BOUNDED AND INHOMOGENEOUS ISING MODELS .I. SPECIFIC-HEAT ANOMALY OF A FINITE LATTICE

The critical-point anomaly of a plane square m×n Ising lattice with periodic boundary conditions (a torus) is analyzed asymptotically in the limit n→ with ξ=mn fixed. Among other results, it is shown that for fixed τ=n(T-Tc)Tc, the specific heat per spin of a large lattice is given by Cmn(T)kBmn=A0lnn+B(τ,ξ)+B1(τ)(lnn)n+B2(τ,ξ)n+O[(lnn)3n2], 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 δ=ΔTTc∼1n, while the maximum in Cmn(T) is displaced from Tc by ε=(Tc-Tmax)Tc∼1n. For ξ0>ξ>ξ0-1, where ξ0=3.13927, the maximum lies above Tc; but for ξ>ξ0 or ξ<ξ0-1, the maximum is depressed below Tc; when ξ=,ξ0,orξ0-1, the relative shift in the maximum from Tc is only of order (lnn)n2. 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. © 1969 The American Physical Society.