Automated construction of $U(1)$-invariant matrix-product operators from graph representations

Submission summary

 As Contributors: Thomas Köhler · Salvatore R. Manmana · Sebastian Paeckel Arxiv Link: http://arxiv.org/abs/1706.05338v2 (pdf) Date accepted: 2017-11-02 Date submitted: 2017-10-06 02:00 Submitted by: Paeckel, Sebastian Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory Quantum Physics Approaches: Theoretical, Computational

Abstract

We present an algorithmic construction scheme for matrix-product-operator (MPO) representations of arbitrary $U(1)$-invariant operators whenever there is an expression of the local structure in terms of a finite-states machine (FSM). Given a set of local operators as building blocks, the method automatizes two major steps when constructing a $U(1)$-invariant MPO representation: (i) the bookkeeping of auxiliary bond-index shifts arising from the application of operators changing the local quantum numbers and (ii) the appearance of phase factors due to particular commutation rules. The automatization is achieved by post-processing the operator strings generated by the FSM. Consequently, MPO representations of various types of $U(1)$-invariant operators can be constructed generically in MPS algorithms reducing the necessity of expensive MPO arithmetics. This is demonstrated by generating arbitrary products of operators in terms of FSM, from which we obtain exact MPO representations for the variance of the Hamiltonian of a $S=1$ Heisenberg chain.

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Published as SciPost Phys. 3, 035 (2017)

Author comments upon resubmission

We thank the referee for his or her careful reading of the manuscript and for the positive assessment "the work is sound and that the results are interesting and will be of use to the MPS community." In the following, we address the issues raised. We have uploaded an updated version including these changes to the arXiv, where the manuscript is accessible under the identifier 1706.05338v2.

Referee:
"I believe the authors should be more careful with their notation. For example, works on tensor methods are often unavoidably awash in a sea of indices, however I found tracking the meaning of indices in this manuscript more difficult than usual. Some of this may be due to typos. In the following I give a list of places where I had issues with the notation."

We thank the referee for this remark. We have carefully taken care of the various points concerning notation and removed some typos. We hope the manuscript is clearer now.

"This work hinges on understanding the operation of finite state machines...."

We have now included a description of finite state machines and sketch the construction for a specific example (Hamiltonian of the XX model).

"The authors should do more to place their work in context with other recent work on efficient construction of generic MPOs, especially Hubig \emph{et al}. PRB 95, 035129 (2017). "

We now explain in more detail how our work relates to prior work in the end of section "Numerical behavior of the variance for antiferromagnetic S=1 Heisenberg chains". We state that the approach of Hubig et al is complementary to ours. The additional advantages of our FSM approach is that the MPO is generated from an abstract graph operation, so that the representation is exact, leading to the higher accuracy.
Note that to our understanding, Hubig et al. make use of non-abelean symmetries in their paper, so that it is not possible to directly compare to the bond dimensions used by us.

"Also in Sec. 5, it would be useful to know if the chain had open or periodic boundary conditions, and if the ground state was obtained using an iterative (i.e. sweeping) MPS method such as DMRG rather than "a standard Lanczos algorithm". At the end of the second paragraph of that section the authors refer to "perfect linear scaling", but the choice of axes in Fig. 8b makes this far from clear."

We apply open boundary conditions, and the ground state is obtained by sweeping through the lattice and locally optimizing the site tensors using a Lanczos algorithm. Both is now stated in the text.
We now also show linear fits in Fig. 8b, which clearly demonstrate the linear scaling.

"In the conclusion the authors refer to their algorithm as "optimized". It might be worth saying what quantity is extremal here?"
In the main text we now address this point - the 'optimization' refers to minimizing the number of floating point operations when calculating expectation values of operator products, as needed in the evaluation of the variance.

"Finally there are some other typos:"
We have corrected typos throughout the manuscript.

Requested changes:
All requested changes are included.

List of changes

We have included several changes throughout the paper in reply to the referee's request, as detailed in our reply above. In more detail, the major changes are:

- We have added numbers to the equations, unified the notation of the tensors, and corrected typos throughout the paper.

- Figs. 1 and 2 have been updated for clarity

- Sec. 3.1 / Fig. 3 provide the example of constructing a FSM for the XX model

- We modified the discussion in Sec. 5 to better relate to the work of Hubig et al.

Submission & Refereeing History

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Resubmission 1706.05338v2 on 6 October 2017
Submission 1706.05338v1 on 23 June 2017

Reports on this Submission

Anonymous Report 1 on 2017-10-19 (Invited Report)

• Cite as: Anonymous, Report on arXiv:1706.05338v2, delivered 2017-10-19, doi: 10.21468/SciPost.Report.266

Report

I think the changes made in response to my previous report are satisfactory, particularly the discussion at the end of section 5 on the context of this work in relation to that of Hubig et al., and I recommend publication.
There are two small typos.

Requested changes

1. 3rd paragraph of the introduction: "capable to efficiently" -> "capable of efficiently".
2. Footnote 8, "non-abelean" -> "non-abelian".

• validity: high
• significance: high
• originality: high
• clarity: high
• formatting: perfect
• grammar: perfect