The amplitude, $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{A}(x,\ensuremath{\tau})$, of the logarithmic specific-heat singularity of a square Ising lattice with bond defects regularly spaced on an $m\ifmmode\times\else\texttimes\fi{}n$ grid (with $x=\frac{1}{\mathrm{mn}}$ and $\ensuremath{\tau}=\frac{n}{m}$) is calculated asymptotically with the result $\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{A}(x,\ensuremath{\tau})={A}_{0}\ensuremath{-}{A}_{1}x\mathrm{ln}{x}^{\ensuremath{-}1}\ensuremath{-}|{A}_{2}(\ensuremath{\tau})x\ensuremath{-}{A}_{3}{x}^{2}{\mathrm{ln}}^{2}x+O({x}^{2}\mathrm{ln}x)$, as $x\ensuremath{\rightarrow}0$. The coefficient ${A}_{1}$ verifies the scaling prediction ${A}_{1}={C}_{1}$, where ${C}_{1}$ is the amplitude of the ${(\mathrm{ln}|t|)}^{2}$ term in the incremental specific heat due to a single, isolated defect.
The amplitude, A\ifmmode˙\else˙\fi(x,\ensuremathτ), of the logarithmic specific-heat singularity of a square Ising lattice with bond defects regularly spaced on an m\ifmmode×\else\texttimes\fin grid (with x=mn1 and \ensuremathτ=mn) is calculated asymptotically with the result A\ifmmode˙\else˙\fi(x,\ensuremathτ)=A0\ensuremath−A1xlnx\ensuremath−1\ensuremath−∣A2(\ensuremathτ)x\ensuremath−A3x2ln2x+O(x2lnx), as x\ensuremath→0. The coefficient A1 verifies the scaling prediction A1=C1, where C1 is the amplitude of the (ln∣t∣)2 term in the incremental specific heat due to a single, isolated defect.
The critical-point anomaly of a plane square $m\ifmmode\times\else\texttimes\fi{}n$ Ising lattice with periodic boundary conditions (a torus) is analyzed asymptotically in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ with $\ensuremath{\xi}=\frac{m}{n}$ fixed. Among other results, it is shown that for fixed $\ensuremath{\tau}=\frac{n(T\ensuremath{-}{T}_{c})}{{T}_{c}}$, the specific heat per spin of a large lattice is given by $\frac{{C}_{\mathrm{mn}}(T)}{{k}_{\mathrm{B}}\mathrm{mn}}={A}_{0}\mathrm{ln}n+B(\ensuremath{\tau}, \ensuremath{\xi})+{B}_{1}(\ensuremath{\tau})\frac{(\mathrm{ln}n)}{n}+\frac{{B}_{2}(\ensuremath{\tau}, \ensuremath{\xi})}{n}+O[\frac{{(\mathrm{ln}n)}^{3}}{{n}^{2}}],$ where explicit expressions can be given for A0 and for the functions B, B1, and B2. It follows that the specific-heat peak of the finite lattice is rounded on a scale $\ensuremath{\delta}=\frac{\ensuremath{\Delta}T}{{T}_{c}}\ensuremath{\sim}\frac{1}{n}$, while the maximum in Cmn(T) is displaced from Tc by $\ensuremath{\epsilon}=\frac{({T}_{c}\ensuremath{-}{T}_{max})}{{T}_{c}}\ensuremath{\sim}\frac{1}{n}$. For ${\ensuremath{\xi}}_{0}>\ensuremath{\xi}>{{\ensuremath{\xi}}_{0}}^{\ensuremath{-}1}$, where ${\ensuremath{\xi}}_{0}=3.13927\ensuremath{\cdots}$, the maximum lies above Tc; but for $\ensuremath{\xi}>{\ensuremath{\xi}}_{0}$ or $\ensuremath{\xi}<{{\ensuremath{\xi}}_{0}}^{\ensuremath{-}1}$, the maximum is depressed below Tc; when $\ensuremath{\xi}=\ensuremath{\infty}, {\ensuremath{\xi}}_{0}, or {{\ensuremath{\xi}}_{0}}^{\ensuremath{-}1}$, the relative shift in the maximum from Tc is only of order n2(lnn). Detailed graphs and numerical data are presented, and the results are compared with some for lattices with free edges. Some heuristic arguments are developed which indicate the possible nature of finite-size critical-point effects in more general systems.
