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

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 $H={-}J({Σ}{xy}{〈ij〉}{s}ᵢ{s}ⱼ+R{Σ}{z}{〈ij〉}{s}ᵢ{s}ⱼ){-}{μ}H{Σ}{}{i}{s}ᵢ{-}{μ}{H}^{{'}}{({-}1)}^{{η}}{Σ}{}{i}{s}ᵢ,$ where $J>0$, ${s}ᵢ=±{}1$, and ${μ}$ 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 ${H}^{{'}}$ is a staggered magnetic field which acts oppositely on adjacent planes of constant $z$ (${η}=0$ on even planes, +1 on odd planes). The Hamiltonian is invariant under $R{→}{-}R$, $H{→}{H}^{{'}}$, and ${s}ᵢ{→}{({-}1)}^{{η}}{s}ᵢ$, so that the Gibbs potential is also invariant, $G(T, H, {H}^{{'}}, R)=G(T, {H}^{{'}}, H, {-}R)$. Using this symmetry, we make a scaling hypothesis about the special point ${τ}[{≡}T{-}{T}c (R=0)]=H={H}^{{'}}=R=0$, namely, that the Gibbs potential obeys the functional equation $G({{λ}}^{{a}_{{τ}}}{τ}, {{λ}}aHH, {{λ}}^{a{H}^{{'}}}{H}^{{'}}, {{λ}}aRR)={λ}G({τ}, H, {H}^{{'}}, R)$; the four scaling powers are found to be ${a}_{{τ}}=1/2$, ${a}H={a}H=15/16$, ${a}R=7/8$.