Arnoldi solver of fixed-point equation

[GlebFedorovich]

Hi all!
I am trying to do some PEPS optimization and use Arnoldi Eigsolver that turns us to be the most robust and fast at least in my case. However, I quite often get the following kind of warnings:

Fixed-point gradient computation using Arnoldi failed: auxiliary component should be finite
but was 4.463425221273398e-11 + 
0.0im. Possibly the Jacobian does not have a unique eigenvalue 1.

in

/PEPSKit/src/algorithms/optimization/fixed_point_differentiation.jl:394

After that it seems that the gradient grows exponentially large and reasonably wrong, so, the further optimization makes no sense. So, I was wondering is there anything I could try/push further/play with some hidden parameters or should I just switch to LinSolver method? I actually tried it, but it quite often goes on plateau and doesn't want to converge at all.

Thanks in advance!

[lkdvos]

Hi Gleb, do you know if the thing you are trying to optimize might have any degeneracies in the spectrum (transfer spectrum for example)? From the warning message, and given quite a lot of the math involved here mostly assumes non-degenerate spectra, this might be something to try and investigate.

[GlebFedorovich]

Indeed, I am trying to turn on SU(2) symmetry and see now warnings in projectors:

└ @ PEPSKit ~/.julia/packages/PEPSKit/yLD3R/src/algorithms/ctmrg/projectors.jl:169
┌ Warning: degenerate singular values detected: 
│   svals =
│    TensorKit.SortedVectorDict{SU2Irrep, Vector{Float64}} with 3 entries:
│      0   => [3.01e-6, 3.01e-6, 2.8829e-7, 2.8829e-7, 1.54148e-7, 1.54148e-7]
│      1/2 => [1.03067e-5, 5.23191e-6, 4.8239e-7, 4.39325e-7, 3.55186e-7, 2.45284e-7, 2.2851e-7, 1.56606e-7, 9.8007e-8, 9.50648e-8, 6.83116e-8, 6.624…
│      1   => [2.97343e-6, 2.97343e-6, 2.80091e-7, 2.80091e-7, 1.45401e-7, 1.45401e-7]

so, you think that this might be the reason why it fails? I thought there is some broadening to take care of degeneracies... or am I missing something completely?

[lkdvos]

I'm not entirely sure if that's turned on everywhere, especially for all of the different ways of computing gradients...

This being said, the fact that your dominant value in the trivial sector is degenerate (in fact all of them seem to be degenerate for integer sectors), seems to indicate some kind of non-injectivity that might point at a different kind of issue. I would have to investigate more carefully, but do you think there might be some unit cell or virtual space choices that could affect this?

[GlebFedorovich]

Well, that could be for sure. The way I am trying to assign sectors both in peps virtual spaces and in environment are quite random now since I have no clue what strategy to follow. I am now trying to have some Simple update at first, maybe it could help to define the spaces. The unit cell I picked now is 2by2, but I don't have a solid argument why I need this :)

P.s. I definitely notice that not for all distribution of charges in SU2 sectors either Simultaneous or Sequential CTMRGs converges, I just somehow guessed the ones that at least allows to converge env at first steps and proceed with optimization

[lkdvos]

I think here it would probably be best to start from trivial spaces for the CTMRG environments and grow from there, that sounds like it should at least be some physically motivated distribution of sectors. (The idea being that you start from the vacuum at infinity, and gradually grow from there).

[pbrehmer]

I'm not entirely sure if that's turned on everywhere, especially for all of the different ways of computing gradients...

By default, the broadening is turned on. However, I don't think this will affect this issue because the broadening only prevents certain divergences in the gradient of the SVD. The Jacobian that is being iterated here is that of one CTMRG iteration w.r.t. the environments I think and that might still have degeneracies, regardless of the SVD broadening.