Transfer matrix in statistical mechanics

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imported>Pythagoras0님의 2013년 2월 3일 (일) 08:55 판 (→‎정의)
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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

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

  • 자유에너지는 다음과 같다

$$ f=-\frac{1}{\beta}\lim_{N\to \infty}\frac{\ln \Lambda_0^N}{N}=-\frac{1}{\beta}\ln \Lambda_0 $$ 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


transfer matrix of the six-vertex model


 

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