dft_tools/doc/guide/SrVO3.rst

.. _SrVO3:

SrVO3 (single-shot)
===================

We will discuss now how to set up a full working calculation,
including the initialization of the :ref:`CTHYB solver <triqscthyb:welcome>`.
Some additional parameter are introduced to make the calculation
more efficient. This is a more advanced example, which is
also suited for parallel execution. The conversion, which
we assume to be carried out already, is discussed :ref:`here <conversion>`.

For the convenience of the user, we provide also two
working python scripts in this documentation. One for a calculation
using Kanamori definitions (:download:`dft_dmft_cthyb.py
<images_scripts/dft_dmft_cthyb.py>`) and one with a
rotational-invariant Slater interaction Hamiltonian (:download:`dft_dmft_cthyb_slater.py
<images_scripts/dft_dmft_cthyb_slater.py>`). The user has to adapt these
scripts to his own needs. How to execute your script is described :ref:`here<runpy>`.

Loading modules
---------------

First, we load the necessary modules::

  from triqs_dft_tools.sumk_dft import *
  from pytriqs.gf import *
  from pytriqs.archive import HDFArchive
  from pytriqs.operators.util import *
  from triqs_cthyb import *
  import pytriqs.utility.mpi as mpi

The last two lines load the modules for the construction of the
:ref:`CTHYB solver <triqscthyb:welcome>`.

Initializing SumkDFT
--------------------

We define some parameters, which should be self-explanatory::

  dft_filename = 'SrVO3'          # filename
  U = 4.0                         # interaction parameters
  J = 0.65
  beta = 40                       # inverse temperature
  loops = 15                      # number of DMFT loops
  mix = 0.8                       # mixing factor of Sigma after solution of the AIM
  dc_type = 1                     # DC type: 0 FLL, 1 Held, 2 AMF
  use_blocks = True               # use bloc structure from DFT input
  prec_mu = 0.0001                # precision of chemical potential


And next, we can initialize the :class:`SumkDFT <dft.sumk_dft.SumkDFT>` class::

  SK = SumkDFT(hdf_file=dft_filename+'.h5',use_dft_blocks=use_blocks)

Initializing the solver
-----------------------

We also have to specify the :ref:`CTHYB solver <triqscthyb:welcome>` related settings.
We assume that the DMFT script for SrVO3 is executed on 16 cores. A sufficient set
of parameters for a first guess is::

  p = {}
  # solver
  p["random_seed"] = 123 * mpi.rank + 567
  p["length_cycle"] = 200
  p["n_warmup_cycles"] = 100000
  p["n_cycles"] = 1000000
  # tail fit
  p["perform_tail_fit"] = True
  p["fit_max_moment"] = 4
  p["fit_min_n"] = 30
  p["fit_max_n"] = 60

Here we use a tail fit to deal with numerical noise of higher Matsubara frequencies.
For other options and more details on the solver parameters, we refer the user to
the :ref:`CTHYB solver <triqscthyb:welcome>` documentation.
It is important to note that the solver parameters have to be adjusted for
each material individually. A guide on how to set the tail fit parameters is given
:ref:`below <tailfit>`.


The next step is to initialize the
:class:`solver class <triqs_cthyb.Solver>`.
It consist of two parts:

#. Calculating the multi-band interaction matrix, and constructing the
   interaction Hamiltonian.
#. Initializing the solver class itself.

The first step is done using methods of the :ref:`TRIQS <triqslibs:welcome>` library::

  n_orb = SK.corr_shells[0]['dim']
  l = SK.corr_shells[0]['l']
  spin_names = ["up","down"]
  orb_names = [i for i in range(n_orb)]
  # Use GF structure determined by DFT blocks:
  gf_struct = [(block, indices) for block, indices in SK.gf_struct_solver[0].iteritems()]
  # Construct U matrix for density-density calculations:
  Umat, Upmat = U_matrix_kanamori(n_orb=n_orb, U_int=U, J_hund=J)

We assumed here that we want to use an interaction matrix with
Kanamori definitions of :math:`U` and :math:`J`.

Next, we construct the Hamiltonian and the solver::

  h_int = h_int_density(spin_names, orb_names, map_operator_structure=SK.sumk_to_solver[0], U=Umat, Uprime=Upmat)
  S = Solver(beta=beta, gf_struct=gf_struct)

As you see, we take only density-density interactions into
account. Other Hamiltonians with, e.g. with full rotational invariant interactions are:

* h_int_kanamori
* h_int_slater

For other choices of the interaction matrices (e.g Slater representation) or
Hamiltonians, we refer to the reference manual of the :ref:`TRIQS <triqslibs:welcome>`
library.

