"Transfer matrix in statistical mechanics"의 두 판 사이의 차이

2013년 2월 19일 (화) 13:07 판

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

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

@@ 7번째 줄: / 7번째 줄: @@
 * 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
@@ 39번째 줄: / 20번째 줄: @@
 ==transfer matrix of the six-vertex model==
 * [[Six-vertex model and Quantum XXZ Hamiltonian]]