Category:Bethe-Salpeter equations

The formalism of the Bethe-Salpeter equation (BSE) allows for calculating the polarizability with the electron-hole interaction and constitutes the state of the art for calculating absorption spectra in solids.

Theory

The Bethe-Salpeter equation

In the BSE, the excitation energies correspond to the eigenvalues ${\displaystyle \omega_\lambda}$ of the following linear problem

${\displaystyle \left(\begin{array}{cc} \mathbf{A} & \mathbf{B} \\ \mathbf{B}^* & \mathbf{A}^* \end{array}\right)\left(\begin{array}{l} \mathbf{X}_\lambda \\ \mathbf{Y}_\lambda \end{array}\right)=\omega_\lambda\left(\begin{array}{cc} \mathbf{1} & \mathbf{0} \\ \mathbf{0} & -\mathbf{1} \end{array}\right)\left(\begin{array}{l} \mathbf{X}_\lambda \\ \mathbf{Y}_\lambda \end{array}\right)~. }$

The matrices ${\displaystyle A}$ and ${\displaystyle A^*}$ describe the resonant and anti-resonant transitions between the occupied ${\displaystyle v,v'}$ and unoccupied ${\displaystyle c,c'}$ states

${\displaystyle A_{vc}^{v'c'} = (\varepsilon_v-\varepsilon_c)\delta_{vv'}\delta_{cc'} + \langle cv'|V|vc'\rangle - \langle cv'|W|c'v\rangle. }$

The energies and orbitals of these states are usually obtained in a ${\displaystyle G_0W_0}$ calculation, but DFT and Hybrid functional calculations can be used as well. The electron-electron interaction and electron-hole interaction are described via the bare Coulomb ${\displaystyle V}$ and the screened potential ${\displaystyle W}$.

The coupling between resonant and anti-resonant terms is described via terms ${\displaystyle B}$ and ${\displaystyle B^*}$

${\displaystyle B_{vc}^{v'c'} = \langle vv'|V|cc'\rangle - \langle vv'|W|c'c\rangle. }$

Due to the presence of this coupling, the Bethe-Salpeter Hamiltonian is non-Hermitian.

The Tamm-Dancoff approximation

A common approximation to the BSE is the Tamm-Dancoff approximation (TDA), which neglects the coupling between resonant and anti-resonant terms, i.e., ${\displaystyle B}$ and ${\displaystyle B^*}$. Hence, the TDA reduces the BSE to a Hermitian problem

${\displaystyle AX_\lambda=\omega_\lambda X_\lambda~. }$

In reciprocal space, the matrix ${\displaystyle A}$ is written as

${\displaystyle A_{vc\mathbf{k}}^{v'c'\mathbf{k}'} = (\varepsilon_v-\varepsilon_c)\delta_{vv'}\delta_{cc'}\delta_{\mathbf{kk}'}+ \frac{2}{\Omega}\sum_{\mathbf{G}\neq0}\bar{V}_\mathbf{G}(\mathbf{q})\langle c\mathbf{k}|e^{i\mathbf{Gr}}|v\mathbf{k}\rangle\langle v'\mathbf{k}'|e^{-i\mathbf{Gr}}|c'\mathbf{k}'\rangle -\frac{2}{\Omega}\sum_{\mathbf{G,G}'}W_{\mathbf{G,G}'}(\mathbf{q},\omega)\delta_{\mathbf{q,k-k}'} \langle c\mathbf{k}|e^{i(\mathbf{q+G})}|c'\mathbf{k}'\rangle \langle v'\mathbf{k}'|e^{-i(\mathbf{q+G})}|v\mathbf{k}\rangle, }$

where ${\displaystyle \Omega}$ is the cell volume, ${\displaystyle \bar{V}}$ is the bare Coulomb potential without the long-range part

${\displaystyle \bar{V}_{\mathbf{G}}(\mathbf{q})=\begin{cases} 0 & \text { if } G=0 \\ V_{\mathbf{G}}(\mathbf{q})=\frac{4 \pi}{|\mathbf{q}+\mathbf{G}|^2} & \text { else } \end{cases}~, }$

and the screened Coulomb potential ${\displaystyle W_{\mathbf{G}, \mathbf{G}^{\prime}}(\mathbf{q}, \omega)=\frac{4 \pi \epsilon_{\mathbf{G}, \mathbf{G}^{\prime}}^{-1}(\mathbf{q}, \omega)}{|\mathbf{q}+\mathbf{G}|\left|\mathbf{q}+\mathbf{G}^{\prime}\right|}. }$

