Contracting edge indices of MPS in infinite DMRG

Question

Hi Miles,

I have a question regarding the MPS in infinite DMRG calculations. When trying to compute the expectation of an operator O, there are two ways of contracting the edge indices, as shown in the figure (please forgive my drawing..)

I think the lower one makes more sense, by definition of MPS, but the upper one is used in the sample codes. Also, the two methods give close but not exactly equal numerical results. Can you explain a little about the (in)equivalence thereof? Thanks!

Best,
Chengshu

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Additional "experimental" fact

Answer 1

Hi Chengshu,
The upper one is exactly right. Thanks for providing a figure as it helps to discuss things. And see the answer I just posted to your other question.

You can see that the upper figure has to be right because the MPS in principle extends much further beyond the sites you show above, but because of the left and right orthogonality property, the tensors outside of the support of "O" cancel leaving identity matrices on the left and right. These become the lines connecting the top (bra) to the bottom (ket).

The lower one wouldn't be correct for a number of reasons. One reason is what I just wrote above about first writing out the full MPS then canceling most of the tensors (it helps to imagine it not as a truly infinite MPS but as a very large finite MPS with open boundaries some unspecified distance away). Another reason it wouldn't be correct is because it would seem to make the states look as if they are periodic states on a system of size N, whereas they are actually wavefunctions on a much larger, or even infinite, system.

Miles

Comments

Hi Miles,

Thanks for your fast reply. The right orthogonality of the MPS does make the upper one reasonable. But I think it also makes sense to regard the state as "periodic", because when we calculate, say, the correlation function, we simply multiply/contract them together. Also, the states corrsponding to each "Link" index seem to be nicely normalized, see the newly added figure. I can't deduce this result from right orthogonality so I expect itself to be of use. What do you think about this?

Best,
Chengshu

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Hi Chengshu,
It's definitely important to be clear about one thing: the upper one is correct and the lower one is incorrect. If you are working with infinite MPS and iDMRG you don't get a choice between the upper and lower diagrams above. The upper diagram is what results from taking the expectation value of an operator between an infinite MPS (bra and ket) and then canceling the tensors outside of the support of the operator by using left and right orthogonality.

The second diagram is just another diagram that could perhaps be justified in some other context outside of iDMRG but I don't know precisely what context it would be.

Finally, the idea of a system being infinite is an orthogonal concept to it being periodic. One advantage of working with infinite systems, in fact, is that for most purposes one can actually be rather agnostic about what the precise boundary conditions are.

Hope that is helpful -

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Hi Miles,

Thanks very much for the clarifications. I'll stick to the upper one.

Best,
Chengshu