Gauge improvements

[ryanlevy]

This implements two improvements:

• finite_mps can now have its gauge center be on the first site instead of the last site
• orthogonalize ensures a real Hermitian transfer matrix eigenvector thanks to schursolve (shout out @lkdvos for the tip)

The first change is a small speed improvement for long finite_mps calls as correlation_matrix will immediately move the orthogonality center to the first site

julia> @btime corrR = correlation_matrix(finite_mps(ψ,1:24; ortho="right"),"Cdagup","Cup"; sites=2:24+1);
  57.864 ms (900184 allocations: 146.93 MiB)

julia> @btime corrL = correlation_matrix(finite_mps(ψ,1:24; ortho="left"),"Cdagup","Cup"; sites=2:24+1);
  47.981 ms (764616 allocations: 117.28 MiB)

The second change includes the additional "gauge fixing" of the output of vumps in the Hubbard example, which should probably always be done so that the state you want to measure has a more consistent C matrix.

[mtfishman]

Seems a bit funny to make this a question. Can't you check if it is real?

[mtfishman]

If it isn't a definitive check, it would be better to phrase it as "Largest transfer matrix eigenvector might not be real.".

[ryanlevy]

I guess it should be "Non-unique largest eigenvector" (or the eigenvector shouldn't be real)

[lkdvos]

I think what it really means is that it is not unique, this results in complex conjugate eigenvalue pairs (hence checking the TT[2, 1], TT[1:2,1:2] = [real(lambda) imag(lambda); -imag(lambda) real(lambda)] or something like that).

[mtfishman]

Nice to see it get more general and simpler!

[ryanlevy]

@lkdvos it seems that using eager=true causes the vector to be non-hermitian (at least higher than the Arnoldi tolerance)

[lkdvos]

That's interesting... The only difference should be that it is checking every iteration if the tolerance has been reached, instead of waiting until the full krylov subspace is populated. Basically, it's probably not fair to expect that you reach the same tolerance for the eigenvector hermiticity as you would get for the eigenvector itself. If you don't put eager=true, you often just get a higher precision than you asked for, because you are doing additional iterations. However, in our cases the extra iterations are quite expensive, so this strongly affects performance

[mtfishman]

Sounds like we should play around with different tolerances when checking for Hermiticity, I don't think there is anything magical about the current choice besides that it has worked so far for catching issues while not being too strict.

[ryanlevy]

I noticed a bug with orthogonalize, you can't call it twice because it adds tags which interfere with the next run

julia> p = orthogonalize(ψ.AL, : ; tol=1e-14)
       p2 = orthogonalize(p.AL, :)
ERROR: In `isapprox(::ITensor, ::ITensor)`, the indices of the ITensors do not
          match. The first ITensor has indices:

#....

[mtfishman]

I noticed a bug with orthogonalize, you can't call it twice because it adds tags which interfere with the next run

julia> p = orthogonalize(ψ.AL, : ; tol=1e-14)
       p2 = orthogonalize(p.AL, :)
ERROR: In `isapprox(::ITensor, ::ITensor)`, the indices of the ITensors do not
          match. The first ITensor has indices:

#....

Got it, I don't have time to look into that but feel free to make a PR.

[ryanlevy]

@mtfishman is this good to go?

[mtfishman]

Can you bump the package version?

[ryanlevy]

Sure thing!