This is a relatively simple question that I just want confirmation on. In literature, I have seen 2 ways of writing the Heisenberg XXZ Chain:
1.) $H = -J \sum_{n=1}^{N}\left(S_n^xS_{n+1}^x+ S_n^yS_{n+1}^y + \Delta S_n^zS_{n+1}^z\right)$
2.) $H = \sum_{n=1}^{N}\left(S_n^xS_{n+1}^x+ S_n^yS_{n+1}^y + \Delta S_n^zS_{n+1}^z\right)$
I would like to understand how to reconcile the 2 expressions. In 2.), when $\Delta = -1$, we say the chain is ferromagnetic. In expression 1.), the chain is ferromagnetic when $\Delta = 1, J > 0$. Thus, it seems to me that what is most important is actually the relative sign between the ZZ terms and the XX, YY terms? How can one go from expression 1.) to expression 2.)?
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1 Answer
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The two expressions can be shown to be equivalent using a canonical transformation that implements a spin rotation by $\pi$ about the spin $z$ axis on every other site. Explicitly, this transformation may be chosen to take $$ S_{2n}^x \rightarrow S_{2n}^x, \quad S_{2n+1}^x\rightarrow -S_{2n+1}^x,\\ S_{2n}^y \rightarrow S_{2n}^y, \quad S_{2n+1}^y\rightarrow -S_{2n+1}^y,\\ S_{n}^z \rightarrow S_{n}^z $$ This flips the sign of the $S_n^x S_{n+1}^x$ and $S_n^y S_{n+1}^y$ terms.
