I would like a reference on the diagonalization of an open XX spin 1/2 chain with homogenous external magnetic field. I am new to the subject and I haven't been able to find a reference for it.


$$H = h \sum\limits_{j=1}^{N} \sigma_j^z - J \sum\limits_{j=1}^{N-1} (\sigma_j^x \sigma_{j+1}^x + \sigma_j^y \sigma_{j+1}^y) $$ I would like the transformation that allows me to express the Hamiltonian in the form $$H = \sum_k \Lambda_k \eta^{\dagger} _k \eta_k + constants$$ I know the procedure. Just want a reference to double check my results.

  • $\begingroup$ Please add some more details. (E.g., write a Hamiltonian.) It is not even clear in which direction the field is pointing. Also, what exactly do you mean (or: need) when you say "diagonalize"? $\endgroup$ Dec 23, 2015 at 23:00
  • $\begingroup$ In this case, it looks like this can be done by mappipng it to non-interacting fermions. But then you should modify your question: It's not that you want to know how to diagonalize this Hamiltian, but you want a reference. (Or would you be happy with an answer explaining that it can be mapped to free fermions?) You might also consider using the "reference-request" (edit: sorry, "specific-reference") tag in that case. $\endgroup$ Dec 23, 2015 at 23:25
  • $\begingroup$ If you just want to check your results, you could also just do numerical tests on small system sizes. $\endgroup$ Dec 23, 2015 at 23:35
  • $\begingroup$ Some references seem to be linked here physics.stackexchange.com/q/2014 $\endgroup$ Dec 23, 2015 at 23:40


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