Screening Potentials in Xatu

Xatu supports two real-space interaction potentials used in the Bethe-Salpeter Equation:

  1. Coulomb potential

  2. Rytova–Keldysh potential

These govern the electron–hole interaction and are used to build the interaction kernel.

Coulomb Potential

The standard Coulomb interaction in real space is defined as:

\[V(\bm{r}) = \frac{e^2}{4 \pi \varepsilon_0 |\bm{r}|}\]

In the implementation, this interaction is:

  • Regularized at $r = 0$ using a small regularization parameter

  • Truncated beyond a distance cutoff defined from the lattice parameter

This option is appropriate when long-range unscreened interactions are desired.

Rytova–Keldysh Potential

This model captures the effect of environmental screening in 2D materials. The potential reads:

\[V(r) = -\frac{e^2}{8 \varepsilon_0 \bar{\varepsilon} r_0} \left[ H_0\left(\frac{r}{r_0}\right) - Y_0\left(\frac{r}{r_0}\right) \right]\]

where:

  • \(\bar{\varepsilon} = (\varepsilon_m + \varepsilon_s)/2\) is the average surrounding dielectric between the medium \(\varepsilon_m\) and substrate \(\varepsilon_s\)

  • $ r_0 $ is the effective screening length of the 2D material

  • $ H_0 $ is the Struve function

  • $ Y_0 $ is the Bessel function of the second kind

In practice:

  • The interaction is regularized at $r = 0$

  • A cutoff beyond which the interaction vanishes is applied

  • The implementation may treat the screening radius anisotropically, i.e., using different $r_0$ values along different directions. This is an extension not typically found in the literature.

Anisotropic Screening

Xatu supports anisotropic screening in the Rytova–Keldysh model by allowing directional dependence in the screening length. This is implemented by constructing an effective vector \(\mathbf{r}_0 = (r_{0}^{x}, r_{0}^{y}, r_{0}^{z})\) , and rescaling the coordinates accordingly.

This allows the screening environment to be tuned independently along in-plane and out-of-plane directions – a generalization that extends beyond conventional isotropic models.