# 海森堡模型

${\displaystyle H=\sum _{ij}J_{ij}{\vec {S}}_{i}\cdot {\vec {S}}_{j}}$

${\displaystyle H=J\sum _{\langle i,j\rangle }{\vec {S}}_{i}\cdot {\vec {S}}_{j}=J\sum _{\langle i,j\rangle }\left(S_{i}^{x}S_{j}^{x}+S_{i}^{y}S_{j}^{y}+S_{i}^{z}S_{j}^{z}\right)}$

${\displaystyle S^{\pm }=S^{x}\pm iS^{y}}$

${\displaystyle H=J\sum _{\langle i,j\rangle }\left[{\frac {1}{2}}\left(S_{i}^{+}S_{j}^{-}+S_{i}^{-}S_{j}^{+}\right)+S_{i}^{z}S_{j}^{z}\right]}$

## 一維海森堡模型

${\displaystyle H=\sum _{j=1}^{N}{\vec {S}}_{j}\cdot {\vec {S}}_{j+1}=\sum _{j=1}^{N}\left[{\frac {1}{2}}\left(S_{j}^{+}S_{j+1}^{-}+S_{j+1}^{+}S_{j}^{-}\right)+S_{j}^{z}S_{j+1}^{z}\right]}$

Kagome晶格中的自旋液體

## 各向異性

${\displaystyle H=J_{x}\sum _{\langle i,j\rangle }S_{i}^{x}S_{j}^{x}+J_{y}\sum _{\langle i,j\rangle }S_{i}^{y}S_{j}^{y}+J_{z}\sum _{\langle i,j\rangle }S_{i}^{z}S_{j}^{z}}$

${\displaystyle H=\sum _{\langle i,j\rangle }\left(S_{i}^{x}S_{j}^{x}+S_{i}^{y}S_{j}^{y}+\Delta S_{i}^{z}S_{j}^{z}\right)+D\sum _{j}\left(S_{j}^{z}\right)^{2}}$