Is there any way to construct a general two-body interaction term

Question

I recently want to construct a general two-body interaction term like C^{\dag}{i,\sigma}C^{\dag} _ {j,\sigma^{\prime}}C{k,\sigma^{\prime}}C_{l,\sigma}, where i, j, k, l are sites and \sigma, \sigma^{\prime} are spins. Is there any way to construct an operator like that using AutoMPO class? Thank you.

Comments

Ok so the new AutoMPO is live on Github. If you pull the latest master branch and recompile, then use AutoMPO it should now be able to handle terms with more than two operators, such as the term you mentioned. Just as with the sample/exthubbard.cc code, you have to respect fermion anticommutation rules when defining the Hamiltonian, so take care to write the C and Cdag operators in the correct order.

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Thank you very much. This feature is helpful in my research. I have some question about this new feature. First, do I have to insert F operators when I am computing something like <C_iC_jC^{\dag}_kC^{\dag}_l>? Second, is it allowed to put four fermion operators (or more) together such as C_i^{\dag}C_iC_i^{\dag}C_i which all act on the same site? Third, does the fermion anti-commutation rule in your response means the order in original hamiltonian? Do I have to take care about the order of the MPO? I notice that there is a sign difference between ITensor 1 and ITensor 2 for extended Hubbard model

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Good questions. Some of this is covered in the tutorial on fermions if you need to read more about it: http://itensor.org/docs.cgi?page=tutorials/fermions

1. Yes, you do have to explicitly put in F operators when measuring correlation functions still. The reason is that when you are getting operators from the sites.op method, it is returning individual tensors to you and currently these tensors have no way of knowing about the existence of fermions on other sites since they are just local tensors. (In the future I plan to experiment with making fermionic IQIndices automatically anticommute - stay tuned.)

2. You should not put four fermion operators on the same site anyway, since this would be zero by definition when acting on a local Hilbert space that can only have at most one spin up and one spin down electron.

3. Yes, by respecting the fermion anti-commutation I meant writing the order and the signs of the terms you put into the AutoMPO in the same way you would on paper when considering the correct fermion anticommutation rules. So there is more than one correct way to input the terms, but you just have to make sure that whatever choice you make matches what you would write on paper (e.g. for the Hermitian conjugate of the hopping term Cdag_i C_j you could either put + Cdag_j C_i or -C_i Cdag_j  (here assuming i != j).

Here is a snippet of the code I am currently using to input the two-particle terms of a chemistry Hamiltonian, where "V" means a particular V_ijkl:


            if(i != j && k != l)
                {
                ampo += V,"Cdagup",i,"Cdagup",j,"Cup",k,"Cup",l;
                ampo += V,"Cdagdn",i,"Cdagdn",j,"Cdn",k,"Cdn",l;
                }
            ampo += V,"Cdagup",i,"Cdagdn",j,"Cdn",k,"Cup",l;
            ampo += V,"Cdagdn",i,"Cdagup",j,"Cup",k,"Cdn",l;

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Thank you for your explanation in detail. I am now using AutoMPO to construct an IQMPO say C^{\dag}_iC^{\dag}_jC_kC_l on only four sites and call psiHphi to compute the expectation value of a given IQMPS. In this case, do I have explicitly put F operators in AutoMPO? Thank you.

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The rule for AuotMPO is that you never have to put F operators, regardless of what you are using the AutoMPO for.

For operators obtained one at a time from a "sites" (SiteSet) object, you do have to put in F's.

Answer 1

Hi, this is a good question. Right now there is not, but we have a new prototype AutoMPO that can handle an arbitrary-size product of operators (not limited to just two). I'm hoping to release it later this week, actually, but we'll see how it goes. I'll post another reply when it's ready.

Comments

See above comments for discussion of this feature, which is now live ^