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Tangent-space methods for uniform matrix product states
by Laurens Vanderstraeten, Jutho Haegeman, Frank Verstraete
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Submission summary
| Authors (as registered SciPost users): | Jutho Haegeman · Laurens Vanderstraeten · Frank Verstraete |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/1810.07006v1 (pdf) |
| Date submitted: | 2018-10-17 02:00 |
| Submitted by: | Vanderstraeten, Laurens |
| Submitted to: | SciPost Physics Lecture Notes |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Computational |
Abstract
In these lecture notes we give a technical overview of tangent-space methods for matrix product states in the thermodynamic limit. We introduce the manifold of uniform matrix product states, show how to compute different types of observables, and discuss the concept of a tangent space. We explain how to variationally optimize ground-state approximations, implement real-time evolution and describe elementary excitations for a given model Hamiltonian. Also, we explain how matrix product states approximate fixed points of one-dimensional transfer matrices. We show how all these methods can be translated to the language of continuous matrix product states for one-dimensional field theories. We conclude with some extensions of the tangent-space formalism and with an outlook to new applications.
Current status:
Reports on this Submission
Report #1 by Anonymous (Referee 1) on 2019-1-6 (Invited Report)
- Cite as: Anonymous, Report on arXiv:1810.07006v1, delivered 2019-01-06, doi: 10.21468/SciPost.Report.778
Strengths
1- These lecture notes provide a very good introduction to tangent-space techniques for MPS.
2- Despite the technical nature of the notes, the authors provide concise descriptions of the algorithms for practical implementations.
3- Most of the paper is very pedagogical.
Weaknesses
1- It would have been interesting to add a few numerical benchmarks.
Report
The lecture notes give a good introduction to the concept of the tangent-space for uniform MPS. Some sections are technical but overall the document is very pedagogical and accessible for anyone with some knowledge of tensor network methods. This review will be very useful as an introduction to various standard algorithms that are explicitly described. My only remark is that it would have been interesting for the readers to have access to some numerical comparisons/benchmarks between methods using the tangent-space formalism and other algorithms. Therefore I recommend the publication in SciPost Lecture notes. I have a few minor remarks and have found some typos (see below).
Requested changes
1- equation (46) : there is an extra n in the second term of the right-hand site of the first line.
2- In the paragraph below eq (76) there is a typo ("recudes")
3- Just below eq (185) I think q should be replaced by p.
4- It seems that the link to arXiv of ref [57] does not send me to the right webpage.
5- Is it possible to add a comment about the feasibility (or not) of applying these methods to study periodic systems in finite-size ?