The specific heat, energy, and free energy of an infinitely long, square-lattice ferromagnetic Ising strip consisting of n parallel layers with free boundary conditions and a surface magnetic field H1=kBTh1 imposed on the last layer, is analyzed for large n in the light of finite-size scaling theory. It is shown rigorously that the free energy (and, similarly, the energy and specific heat) can be written asymptotically in the scaling form $\ensuremath{\Delta}f(n,t,{h}_{1})\ensuremath{\approx}{n}^{\ensuremath{-}2}(\mathrm{ln}n){W}_{1}(nt,{n}^{\frac{1}{2}}{h}_{1})+{n}^{\ensuremath{-}2}{W}_{2}(nt,{n}^{\frac{1}{2}}{h}_{1})$, where $t=\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$. The scaling functions Wi(x,y) are computed in explicit closed form and shown to verify all the analyticity and asymptotic requirements anticipated by scaling theory. Furthermore, in the limit $n\ensuremath{\rightarrow}\ensuremath{\infty}$ at fixed $t\ensuremath{\ne}0$, the bulk and surface contributions to the thermodynamic properties are found to account for all except a correction of order ${e}^{\frac{\mathrm{pna}}{\ensuremath{\xi}(T)}}$, where a is the lattice spacing and $\ensuremath{\xi}(T)$ is the bulk correlation length; the value of the small rational constant p is interpreted in terms of interference effects between the two opposite boundaries (or "surfaces").
We calculate the scaled constant-magnetization and constant magnetic field specific heats $\frac{[{C}_{M}(\ensuremath{\epsilon},M)\ensuremath{-}{C}_{M}(\ensuremath{\epsilon},0)]}{{M}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{\ensuremath{\beta}}}}$ and $\frac{[{C}_{H}(\ensuremath{\epsilon},H)\ensuremath{-}{C}_{H}(\ensuremath{\epsilon},0)]}{{H}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{\ensuremath{\beta}\ensuremath{\delta}}}}$ as functions of the scaled variable $x\ensuremath{\equiv}\frac{\ensuremath{\epsilon}}{{M}^{\frac{1}{\ensuremath{\beta}}}}$ and $y\ensuremath{\equiv}\frac{\ensuremath{\epsilon}}{{H}^{\frac{1}{\ensuremath{\beta}\ensuremath{\delta}}}}$, respectively [where $\ensuremath{\epsilon}\ensuremath{\equiv}\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and Tc is the critical temperature],for the spin-\textonehalf{} Ising model (bcc lattice) and the spin-\textonehalf{}, $\ensuremath{\infty}$ Heisenberg models (fcc lattice). Our calculations are based on previously calculated scaling functions for the M(H,T) equation of state. We also calculate the amplitudes of the zero-field specific heat for T>Tc and T<Tc. Thus, we obtain the functions ${C}_{M}(\ensuremath{\epsilon},M)$ and ${C}_{H}(\ensuremath{\epsilon},H)$, though for the Heisenberg models we cannot obtain the finite nonzero constant CM(0,0)=CH(0,0). We compare our calculated functions with the data of Connelly, Loomis, and Mapother on nickel.
A previously developed method for diagrammatic expansion of the Ising-model partition function and pair distribution function is applied to calculation of the field-free quadruplet spin averages $〈{\ensuremath{\mu}}_{1}{\ensuremath{\mu}}_{2}{\ensuremath{\mu}}_{3}{\ensuremath{\mu}}_{4}〉$. The central four-vertex function Q generated by these averages is topologically analyzed in standard fashion in terms of an irreducible four-vertex quantity e. Several rigorously necessary conditions that must be satisfied by Q are listed. It is furthermore pointed out that the existence of a logarithmic specific-heat anomaly puts an additional constraint on quadruplet averages (which is verified in an Appendix by direct calculation on the Ising-Onsager two-dimensional square lattice). Upon making a simplifying functional assumption for Q, this latter quantity may be entirely determined above Tc by one of the necessary conditions. This leads at Tc to a k2d spectrum (d=dimensionality) and above Tc to a logarithmic specific heat, both of which were adduced by Abe's Ising-model version of the He4 speculative analysis due to Patashinskii and Pokrovskii (but for different reasons from those in the present analysis). Sinec the Abe-Patashinskii-Pokrovskii spectrum almost certainly exhibits an incorrect exponent, an alternative and more powerful functional assumption for Q is suggested which still yields a soluble theory in principle, but construction of the solution is not attempted here.
The two-dimensional Ising model defined on square lattices with diamond-type bond-decorations is employed to study the nature of the ferromagnetic phase transitions of inhomogeneous systems. The model is studied analytically under the bond-renormalization scheme. For an n-level decorated lattice, the long-range ordering occurs at the critical temperature given by the fitting function (kBTc/J)n=1.6410+(0.6281)exp[−(0.5857)n], and the local ordering inside n-level decorated bonds occurs at the temperature given by the fitting function (kBTm/J)n=1.6410−(0.8063)exp[−(0.7144)n]. The critical amplitude Asing(n) of the logarithmic singularity in specific heat characterizes the width of the critical region, and it varies with the decoration-level n as Asing(n)=(0.2473)exp[−(0.3018)n], obtained by fitting the numerical results. The cross over from a finite-decorated system to an infinite-decorated system is not a smooth continuation. For the case of infinite decorations, the critical specific heat becomes a cusp with the height c(n)=0.639852. The results are compared with those obtained in the cell-decorated Ising model.