what can we understand from Entropy calculations at each DMRG iteration?

General Discussion
1

Hi!

I have a very naive question. I noticed the dmrg template in iTensor prints entanglement entropy calculated at the center of the system, at each iteration step. I was wondering as to what can we understand from the change in entanglement entropy at each step.

For instance, I was performing a dmrg calculation for N=100 Hubbard chain with 80 electrons. I saw a drastic difference between the entanglement entropy for two different initial conditions.

  1. First initial condition: Setting electrons with spin up and down alternatively till N=80. This initial condition resulted in very slow convergence in energy along with oscillatory entanglement entropy (at b=50) around 1. even after 50 sweeps the system did not converge.
  2. Second initial condition: Equally distributing holes in the whole chain (setting i%10==0, 1 as holes) while placing equal number of spin up and spin down. For this initial condition, the convergence in energy is better than previous although I will need to run some more sweeps for accurate result. I notice the entanglement entropy (at b=50) continuously in creases towards 2 and never saturates.

Can you comment on this difference in entanglement entropy for different initial condition?

Thanks

2

Hi thanks for the question. So first of all, the initial condition you tried in (1) is indeed pretty likely to get “stuck” and be harder (though not impossible) to converge. The intuitive reason is that since fermions of the same spin can’t occupy the same site, they can block each other and have a hard time moving from the left half of the system over to the right half.

It makes sense that your approach (2) worked better. In general DMRG for the Hubbard model can take a large number of sweeps to converge well, depending on some details of your sweeping parameters. A good approach is actually to do a large number of initial sweeps at a small maxdim like 10 or 20 because this can be very effective for getting the ‘rough’ details of the state to be correct, then you can do a smaller number of more accurate sweeps at the end.

Finally, the reason the entanglement is different is that the wavefunctions are different. In one or both of your cases, the MPS is not converged to the true ground state yet, so then it is different from the other MPS and thus they have different entanglement. If both were converged they would have the same entanglement.

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