Implementing XXZ model with finite DMRG

[squareroottwo]

Hello,

I am trying to explore the XXZ chain with DMRG to validate the performance of DMRG in an around the gapless phase. I would like to model systems with open boundary conditions and push the system sizes to see whether DMRG breaks down. I've got a few questions regarding this.

1. TeNPy has a built-in model for the XXZ chain at tenpy.models.xxz_chain.XXZChain, but the example for the XXZ model on the repository at https://github.com/tenpy/tenpy/blob/mai ... _length.py utilises tenpy.models.spins.SpinChain. I suspect these are equivalent?

2. I found this worksheet https://ethz.ch/content/dam/ethz/specia ... rcise9.pdf, which points to an example python script

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xxz_groundstate.py

that was part of the TeNPy repository a few years back. I could not find this example on the current repository. Is it available somewhere?

3. More of a general question rather than TeNPy related, but would one expect DMRG to struggle with this problem given I am using finite systems? I've attempted to model the TFI chain similarly, and (finite) DMRG had no trouble capturing the physics around the critical point.

Thank you so much for your time!

[Johannes]

1. Yes - as noted in the comment at the top of the xxz_chain.py implementation.

2. You can always go back in the git history and find it there with

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git log --all -- examples/xxz_groundstate.py

which brings up d1dccf29ee50899451ea95931c5f4216efae8e56.'

3. For finite, critical 1D systems, the famouse Cardy-Calabrese paper arXiv:hep-th/0405152 uses conformal field theory to predict:
\[S(L_A, L) = \frac{c}{6} \log(\frac{L}{a \pi} \sin(\frac{\pi L_A}{L}))+ const \].
Here \(L_A\) is the cut position, and \(L\) is the total system size.

This means in the center of the chain, you'll have to capture a maximal entanglement of $S \propto \log L$.
On the other hand $S \leq \log \chi$ - so for fixed, finite L, you can often converge your results in bond dimension chi, and get quasi-exact results and good convergence with finite DMRG.
If you use infinite DMRG, on the other hand, you would formally need infinite chi to converge. Nevertheless, you can still get a lot of information, like the central charge c, from a finite-chi scaling analysis.