.. _welcome:

The hybridization-expansion solver
**********************************


The :ref:`TRIQS-based <triqslibs:welcome>` hybridization-expansion solver
allows to  solve the generic problem of a **quantum impurity** embedded in a
conduction bath for an arbitrary local interaction vertex.  The "impurity" can
be any set of orbitals, on one or several atoms. To be more specific, the
Hamiltonian of the problem has the form

.. math::

  \hat H  = \sum_{k,\alpha} \epsilon_{k,\alpha} c^\dagger_{k,\alpha} c_{k,\alpha} + \sum_{k,\alpha}
            (V_{k,\alpha} c^\dagger_{k,\alpha} d_{\alpha} + h.c.)
            - \mu \sum_\alpha d^\dagger_\alpha d_\alpha +
            \sum_{\alpha\beta} h_{\alpha\beta} d^\dagger_\alpha d_\beta +
            \frac{1}{2}\sum_{\alpha\beta\gamma\delta} U_{\alpha\beta\gamma\delta}
            d^\dagger_\alpha d^\dagger_\beta d_\delta d_\gamma.

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Here the operators :math:`c^\dagger` construct a fermion in the bath, while
the operators :math:`d^\dagger` construct a fermion on the impurity.
In this problem, the hybridization function :math:`\Delta` between the bath
and the impurity is given by:

.. math::

  \Delta_{\alpha\beta} (i \omega_n) = \sum_k \frac{V_{k,\alpha} V^*_{k,\beta}}{i \omega_n - \epsilon_{k,\alpha}},

so that the non-interacting Green's function of the impurity is:

.. math::

  \hat{G}^{-1}_0 (i \omega_n) = i \omega_n + \mu - \hat h - \hat{\Delta}(i \omega_n).

With the knowledge of :math:`G_0` and the matrix :math:`U_{\alpha\beta\gamma\delta}`,
the quantum impurity solvers find the interacting Green's function :math:`G` of the
problem. Learn how to use it in the :ref:`documentation`.

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.. toctree::
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