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