Is it possible to create a quantum system with different types of sites?

Question

Is it possible to create quantum systems and the associated (states, MPOs, Hamiltonians) for a quantum system with different types of sites, e.g. a chain of alternating spin-1/2 sites and spin-1 sites?

I ask this with specific reference to the Kogut-Susskind staggered formulation of the Schwinger model, shown in Equation (2.3) in https://arxiv.org/abs/1305.3765. This model has alternating fermionic sites and "links" (which, for the sake of my work, can be thought of as spin-L sites, if I restrict the link values to be at most L in magnitude for some positive integer L). So in this case, I would like a chain of alternating fermionic sites and spin-1 sites.

Up until now, I have been using the formulation of the Schwinger model on spin-1/2 sites (which follows from applying the Jordan-Wigner transformation to the above formulation), shown in Equation (2.6) in the same paper. However, I would like to know if it is possible to use ITensor to work with the original lattice formulation.

Answer 1

Hi Sujay,
Thanks for the question: yes this is definitely possible.

An example showing how to do this in the Julia version of ITensor can be found here:
http://itensor.org/docs.cgi?vers=julia&page=formulas/mixed_sites_dmrg
Later we will update this example to show how to include quantum number conservation; it can already be done so please reply below if you have a question about that.

For the C++ version, an example of doing this can be found in the sample code itensor/sample/mixedspin.cc. Also we have an example here:
http://itensor.org/docs.cgi?vers=cppv3&page=formulas/gs_holst_polaron

Best,
Miles

Comments

Hi Miles,

Thank you so much! Just to be sure, is this just in the C++ code, or has this also been implemented in the Julia code? Because I'm pretty sure I don't see it anywhere in the Julia code.

Thanks,
Sujay

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Hi, I see - this is in the C++ version 3 of the code. Please do mention which version of the code you are asking about since we're currently supporting two different langauges, as mentioned in this Guidelines post: http://itensor.org/support/2003/guidelines-for-asking-questions

I'll make a note to update the answer above soon with a Julia version answer.

The short answer is that, yes, it's also supported in Julia. All you have to do is make an array of Index objects which alternate between two different site types. In Julia a site type is denoted by a special Index tag type. So for example the array:

sites = [Index(2,"Site,S=1/2,n=1"), Index(2,"Site,Fermion,n=2"),Index(2,"Site,S=1/2,n=3"),Index(2,"Site,Fermion,n=4")]

would define a 4-site system with alternating S=1/2 and spinless-fermion sites. You can then use this sites array to define MPS; create an MPO with AutoMPO; etc. and our system is smart enough to figure out what operators to use etc. just by inspecting the Index tags ("S=1/2" versus "Fermion" in this case).