Dynamical Structure Factor of 1D Spin-1 chain

DMRG and Numerical Methods
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1

Hi!
I am trying to calculate the dynamical structure factor of 1D spin-1 chain using the following relation:

S^{zz}(k,\omega) = \int dt e^{-i\omega t} \sum_{j} e^{-ikr_j} \langle\psi_0| S^z_j(t)S^z_0|\psi_0\rangle

Here is the way I am approaching to calculate the correlation;

  1. Getting the ground state \psi_0 using DMRG.
  2. Evolving this ground state to some time t.
  3. Applying S^z_j to the ground state \psi_0 to get some \phi_j.
  4. Evolving this \phi to time t.
  5. Applying S^z_j to the time-evolved ground state.
  6. Then taking the inner product: \langle\psi^{\prime}_0(t)\phi_j(t)\rangle.

code snippet;

  for c in 1:1:N
    psi_new = apply_op(psi, "Sz", sites, c)

    for t in 0.0:tau:t_f

      psiv = apply_op(psi, "Sz", sites, 1)
      val = inner(psiv,psi_new)
      push!(corr_val,val)
      println("<Sz_($c)($(t)T) * Sz_($c)(0)> = $val")
      t≈t_f && break

      psi = apply(gates, psi; cutoff)
      psi_new = apply(gates, psi_new; cutoff)
    end
  end

I have verified the ground state obtained from DMRG, it’s correct. But still after Fourier trnasform, I am unable to get the correct DSF spectra(
arXiv:2501.13059). Am I doing anything wrong in this?

Any suggestion would be really helpful.

Thank you in advance.

2

Perhaps you are missing the conjugate? One of those times should be “backward”. You also can make it so you don’t have to apply S^z_j twice, but instead measure \langle A|S^z_j|B\rangle so you can obtain all values quickly

There are number of tricks one has to do as well I believe, like linear prediction in [0901.2342] Spectral functions in one-dimensional quantum systems at T>0

See also Artifacts in Dynamical structure factor plots - #2 by miles