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Abelian and non-abelian symmetries in infinite projected entangled pair states

by Claudius Hubig

- This is not the current version.

Submission summary

As Contributors: Claudius Hubig
Arxiv Link: https://arxiv.org/abs/1808.10804v1
Date submitted: 2018-09-03
Submitted by: Hubig, Claudius
Submitted to: SciPost Physics
Domain(s): Computational
Subject area: Condensed Matter Physics - Theory

Abstract

We show how to implement arbitrary abelian and non-abelian symmetries in the setting of infinite projected entangled pair states on the two-dimensional square lattice. We observe a large computational speed-up obtained through the increased sparsity and reduced tensor sizes; easily allowing bond dimensions $D = 10$ in the square lattice Heisenberg model at computational effort comparable to calculations at $D = 6$ without symmetries. We also find that absent spontaneous symmetry breaking, implementing a symmetry does not negatively affect the representative power of the state and leads to identical ground-state energies. Furthermore, we point out how we can use symmetry implementations to detect such spontaneous symmetry breaking.

Current status:
Has been resubmitted


Invited Reports on this Submission

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Anonymous Report 2 on 2018-10-15

Strengths

1-written in good and clear language
2-some interesting results

Weaknesses

1-Seems to aim to be a method paper, but doesn't explain the method clearly enough for those that do not already know a great deal about it
2-main message not clear

Report

I think the paper should probably be published, eventually, but while reading it I was struggling a bit to figure out what the main message, or point of the paper is. From the abstract and introduction, I expected it to be a method paper, explaining the method and then applying it to some examples. However, I do not find the methods to be very clearly explained. The last part of the paper makes it sound a bit like the detection of spontaneous symmetry breaking is the main point. I think maybe the paper should be a method paper, but in this case I would recommend some modifications that clarify the method to those that don't know it. I make some suggestions and comments below on this point.

In the discussion of the Kagome lattice, I believe there are some more works that could and probably should be cited. The work of He et al (Phys. Rev. X 7, 031020 (2017)) for example, seems to me to be relevant.

Requested changes

1-Eq. (2) introduces the $c_i^\gamma$ tensor, but I find it hard from the discussion to understand (without prior knowledge) what they are. It would be good to clarify a bit better what they are, for example by taking a simple example for U(1) and SU(2). In particular, I'm not sure the definition given in terms of addition rule is very clear.

2-The abbreviation FFU used in caption of Fig. 2 is never defined. The same is true about CTM used in several places.

3-I would suggest to define D explicitly, for example in the context of Fig. 1. Of course, people that know will know, but it's good to actually spell it out somewhere.

4-The definition of the remover tensor R is not fully clear. I would be more explicit and maybe give an example.

5-There have been many papers, as cited by the author, on implementing SU(2) symmetries. It would be useful to be even more explicit about what is different or new in the case if iPEPS, as presented in this work.

  • validity: good
  • significance: good
  • originality: good
  • clarity: ok
  • formatting: good
  • grammar: excellent

Author Claudius Hubig on 2018-10-17

(in reply to Report 2 on 2018-10-15)
Category:
remark
answer to question

I would like to thank the referee for their thorough reading of the manuscript and the helpful comments, all of which have been incorporated into the second version.

The paper is indeed intended as a method paper. First, it explains both the implementation of symmetries in tensor networks though a complete description would be too large and is hence referenced and in particular the adaptations necessary to use such an implementation in iPEPS calculations. Second, the effect of the symmetry implementation is then explored in detail regarding the computational speed-up, obtainable bond dimensions and obtained energies which largely confirm previous widespread experience from 1D MPS-DMRG settings. Furthermore, as spontaneous symmetry breaking is more prevalent in 2D than 1D, the paper then points out how symmetry implementations can help diagnose such cases using only the local energy. To this end, the abstract and introduction have been clarified.

I agree that the initial description of the $c^\gamma_i$ tensors was not sufficient and have extended this section with an illustrative example (also after receiving comments to this end by e-mail).

Anonymous Report 1 on 2018-10-2

Strengths

See report.

Weaknesses

See report.

Report

The paper by Claudius Hubig on on "Abelian
and non-abelian symmetries in infinite projected entangled
pair states" is a well-presented and balanced study of the application of
SU(2) symmetries in the 2D-iPEPS context; therefore I recommend
publication of this article in SciPost, after the following
comments have been clearly addressed in the paper.

It may have escaped the author that an SU(2) iPEPS study was
already also performed by Liu et al. on the spin-1 Kagome
Heisenberg antiferromagnet [see PRB 91, 060403(R) (2015)].
There (also) comparable ground state energies were observed,
with and without SU(2) symmetries at sufficiently large D.

Fig. 5 shows data for U(1) and SU(2) on the Kagome lattice,
both apparently from the present algorithm based on a square
lattice tensor network representation of the Kagome lattice.
The data appears to converge to significantly different values
for larger D, which appears to contradict the sentence in the
abstract: "We also find that [in the absence of spontaneous
symmetry breaking] implementing a symmetry does not negatively
affect the representative power of the state and leads to
identical ground-state energies."

What does the author mean by "avoiding inadequate truncations"?
While truncating into degenerate multiplets leads to error
at insufficient D, if an algorithm is deterministic, it should
converge to the same result, once D is large enough.
But again, the data in Fig. 5 for U(1) and SU(2) converges
to significantly different values. So what does this imply
on the reliability of the implemented iPEPS algorithm itself,
if the ground state depends on the history in imaginary time
evolution?

It appears to me that "inadequate truncations" leads to
loss of (some generalized) "orthonormality" in the tensor network, i.e.
the deterioration of the conditioning of the environment while
truncating. I was under the impression that gauge fixing
was designed to deal with this problem. Apparently not?

It surprises me, that the energy of the U(1) data in Fig. 5
lies that far above the SU(2) data, throughout, for all D:
one would have expected that for small(er) D, the U(1) or
no-symmetry iPEPS would be superior because for the same D,
it has a larger variational parameter space available.
Why isn't this so?

The data for `None' in Fig. 5 is barely visible hidden
underneath the other symbols; one may mistake the red
horizontal line for this data.

Requested changes

See report.

  • validity: good
  • significance: good
  • originality: good
  • clarity: good
  • formatting: good
  • grammar: excellent

Author Claudius Hubig on 2018-10-17

(in reply to Report 1 on 2018-10-02)
Category:
remark
answer to question

I would like to thank the referee for their detailed reading of the manuscript and the helpful comments. In particular the work on spin-1 Kagomé chains has escaped me, but I am very happy to see that previous results also support the implementation of symmetries in iPEPS.

Regarding the surprisingly good results from an SU(2)-invariant initial state for the spin-½ Kagomé model, I have reworked the discussion extensively. My main explanation is that the fast full update with imaginary time evolution depends to a relatively large degree on the initial state which I also found later on the square lattice Heisenberg at Δ<1 and seems to be in agreement with the experiences of others. Then, starting from a SU(2)-invariant state may guide the calculation into a much better direction than starting from another random state which first has to restore the SU(2)-invariance. This is contrary to the general – and of course true - expectation that less constrained states generally provide better energies at identical effective (small) bond dimensions.

The square lattice mapping likely exacerbated the convergence problems of the imaginary time evolution which might explain why a similarly large difference was not observed in the work by Liu et al. Switching to another update mechanism, like gradient descent with boundary MPS or the variational update using the CTM should help guide also the U(1)-symmetric calculations in the right direction, but the underlying problem is quite orthogonal to symmetries and only shows up here due to the initial state which is either SU(2) symmetric or breaks that symmetry.

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