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77 lines
2.8 KiB
ReStructuredText
77 lines
2.8 KiB
ReStructuredText
.. _plovasp:
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Numerical Treatment of the Sign-Problem: Basis Rotations
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=======
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When performing calculations with off-diagonal complex hybridisation or local Hamiltonian, one is
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often limited by the fermionic sign-problem. However, as the sign is no
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physical observable, one can try to improve the calculation by rotating
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to another basis.
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While the choice of basis to perform the calculation in can be chosen
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arbitrarily, two choices which have shown good results are the basis
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which diagonalizes the local Hamiltonian or the density matrix of then
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system.
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The transformation matrix can be stored in the :class:`BlockStructure` and the
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transformation is automatically performed when using :class:`SumkDFT`'s :meth:`extract_G_loc`
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and :meth:`put_Sigma` (see below).
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Finding the Transformation Matrix
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-----------------
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The :class:`TransBasis` class offers a simple mehod to calculate the transformation
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matrices to a basis where either the local Hamiltonian or the density matrix
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is diagonal::
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from triqs_dft_tools.trans_basis import TransBasis
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TB = TransBasis(SK)
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TB.calculate_diagonalisation_matrix(prop_to_be_diagonal='eal', calc_in_solver_blocks = True)
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SK.block_structure.transformation = [{'ud':TB.w}]
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Transforming Green's functions manually
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-----------------
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One can transform Green's functions manually using the :meth:`convert_gf` method::
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# Rotate a Green's function from solver-space to sumk-space
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new_gf = block_structure.convert_gf(old_gf, space_from='solver', space_to='sumk')
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Automatic transformation during the DMFT loop
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-----------------
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During a DMFT loop one is switching back and forth between Sumk-Space and Solver-Space
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in each iteration. Once the block_structure.transformation property is set, this can be
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done automatically::
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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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One must not forget to also transform the interaction Hamiltonian to the diagonal basis!
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This can be done with the :meth:`transform_U_matrix` method. However, due to different
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conventions in this method, one must pass the conjugated version of the transformation matrix::
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U_trans = transform_U_matrix(U, SK.block_structure.transformation[0]['ud'].conjugate())
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