# Is there a name for a Heisenberg-like model, but instead of the ZZ operator, we have one that favor only spin-up-spin-up configurations?

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!

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

As pointed out by @LPZ, the term you mention is $$$$J_{\mathrm{new}}(4\sigma^z_i\sigma^z_{j}+2\sigma^z_i+2\sigma^z_j+1)/4,$$$$ and therefore, if you plug this in to where you have $$\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.