Tricriticality and the failure of scaling in the many-component limit

Abstract

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

Universality of TricriticalHe3-He

“…of the data with Aw=A(l-T/T t )* 9 where A, T t , and ]3 are fit parameters, which yielded T t =0.867 K at SVP and 0.745 K at 22.6 bars. The critical exponent j3 derived from these data is V-__ )0.99±0.02 at SVP, 1.02±0.03at 22.6 bars; (2) similar values were also found at intermediate pressures.…”

Mean-field expansion for spin models with medium-range interactions

“…The scaling behaviour of β is obtained requiring t ∼ O(R −6 ). If we introduce a variable t related to β by 56) and define the critical crossover limit as the limit β → 0, R → ∞ with t fixed, we find that indeed t ∼ R −6 . More precisely, using Eqs.…”

“…3 in the large-N limit following the strategy of Refs. [56][57][58]. We write V (ϕ) = NW (ϕ 2 ) and study the limit N → ∞ with β and W (x) fixed.…”

The O(N ) model with ϕ6 potential in ℝ2 × ℝ+

“…3 The beta function naively indicates that in the strict large N limit there is a flow from an interacting UV fixed point with g = 192 to a free IR fixed point. In fact, the theory appears to be unstable for g > 16π 2 ≈ 158 [12,18,19]. We will find some additional evidence for this instability from our boundary field theory perspective.…”