Jordan wigner transform and periodic boundary condition

[wdlang]

i think jordan wigner transform, when applied to open boundary system, can simplify a spin 1/2 system to a free fermion system

but there is a difficulty in the case of periodic boundary condition

in this case, we have to deal with terms like

$$S_N^+S_1^-=(-)^{\sum_{k=1}^{N-1}n_k} a_N^\dagger a_1$$

the phase term cannot drop out!

 

[peteratcam]

E.H. Lieb, T.D. Schultz, and D.C. Mattis, Ann. Phys. (N.Y.) 16, 407 (1961).
http://dx.doi.org/10.1016/0003-4916(61)90115-4

 

[Sillyboy]

I thought Nielson's would help you!
http://www.qinfo.org/people/nielsen/blog/archive/notes/fermions_and_jordan_wigner.pdf
Yes, it is just an idea, you should calculate it by youself!

 

[Sillyboy]

And I cannot understand the relationship between the periodicity(periodic boundary or antiperiodic boundary) and the parity!

 

[wdlang]

E.H. Lieb, T.D. Schultz, and D.C. Mattis, Ann. Phys. (N.Y.) 16, 407 (1961).
http://dx.doi.org/10.1016/0003-4916(61)90115-4

yes, they also have to deal with this problem

but to my surprise, they can handle it!