## 01_dmrg.ipynb but with periodic boundary conditions

How do I use this algorithm? What does that parameter do?
nick
Posts: 2
Joined: 21 Sep 2022, 18:40

### 01_dmrg.ipynb but with periodic boundary conditions

I am trying out DMRG for the first time in Google Colab with Tenpy. After adding a cell with

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!pip install physics-tenpy,
I am able to run 01_dmrg.ipynb from https://tenpy.readthedocs.io/en/latest/ ... _dmrg.html. It runs well and it's really nice to see the output!

One thing I noticed is that the central charge fitting, using the default exclude=None to drop 100/4=25 sites from the left and right boundaries, gives a central charge of .508 instead of .5. That's of course a small enough error to confidently know that the central charge is .5, but I've noticed that in exact-diagonalization of the transverse-field ising model in periodic chains, I can get closer to .5 at smaller system sizes.

For example, for L=20 with sparse extraction of the ground state in a periodic TFIM, I get that fitting $c/3 \log(\frac{L}{\pi} \sin(\pi l/L)+\text{const}$ gives me c=.5008 - an order of magnitude better error for a smaller system size. The smallest subsystem size I include has three spins, which is also fractionally smaller than the default exclude of tenpy.tools.fit.central_charge_from_S_profile.

Since the 01_dmrg code is for an open chain, I want to see whether I can get even closer to c=.5 with a periodic chain and larger system sizes than I access with exact diagonalization. I recognize that Tenpy is primarily designed for open MPS which can be put in canonical form, and so that periodic boundary conditions involve long-range couplings and bond dimensions being squared. Even given this, I would like to try to make a version of 01_dmrg for periodic boundary conditions with the dream of L=100 but maybe more realistically L=40. I read that I will need to put bc_x in the model params to 'periodic', and so I am trying to run the following code:

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L = 100
model_params = {
'J': 1. , 'g': 1.,  # critical
'L': L,
'bc_x': 'periodic',
'bc_MPS': 'finite',
}

M = TFIChain(model_params)
However, I receive the error

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ValueError: Can't give nearest neighbor H_bond for long-range 0-99
How can I fix this error? I anticipate I maybe need to set some additional parameters in the model_params or perhaps make my own model.
nick
Posts: 2
Joined: 21 Sep 2022, 18:40

### Re: 01_dmrg.ipynb but with periodic boundary conditions

I figured out my confusion. Although there's in principle the option optionCouplingMPOModel.bc_x which takes open or periodic as options, from the inheritance diagram on https://tenpy.readthedocs.io/en/latest/ ... Chain.html, TFIChain is a "NearestNeighborModel". The long-range coupling to do periodic boundary conditions violates the NearestNeighborModel assumptions. Instead, the following code works well:

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class PBC_TI(CouplingModel, MPOModel):
def __init__(self, L=2,S=.5, over=1, h=1, lam=.1):
spin = SpinSite(S=S, conserve="None")
# the lattice defines the geometry
lattice = Chain(L, spin, bc="periodic", bc_MPS="finite")
CouplingModel.__init__(self, lattice)
# add terms of the Hamiltonian

# finish initialization
# generate MPO for DMRG
MPOModel.__init__(self, lattice, self.calc_H_MPO())
Running DMRG for L=30, over=-1, h=1, and max bond-dimension of 500, I got a central charge fit (lazily using tenpy.tools.fit.central_charge_from_S_profile and dividing the resulting c by two) of .5002, so I'm very happy.
Johannes
You don't even need to define a custom model, since there is also the tenpy.models.tf_ising.TFIModel, which is the same as the TFIChain except that it's not a NearestNeighborModel and works also for other lattices. You just need to add the model parameter 'lattice': 'Chain' to actually select the lattice, and can force the periodic boundary conditions of H with 'bc_x': 'periodic' (but still keep the default 'bc_MPS': 'finite'.