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Optical Conductivity from Excitonic States
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Xatu can compute the linear optical conductivity using the excitonic states obtained from the BSE. The calculation is performed in the **independent-particle approximation (IPA)** or the **Bethe-Salpeter equation (BSE)** formalism, based on the eigenstates previously computed.

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General Expression
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The optical conductivity tensor is computed from the excitonic eigenstates and their coupling to the electromagnetic field, following the Kubo-Greenwood formalism.

The real part of the conductivity tensor :math:`\sigma_{ab}(\omega)` is given by:

.. math::

   \sigma_{ab}(\omega) = \frac{\pi e^2 \hbar}{V} \sum_{\bm{k}}^{N_X} \frac{1}{E_{\bm{k}}}
   \left[ V_{\bm{k}}^a (V_{\bm{k}}^b)^* \right] \delta(\hbar\omega - E_{\bm{k}})

where:

* $ V $ is the system volume
* :math:`E_{\mathbf{k}}` is the exciton energy at momentum :math:`\mathbf{k}`
* :math:`V_{\mathbf{k}}^a` velocity matrix elements :math:`\langle GS| \hat{v}^{a} | X_{\mathbf{k}} \rangle`
* $ N_X $ is the number of computed exciton states
* The delta function is broadened numerically using a specified kernel

This expression is implemented directly in Xatu when the linear response spectrum is requested using the ``-a`` flag and a valid ``kubo_w.in`` file is provided.

Excitonic Absorption Spectrum
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The absorption spectrum is computed by convoluting the excitonic delta functions with a chosen broadening. The user can specify:

* Broadening type: `lorentzian`, `gaussian`, or `exponential`
* Broadening width (in eV)
* Frequency range and resolution

This is controlled by the ``kubo_w.in`` input file. The computed spectra include:

* **Independent-particle spectrum** (IPA)
* **Excitonic spectrum** (BSE)

Each output is saved to separate `.dat` files for plotting, with names defined in the exciton configuration file.

Reference
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For details, see:

`Efficient computation of optical excitations in two-dimensional materials with the Xatu code, Computer Physics Communications, 2024` <https://doi.org/10.1016/j.cpc.2023.109001>