Phase Transition of a Two-Dimensional Continuum—Ising Model

We decorate a plane two‐dimensional Ising model by placing on each bond of the lattice a continuum spin which is allowed to interact only with the Ising spins at the ends of the bonds. The continuum spins are chosen to be either Gaussian (i.e., normally distributed with zero mean and unit variance) or spherical (i.e., constrained to lie on the surface of a sphere) after Berlin and Kac, and by integrating out the continuum spins, the partition function of the decorated lattice is expressed in terms of the Onsager partition function of the plane two‐dimensional Ising model. The critical behavior of the model is as follows: For the Gaussian case, as for the plane Ising model, the specific heat has a logarithmic singularity at the critical point T_c^G given by $$\text{2\hspace{0.17em}}{\tanh }^2{\text{\hspace{0.17em}[2(J/kT}}_{\mathrm{c}}^G{)}^2\text{]=1}$$ and as ${\mathrm{t=T}}_{\mathrm{c}}^G{\text{−T→0}}^{+}$ , the spontaneous magnetization goes to zero like t^1/8. For the spherical case, the specific heat is continuous and has a cusp at the critical point T_c^s given by ${\mathrm{J/kT}}_{\mathrm{c}}^{\mathrm{s}}{\mathrm{=[z}}_{\mathrm{c}}{\text{+(2+2}}^{\frac{1}{2}}{\mathrm{)z}}_{\mathrm{c}}^2{]}^{\frac{1}{2}}\text{,\hspace{0.17em}2\hspace{0.17em}}{\tanh }^2{\text{\hspace{0.17em}(2z}}_{\mathrm{c}}\text{)=1}$ , with slope going to ± ∞ like $t^{\text{−1}}\mathrm{\hspace{0.17em}[}\ln {\mathrm{|t|]}}^{\text{−2}}$ as ${\mathrm{t=T}}_{\mathrm{c}}^{\mathrm{s}}{\text{−T→0}}^{\pm }$ , and as t → 0⁺, the spontaneous magnetization goes to zero like [t/ln t]^1/8.