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An introduction to infinite projected entangled-pair state methods for variational ground state simulations using automatic differentiation
by Jan Naumann, Erik Lennart Weerda, Matteo Rizzi, Jens Eisert, Philipp Schmoll
Submission summary
| Authors (as registered SciPost users): | Jan Naumann · Philipp Schmoll |
| Submission information | |
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| Preprint Link: | scipost_202405_00004v1 (pdf) |
| Code repository: | https://github.com/variPEPS |
| Date submitted: | 2024-05-04 11:10 |
| Submitted by: | Naumann, Jan |
| Submitted to: | SciPost Physics Lecture Notes |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Computational |
Abstract
Tensor networks capture large classes of ground states of phases of quantum matter faithfully and efficiently. Their manipulation and contraction has remained a challenge over the years, however. For most of the history, ground state simulations of two-dimensional quantum lattice systems using (infinite) projected entangled pair states have relied on what is called a time-evolving block decimation. In recent years, multiple proposals for the variational optimization of the quantum state have been put forward, overcoming accuracy and convergence problems of previously known methods. The in-corporation of automatic differentiation in tensor networks algorithms has ultimately enabled a new, flexible way for variational simulation of ground states and excited states. In this work we review the state-of-the-art of the variational iPEPS framework, providing a detailed introduction to automatic differentiation, a description of a general foundation into which various two-dimensional lattices can be conveniently incorporated, and demonstrative benchmarking results.
Current status:
Reports on this Submission
Strengths
* a proper introduction with a lot of references, showing the versatility of variational PEPS algorithms for simulating ground states of strongly-correlated quantum lattice systems in two dimensions
* a pedagogical explanation of all the necessary steps for implementing the CTMRG+AD procedure for optimizing PEPS
* a description of possible pitfalls in applying this methodology
* illustrative benchmarks
* an accompanying software package, both in Python and Julia
* the language is good and easy to follow
Weaknesses
* alternative methods for contracting PEPS are not discussed, although these are common practice in the community and can be combined with AD in a very similar way as CTMRG-based contractions
* the use of symmetries in PEPS representations (either internal or spatial symmetries) is not discussed, although this is crucial to obtain state-of-the-art results in many cases
Report
In this work, the authors give a pedagogical introduction to the variational optimization of infinite projected entangled pair states by combining the CTMRG algorithm with automatic differentiation (AD). These lecture notes will be very valuable for newcomers in the field, allowing them to use existing software packages with the proper insight into the inner workings of the method.
The CTMRG+AD methodology is a very good choice for optimizing PEPS representations, but it is not the only viable option. In particular, a comparison between boundary MPS and CTMRG methods shows that the methods are competitive for contracting PEPS [1]. Furthermore, boundary MPS contraction can be combined with AD to yield an efficient optimization code [2]. It is a legitimate choice for these lecture notes to focus on one method, but the authors should state explicitly that other methods are equally viable options.
A crucial aspect in obtaining PEPS results for challenging problems is the use of symmetries, both spatial and internal. Regarding the former, the authors discuss different lattice structures, mapping each instance to the square lattice. This mapping typically violates the symmetries of the original lattice, potentially leading to severe artefacts. This is not discussed. Moreover, in many works the use of spatial symmetries such as reflection or rotation have been used to constrain the number of variational parameters in the PEPS tensor. The CTMRG+AD methodology should be well suited to optimize within this constrained space.
Regarding the internal symmetries. It should be noted that there are many software packages that can optimize PEPS with internal symmetries (abelian and non-abelian), and there are many works where this is used with great success. The authors should discuss this feature.
[1] Phys. Rev. B 105, 195140 (2022)
[2] Phys. Rev. B 108, 085103 (2023)
Requested changes
1- An explicit discussion of alternative methodologies for contracting and optimizing PEPS should be included, leaving the reader to make an informed choice.
2- A discussion of the use of symmetries in PEPS, and how these can be readily integrated in the CTMRG+AD methodology
Recommendation
Ask for minor revision