Nonstandard Finite-Size Scaling at First-Order Phase Transitions

We note that the standard inverse system volume scaling for finite-size corrections at a first-order phase transition (i.e., $1/L^3$ for an $L\times L\times L$ lattice in 3D) is transmuted to $1/L^2$ scaling if there is an exponential low-temperature phase degeneracy. The gonihedric Ising model which has a four-spin interaction, plaquette Hamiltonian provides an exemplar of just such a system. We use multicanonical simulations of this model to generate high-precision data which provide strong confirmation of the nonstandard finite-size scaling law. The dual to the gonihedric model, which is an anisotropically coupled Ashkin-Teller model, has a similar degeneracy and also displays the nonstandard scaling.

Figure 1

Best fits using the leading $1/L^2$ scaling obtained for the original model in Eq. (8) using the (finite lattice) peak locations for the specific heat $C_V^{\max }$, Binder’s energy parameter $B^{\min }$; or inverse temperatures ${\beta }^{\text{eqw}}$ and ${\beta }^{\text{eqh}}$, where the two peaks of the energy probability density are of same weight or have equal height, respectively. The values for ${\beta }^{\text{eqw}}$ and ${\beta }^{C_V^{\max }}$ are indistinguishable in the plot. The omitted corrections which we discuss in detail in Ref. [15] give the slightly different effective slopes. The inset shows the energy probability density $p(e)$ over $e=E/L^d$ at ${\beta }^{\text{eqh}}$ for lattices with linear length $L\in \{13,14,\dots ,26,27\}$.

Figure 2

Same as Fig. 1 but for the dual model, Eq. (9). Here the inset shows the energy probability density $p(e)$ at ${\beta }^{\text{eqh}}$ for lattices with linear length $L\in \{12,14,\dots ,22,24\}$.

Figure 3

Plot of the goodness-of-fit parameter $Q$ for fits on the extremal locations of the specific heat, ${\beta }^{C_V^{\max }}$, and Binder’s energy parameter, ${\beta }^{B^{\min }}$ of the original model for different fitting ranges $L_{\min }$–$L_{\max }$. Upper row: Standard ($1/L^3$) finite-size scaling ansatz. Lower row: Transmuted ($1/L^2$) finite-size scaling.