Transfer matrix in statistical mechanics

introduction

• transfer matrix is builtup from matrices of  Boltzmann weights
• trace of monodromy matrix is the transfer matrix
• finding eigenvalues and eigenvectors of transfer matrix is crucial
• Bethe ansatz equation is used to find the eigenvectors and eigenvalues of the transfer matrix
• partition function = trace of power of transfer matrices
• so the partition function  is calculated in terms of the eigenvalues of the transfer matrix
• then the problem of solving the model is reduced to the computation of this trace

transfer matrix of the 1D Ising model

• 1d Ising model

transfer matrix of the 2D Ising model

• Ising model on rectangular lattice

transfer matrix of the six-vertex model

• Six-vertex model and Quantum XXZ Hamiltonian

 

related items

• S-matrix or scattering matrix

 

expositions

• Anon.1980. The Ruelle-Araki Transfer Operator in Classical Statistical Mechanics. Vol. 123. Berlin/Heidelberg: Springer-Verlag.[1]http://www.springerlink.com/content/f12j034740601kjx/.