We propose a scaling relation for critical phenomena in which a symmetry-breaking field is dengerously irrelevant. We confirm its validity on the 6-state clock model in three and four dimensions by numerical simulation. In doing so, we point out the problem in the previously-used order parameter, and present an alternative evidence based on the mass-dependent fluctuation. PACS numbers: 75.40.Cx, 05.70.Fh, 75.10.Hk, 75.40.Mg Irrelevant scaling fields are ubiquitous. While they play minor roles in most cases, some of them are quite relevant in the usual sense of the word. A text-book example is the φ 4 term in the φ 4 theory above the upper critical dimension [1]. In the present Letter, we discuss cases where such a dangerously-irrelevant scaling field reduces the symmetry of the system, and demonstrate that it yields a new scaling relation.Consider a renormalization-group flow diagram including two fixed points; one describing the critical point and the other the ordered phase. In principle it is possible that some irrelevant perturbative field at the critical fixed point contains some scaling field that is relevant at the one of the two. In particular, when the perturbation is symmetry-reducing, it can happen that both fixed points lie on the same manifold characterized by zero of the perturbative field as illustrated in Fig. 1. In such cases, even if the perturbation almost dies out at some length scale, say ξ, it may recover its amplitude at larger length scale, say ξ ′ . When the system size is between the two scaling lengths, ξ ≪ L ≪ ξ ′ , the system may look ordered but still no effect of the symmetry breaking is visible. It may then appear that an intermediate phase exists where the system acquires an emergent symmetry. A classical example of this type of renormalization group flow is the q-state clock model in three dimensions [2], and its continuous-spin counterpart.In fact, such an intermediate phase really exists in two dimensions [3]. However, based on the Monte Carlo simulation results, Miyashita [4] suggested a simpler scenario for the three dimensional case. Furthermore, Oshikawa [2] pointed out that the existence of the intermediate phase is very unlikely because the low-temperature phase is already ordered in the pure model in three dimensions, and that the whole low-temperature phase is controlled by the zero-temperature fixed point, in contrast to the two-dimensional case. The two-dimensional quantum SU(N ) Heisenberg model may offer a quantummechanical example. While the ground state of this model is the Neèl state upto N = 4, the valence bond solid state takes over for N ≥ 5 [5]. When described in terms of effective spins representing the direction of the ordered valence bond pattern, the system can be regarded as a model analogous to the clock model. It was discovered that the order parameter distribution function is almost circular symmetric, indicating the extremely small effect of the anisotropy. Later, an additional term was introduced [6-8] to control the quantum fluctuation and driv...
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