1D Ising Model — Exact Transfer Matrix Solution

Transfer matrix method: The 1D Ising Hamiltonian H = −J Σ sᵢsᵢ₊₁ − h Σ sᵢ is solved exactly. Transfer matrix T = [[e^{β(J+h)}, e^{-βJ}],[e^{-βJ}, e^{β(J-h)}]]. Eigenvalues λ± = e^{βJ}[cosh(βh) ± √(sinh²(βh)+e^{-4βJ})]. Free energy per spin: f = −kT ln λ+. Correlation length: ξ = −1/ln(λ−/λ+). No phase transition in 1D at finite T.