How to define a custom two-site operator for autoMPO?

Question

I want to use ITensor to obtain thermal states for the 1D Fermi-Hubbard model using the purification method and to do so I need to define an entangler Hamiltonian but I'm having some problems.

Right now I'm trying to use autoMPO to create the Hamiltonian and that requires me to define the operator |1 - Nupi - Ndni'| acting on a site i and its ancilla i' (as in Eq. 26 and 29 of https://journals.aps.org/prb/abstract/10.1103/PhysRevB.93.045137 ). It seems that this operator just projects away the states |Up Up> and |Dn Dn> so I could define a tensor with two indices going and two going in out that does that(with something like to "Op.set()..." I imagine) but is there a way to use this new tensor with the autoMPO system?

If that is not possible then what would be the best way to construct the Hamiltonian MPO?

Answer 1

Hi, the way to get custom operators to use with AutoMPO is to define a custom site set. The best starting point would be to build off the code for the Hubbard site set, which is defined in the file itensor/mps/sites/hubbard.h. If you make your own copy of this file, you can new operator definitions alongside the current ones. All you have to do is give your new operator a name and specify its non-zero matrix elements. (If the operator happens to be fermionic, by which I mean consisting of an odd number of creation/annihilation operators, let me know as this requires extra steps to work correctly with the AutoMPO system; but your density/projection operator looks to be bosonic.)

Finally, you may want to consider an alternative approach to using an entangler Hamiltonian. That idea was based on the assumption that one is not using an MPS based code. But since ITensor is MPS/MPO based, it may be possible to directly define the starting wavefunction that would be the ground state of the entangler Hamiltonian without needing to do any optimization.

Comments

Hey Miles,

First of all, thank you for the quick reply and all the awesome work done in ITensor.

The operator is indeed bosonic so that shouldn't be a problem and I think I understand how to modify the Hubbard site based on what you've said. I will try to implement this and see if it works.

I haven't thought about writing the wavefunction directly as a MPS but it does seem to be a better idea, especially since the entangler hamiltonian is highly non-local. I will take a look at how to do this, thanks for the suggestion.

Best,
Ricardo

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Yes it might be non-trivial to write the entangled MPS, but I think it should be possible. In the end it's really just a projection operator written as an MPS by "folding" the bra index lines over and treating them as extra ket index lines, similar to how the starting point for ancilla/purification is just an identity matrix similarly "folded" into a set of entangled pairs as I'm sure you know.