Exact 1D solution via T, partition function Z = Tr(TN), Onsager 2D
1D Transfer Matrix
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Transfer matrix: T = [[e^(β(J+h)), e^(-βJ)], [e^(-βJ), e^(β(J-h))]]
Eigenvalues: λ± = e^(βJ)cosh(βh) ± √(e^(2βJ)sinh²(βh)+e^(-2βJ)) Z = λ+^N + λ-^N, F = -kT ln Z
Free Energy & Correlation Length vs T
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Correlation length: ξ = -1/ln(λ-/λ+)
Diverges as T→0: ξ ~ e^(2J/kT) (no phase transition in 1D at finite T by Peierls argument).
Correlation Function ⟨σ₀σᵣ⟩
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Exact: ⟨σ₀σᵣ⟩ = (λ-/λ+)^r = tanh^r(βJ) (h=0)
Exponential decay with correlation length ξ. At T→0: tanh→1, infinite correlation.
Onsager 2D Exact Solution
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Onsager (1944): T_c = 2J/[k·ln(1+√2)] ≈ 2.269 J/k
Specific heat: C ~ -A·ln|T-T_c| (logarithmic divergence, α=0).
Spontaneous magnetization: M = (1-sinh^(-4)(2βJ))^(1/8)