How to use MPS.from_Bflat

[ksrai]

This question concerns with the correct usage of the option [tenpy] MPS.from_Bflat[/tenpy]

I am trying to create the AKLT MPS form from given tensors for the AKLT state, check section "Matrix product state representation" of https://en.wikipedia.org/wiki/AKLT_model

The shape of tensor on each site is (3,2,2), where physical dimension is 3 and the bond dimension is 2. The following code shows the implementation:

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tensor = aklt_tensors()
nsites = len(model.lat.mps_sites())
Bs = [tensor.copy() for i in range(nsites)]
print(f"nsites = {nsites}")
final_mps = MPS.from_Bflat(model.lat.mps_sites(),Bs,bc=model.lat.bc_MPS)

Here the model is a CouplingMPOModel, and the

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aklt_tensors()

is the (3,2,2) shape tensor.

However, I get the error:

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shape of B incompatible with len of singular values

What could be the issue? Also, note that the AKLT state I want to prepare in this way is translation-invariant, so in the MPS description it has periodic boundary conditions.

Thanks in advance!

[Johannes]

The finite mps in Tenpy always has open boundary conditions.
In other words, in finite systems, the vL on the first tensor should be trivial of dimension 1, and so should be the vR leg of the last tensor.
Closing the loop on the virtual legs with a trace is nasty for numerical simulations, since it means that we can't define a canonical form g but we need that for efficient optimization and expectation values.

Is perfektly okay to use those tensor of shape (3,2,2) with infinite bc_MPS.

If you really want to consider a finite Ring, you can still "bend over" that extra leg on the first and last tensor and manually group them,
I.e. Reshape the first tensor to (3, 1,4), the last to (3,4,1), and the bulk tensor to (3,4,4) by taking a kronecker product with the 2x2 identity on the last virtual legs.
As you see, that MPS has a larger bond dimension - that's another reason to avoid periodic boundary conditions, even if you can formally handle them with one long range coupling from the first to the last side.

[ksrai]

Thanks for the quick response!
I also tried the infinite bc_MPS and it works.
However, when I check the overlap of the AKLT MPS constructed using infinite bc_MPS

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infinite_aklt_mps

with the ground state

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gs_ed_mps

obtained using ED for the AKLT Hamiltonian (for a simple 4 site system), the fidelity is not equal to 1.
I am assuming that the ground state I obtain from the ED is the correct one for AKLT.
But I am also not sure why the infinite bc_MPS doesn't give fidelity 1. To calculate the fidelity I used:

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(np.abs(infinite_aklt_mps.overlap(gs_ed_mps,ignore_form=False)))**2

Could it be something to do with the canonical forms while using overlap of MPS?

[Johannes]

I'm confused. ED should give you an MPS on a finite system, possibly with PBC (?). How did you map that to the infinite system for comparison?

If you want to compare 1:1, you need to compare finite MPS on PBC to ED on PBC, as I explained above.