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
• 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

Bethe ansatz

• Bethe ansatz equation is used to find the eigenvectors and eigenvalues of the transfer matrix

transfer matrix of the six-vertex model

• Six-vertex model and Quantum XXZ Hamiltonian

 

related items

• S-matrix or scattering matrix
• 1d Ising model
• Ising model on rectangular lattice

expositions

• “The Kramers-Wannier Transfer Matrix.” 1980. In The Ruelle-Araki Transfer Operator in Classical Statistical Mechanics, 13–39. Lecture Notes in Physics 123. Springer Berlin Heidelberg. http://link.springer.com/chapter/10.1007/BFb0017921.