Bounded and Inhomogeneous Ising Models. I. Specific-Heat Anomaly of a Finite Lattice

The critical-point anomaly of a plane square m Ising lattice with periodic boundary conditions (a torus) is analyzed asymptotically in the limit n with =mn fixed. Among other results, it is shown that for fixed =n (T-{T₂) }{T₂}, the specific heat per spin of a large lattice is given by C₌₍ (T) {k₁mn}=A₀lnn+B (, ) +B₁ () (lnn) n+B₂ (, ) n+O (lnn) ^{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 ={T₂}1n, while the maximum in C₌₍ (T) is displaced from T₂ by = (T₂-{T₌₀ₗ) }{T₂}1n. For ₀>>{₀}^-1, where ₀=3. 13927, the maximum lies above T₂; but for >₀ or <{₀}^-1, the maximum is depressed below T₂; when =, ₀, or {₀}^-1, the relative shift in the maximum from T₂ is only of order (lnn) {n^2}. 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.