.. _dmft_susceptibility_dbse: Spin susceptibility in Sr2RuO4 ============================== In this tutorial we will compute the static magnetic susceptibilitly :math:`\chi_{S_z S_z}(\mathbf{q})` of the correlated Hund's metal Sr2RuO4 withn dynamical mean field theory (DMFT), reproducing the results of `PRB 100, 125120 (2019) `_. We will use the reformulation of the (DMFT) lattice susceptibility in terms of the dual Bethe-Salpeter equation (DBSE) with :math:`1/N_\nu^3` convergence with respect to the number of fermionic Matsubara frequencies :math:`N_\nu`, for details see `arXiv 2306.05157 `_. The calculation is based on a Wannier model for the three bands crossing the Fermi level in Sr2RuO4. These bands have Ru-4d t2g symmetry and a Wannier interpolation with Wannier90 converges in just a few iterations, giving the band structure .. image:: figure_sro_band_structure.svg :align: center see :download:`tight_binding_model.py ` using the Wannier90 output :download:`sro_hr.dat <./calc_dft/wannier_fit/sro_hr.dat>`, :download:`sro.wout <./calc_dft/wannier_fit/sro.wout>`. The setup for producing these files using Quantum Espresso and Wannier90 is available in `the TPRF repository `_ under `./doc/user_guide/dmft_susceptibility_dbse/calc_dft`. The Wannier Hamiltonian is combined with a local Kanamori interaction with Hubbard :math:`U=2.4` eV and Hund's :math:`J=0.4` eV and the self-consistent DMFT solution is determined using TRIQS/cthyb as impurity solver. The scripts for the DMFT solution are :download:`common.py ` and :download:`calc_sc_dmft.py `. Dual Bethe-Salpeter equation ---------------------------- In order to use the dual Bethe-Salpeter equation for computing the lattice susceptiblity we need to sample not one but three different kinds of two-particle correlators of the DMFT impurity problem. 1. The three frequency two particle Green's function :math:`g^{(4)}_{abcd}(\omega, \nu, \nu')` 2. The two frequency two particle Green's function :math:`g^{(3)}_{abcd}(\omega, \nu)` 3. The one frequency two particle Green's function, a.k.a. the susceptiblity :math:`g^{(2)}_{abcd}(\omega) = X_{abcd}(\omega)` Since the hybridization function of the Sr2RuO4 impurity problem is diagonal due to symmetry, it is not possible to sample all spin-orbital components :math:`abcd` of these correlators using partition function sampling Monte Carlo. Therefore we use the hybridization expansion with worm sampling as implemented in `W2Dynamics `_ to sample these correlators, using the `TRIQS/w2dynamics_interface `_. The example scripts for the sampling are :download:`calc_g2.py `, :download:`calc_tri.py `, and :download:`calc_chi.py `. From :math:`g^{(4)}_{abcd}(\omega, \nu, \nu')` we compute the impurity reducible vertex function :math:`F_{abcd}(\omega, \nu, \nu')` and from :math:`g^{(3)}_{abcd}(\omega, \nu)` the three point vertex function :math:`L_{abcd}(\omega, \nu)` is obtained, see `arXiv 2306.05157 `_. Using the impurity susceptibility :math:`X_{abcd}(\omega)`, :math:`F`, and :math:`L` the lattice susceptibility :math:`\chi` is given by .. math:: \chi = X + L \frac{\tilde{\chi}^0}{1 - \tilde{\chi}^0 F} L where :math:`\tilde{\chi}^0` is the dual bubble propagator constructed from the non-local part of the single particle Green's function. Here is an example scirpt that performs these steps starting from the sampled propagators from W2Dynamics: .. literalinclude:: calc_dbse.py :lines: 23- and also solves the traditional Bethe-Salpeter equation using the irreducible vertex :math:`\Gamma` for comparison. Solving both the dual Bethe-Salpeter equation (DBSE) and the Bethe-Salpeter equation (BSE) for a range of the Fermionic cut-off frequencies :math:`N_\nu` (the number of frequencies :math:`\nu` and :math:`\nu'` used) shows the superior convergence property of the dual Bethe-Salpeter equation .. image:: figure_sro_chi_bandpath.svg :align: center see :download:`plot_dbse.py ` for the plot script. Since the standard Bethe-Salpeter equation (BSE) only converges as :math:`1/N_\nu` the calculations at :math:`N_\nu = 4, 8, 16` are far from the :math:`N_\nu \rightarrow \infty` limit and requires extrapolation in order to obtain a quantiative correct result. However, using the dual Bethe-Salpeter equation (DBSE) implementation we observe a drastically improved convergence rate and already at :math:`N_\nu=4` the result is within 5% of the converged solution. If you use the dual Bethe-Salpeter equation formulation in your work please cite `arXiv 2306.05157 `_.