Bethe-Salpeter Equation in Xatu

The Bethe-Salpeter Equation (BSE) governs the formation of excitons in semiconductors and insulators. Xatu solves the BSE using localized orbitals and static screened interactions.

BSE Formalism

Starting from the full interacting Hamiltonian projected onto electron-hole pairs:

\[\sum_{v',c',\bm{k}'} H_{vc,v'c'}(\bm{k},\bm{k}',Q) A^Q_{v'c'}(\bm{k}') = E_X A^Q_{vc}(\bm{k})\]

we define the interaction kernel and simplify the problem by transforming into the Hartree-Fock (HF) band basis. This incorporates self-energy corrections into the quasiparticle energies.

The resulting working form of the BSE solved in Xatu is:

\[\left( \varepsilon_{c,\bm{k+Q}} - \varepsilon_{v,\bm{k}} \right) A^Q_{vc}(\bm{k}) + \sum_{v',c',\bm{k}'} K_{vc,v'c'}(\bm{k}, \bm{k}', Q) A^Q_{v'c'}(\bm{k}') = E_X A^Q_{vc}(\bm{k})\]

where:

• \(\varepsilon_{n,\mathbf{k}}\) are the HF (or DFT/GW) quasiparticle energies

• \(A^{Q}_{vc}(\mathbf{k})\) are the exciton amplitudes

• $ K = -(D - X) $ is the interaction kernel with:

  • $ D $ : direct interaction between electron and hole

  • $ X $ : exchange interaction (optional)

This is the Tamm-Dancoff approximation (TDA) form of the BSE.

Solution Methods

The BSE matrix is constructed and diagonalized using Armadillo linear algebra routines. For large systems, the following methods are available:

• diag: full diagonalization (default)

• davidson: iterative solver for low-lying states

• sparse: Lanczos-based sparse diagonalization

Output includes exciton energies, wavefunctions, real* and reciprocal-space densities, and optical matrix elements.