Exact Diagonalization

How Exact Diagonalization/Arpack/Lanczos is performed

Quick access

Routines:

diagonalize_impurity_normal()

Used modules

  • sf_constants

  • sf_linalg

  • sf_timer

  • sf_iotools

  • sf_stat

  • sf_sp_linalg

  • 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_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_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_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_diag_normal/diagonalize_impurity_normal()

This procedure performs the diagonalization of the Hamiltonian in each symmetry sector for ed_mode = normal , i.e. for quantum number \(\vec{Q}=[\vec{N}_\uparrow,\vec{N}_\downarrow]\).

The diagonalization proceeds in three steps.

  1. Setup the diagonalization of selected sectors only if the file sectorfile exists.

  2. Perform a cycle over all the symmetry sectors \({\cal S}(\vec{Q})\)

  • Decide if the sector should be diagonalized, i.e. if this is a twin sector and ed_twin is True.

  • Setup the Arpack/Lanczos parameters neigen , nblock , nitermax

  • Construct the sector Hamiltonian and/or just associate the correct matrix-vector product in build_hv_sector_normal

  • call sp_eigh (Arpack) or sp_lanc_eigh (Lanczos) or eigh (Lapack) according to the dimension of the sector and the value of lanc_method

  • Retrieve the eigen-states and save them to state_list

  1. Analyze the state_list , find the overall groundstate \(|\psi_0\rangle\) , trim the list using the cutoff \(\epsilon\) to that \(e^{-\beta E_{max}} < \epsilon\) , create an histogram of the states requested to each sector and use it to increase or decrease those number according to the contribution of each sector to the spectrum.