Nonuniversal finite-size scaling in anisotropic systems

We study the bulk and finite-size critical behavior of the $\mathrm{O}\left(n\right)$ symmetric ${\phi }^4$ theory with spatially anisotropic interactions of noncubic symmetry in $d<4$ dimensions. In such systems of a given $(d,n)$ universality class, two-scale factor universality is absent in bulk correlation functions, and finite-size scaling functions including the Privman-Fisher scaling form of the free energy, the Binder cumulant ratio, and the Casimir amplitude are shown to be nonuniversal. In particular it is shown that, for anisotropic confined systems, isotropy cannot be restored by an anisotropic scale transformation.

Figure 1

Cumulant ratio $1-U\left(w\right)∕U\left(0\right)$ vs coupling ratio $w=J^{\prime }∕(J+2J^{\prime })$ of the field-theoretic model, Eq. (7), in three dimensions for $n=1$, 2, 3 (solid, dotted, dashed lines) according to Eqs. (53, 54, 55, 56, 57).