Scaling relation for dangerously irrelevant symmetry-breaking fields

Abstract

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...

Nature of the possible magnetic phases in a frustrated hyperkagome iridate

“…To obtain them, we fixed the critical exponents β and ν to be those for the 3D XY universality class (ν = 0.672 and β = 0.348) 37 , while finetuned the critical temperature T c such that all the data points for m fall into one scaling function f (x). With T c thus obtained, we further fine-tuned the crossover exponent for m 6 , ν 6 , such that all the data points for The former value of the crossover exponent ν 6 (1.45 ±0.05) is consistent with previous estimation in the Potts model 35,36 , while the latter value (1.85 ±0.05) is relatively larger. The discrepancy stems from the presence of a high symmetric point at (J, D, G) = (0, 0, −1) near the latter parameter point.…”

“…In the effective model, the quantum effect is taken into account as an addition of the imaginary time dimension (+1) to the spatial dimension (d = 3). The 4D Z 6 Potts model exhibits the same kind of finite-size crossover phenomena with different crossover exponent 36 . Moreover, in the (3+1)D model, finite temperature leads to a non-trivial 'finite-size' effect along the imaginary time direction, in the same way as the finite system size does along the spatial direction.…”

“…The crossover system size and temperature can be evaluated from the FNS analysis on the XY order parameter m and Z 6 order parameter m 6 . The scaling argument [34][35][36] suggests that these two follow single-parameter scalings;…”

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“…We again find the same conclusions as that from visual inspection of W 3 versus L behavior: there is a finite value of W 3c = g W3 (0) at the critical point for all Q 2 , which furthermore seems to slightly increase with Q 2 . Finally, within our precision, it is not possible to positively confirm that the extracted value of ν 3 is larger than ν (the two exponents are essentially equal within error bars): independent of the exact relation between the two 26,64 , this indicates that 3−fold anisotropy is only very slightly irrelevant, consistent with a non-vanishing W 3c within our system size range.…”

“…It is also possible to analyze our data using scaling theories 25,26,64 to capture the finite-size behavior of W 3 near criticality. We have used such a scaling analysis to fit our numerical data as detailed in Appendix A, but we prefer to display the bare numerical data for the anisotropy measure W 3 in Sec.…”