Transfer matrix in statistical mechanics
imported>Pythagoras0님의 2013년 2월 19일 (화) 12:56 판
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
transfer matrix of the 2D Ising model
transfer matrix of the six-vertex model
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/.