SciPost Submission Page
Automatic, highorder, and adaptive algorithms for Brillouin zone integration
by Jason Kaye, Sophie Beck, Alex Barnett, Lorenzo Van Muñoz, Olivier Parcollet
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Sophie Beck · Jason Kaye 
Submission information  

Preprint Link:  https://arxiv.org/abs/2211.12959v2 (pdf) 
Date accepted:  20230522 
Date submitted:  20230330 20:06 
Submitted by:  Kaye, Jason 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Computational 
Abstract
We present efficient methods for Brillouin zone integration with a nonzero but possibly very small broadening factor $\eta$, focusing on cases in which downfolded Hamiltonians can be evaluated efficiently using Wannier interpolation. We describe robust, highorder accurate algorithms automating convergence to a userspecified 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, treebased 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 treebased schemes. We illustrate the algorithms by calculating the spectral function of SrVO$_3$ with broadening on the meV scale.
Author comments upon resubmission
In addition, we wish to provide a brief response to the main point raised by the second referee concerning $k$dependent selfenergies. As long as the selfenergy 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 selfenergy 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 pretabulate a function on a very dense grid, and use a local linear polynomial interpolant for rapid evaluation). This preparation of the selfenergy is a precomputation. Using such a scheme, the efficiency of integration can be made as good as the underlying structure of the selfenergy allows. Second, the most natural fasttoevaluate representation is problemdependent. 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 piecewisepolynomial 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 highorder 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 zeropadded 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 selfenergies.
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
Report 1 by Stepan Tsirkin on 202344 (Invited Report)
Report
The autoors have adequately responded to the previous comments and made corresponding changes in the manuscript. I recommend the paper for publication