하이젠베르크 스핀 1/2 XXZ 모형

개요

• 해밀토니안

<math>\hat H = \sum_{j=1}^{L} (\sigma_j^x \sigma_{j+1}^x +\sigma_j^y \sigma_{j+1}^y + \Delta \sigma_j^z \sigma_{j+1}^z+1)=\sum_{j=1}^{L-1}P_{i,i+1}+P_{L,1}</math>

• <math>\Delta=1</math>이면 하이젠베르크 스핀 1/2 XXX 모형(Heisenberg model)에 해당
• two body scattering term

<math>s_{j,l}=1-2\Delta e^{ik_l}+ e^{ik_l+ik_j}</math>

• phase shift term <math>\theta(p,q)</math>

<math>\exp(-i\theta(k_j,k_l))=\frac{s_{l,j}}{s_{j,l}}=\frac{1-2\Delta e^{ik_j}+e^{i(k_j+k_l)}}{1-2\Delta e^{ik_l}+e^{i(k_j+k_l)}}</math>

• 베테 안싸쯔 방정식

<math>\exp(ik_jL)=(-1)^{n-1}\prod_{l=1, l\neq j}^{n}\exp(-i\theta(k_j,k_l))=(-1)^{n-1}\prod_{l=1, l\neq j}^{n}\frac{s_{l,j}}{s_{j,l}}, \quad j=1,\cdots, n</math>

• fundamental equation

<math>k_jL=2\pi I(k_j)+\sum_{l=1}^{n}\theta(k_j,k_l), \quad j=1,\cdots, n</math>