[Feature] Degeneracy-aware truncation

[Yue-Zhengyuan]

According to 2510.04756, in the context of CTMRG:

Then, we do singular value decomposition (SVD) as in Fig. 7(C). The diagonal $S$ has $χ_0$ singular values sorted in the non-increasing order: $S_1 ≥ S_2 ≥ · · · ≥ S_{χ_0}$ . Instead of simply truncating $S$ to a predefined number of singular values, we noticed that $S$ has a multiplet structure that needs to be dealt with carefully when it comes to the truncation. To do this, we initially truncate S to $χ_i$ such that $χ_i$ is the largest number satisfying $S_{χ_i} / S_1 ≥ ϵ_C$ and $χ_i ≤ χ$. Next, we find the largest $χ_f$ such that $χ_f ≥ χ_i$ and $1 − S_{χf} /S_{χi} < r_m$. By including a complete multiplet during truncation, we improve the stability of CTMRG.

The idea is: Sometimes a hard truncrank(χ) cuts in the middle of a degenerate part of the singular value spectrum. It benefits to provide a truncation strategy that automatically increases χ (when these degenerate singular values are not too small) so that the entire degenerate part is kept.

[Confusio]

The following overload of TensorKit._compute_truncdim maybe useful and I have tested intensively and find it works well.

import TensorKit: TruncationDimension, SectorDict, keytype, dim, eps, scalartype, _findnexttruncvalue
function TensorKit._compute_truncdim(Σdata, trunc::TruncationDimension, p=2)
    I = keytype(Σdata)
    truncdim = SectorDict{I,Int}(c => length(v) for (c, v) in Σdata)

    S = [v for (c, v) in Σdata for _ in 1:dim(c)]
    S = sort(vcat(S...); by=x -> -x)

    n = trunc.dim

    cutoff = eps(scalartype(S)) #cutoff here maybe useful for stablity

    deg_threshold = 1.0 - 1.0e-4 # 1.0e-4 usually enough. If not, decrease it. 
    n_above_cutoff = count(>=(cutoff), S / S[1])
    n = Int(min(n, n_above_cutoff))
    if n > 0 && n < length(S)


        last_index = n
        for i in n:(length(S)-1)

            if S[i+1] >= S[i] * deg_threshold
                last_index = i + 1
            else

                break
            end
        end
        n = last_index
    end
    while sum(dim(c) * d for (c, d) in truncdim) > n
        cmin = TensorKit._findnexttruncvalue(Σdata, truncdim, p)
        isnothing(cmin) && break
        truncdim[cmin] -= 1
    end
    return truncdim
end

[Yue-Zhengyuan]

@Confusio I think n, cutoff and 1 - deg_threshold play the role of $\chi$, $\epsilon_C$ and $r_m$ respectively, right?

[Confusio]

Yes, after a quick check, I found that they are exactly the same, although I hadn’t noticed that paper before. It seems the authors also struggled with the degeneracy issue, just as I did with the symmetric PEPS.

[lkdvos]

Also linking this here: https://github.com/ITensor/TensorAlgebra.jl/blob/51b5c56c0fd6f7fb9c790c388222d4da7af23ff0/src/MatrixAlgebra.jl#L145

It's definitely something that has been on my to do list, just haven't gotten around to it yet. Thanks for opening an issue though, that does help reminding us :)