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dft_tools/doc/tour/ctqmc.rst
tayral edd1ff4529 Restructuring documentation.
A first general restructuration of the doc according to the pattern [tour|tutorial|reference].
In the reference part, objects are documented per topic.
In each topic, [definition|c++|python|hdf5] (not yet implemented)
2014-10-18 12:21:08 +01:00

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Solving a quantum impurity model with CTQMC
-------------------------------------------
.. note::
Requires TRIQS and the :doc:`application cthyb_matrix <../../applications>`
Free electrons are nice, but the `I` in TRIQS means `interacting`.
So let us solve a simple one-band Anderson impurity model
.. math::
\mathcal{H}_\mathrm{loc} = U n_\uparrow n_\downarrow,
where the non-interacting Green's function is:
.. math::
G^{-1}_{0,\sigma} (i \omega_n) = i \omega_n - \epsilon_f - V^2 \Gamma_\sigma(i \omega_n).
In this example, an impurity with the non-interacting level position at energy :math:`\epsilon_f` and on-site Coulomb repulsion :math:`U` is embedded into an electronic bath.
The
electronic bath has a flat density of states over the interval
:math:`[-1,1]` and hybridizes with the impurity with the amplitude :math:`V`.
We solve this model using the hybridization expansion Continuous Time Quantum Monte Carlo method (CT-Hyb)
proposed by `P. Werner et al. <http://link.aps.org/doi/10.1103/PhysRevLett.97.076405>`_
To this end we first initialize the ``Solver`` class of the TRIQS CT-Hyb implementaion
``pytriqs.applications.impurity_solvers.cthyb_matrix``.
Then, after having constructed the non-interacting Green's function :math:`G^{-1}_{0,\sigma}`,
we launch the impurity solver calculations by calling the ``Solve`` method.
Finally, the resulting interacting Green's function as well as average impurity occupancy
is stored in the :ref:`HDF5 format<hdf5_base>`.
.. runblock:: python
from pytriqs.gf.local import *
from pytriqs.operators import *
from pytriqs.applications.impurity_solvers.cthyb_matrix import Solver
# Parameters
D, V, U = 1.0, 0.2, 4.0
e_f, beta = -U/2.0, 50
# Construct the impurity solver with the inverse temperature
# and the structure of the Green's functions
S = Solver(beta = beta, gf_struct = [ ('up',[1]), ('down',[1]) ])
# Initialize the non-interacting Green's function S.G0
for spin, g0 in S.G0 :
g0 <<= inverse( iOmega_n - e_f - V**2 * Wilson(D) )
# Run the solver. The result will be in S.G
S.solve(H_local = U * N('up',1) * N('down',1), # Local Hamiltonian
quantum_numbers = { # Quantum Numbers
'Nup': N('up',1), # Operators commuting with H_Local
'Ndown': N('down',1) },
n_cycles = 500000, # Number of QMC cycles
length_cycle = 200, # Length of one cycle
n_warmup_cycles = 10000, # Warmup cycles
n_legendre = 50, # Number of Legendre coefficients
random_name = 'mt19937', # Name of the random number generator
use_segment_picture = True, # Use the segment picture
measured_operators = { # Operators to be averaged
'Nimp': N('up',1)+N('down',1) }
)
# Save the results in an hdf5 file (only on the master node)
from pytriqs.archive import HDFArchive
import pytriqs.utility.mpi as mpi
if mpi.is_master_node():
Results = HDFArchive("solution.h5",'w')
Results["G"] = S.G
Results["Gl"] = S.G_legendre
Results["Nimp"] = S.measured_operators_results['Nimp']
The result can be then read from the ``HDF5`` file and plotted using the ``oplot`` function:
.. literalinclude:: aim_plot.py
.. image:: aim_plot.png
:width: 700
:align: center
We go through this example in more details in the documentation of the cthyb_matrix application.