Fermionic correlation function and jordan-wigner string

Question

Hi,

I am manually writing a correlation function for Fermions (that is, not using AutoMPO).

In order to include the Jordan-Wigner string, will I need to use bosonic operators (i.e. A and A^dag) instead of fermionic operators (C and C^dag), along with Jordan-wigner string operators, F?

Or, do I write the correlation function in terms of C operators along with F operators? That is, since the MPS that this correlation function acts on is for spinful fermions, I must use C operators, and then also need F operators because they don't automatically include the F operators?

Sorry if this has been covered before, but this is a point of significant confusion for me.

Jon

Answer 1

Ok I answered my own question: Yes, use the C operators plus F operators.

Comments

Hi Jon, you can certainly do that, and it might be easiest for the case of fermions with spin. However I personally prefer to use the "A" operators along with the string. Otherwise it's sort of a "half way" Jordan-Wigner transformation. To me it's cleaner to go all the way to the bosonic operators.

By the way, there is the Code Formula about this which you and I had worked on before. It only covers the spinless case but I believe it's correct for that case:
http://itensor.org/docs.cgi?page=formulas/spinless_correlator_mps
We should work on a second one, or an extra section on this page, for the case of fermions with spin.

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Yes I know, I had planned to do it someday. Maybe that day is coming soon, I'll let you know. This is good timing since I have a student helping me program correlation functions.

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Yeah I'll start writing an additional section for fermions with spin. My student helper will need it.

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Ok great! The main difference is just getting the operators at each end correctly defined. Using the C operators ought to work I think. Also you can use the tutorial on fermions and Jordan-Wigner string as a reference.

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Won't this be outdated though, since the MPO version is easier to implement?

I was only going to have my assistant try this as a verification of the MPO version.

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Or is this important so that people can develop custom code that is not possible with MPO?

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Hi Jon, the reason is efficiency. Consider the case where one want to compute <Cdag_i C_j> for many different values of i and j (in some cases all i<j to get the full "correlation matrix"). Then doing it by the explicit code such as in the code formula scales as the square of the system size, where as using an MPO expectation value (with each evaluation of the MPO scaling as N, the system size) give an overall scaling that is cubic in system size. But for just small number of i, j values using MPOs is fine and is more convenient.