Ising model: Difference between revisions

Revision as of 03:18, 11 January 2025

Revision as of 04:01, 11 January 2025

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In matrix formalism

<math display="block">Z(\beta) = \operatorname{Tr} \left(V^L\right) = \lambda_1^L + \lambda_2^L = \lambda_1^L \left[1 + \left(\frac{\lambda_2}{\lambda_1}\right)^L\right],</math>

where λ₁ is the highest eigenvalue of ''V'', while λ{{mvar|&lambda;₂}} is the other eigenvalue:

<math display="block">\lambda_1 = e^{\beta J} \cosh \beta h + \sqrt{e^{2\beta J} (\cosh \beta h)^2 -2 \sinh 2 \beta J}=e^{\beta J} \cosh \beta h + \sqrt{e^{2\beta J} (\sinh \beta h)^2 -e^{-2\beta J}},</math>

and $|\frac{\lambda_2}{\lambda_1}mvar|&lambda;₂ \leq< &lambda;_1$}}. This gives the formula of the free energy above. In the thermodynamics limit for the non-interaction case (J = 0), we got

<math display="block">Z_N \to (\lambda_1)^N = (2\cosh \beta h)^N,</math>

as the answer for the open-boundary Ising model.