Bath dynamical functions

Description

A comprehensive set of procedures to evaluate the non-interacting impurity Green's functions \(\hat{G}^{\rm And}\) and hybridizations \(\hat{F}^{\rm And}\) in the complex frequency domain given the effective_bath instance.

Quick access

Routines:

delta_bath_function(), ed_get_delta(), ed_get_g0and(), f0and_bath_function(), fdelta_bath_function(), g0and_bath_function(), invf0_bath_function(), invg0_bath_function()

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_aux: Auxiliary routines for bath creation and manipulation

• ed_bath_replica: Routines for replica bath manipulation

• ed_bath_dim: Routines for bath dimension checks

• ed_bath_user: Routines for bath symmetrization

• ed_bath_dmft: Routines to allocate, read and write the bath

External modules

• sf_constants

  • zero

• sf_iotools

  • free_unit()

  • reg()

  • file_length()

  • txtfy()

  • str()

  • to_lower()

• sf_linalg

  • eye()

  • inv()

  • diag()

  • zeye()

  • inv_her()

  • kron()

  • det()

• sf_spin

  • pauli_sigma_z

Subroutines and functions

interface  ed_bath_functions/delta_bath_function(x[, axis])

Evaluates the normal hybridization function \(\Delta(x)\).

Output:

• delta : complex rank-5 array with dimension [ nspin , nspin , norb , norb , size(x) ]

Parameters:

x (•) [complex, in] – complex array for the frequency

Options:

axis [character(len=*)] – string indicating the desired axis, 'm' for Matsubara (default), 'r' for Real-axis

Result:

delta (nspin, nspin, norb, norb, size(x)) [complex]

interface  ed_bath_functions/fdelta_bath_function(x[, axis])

Evaluates the anomalous hybridization function \(\Theta(x)\).

Output:

• fdelta : complex rank-5 array with dimension [ nspin , nspin , norb , norb , size(x) ]

Parameters:

x (•) [complex, in] – complex array for the frequency

Options:

axis [character(len=*)] – string indicating the desired axis, 'm' for Matsubara (default), 'r' for Real-axis

Result:

fdelta (nspin, nspin, norb, norb, size(x)) [complex] – = superc. Return zero"

interface  ed_bath_functions/g0and_bath_function(x[, axis])

Evaluates the normal non-interacting Green's function \(G_0(x)\).

Output:

• g0and : complex rank-5 array with dimension [ nspin , nspin , norb , norb , size(x) ]

Parameters:

x (•) [complex, in] – complex array for the frequency

Options:

axis [character(len=*)] – string indicating the desired axis, 'm' for Matsubara (default), 'r' for Real-axis

Result:

g0and (nspin, nspin, norb, norb, size(x)) [complex]

interface  ed_bath_functions/f0and_bath_function(x[, axis])

Evaluates the anomalous non-interacting Green's function \(F_0(x)\).

Output:

• f0and : complex rank-5 array with dimension [ nspin , nspin , norb , norb , size(x) ]

Parameters:

x (•) [complex, in] – complex array for the frequency

Options:

axis [character(len=*)] – string indicating the desired axis, 'm' for Matsubara (default), 'r' for Real-axis

Result:

f0and (nspin, nspin, norb, norb, size(x)) [complex] – = superc. Return zero"

interface  ed_bath_functions/invg0_bath_function(x[, axis])

Evaluates the inverse of the normal non-interacting Green's function \(G^{-1}_0(x)\).

Output:

• g0and : complex rank-5 array with dimension [ nspin , nspin , norb , norb , size(x) ]

Parameters:

x (•) [complex, in] – complex array for the frequency

Options:

axis [character(len=*)] – string indicating the desired axis, 'm' for Matsubara (default), 'r' for Real-axis

Result:

g0and (nspin, nspin, norb, norb, size(x)) [complex]

interface  ed_bath_functions/invf0_bath_function(x[, axis])

Evaluates the inverse of the anomalous non-interacting Green's function \(F^{-1}_0(x)\).

Output:

• f0and : complex rank-5 array with dimension [ nspin , nspin , norb , norb , size(x) ]

Parameters:

x (•) [complex, in] – complex array for the frequency

Options:

axis [character(len=*)] – string indicating the desired axis, 'm' for Matsubara (default), 'r' for Real-axis

Result:

f0and (nspin, nspin, norb, norb, size(x)) [complex] – = superc. Return zero"

interface  ed_bath_functions/ed_get_g0and(x, bath_, g0and[, axis, type])

This subroutine returns to the user the normal non-interacting Green's function \(G_0(x)\) and the anomalous non-interacting Green's function \(F_0(x)\) on a given set of frequencies. It does so by calling g0and_bath_function() and g0and_bath_function().

The non-interacting Green's function is an array having the following possible dimensions:

• [nspin \(\cdot\) norb, nspin\(\cdot\)norb, size(x)]

• [nspin, nspin, norb, norb, size(x)]

Parameters:

• x (•) [complex, in] – complex array of frequencies

• bath_ (•) [real] – user-accessible bath array

• g0and (various shapes) [complex] – non-interacting Green's function

Options:

• axis [character(len=*)] – string indicating the desired axis, 'm' for Matsubara (default), 'r' for Real-axis

• type [character(len=*)] – string indicating the desired function, 'n' for normal (default), 'a' for anomalous

interface  ed_bath_functions/ed_get_delta(x, bath_, delta[, axis, type])

This subroutine returns to the user the normal hybridization function \(\Delta(x)\) and the anomalous hybridization function \(\Theta(x)\) on a given set of frequencies. It does so by calling delta_bath_function() and fdelta_bath_function().

The hybridization function is an array having the following possible dimensions:

• [nspin \(\cdot\) norb, nspin\(\cdot\)norb, size(x)]

• [nspin, nspin, norb, norb, size(x)]

Parameters:

• x (•) [complex, in] – complex array of frequencies

• bath_ (•) [real] – user-accessible bath array

• delta (various shapes) [complex] – hybridization function

Options:

• axis [character(len=*)] – string indicating the desired axis, 'm' for Matsubara (default), 'r' for Real-axis

• type [character(len=*)] – string indicating the desired function, 'n' for normal (default), 'a' for anomalous