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Automatic, high-order, and adaptive algorithms for Brillouin zone integration

by Jason Kaye, Sophie Beck, Alex Barnett, Lorenzo Van Muñoz, Olivier Parcollet

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Submission summary

Authors (as registered SciPost users): Sophie Beck · Jason Kaye
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-05-22
Date submitted: 2023-03-30 20:06
Submitted by: Kaye, Jason
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Computational
Approach: Computational


We present efficient methods for Brillouin zone integration with a non-zero but possibly very small broadening factor $\eta$, focusing on cases in which downfolded Hamiltonians can be evaluated efficiently using Wannier interpolation. We describe robust, high-order accurate algorithms automating convergence to a user-specified error tolerance $\varepsilon$, emphasizing an efficient computational scaling with respect to $\eta$. After analyzing the standard equispaced integration method, applicable in the case of large broadening, we describe a simple iterated adaptive integration algorithm effective in the small $\eta$ regime. Its computational cost scales as $\mathcal{O}(\log^3(\eta^{-1}))$ as $\eta \to 0^+$ in three dimensions, as opposed to $\mathcal{O}(\eta^{-3})$ for equispaced integration. We argue that, by contrast, tree-based adaptive integration methods scale only as $\mathcal{O}(\log(\eta^{-1})/\eta^{2})$ for typical Brillouin zone integrals. In addition to its favorable scaling, the iterated adaptive algorithm is straightforward to implement, particularly for integration on the irreducible Brillouin zone, for which it avoids the tetrahedral meshes required for tree-based schemes. We illustrate the algorithms by calculating the spectral function of SrVO$_3$ with broadening on the meV scale.

Author comments upon resubmission

We again thank the referees for their thorough comments on our manuscript. We have made a variety of modifications in response to the first referee report, most of which were mentioned in our previous response. We provide a detailed list of all changes to the manuscript below.

In addition, we wish to provide a brief response to the main point raised by the second referee concerning $k$-dependent self-energies. As long as the self-energy can be evaluated rapidly on the fly, the generalization to this case is straightforward. There are then two points to make about this issue. First, although the self-energy may be expensive to evaluate in its given form, many standard tools exist to replace a representation of a function which is expensive to evaluate by another which can be made essentially as fast to evaluate as needed (to take an extreme case, for example, one could pre-tabulate a function on a very dense grid, and use a local linear polynomial interpolant for rapid evaluation). This preparation of the self-energy is a precomputation. Using such a scheme, the efficiency of integration can be made as good as the underlying structure of the self-energy allows. Second, the most natural fast-to-evaluate representation is problem-dependent. For example, the most efficient representation might be on an adaptive grid (like the adaptive Chebyshev interpolant described in the manuscript); it might be a Fourier series in $k$ with $\omega$-dependent coefficients represented by a polynomial interpolant; or it might be a local piecewise-polynomial interpolant on a uniform grid. We have added a remark to the conclusion which summarizes these points.

We welcome feedback from the referees on whether our modifications address their concerns satisfactorily.

List of changes

Changes in response to first referee report, using the referee's numbering:

(1) In Section IIIB we have given a comparison between the PTR and adaptive integration for the 1D example described in Section II.
(2) We have added a sentence in the introduction explaining why our manuscript focuses on high-order methods.
(2) We have added a sentence at the end of Section IIIA noting that the PTR is superior to Gauss quadrature for periodic function, with a reference.
(3) In Section IIB we have specified the initial value of $N_{\text{trial}}$ that we use.
(4) We have added a remark in Section IIB, as well as an Appendix describing the suggested algorithm. We have also modified Algorithm 1 and the surrounding discussion to make it clear that it is not necessary to specify the list of evaluation frequencies in advance.
(5) We have modified Appendix C. In particular, we added a discussion on the pruned FFT, with references, and have removed the statement that there is no efficient algorithm to restrict a zero-padded FFT to the irreducible Brillouin zone.
(7) We have added a sentence in the caption to Fig. 5 explaining that the time to evaluate $H(k)$ is not included in the timings reported in the figure.
(8) We have modified Fig. 2 to improve the contrast and clarity of the grid points.

Changes in response to the second referee report:

We have added a short remark in the introduction, and a longer explanation in the conclusion, addressing the generalization to $k$-dependent self-energies.

Other changes

- We have corrected typos in the definition of the function used in Fig. 2, and in Eq. 7.
- We have made a few minor notational changes and remarks.

Published as SciPost Phys. 15, 062 (2023)

Reports on this Submission

Anonymous Report 2 on 2023-4-12 (Invited Report)


I fully agree with my colleague and recommend the paper for publication.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Report 1 by Stepan Tsirkin on 2023-4-4 (Invited Report)


The autoors have adequately responded to the previous comments and made corresponding changes in the manuscript. I recommend the paper for publication

  • validity: top
  • significance: high
  • originality: high
  • clarity: top
  • formatting: excellent
  • grammar: excellent

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