Exact solutions and duality

Exact results for the two-dimensional Ising model: Kramers–Wannier duality between high- and low-temperature expansions, and Onsager's formula for the free energy through Kac–Ward determinants, Sherman's loop identities and the homology of the torus.

• Kramers–Wannier duality
• Kac–Ward matrices
• Sherman's loop cancellations and product formula
• Homology and turning of loops on the square-lattice torus
• Trivalent decoration of the torus
• Spin Kac–Ward identities
• Onsager's solution of the two-dimensional Ising model

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