the X line (H = H, = 0, T & T,) it is essential to allow for a dangerous irrelevant scaling variable p af: 1/Ro, where Ro is the range of the pair interactions. The appropriate crossover exponent is Q~= 3d so that p is marginal for d = 3: orthodox scaling is then possible but the scaling functions are nonuniversal. On the disordered symmetry plane (H = H3 = 0) only the corrections to scaling survive and describe crossover to Gaussian tricritical behavior. For T & T, bicritical crossover from spherical to classical critical behavior occurs when H3 varies but scaling is fully obeyed. Some inferences for systems with finite n are drawn.Landau-Ginzburg-Wilson-type Hamiltonian).At first sight, the results of this analysis are in good agreement with the general scaling theory. However, we will show that this impression is erroneoust Indeed, some difficulties in a naive application of scaling have been noted previously in connection with certain exactly soluble models exhibiting tricritical behavior. The models are fairly special, but quite subtle analyses, sometimes entailing the 18
The asymptotic tricritical equation of state, including the three=phase coexistence monohedron, is analyzed in detail for the exactly soluble multicomponent or spherical limit, n~, of the continuous-spin model with terms of order s, s, and s, in d =3 spatial dimensions. Various nonuniversal scaling functions and amplitude ratios, depending on the range parameter z~1/Ro, are evaluated explicitly and reveal the nature and magnitude of the deviations from the classical, phenomenological theory of tricriticality (which is developed systematically in an Appendix). The relationship to results for finite n is discussed briefly.
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