Exciton Susceptibility

In ed_chi_exct we evaluate the impurity exciton-exciton susceptibility, defined as:

\[\chi^{X}_{ab}(\omega) = \langle {X}^\dagger_{ab}(\omega) X_{ab}(\omega) \rangle = \frac{1}{\cal Z}\sum_m e^{-\beta E_m} \langle m | X^\dagger_{ab} [\omega-H]^{-1} X_{ab} | m \rangle\]

where \(X_{ab}=S_{ab},T^x_{ab},T^y_{ab},T^z_{ab}\) are, respectively, the singlet and triplet \(x,y,z\) exciton operators:

\[\begin{split}S_{ab} & = \sum_{rs} c^\dagger_{ar} \sigma^0_{rs} c_{bs}\\ T^x_{ab} & = \sum_{rs} c^\dagger_{ar} \sigma^x_{rs} c_{bs}\\ T^y_{ab} & = \sum_{rs} c^\dagger_{ar} \sigma^y_{rs} c_{bs}\\ T^z_{ab} & = \sum_{rs} c^\dagger_{ar} \sigma^z_{rs} c_{bs}\end{split}\]

and \(\omega \in {\mathbb C}\). As for the Green's functions, the susceptibility is evaluated using the dynamical Lanczos method: a) the partial tridiagonalization of the sector Hamiltonian \(H\) with quantum numbers \(\vec{Q}=[\vec{N}_\uparrow,\vec{N}_\downarrow]\) on the Krylov basis of \(X_{ab}|m\rangle\) is obtained; b) the resulting tridiagonal matrix is further diagonalized to obtained excitations amplitudes or weights \(\langle p | X_{ab} | m \rangle\) for any state \(| p \rangle\) in the spectrum (without knowing the state itself ) and the excitations energies \(\delta E = E_p - E_m\) or poles; c) an controlled approximation to the Kallen-Lehmann sum is constructed for \(a,b=1,\dots,N_{\rm orb}\).

Description

Evaluates the impurity excitonc susceptibility.

Quick access

Routines:

build_exctchi_normal(), get_exctchi_normal()

Used modules

• ed_input_vars: User-accessible input variables

• ed_vars_global: Global variable accessible throughout the code

• ed_eigenspace: Data types for the eigenspace

• ed_bath: Routines for bath creation and manipulation

• ed_setup: Routines for solver environment initialization and finalization

• ed_sector: Routines for Fock space sectors creation and manipulation

• ed_hamiltonian_normal: Routines for Hamiltonian construction, NORMAL case

• ed_aux_funx: Assortment of auxiliary procedures required throughout the code

External modules

• sf_constants

  • one

  • xi

  • zero

  • pi

• sf_timer

• sf_iotools

  • str()

  • free_unit()

  • reg()

  • free_units()

  • txtfy()

• sf_linalg

  • inv()

  • eigh()

  • eye()

• sf_sp_linalg

  • sp_lanc_tridiag()

Subroutines and functions

subroutine  ed_chi_exct/build_exctchi_normal()

Evaluates the impurity exciton-exciton susceptibility \(\chi^{X}_{ab}=\langle T_\tau X^\dagger_{ab}(\tau) X_{ab}\rangle\) in the Matsubara \(i\omega_n\) and Real \(\omega\) frequency axis, the imaginary time \(\tau\) as well as the singlet and triplet components of the operator. As for the Green's function, the off-diagonal component of the the susceptibility is determined using an algebraic manipulation to ensure use of Hermitian operator in the dynamical Lanczos.

Use :

ed_input_vars (nspin, norb)

function  ed_chi_exct/get_exctchi_normal(zeta[, axis])

Reconstructs the system impurity electrons Green's functions using impgmatrix to retrieve weights and poles.

Parameters:

zeta (•) [complex, in] – Array of frequencies or imaginary times

Options:

axis [character(len=*)] – Axis: can be m for Matsubara, r for real, t for imaginary time

Result:

chi (3, norb, norb, size(zeta)) [complex] – Excitonic susceptibility matrix

Use :

ed_input_vars (nspin, norb)