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82 lines
4.9 KiB
ReStructuredText
82 lines
4.9 KiB
ReStructuredText
.. _basisrotation:
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Automatic basis rotations in DFT+DMFT
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=====================================
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When performing calculations with off-diagonal terms in the hybridisation function or in the local Hamiltonian, one is
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often limited by the fermionic sign-problem slowing down the QMC calculations significantly. This can occur for instance if the crystal shows locally rotated or distorted cages, or when spin-orbit coupling is included. The examples for this are included in the :ref:`tutorials` of this documentation.
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However, as the fermonic sign in the QMC calculation is no
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physical observable, one can try to improve the calculation by rotating
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to another basis. While this basis can in principle be chosen arbitrarily,
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two choices which have shown good results; name the basis sets that diagonalize the local Hamiltonian or the local density matrix of the
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system.
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As outlined in section :ref:`blockstructure`, the :class:`BlockStructure` includes all necessary functionalities. While it is possible to manually transform each Green's functions and self energies between the *sumk* and the *solver* basis, this leads to cumbersum code and is discouraged. Instead, in order to facilitate the block-structure manipulations for an actual DFT+DMFT calculation, some of the necessary steps are automatically included automatically. As soon as the
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transformation matrix is stored in the :class:`BlockStructure`, the
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transformation is automatically performed when using :class:`SumkDFT`'s :meth:`extract_G_loc`,
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:meth:`put_Sigma`, and :meth:`calc_dc` (see below).
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Setting up the initial solver structure from DFT
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------------------------------------------------
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Before the actual calculation one has to specify the *solver* basis structure, in which the impurity problem will be tackled. The different available approximation were introduced in section :ref:`blockstructure`. An important feature of DFTTools is that there is an automatic way to determine the entries of the Green's function matrix that are zero by symmetry, when initialising the class::
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from triqs_dft_tools.sumk_dft import *
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SK = SumkDFT(hdf_file,use_dft_blocks='True')
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The flag *use_dft_blocks=True* analysis the local density matrix, determines the zero entries, and sets up a minimal *solver* structure. Alternatively, this step can be achieved by (see the reference manual for options)::
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SK.analyse_block_structure()
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Finding the transformation matrix
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---------------------------------
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The SumkDFT class offers a method that can determine transformation matrices to certain new basis. It is called as follows::
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SK.calculate_diagonalization_matrix(prop_to_be_diagonal='eal')
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Possible option for *prop_to_be_diagonal* are *eal* (diagonalises the local hamiltonian) or *dm* (diagonalises the local density matrix). This routine stores the transformation matrix in the :class:`SK.block_structure` class, such that it can be used to rotate the basis.
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Automatic transformation during the DMFT loop
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---------------------------------------------
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During a DMFT loop one is often switching back and forth between the unrotated basis (Sumk-Space) and the rotated basis that is used by the QMC Solver.
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Once the SK.block_structure.transformation property is set as shown above, this is
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done automatically, meaning that :class:`SumkDFT`'s :meth:`extract_G_loc`, :meth:`put_Sigma`, and :meth:`calc_dc` are doing the transformations by default::
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for it in range(iteration_offset, iteration_offset + n_iterations):
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# every GF is in solver space here
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S.G0_iw << inverse(S.Sigma_iw + inverse(S.G_iw))
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# solve the impurity in solver space -> hopefully better sign
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S.solve(h_int = H, **p)
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# calc_dc(..., transform = True) by default
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SK.calc_dc(S.G_iw.density(), U_interact=U, J_hund=J, orb=0, use_dc_formula=DC_type)
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# put_Sigma(..., transform_to_sumk_blocks = True) by default
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SK.put_Sigma([S.Sigma_iw])
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SK.calc_mu()
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# extract_G_loc(..., transform_to_solver_blocks = True) by default
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S.G_iw << SK.extract_G_loc()[0]
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.. warning::
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Before doing the DMFT self-consistency loop, one must not forget to also transform the
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interaction Hamiltonian to the diagonal basis!
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This can be also be done with a method of the :class:`BlockStructure` class,
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namely the :meth:`convert_operator` method. Having set up a Hamiltonian in the *sumk* structure, it can easily
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be transformed to the *solver* structure (including rotations of basis, picking of orbitals,
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making matrices diagonal, etc) by::
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H_solver = SK.block_structure.convert_operator(H_sumk)
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We refer to the tutorials on how to set up the Hamiltonian H_sumk in selected cases.
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Note that this transformation might generally lead to complex values in the
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interaction Hamiltonian. Unless you know better and can make everything real,
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you should take care of using the correct version of the TRIQS CTQMC solver. |