Here, the dielectric function ${\displaystyle \epsilon_\mathbf{G,G'}(\mathbf{q})}$ describes the screening in ${\displaystyle W}$ within the random-phase approximation (RPA)

${\displaystyle \epsilon_{\mathbf{G}, \mathbf{G}^{\prime}}^{-1}(\mathbf{q}, \omega)=\delta_{\mathbf{G}, \mathbf{G}^{\prime}}+\frac{4 \pi}{|\mathbf{q}+\mathbf{G}|^2} \chi_{\mathbf{G}, \mathbf{G}^{\prime}}^{\mathrm{RPA}}(\mathbf{q}, \omega). }$

Although the dielectric function is frequency-dependent, the static approximation ${\displaystyle W_{\mathbf{G}, \mathbf{G}^{\prime}}(\mathbf{q}, \omega=0)}$ is considered a standard for practical BSE calculations.

The macroscopic dielectric which account for the excitonic effects is found via eigenvalues ${\displaystyle \omega_\lambda}$ and eigenvectors ${\displaystyle X_\lambda}$ of the BSE

${\displaystyle \epsilon_M(\mathbf{q},\omega)= 1+\lim_{\mathbf{q}\rightarrow 0}v(q)\sum_{\lambda} \left|\sum_{c,v,k}\langle c\mathbf{k}|e^{i\mathbf{qr}}|v\mathbf{k}\rangle X_\lambda^{cv\mathbf{k}}\right|^2 \times \left(\frac{1}{\omega_\lambda - \omega - i\delta} + \frac{1}{\omega_\lambda+\omega + i\delta}\right)~. }$

Scaling

The scaling of the BSE equation strongly limits its application for large systems. The main limiting factor is the diagonalization of the BSE Hamiltonian. The rank of the Hamiltonian is

${\displaystyle N_{\rm rank} = N_k\times N_c\times N_v}$,

where ${\displaystyle N_k}$ is the number of k-points in the Brillouin zone and ${\displaystyle N_c}$ and ${\displaystyle N_v}$ are the number of conduction and valence bands, respectively. The diagonalization of the matrix scales cubically with the matrix rank, i.e., ${\displaystyle N_{\rm rank}^3}$.

Despite the fact that this matrix diagonalization is usually the bottleneck for bigger systems, the construction of the BSE Hamiltonian also scales unfavorably and can play a dominant role in big systems, i.e.,

${\displaystyle N_k\times N_q\times (N_v\times N_v\times N_G\times N_c\times N_c)}$,

where ${\displaystyle N_q}$ is the number of q-points and ${\displaystyle N_G}$ number of G-vectors.

Solution of the BSE

Diagonalization

The exact diagonalization of the BSE Hamiltonian can be perform using various eigensolvers provided in ScaLAPACK, ELPA, and cuSolver libraries. The advantage of this approach is that the eigenvectors can be directly obtained and used for the analysis of the excitons.

The following features are currently supported:

• Obtaining the spectra and eigenvectors
• Calculations beyond Tamm-Dancoff approximation
• Calculations of ${\displaystyle \varepsilon(\mathbf{q},\omega)}$ for ${\displaystyle \mathbf{q}\neq0}$
• Fatband plot

Time evolution

The alternative approach is to formulate the BSE as the initial-value problem for the macroscopic polarizability. This approach converges to the same solution as the exact diagonalization and can be used for obtaining the absorption spectrum, but does not yield the eigenvectors, which can be limiting for the analysis of the excitons. The advantage of this approach is the scaling with the size of the BSE Hamiltonian which is ${\displaystyle N_{rank}^2}$.

The following features are currently supported:

• Obtaining the spectra
• Calculations beyond Tamm-Dancoff approximation
• Calculations of ${\displaystyle \varepsilon(\mathbf{q},\omega)}$ for ${\displaystyle \mathbf{q}\neq0}$

How to

• Practical guide for solving the BSE via diagonalization BSE calculations

References