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

정의

• 스핀 $s_i, i=1,\cdots, N$과 주기조건 $s_{N+1}=s_1$을 가정
• 스핀 $s_i$과 $s_{i+1}$의 상호작용 $E(s_i,s_{i+1})$
• 해밀토니안이 $H=\sum_{i=1}^{N} E(s_i,s_{i+1})$ 꼴로 쓰여지는 경우
• 전달행렬은 $T_{s_i,s_{i+1}}=\exp(-\beta E(s_i,s_{i+1}))$ 꼴로 쓸 수 있으며, 분배함수는 다음과 같이 주어진다

$$ Z_N=\sum_{s_1,\cdots,s_N}T_{s_1,s_2}\cdots,T_{s_N,s_1}=\operatorname{Tr} T^N $$

• 자유에너지(per site) 는 다음과 같다

$$ F=-\frac{1}{\beta}\lim_{N\to \infty}\frac{\ln \Lambda_0^N}{N}=-\frac{1}{\beta}\ln \Lambda_0, $$ 또는 $$ F=-\frac{1}{k T}\ln \Lambda_0, $$ 이 때 $\Lambda_0$는 $T$의 최대인 고유값

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/.