System Exhibiting a Critical Point of Order Four: Ising Planes with Variable Interplanar Interactions

Abstract

The first phase diagram explicitly exhibiting intersecting lines of tricritical points is presented. Their point of intersection is a critical point of order 4. The system is a simple assembly of ferromagnetic Ising planes coupled with an arbitrary interplanar interaction, with Hamiltonian $\mathcal{H}=-J\left(\Sigma \stackrel{\mathrm{xy}}{〈\mathrm{ij}〉}{s}_i{s}_j+\mathcal{R}\Sigma \stackrel{z}{〈\mathrm{ij}〉}{s}_i{s}_j\right)-\mu H\Sigma \stackrel{}{i}{s}_i-\mu H^{\prime }{\mathrm{}(-1\mathrm{})\mathrm{}}^{\eta }\Sigma \stackrel{}{i}{s}_i,$ where $J>\mathrm{}0\mathrm{}$, $s_i=\pm 1$, and $\mu$ is the magnetic moment per spin. The first sum is over nearestneighbor (nn) spins in an $x-y$ plane, the second sum over nn spins coupled along the $z$ direction, $H$ is the magnitude of a uniform magnetic field, and $H^{\prime }$ is a staggered magnetic field which acts oppositely on adjacent planes of constant $z$ ($\eta =0\mathrm{}$ on even planes, +1 on odd planes). The Hamiltonian is invariant under $\mathcal{R}\to -\mathcal{R}$, $H\to H^{\prime }$, and $s_i\to {\mathrm{}(-1\mathrm{})\mathrm{}}^{\eta }{s}_i$, so that the Gibbs potential is also invariant, $G(T, H, H^{\prime }, \mathcal{R})=G(T, H^{\prime }, H, -\mathcal{R})$. Using this symmetry, we make a scaling hypothesis about the special point $\tau [\equiv T-T_c \mathrm{}(R=0\mathrm{}\left)\right]=H=H^{\prime }=R=0\mathrm{}$, namely, that the Gibbs potential obeys the functional equation $G({\lambda }^{a_{\tau }}\tau , {\lambda }^{\mathrm{aH}}H, {\lambda }^{a{H}^{\prime }}{H}^{\prime }, {\lambda }^{a\mathcal{R}}\mathcal{R})=\lambda G(\tau , H, H^{\prime }, \mathcal{R})$; the four scaling powers are found to be $a_{\tau }=\frac{1}{2}$, $a_H=a_H=\frac{15}{16}$, $a_{\mathcal{R}}=\frac{7}{8}$.