Overview of CyGutz with WIEN2k and model interfaces
===================================================

DFT-LDA
-------

The Density Functional Theory implementation within WIEN2k 
consists of several steps depicted in the figure 
(Check http://www.wien2k.at/) 

.. image:: _images/WIEN2kFC.png
   :alt: alternate text
   :scale: 100 %
   :align: center

The steps are:

* init_lapw:

    * NN: lists nearest neighbor distances and determine atomic sphere radii.
    * SGROUP: determines the spacegroup of the structure defined 
      in your case.struct file.
    * SYMMETRY: generates from a raw case.struct file 
      the space group symmetry operations, 
      determines the point group of the individual atomic sites, 
      generates the LM expansion for the lattice harmonics 
      and determines the local rotation matrices.
    * LSTART: generates free atomic densities 
      and determines how the different orbitals are treated 
      in the band structure calculations.
    * KGEN: generates a k-mesh in the irreducible part of the BZ.
    * DSTART: generates a starting density for the scf cycle 
      by a superposition of atomic densities generated in LSTART.

* run_lapw:
  
    * LAPW0: (POTENTIAL) generates potential from density.
    * LAPW1: (BANDS) calculates valence bands (eigenvalues and eigenvectors).
    * LAPWSO: adds spin-orbit coupling.
    * LAPW2: (RHO) computes valence densities from eigenvectors.
    * LCORE: computes core states and densities.
    * MIXER: mixes input and output densities.

LDA + Gutzwiller-Slave-Boson
----------------------------

The LDA+G inserts two new steps (dmft1 and CyGutz) 
and replaces lapw2 with dmft2 step, 
following closely the DFT+DMFT code 
(Check http://hauleweb.rutgers.edu/). 
The process looks like that: 

.. image:: _images/WIENGAFC.png
   :alt: alternate text
   :scale: 100 %
   :align: center

The LDA+G steps (``run_ga.py``) are:

* LAPW0:  (POTENTIAL) generates potential from density.
* LAPW1:  (BANDS) calculates valence bands (eigenvalues and eigenvectors).
* LAPWSO: adds spin-orbit coupling (optional).
* DMFT1:  calculates the local correlated orbitals (projectors) 
  using the Kohn-Sham bands as the basis.
* CYGUTZ: Solve the generic Kohn-Sham-Hubbard Hamiltonian 
  using the Gutzwiller Slave-Boson method.
* dmft2:  calculates LDA+G renormalized valence electron density.
* LCORE:  computes core states and densities.
* MIXER:   mixes input and output densities.

CyGutz
------

CyGutz is a stand-alone program to solve a generic Hubbard model 
(including local correlated orbitals and nonlocal orbitals) 
using Gutzwiller-Slave-Boson method 
(See Ref. [prx2015]_ and [prl2017]_).
The steps to reach self-consistent solutions are as follows:


.. image:: _images/CyGutzFC.png
   :alt: alternate text
   :scale: 100 %
   :align: center

CyGutz starts with the setup of a generic [Kohn-Sham]-Hubbard model 
based on the input bare band energies 
and the specified local correlated orbitals, 
which are expressed in terms of the bare band wave functions. 
Here the bare bands can be obtained from LDA/quasi-particle GW calculations 
or tight-binding models. 
The solution of the generic Hubbard model is cast to a root-finding problem
within the Gutzwiller-Slave-Boson method. 
Check Ref. [prx2015]_ and [prl2017]_ for details.


.. [prx2015] Lanatà, Nicola, Yongxin Yao, Cai-Zhuang Wang, Kai-Ming Ho, 
   and Gabriel Kotliar. 2015. 
   ''Phase Diagram and Electronic Structure of Praseodymium and Plutonium.''
   Physical Review X 5 (1): 011008. 
   doi:10.1103/PhysRevX.5.011008.
.. [prl2017] Lanatà, Nicola, Yongxin Yao, Xiaoyu Deng, Vladimir Dobrosavljević,
   and Gabriel Kotliar. 2017. 
   ''Slave Boson Theory of Orbital Differentiation with Crystal Field Effects:
   Application to UO :sub:`2`.'' 
   Physical Review Letters 118 (12): 126401. 
   doi:10.1103/PhysRevLett.118.126401.