Ising Transfer Matrix — Exact Solution

The 1D Ising model is exactly solvable via the transfer matrix T. Its largest eigenvalue λ₁ gives the free energy; the gap λ₁−λ₂ determines the correlation length ξ = 1/ln(λ₁/λ₂). Compare the exact solution with Monte Carlo simulation to see how they agree perfectly.

Temperature T/J: 2.0 External field h/J: 0.0 Chain length N: 50

T = [[e^{K+h}, e^{-K}],
[e^{-K}, e^{K-h}]]
K = J/T, Z = Tr(T^N)
f = -T·ln(λ₁)/N

No phase transition in 1D (Ising 1925) — ξ diverges only at T→0. The 2D Ising model (Onsager 1944) has T_c = 2J/ln(1+√2) ≈ 2.27J. Here we see exact thermodynamics from 2×2 eigenvalues.