Spin Susceptibility

In ed_chi_spin we evaluate the impurity spin-spin susceptibility, defined as:

\[\chi^{z}_{ab}(\omega) = \langle S^z_a(\omega) S^z_b(\omega) \rangle = \frac{1}{\cal Z}\sum_n e^{-\beta E_n} \langle n | S^z_a [\omega-H]^{-1} S^z_b | n \rangle\]

where \(S^z_a\) is the z-component of the spin operator of the orbital \(a\) 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 \(S^z_a|n\rangle\) is obtained; b) the resulting tridiagonal matrix is further diagonalized to obtained excitations amplitudes or weights \(\langle m | S^z_a | n \rangle\) for any state \(| m \rangle\) in the spectrum (without knowing the state itself ) and the excitations energies \(\delta E = E_m - E_n\) or poles; c) an controlled approximation to the Kallen-Lehmann sum is constructed for \(a,b=1,\dots,N_{\rm orb}\).

Note

A more general susceptibility function for the other components of the spin operators \(S_x, S_y\) should be implemented.

Description

Evaluates the impurity spin-spin susceptibility.

Quick access

Routines:

build_chi_spin_normal()

Used modules

• sf_constants

  • one()

  • xi()

  • zero

  • pi()

• sf_timer

• sf_iotools

  • str()

  • reg()

  • txtfy()

• sf_linalg

  • inv()

  • eigh()

  • eye()

• ed_input_vars: Contains all global input variables which can be set by the user through the input file. A specific preocedure ed_read_input() should be called to read the input file using parse_input_variable() procedure from SciFortran. All variables are automatically set to a default, looked for and updated by reading into the file and, sequentially looked for and updated from command line (std.input) using the notation variable_name=variable_value(s) (case independent).

• ed_vars_global: Contains all variables, arrays and derived types instances shared throughout the code. Specifically, it contains definitions of the effective_bath, the gfmatrix and the sector data structures.

• ed_aux_funx: Hosts a number of auxiliary procedures required in different parts of the code. Specifically, it implements: creation/annihilation fermionic operators, binary decomposition of integer representation of Fock states and setup the local impurity Hamiltonian

• ed_eigenspace: A class implementing a data structure to efficiently store the low part of the Fock space spectrum, automatically spreading and retrieving the eigenstates among/from MPI threads.

• ed_bath: Contains routines for setting, accessing, manipulating and clearing the bath of the Impurity problem.

• ed_setup: Contains procedures to set up the Exact Diagonalization calculation, executing all internal consistency checks and allocation of the global memory.

• ed_sector: Contains procedures to construct the symmetry sectors corresponding to a given set of quantum numbers \(\vec{Q}\), in particular it allocated and build the sector_map connecting the states of a given sector with the corresponding Fock ones.

• ed_hamiltonian_normal: Setup and build the sector Hamiltonian, returns the correct dimension of the vectors in the Arpack/Lanczos procedure in each thread and provides an interface to Tri-Diagonalize the Hamiltonian on a Krylov basis given a starting vector.

Subroutines and functions

subroutine  ed_chi_spin/build_chi_spin_normal()

Evaluates the impurity Spin susceptibility \(\chi^z=\langle T_\tau S^z_a(\tau) S^z_b\rangle\) in the Matsubara \(i\omega_n\) and Real \(\omega\) frequency axis as well as imaginary time \(\tau\).

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.