I understand that the Quantum Heisenberg XXZ model in 1D has the form:

$$\hat H = \frac{1}{2} \sum_{j=1}^{N} (J_x \sigma_j^x \sigma_{j+1}^x + J_y \sigma_j^y \sigma_{j+1}^y + J_z \sigma_j^z \sigma_{j+1}^z + h\sigma_j^{z})$$

with the term $J_z \sigma_j^z \sigma_{j+1}^z$ disfavoring/favoring nearby aligned spins depends on the sign of $\hat H$. However, I was wondering if there's a name for a model with the $J_z \sigma_j^z \sigma_{j+1}^z$ replaced by something that only cares for up-up nearby spins, but don't care about down-down alignment? Something like that could be implemented by a $J_z n_j n_{j+1}$ for example, with $n_j$ as the number operator at site $j$.

If there's a solution for it that'd be even better. Thank you!

  • 2
    $\begingroup$ By adjusting $h$ accordingly, you should recover your model, assuming what you call $n=\frac{\sigma_z+1}2$ $\endgroup$
    – LPZ
    Feb 17 at 23:13
  • $\begingroup$ I think it's $n = \sigma_z + \frac{1}{2}$, but in general you're correct. Thank you! $\endgroup$
    – Kim Dong
    Feb 18 at 13:59

1 Answer 1


As pointed out by @LPZ, the term you mention is \begin{equation} J_{\mathrm{new}}(4\sigma^z_i\sigma^z_{j}+2\sigma^z_i+2\sigma^z_j+1)/4, \end{equation} and therefore, if you plug this in to where you have $\sigma^z_i\sigma^z_j$ now and calculate the result, it ends up being just another XXZ model with $J_z^{\prime}=J_{\mathrm{new}}$ and $h^{\prime}=h+J_{\mathrm{new}}$, with a constant $NJ_{\mathrm{new}}/8$ shift in the baseline energy. So, that slightly different version ultimately is just the same as the usual XXZ Hamiltonian.

XXZ chains are known to have Bethe Ansatz solutions (if $h=0$), which is a beast on its own. There are many papers on it so I'd recommend searching for them.


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