Scaling relation for dangerously irrelevant symmetry-breaking fields

We propose a scaling relation for critical phenomena in which a symmetry-breaking field is dangerously irrelevant. We confirm its validity on the six-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.

Figure 1

The generic renormalization flow diagram with four fixed points: P, Q, X, and Y.

Figure 2

The estimated scaling eigenvalues from Ref. [12] (symbols), the second-order polynomial fitting to them (dashed curve), and the result of the first-order $\varepsilon$ expansion (solid curve).

Figure 3

The finite size scaling of ${\varphi }_6$ for the $Z_6$-anisotropic $XY$ model with ${\lambda }_6/J=5$ in three dimensions. (a) ${\nu }^{\prime }=1.45$, (b) ${\nu }^{\prime }=1.23$. For the scaling we used the data set of $(T_c-T)/J\le 0.4$.

Figure 4

The angle dependent spin structure factor $S_{\mathit{k}}\left({\theta }_0\right)$ (filled circle) and the probability distribution of ordering angle ${\theta }_0$ (open circle) for the $Z_6$-anisotropic $XY$ model with ${\lambda }_6/J=5, L=64$ at $T/J=2.0<T_c/J$ ($T_c/J=2.202$) in three dimensions.

Figure 5

Log-log plot of the angular Fourier transform of the angle dependent spin structure factor $\tilde{S}$ for the $Z_6$-anisotropic $XY$ model with ${\lambda }_6/J=5$ in three dimensions (a) and in four dimensions (b). (a) The finite-size scaling plot assuming $\mu =1.983$ and ${\nu }^{\prime }=1.511$. (b) The finite-size scaling plot assuming $\mu =2.0$ and ${\nu }^{\prime }=1$ in four dimensions. The critical temperatures $T_c/J=2.202$ for three dimension and $T_c/J=3.3143$ for four dimensions are estimated from the binder ratios of the magnetization. The solid curves are the fittings of the scaling function (9) assuming $\nu =0.6717, x_h=0.519, y_{\lambda }=-2.5$ for three dimensions and $\nu =0.5, x_h=1, y_{\lambda }=-2$ for four dimensions with the relation ${\nu }^{\prime }/\nu =1-y_{\lambda }/2$.