Ising 1D: Exact Transfer Matrix

The transfer matrix method solves the 1D Ising model exactly. No phase transition at finite temperature — the partition function Z = (λ₊^N + λ₋^N) where λ_± = e^J/Tcosh(H/T) ± √(e^2J/Tsinh²(H/T) + e^−2J/T).

Parameters

J (coupling) 1.00 H (field) 0.00 T (temperature) 2.00 N (chain length) 40

Transfer matrix:
T = [[e^(J+H)/T, e^(-J)/T],
[e^(-J)/T, e^(J-H)/T]]

λ±: eigenvalues (shown below)
F = −NT ln(λ+) as N→∞

No phase transition in 1D
(Ising 1925, Peierls argument)

Phase transition only for d ≥ 2.

λ+ = —
λ− = —
ξ = — (corr. length)