Slow convergence of spin 1 AF Heisenberg chain

DMRG and Numerical Methods
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1

Dear friend,

I’m testing the sample code given in the documentation. Simply run a DMRG sweep for a spin one antiferromagnetic Heisenberg chain. Strangely, this takes me a huge number of sweeps to get converged.

Here is the code, I run DMRG to get the ground state and output entanglement entropy every 30 sweeps. While typically 30 sweeps is sufficient for a normal DMRG calculation to get converged. For such a simple system it takes 150 sweeps.

using ITensors, ITensorMPS
using CairoMakie
function entropy_von_neumann(psi::MPS,N)
    s = siteinds(psi)  
    SvNlist = []
    for b in 2:N
        orthogonalize!(psi, b)
        _,S = svd(psi[b], (linkind(psi, b-1), s[b]))
        SvN = 0.0
        for n in 1:dim(S, 1)
            p = S[n,n]^2
            SvN -= p * log(p)
        end
        push!(SvNlist, SvN)
    end
    return SvNlist
  end
let
  N = 48
  sites = siteinds("S=1",N;conserve_qns=true)
  # sites = siteinds("S=1",N)

  os = OpSum()
  for j=1:N-1
    os +=  1, "Sz",j,"Sz",j+1
    os += 0.5,"S+",j,"S-",j+1
    os += 0.5,"S-",j,"S+",j+1
  end
  H = MPO(os,sites)

  state = [isodd(n) ? "Up" : "Dn" for n=1:N]
  psi = randomMPS(sites,state,120)
  @show flux(psi)
  fig = Figure(size = (800, 600))
  ax = Axis(fig[1, 1]; xlabel = "x", ylabel = "Von Neumann Entropy", title = "Entanglement Entropy vs x")
  for iter in 1:10
    nsweeps = 30
    maxdim = [120]
    cutoff = [1E-10]

    energy, psi = dmrg(H,psi; nsweeps, maxdim, cutoff)

    ee = entropy_von_neumann(psi,N)
    ee_all = vcat([0],ee)

  
    scatterlines!(ax, 1:N-1, ee;  marker = :circle, label = "Iteration $iter")
  end
  axislegend(ax, position = :rt, fontsize = 10)
  display(fig)
end

The output is

image
image725×493 61.6 KB

I’m wondering if there are physical reasons for such slow convergence?

Thank you

2

What’s your criteria for convergence? Gapless systems will generally converge slower. It may help to increase the Krylov dimension at each step of the solver (see the note on the eigsolve_krylovdim keyword argument to dmrg on this page of the docs: DMRG FAQs · ITensorMPS.jl ).

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3

Thanks for the reply!

My criterion for convergence is the saturation of Entanglement entropy; it seems that this quantity converges much more slowly than the ground-state energy.

I changed the eigsolve_krylovdimto 5, and the convergence is much faster. I remember the spin one chain is gapped according to Haldane conjecture, so I’m wondering if the gapless mode in my actual calculation comes from the fractionalized excitations from the open boundary. Then has 4-fold degeneracy.

Thank you again!

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