Anisotropy in spin chain hamiltonian

Question

The Hamiltonian of XY Spin Chain on a lattice of N sites can be written as $$ H = -J\sum_{i=1}^N \left(\frac{1+\gamma}{2}\sigma_i^x\sigma_{i+1}^x + \frac{1-\gamma}{2}\sigma_i^y\sigma_{i+1}^y + \lambda \sigma_i^z\right) $$ where $J$ is exchange interaction and $\gamma$ is anisotropy. I am not able to understand why there is a anisotropy parameter in the hamiltonian. Why cant anisotropy be encapsulated in exchange interaction?

Answer 1

You can write the Hamiltonian for the Heisenberg model as:

$H = - J_x \sum_i \sigma_i^x \sigma_{i+1}^x - J_y \sum_i \sigma_i^y \sigma_{i+1}^y - J_z \sum_i \sigma_i^z \sigma_{i+1}^z$

where $J_x, \ J_y$ and $J_z$ are the three exchange terms. In the isotropic Heisenberg model (or the $XXX$ model), all three of these terms would be equal to $J$. In the expression you have given, $J_x = J \frac{1+\gamma}{2}$, $J_y = J \frac{1-\gamma}{2}$, and $J_z = J \lambda$. So you are just using the parameter $\gamma$ to tune the anisotropy between $J_x$ and $J_y$.