Fredholm approximations to the Bethe-Salpeter equation with spherical-Bessel kernels for Wannier-Mott excitons
Abstract
We develop a deterministic radial approximation to the static Bethe-Salpeter equation (BSE), designed for rapid preliminary screening of Wannier-Mott exciton binding across direct-gap bulk materials. The approximation replaces the electron-hole transition problem by small radial eigenvalue problems that require no stochastic sampling. Starting from the four-point BSE and its matrix representation in a Bloch electron-hole basis, we retain one valence and one conduction band in the effective-mass limit within the Tamm-Dancoff approximation. We introduce a central projection of the screened direct interaction and omit bare electron-hole exchange. The resulting model applies to spatially extended excitons whose envelopes vary slowly on the scale of the unit cell, provided that the effective-mass description and central screened interaction adequately represent the exciton of interest. A partial-wave decomposition separates the angular-momentum channels and expresses their interaction kernels as spherical-Bessel transforms of the real-space screened Coulomb interaction. Nyström discretization produces real symmetric radial matrices suitable for standard eigensolvers. Their dimension is set by the radial grid independently of the Brillouin-zone sampling used to calculate screening. We implement the formulation in the all-electron, full-potential code exciting. For bulk GaAs, the compact radial problem reproduces the corresponding conventional matrix-BSE binding energy within (1 meV) at matched sampling. Further refinement achieves finite-mesh stability within the same tolerance on denser screening meshes without increasing the radial matrix size.
Related
- Magnons and excitons from Bethe-Salpeter equation and its application to graphene-based magnets
- Nonuniform Screening Reshapes Collective Excitons in Molecular Aggregates from Stochastic Bethe-Salpeter Theory
- GW and Bethe-Salpeter Theory for Molecular Polaritons, Quasiparticles, and Excitons
- Baryon Bethe–Salpeter equation in Minkowski-space QCD2
- Parity-selected Bethe-Salpeter equation: symmetry-adapted regularization of the exciton-phonon self-energy in polar semiconductors
BibTeX
@misc{tantardini2026fredholm,
title = {{Fredholm approximations to the Bethe-Salpeter equation with spherical-Bessel kernels for Wannier-Mott excitons}},
author = {Christian Tantardini and Benedikt Maurer and Claudia Draxl},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01570},
eprint = {2611.01570},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01570},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Fredholm approximations to the Bethe-Salpeter equation with spherical-Bessel kernels for Wannier-Mott excitons
AU - Christian Tantardini
AU - Benedikt Maurer
AU - Claudia Draxl
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01570
UR - https://arxiv.org/abs/2611.01570
AB - We develop a deterministic radial approximation to the static Bethe-Salpeter equation (BSE), designed for rapid preliminary screening of Wannier-Mott exciton binding across direct-gap bulk materials. The approximation replaces the electron-hole transition problem by small radial eigenvalue problems that require no stochastic sampling. Starting from the four-point BSE and its matrix representation in a Bloch electron-hole basis, we retain one valence and one conduction band in the effective-mass limit within the Tamm-Dancoff approximation. We introduce a central projection of the screened direct interaction and omit bare electron-hole exchange. The resulting model applies to spatially extended excitons whose envelopes vary slowly on the scale of the unit cell, provided that the effective-mass description and central screened interaction adequately represent the exciton of interest. A partial-wave decomposition separates the angular-momentum channels and expresses their interaction kernels as spherical-Bessel transforms of the real-space screened Coulomb interaction. Nyström discretization produces real symmetric radial matrices suitable for standard eigensolvers. Their dimension is set by the radial grid independently of the Brillouin-zone sampling used to calculate screening. We implement the formulation in the all-electron, full-potential code exciting. For bulk GaAs, the compact radial problem reproduces the corresponding conventional matrix-BSE binding energy within (1 meV) at matched sampling. Further refinement achieves finite-mesh stability within the same tolerance on denser screening meshes without increasing the radial matrix size.
ER -