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Pymablock: an algorithm and a package for quasi-degenerate perturbation theory
by Isidora Araya Day, Sebastian Miles, Hugo K. Kerstens, Daniel Varjas, Anton R. Akhmerov
Submission summary
| Authors (as registered SciPost users): | Anton Akhmerov · Isidora Araya Day · Hugo Kerstens · Sebastian Miles |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2404.03728v1 (pdf) |
| Code repository: | https://gitlab.kwant-project.org/qt/pymablock |
| Data repository: | https://zenodo.org/records/10629243 |
| Date submitted: | 2024-06-12 10:56 |
| Submitted by: | Araya Day, Isidora |
| Submitted to: | SciPost Physics Codebases |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
A common technique in the study of complex quantum-mechanical systems is to reduce the number of degrees of freedom in the Hamiltonian by using quasi-degenerate perturbation theory. While the Schrieffer--Wolff transformation achieves this and constructs an effective Hamiltonian, its scaling is suboptimal, and implementing it efficiently is both challenging and error-prone. We introduce an algorithm for constructing an equivalent effective Hamiltonian as well as a Python package, Pymablock, that implements it. Our algorithm combines an optimal asymptotic scaling with a range of other improvements. The package supports numerical and analytical calculations of any order and it is designed to be interoperable with any other packages for specifying the Hamiltonian. We demonstrate how the package handles constructing a k.p model, analyses a superconducting qubit, and computes the low-energy spectrum of a large tight-binding model. We also compare its performance with reference calculations and demonstrate its efficiency.
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Report for “Pymablock: an algorithm and a package for quasi-degenerate perturbation theory” by Day et al.
The Python package “Pymablock” implements a unitary transformation for the block diagonalization of a Hermitian matrix using a perturbative approach. This method allows for symbolic evaluation and uses a recursive approach for the operator series expressions, equivalent to the Schrieffer-Wolff transformation, with the primary step requiring the solution of a Sylvester equation for each order. The package’s capabilities are demonstrated through examples including a k.p model for AB-stacked Graphene, a transmon qubit coupled to a resonator, and a double quantum dot.
I have put the package to the test by investigating a bosonic Hamiltonian with a quartic perturbation, see Eq. (11) in Phys. Rev. B 86, 125113 (2012). First observation: Pymablock cannot directly work with a Hamiltonian defined in second quantization. Indeed, as the authors themselves do it in 2.2 I had to translate the terms into matrix form and truncate the Hilbert space in order for the approach to work. If I chose to block diagonalize the ground state (A) end decouple it from all the other states (B) the approach worked well and provides the correct series expansion. For the first excited state the first order correction (although trivial) was also correct. The second order correction yielded a value of $-30x^2/\omega$ which first confused me, since the corresponding bilinear operator b^dagger b has a series expansion of $-24x^2/\omega$, see table I in above reference. But it is explained by the fact, that Pymablock is not diagonalizing the ground state at this point anymore, thus one would need to also subtract the value $-6x^2/\omega$ from the ground state energy to make connection to previous results. Thus the package works but the drawback of not blockdiagonalizing several blocks becomes apparent.
SciPost Physics Codebase requires the following for acceptance:
1. Benchmarking tests must be provided.
I think this criteria is fulfilled in section 5, but only marginally so. It indeed seems like there are no open source packages available that implement high order series expansion methods.
2. At least one example application must be presented in detail
This criteria is perfectly fulfilled, as the authors discuss 3 examples in their paper.
3. High-level programming standards must be followed throughout the source code
The source code is well documented and from my review seems to be in good condition.
4. The userguide must properly contextualize the software, describe the logic of its workings and highlight its added value as compared to existing software
Also this criteria is fulfilled.
5. The software must address a demonstrable need for the scientific community
This is the biggest issue I have with the paper, see comment below.
6. The documentation must be complete, including detailed instructions on downloading, installing and running the software
This criteria is fulfilled, the package worked out of the box after installing with conda/pip.
Overall I think the packages provides a good first start for a python package aiming at perturbatively decoupling Hamiltonians. However the paper is missing a huge portion of literature that aims exactly at the same issue: continuous unitary transformations (CUT) are a tool that provides effective models based on perturbative expansion. A good overview (although a bit outdated) is given in sections I and II of Phys. Rev. B 86, 125113 (2012). Having in mind the capabilities of these approaches the main drawbacks I see with pymablock are:
1) Lack of simultaneous Block-Diagonalization beyond 2 blocks.
2) Lack of support for second quantized Hamiltonians (i.e. true expansion in terms of prefactors for the operators instead of truncated Matrix representation.
I still think that due to the missing open source implementations of above methods, pymablock fills this gap.
Minor point:
So far only two subspaces are block diagonalized, hence Fig 1 is misleading, as it looks like a block diagonalization of more than 2 blocks!
Therefore my recommendation is to publish, after the authors add a more comprehensive literature review on above mention methods in order to derive perturbative expansions for effective models. In the long-term (for future releases) I recommend the authors to address the points about simultaneous Block-Diagonalization and support for second quantized Hamiltonians.
Requested changes
Add comprehensive literature review on CUT methods for effective perturbative Hamiltonians
Recommendation
Ask for minor revision