하이젠베르크 스핀 1/2 XXZ 모형

개요

• 해밀토니안

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

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

\[s_{j,l}=1-2\Delta e^{ik_l}+ e^{ik_l+ik_j}\]

• phase shift term \(\theta(p,q)\)

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

• 베테 안싸쯔 방정식

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

• fundamental equation

\[k_jL=2\pi I(k_j)+\sum_{l=1}^{n}\theta(k_j,k_l), \quad j=1,\cdots, n\]