\(\chi^2\) Fit

In ED_BATH_FIT we perform an optimisation of the user bath by minimizing the projection of a user supplied function with respect to the corresponding quantum impurity problem, bath dependent, one \(\min_{\vec{b}}\chi^2(\vec{b})\) with:

\[\chi^2(\vec{b}) = || F(z) - F^{And}(z;\vec{b}) ||\]

\(F(z)\) is either the Weiss field \({\cal G}_0\) or the hybridization function \(\Delta\) as evaluated by the user. \(F^{And}(z;\vec{b})\) is, respectively, the quantum impurity non-interacting Green's function \(G^{And}_0(z;\vec{b})=[z+\mu-h_0-\Delta(z;\vec{b})]^{-1}\) or the hybrization function \(\Delta(z,c)=\sum_p V_p[z-h^p]^{-1}V_p\). Finally \(\vec{b}\) is the array containing the discretized bath parameters.

The minimization with respect to \(\vec{b}\) is performed using conjugate gradient algorithm.

Description

Contains routines that fit the Impurity model bath

Quick access

Routines:

ed_chi2_fitgf()

Used modules

• ed_input_vars: User-accessible input variables

• ed_vars_global: Global variable accessible throughout the code

• ed_aux_funx: Assortment of auxiliary procedures required throughout the code

• ed_bath: Routines for bath creation and manipulation

• ed_fit_common: Common variables and types for bath fitting

• ed_fit_normal: Routines for bath fitting, NORMAL case

• ed_fit_hybrid: Routines for bath fitting, HYBRID case

• ed_fit_replica: Routines for bath fitting, REPLICA case

• ed_fit_general: Routines for bath fitting, GENERAL case

External modules

• sf_constants

• sf_optimize

  • fmin_cg()

  • fmin_cgplus()

  • fmin_cgminimize()

• sf_linalg

  • eye()

  • zeye()

  • inv()

  • inv_her()

  • operator(.x.)()

• sf_iotools

  • reg()

  • free_unit()

  • txtfy()

• sf_arrays

  • arange()

• sf_misc

  • assert_shape()

• mpi

• sf_mpi

Subroutines and functions

interface  ed_bath_fit/ed_chi2_fitgf(g, bath, f[, ispin, iorb, fmpi])

This subroutine realizes the \(\chi^2\) fit of the Weiss field or hybridization function via an impurity model non-interacting Green's function. The bath levels (levels/internal structure and hybridization strength) are supplied by the user in the bath array and are the parameters of the fit. The function(s) to fit can have different shapes:

• [nspin \(\cdot\) norb, nspin\(\cdot\)norb, lfit ]

• [nlat \(\cdot\) nspin \(\cdot\) norb, nlat \(\cdot\) nspin \(\cdot\) norb, lfit ]

• [nlat, nspin \(\cdot\) norb, nspin \(\cdot\) norb, lfit ]

• [nspin, nspin, norb, norb, lfit ]

• [nlat, nspin, nspin, norb, norb, lfit ]

where nlat is the number of impurity sites in real-space DMFT. Accordingly, the bath array or arrays have rank 2 or 3. Some global variables directly influence the way the fit is performed and can be modified in the input file. See ed_input_vars for the description of lfit, cg_method , cg_grad, cg_ftol, cg_stop , cg_niter , cg_weight , cg_scheme , cg_pow , cg_minimize_ver , cg_minimize_hh .

Parameters:

• g (various shapes) [complex] – normal Weiss field or hybridization function to fit

• bath (•) [real] – bath parameters array

• f (various shapes) [complex] – anomalous Weiss field or hybridibazion function to fit (only if ed_mode = superc )

Options:

• ispin [integer] – spin component to be fitted (default = 1 ). Only used if ed_mode = normal and bath_type = normal, hybrid

• iorb [integer] – orbital to be fitted

• fmpi [logical] – flag to automatically broadcast the fit over the MPI communicator (default = .true. )