.. _kernels:

====================================
Available kernels
====================================

:math:`\mu`-Kernels
====================

In :mod:`~py4mulas.mu_kernels` the following kernels are implemented:

Kubo kernels
-------------

Different Kubo kernels can be accessed from :class:`~py4mulas.mu_kernels.KuboKernel`.
These include ``inter_band``, ``intra_band``, ``anomalous``, ``even`` and ``odd`` kernels.
For instance, to access the ``inter_band`` kernel:

.. code-block:: python

    from py4mulas.mu_kernels import KuboKernel 
    mu_kernel = KuboKernel(name='inter_band')

* Anomalous Hall kernel ``name='anomalous'``

  :math:`-2i\frac{f_n}{(E_n - E_m)(E_n - E_m + i\eta)}`

* Interband Kubo kernel ``name='inter_band'``
    :math:`i\frac{f_m - f_n}{(E_n - E_m)(E_n - E_m + i\eta)}`

    See. [1]

* Intraband Kubo kernel ``name='intra_band'``
    :math:`\frac{i}{\eta}\partial_E f|_{E=E_n}`

* Constant broadening kernels

    The effect of temperature is included in the constant broadening :math:`\eta`. Thetse kernels are 
    therfore :math:`T` independent.

    * ``name='even'`` Eq(4) of [2]
    * ``name='odd'`` Eq(5) of [2]

        

Kubo-Bastin kernels
-------------------

Other ``mu_kernel`` s are also available from :class:`~py4mulas.mu_kernels.KuboBastinKernel`.
These kernels are originally defined within the Kubo-Bastin framework. Using the recently proposed 
approach to analytically handle the Kubo-Bastin energy integration [3], the formula becomes easier to implement
as no energy integration is required anymore. This formulation is useful when we are interested in
separating different response components such as Fermi-surface and Fermi-sea contributions.

This kernel class takes the following names:
``StredaI``, ``StredaII``, ``BMI``, ``Overlap`` and ``BMII``.
  
* ``StredaI`` and ``StredaII`` correspond to the Streda decomposition of the Kubo-Bastin formula.
* ``BMI`` and ``BMII`` are Fermi-surface and Fermi-sea contributions recently suggested by Bonbien and Manchon [4]. This decomposition corrects the original Streda decomposition. 
* ``Overlap`` corresponds to the overlap between these two contributions.

These kernels can be accesses by specifiying their names in the underlying kernel class

.. code-block:: python

    from py4mulas.mu_kernels import KuboBastinKernel 
    mu_kernelBMI = KuboBastinKernel(name='BMI')

Finally, the kernel should be passed to a formula. For instance to the built-in KuboFormula as 

.. code-block:: python

    from py4mulas.responses import Kubo 
    formula = Kubo(some_model, kspace_options=kspace_options, mu_kernel=mu_kernelBMI)

.. only:: builder_html 

    See. :download:`rashbaModel.py <../../../examples/rashbaModel.py>` where these kernels ar explored. 

[1] `Crepieux et Bruno, [PRB 64, 014416 (2001)] <https://journals.aps.org/prb/abstract/10.1103/PhysRevB.64.014416>`_.

[2] `Freimuth et al, [PRB 90, 174423 (2014)] <https://journals.aps.org/prb/abstract/10.1103/PhysRevB.90.174423>`_.

[3] `Ly, [arXiv.2605.19670] <https://doi.org/10.48550/arXiv.2605.19670>`_.

[4] `Bonbian and Manchon [PRB 102, 085113 (2020)] <https://journals.aps.org/prb/abstract/10.1103/PhysRevB.102.085113>`_.