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 $\mathcal{H}=\ensuremath{-}J\left(\ensuremath{\Sigma}\stackrel{\mathrm{xy}}{〈\mathrm{ij}〉}{s}_{i}{s}_{j}+\mathcal{R}\ensuremath{\Sigma}\stackrel{z}{〈\mathrm{ij}〉}{s}_{i}{s}_{j}\right)\ensuremath{-}\ensuremath{\mu}H\ensuremath{\Sigma}\stackrel{}{i}{s}_{i}\ensuremath{-}\ensuremath{\mu}{H}^{\ensuremath{'}}{(\ensuremath{-}1)}^{\ensuremath{\eta}}\ensuremath{\Sigma}\stackrel{}{i}{s}_{i},$ where $J>0$, ${s}_{i}=\ifmmode\pm\else\textpm\fi{}1$, and $\ensuremath{\mu}$ is the magnetic moment per spin. The first sum is over nearestneighbor (nn) spins in an $x\ensuremath{-}y$ plane, the second sum over nn spins coupled along the $z$ direction, $H$ is the magnitude of a uniform magnetic field, and ${H}^{\ensuremath{'}}$ is a staggered magnetic field which acts oppositely on adjacent planes of constant $z$ ($\ensuremath{\eta}=0$ on even planes, +1 on odd planes). The Hamiltonian is invariant under $\mathcal{R}\ensuremath{\rightarrow}\ensuremath{-}\mathcal{R}$, $H\ensuremath{\rightarrow}{H}^{\ensuremath{'}}$, and ${s}_{i}\ensuremath{\rightarrow}{(\ensuremath{-}1)}^{\ensuremath{\eta}}{s}_{i}$, so that the Gibbs potential is also invariant, $G(T, H, {H}^{\ensuremath{'}}, \mathcal{R})=G(T, {H}^{\ensuremath{'}}, H, \ensuremath{-}\mathcal{R})$. Using this symmetry, we make a scaling hypothesis about the special point $\ensuremath{\tau}[\ensuremath{\equiv}T\ensuremath{-}{T}_{c} (R=0)]=H={H}^{\ensuremath{'}}=R=0$, namely, that the Gibbs potential obeys the functional equation $G({\ensuremath{\lambda}}^{{a}_{\ensuremath{\tau}}}\ensuremath{\tau}, {\ensuremath{\lambda}}^{\mathrm{aH}}H, {\ensuremath{\lambda}}^{a{H}^{\ensuremath{'}}}{H}^{\ensuremath{'}}, {\ensuremath{\lambda}}^{a\mathcal{R}}\mathcal{R})=\ensuremath{\lambda}G(\ensuremath{\tau}, H, {H}^{\ensuremath{'}}, \mathcal{R})$; the four scaling powers are found to be ${a}_{\ensuremath{\tau}}=\frac{1}{2}$, ${a}_{H}={a}_{H}=\frac{15}{16}$, ${a}_{\mathcal{R}}=\frac{7}{8}$.