DMFT cycle
----------

Now we can go to the definition of the self-consistency step. It consists again
of the basic steps discussed in the :ref:`previous section <singleshot>`, with
some additional refinements::

  for iteration_number in range(1,loops+1):
      if mpi.is_master_node(): print "Iteration = ", iteration_number

      SK.symm_deg_gf(S.Sigma_iw,orb=0)                        # symmetrizing Sigma
      SK.set_Sigma([ S.Sigma_iw ])                            # put Sigma into the SumK class
      chemical_potential = SK.calc_mu( precision = prec_mu )  # find the chemical potential for given density
      S.G_iw << SK.extract_G_loc()[0]                         # calc the local Green function
      mpi.report("Total charge of Gloc : %.6f"%S.G_iw.total_density())

      # Init the DC term and the real part of Sigma, if no previous runs found:
      if (iteration_number==1 and previous_present==False):
          dm = S.G_iw.density()
          SK.calc_dc(dm, U_interact = U, J_hund = J, orb = 0, use_dc_formula = dc_type)
          S.Sigma_iw << SK.dc_imp[0]['up'][0,0]

      # Calculate new G0_iw to input into the solver:
      S.G0_iw << S.Sigma_iw + inverse(S.G_iw)
      S.G0_iw << inverse(S.G0_iw)

      # Solve the impurity problem:
      S.solve(h_int=h_int, **p)

      # Solved. Now do post-solution stuff:
      mpi.report("Total charge of impurity problem : %.6f"%S.G_iw.total_density())

      # Now mix Sigma and G with factor mix, if wanted:
      if (iteration_number>1 or previous_present):
          if mpi.is_master_node():
              ar = HDFArchive(dft_filename+'.h5','a')
              mpi.report("Mixing Sigma and G with factor %s"%mix)
              S.Sigma_iw << mix * S.Sigma_iw + (1.0-mix) * ar['dmft_output']['Sigma_iw']
              S.G_iw << mix * S.G_iw + (1.0-mix) * ar['dmft_output']['G_iw']
              del ar
          S.G_iw << mpi.bcast(S.G_iw)
          S.Sigma_iw << mpi.bcast(S.Sigma_iw)

      # Write the final Sigma and G to the hdf5 archive:
      if mpi.is_master_node():
          ar = HDFArchive(dft_filename+'.h5','a')
          ar['dmft_output']['iterations'] = iteration_number
          ar['dmft_output']['G_0'] = S.G0_iw
          ar['dmft_output']['G_tau'] = S.G_tau
          ar['dmft_output']['G_iw'] = S.G_iw
          ar['dmft_output']['Sigma_iw'] = S.Sigma_iw
          del ar

      # Set the new double counting:
      dm = S.G_iw.density() # compute the density matrix of the impurity problem
      SK.calc_dc(dm, U_interact = U, J_hund = J, orb = 0, use_dc_formula = dc_type)

      # Save stuff into the user_data group of hdf5 archive in case of rerun:
      SK.save(['chemical_potential','dc_imp','dc_energ'])


This is all we need for the DFT+DMFT calculation.
You can see in this code snippet, that all results of this calculation
will be stored in a separate subgroup in the hdf5 file, called `dmft_output`.
Note that this script performs 15 DMFT cycles, but does not check for
convergence. Of course, it would be possible to build in convergence criteria.
A simple check for convergence can be also done if you store multiple quantities
of each iteration and analyse the convergence by hand. In general, it is advisable
to start with a lower statistics (less measurements), but then increase it at a
point close to converged results (e.g. after a few initial iterations). This helps
to keep computational costs low during the first iterations.

Using the Kanamori Hamiltonian and the parameters above (but on 16 cores),
your self energy after the **first iteration** should look like the
self energy shown below.

.. image:: images_scripts/SrVO3_Sigma_iw_it1.png
    :width: 700
    :align: center

.. _tailfit:

Tail fit parameters
-------------------

A good way to identify suitable tail fit parameters is by "human inspection".
Therefore disabled the tail fitting first::

    p["perform_tail_fit"] = False

and perform only one DMFT iteration. The resulting self energy can be tail fitted by hand::

    Sigma_iw_fit = S.Sigma_iw.copy()
    Sigma_iw_fit << tail_fit(S.Sigma_iw, fit_max_moment = 4, fit_min_n = 40, fit_max_n = 160)[0]

Plot the self energy and adjust the tail fit parameters such that you obtain a
proper fit. The :meth:`fit_tail function <pytriqs.gf.tools.tail_fit>` is part
of the :ref:`TRIQS <triqslibs:welcome>` library.

For a self energy which is going to zero for :math:`i\omega \rightarrow 0` our suggestion is
to start the tail fit (:emphasis:`fit_min_n`) at a Matsubara frequency considerable above the minimum
of the self energy and to stop (:emphasis:`fit_max_n`) before the noise fully takes over.
If it is difficult to find a reasonable fit in this region you should increase
your statistics (number of measurements). Keep in mind that :emphasis:`fit_min_n`
and :emphasis:`fit_max_n` also depend on :math:`\beta`.