Quantum Numbers Sectors

The direct diagonalization of the Hamiltonian for a finite system becomes quickly impossible due to the exponential increase of its size \(D^{Ns}\), where \(D\) is the dimension of the local Hilbert space (\(D=4\) for spin-\(1/2\) electrons).

A way to partially soften such severe behavior is to take into account conserved quantities. For any quantity \({\cal Q}\) such that \([H,{\cal Q}]=0\), we can separate the structure of the Hamiltonian matrix into different blocks, each corresponding to a discrete value \(q\) of the spectrum of \(Q\). Analyzing one-by-one such symmetry sector it becomes possible to construct at least part of the energy spectrum.

The ED_SECTOR represents a key part of the ED code. Here the symmetry sectors construction essentially consist in the determination of the injective map \({\cal M}:{\cal S}(\vec{Q})\rightarrow{\cal F}\) relating the states of the sectors \(|i\rangle\in{\cal S}(\vec{Q})\) to the corresponding ones \(|I\rangle\in{\cal F}\) in the Fock space.

The map is constructed iterating over the states of the Fock spaces while verified the quantum number conditions. If verified for a given state \(|I\rangle\) a new state is found and it is added to the list. The following code snippet illustrates the case for a system conserving the total magnetization \(S_z=N_\uparrow-N_\downarrow\)

dim=0
! Iterate over down spin configurations
do idw=0,2**Ns-1
   ! use bit operation to get :math:`N_\downarrow`
   ndw_= popcnt(idw)
   ! Iterate over up spin configurations
   do iup=0,2**Ns-1
      ! use bit operation to get :math:`N_\uparrow`
      nup_ = popcnt(iup)
      ! Get :math:`S_z=N_\uparrow-N_\downarrow`
      sz_  = nup_ - ndw_
      ! if the evaluated magnetization is the required one:
      ! increase counter and store the Fock state into the map
      if(sz_ == self%Sz)then
         dim=dim+1
         self%H(1)%map(dim) = iup + idw*2**Ns
      endif
   enddo
enddo

The dimension of the symmetry sector, i.e. the size of the map \({\cal M}(\vec{Q})\), is determined by a simple combinatorial algorithm (\(N\) being the total number of electronic levels):

Symmetry Sectors Dimensions

ed_mode	Quantum Numbers	Sector Dimension
normal	\([\vec{N}_\uparrow,\vec{N}_\downarrow]\)	\(\prod_{\alpha}\binom{N}{N_{\alpha\uparrow}}\binom{N}{N_{\alpha\downarrow}}\)
superc	\(S_z=N_\uparrow-N_\downarrow\)	\(\sum_i 2^{N-S_z-2i}\binom{N}{N-S_z-2i}\binom{S_z+2i}{i}\)
nonsu2	\(N_{tot}=N_\uparrow+N_\downarrow\)	\(\binom{2N}{N_{tot}}\)

Description

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.

Quick access

Routines:

build_sector(), get_dim(), get_quantumnumbers(), get_sector(), map_allocate()

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

External modules

• sf_timer

• sf_iotools

  • free_unit()

  • reg()

  • file_length()

• mpi

• sf_mpi

Subroutines and functions

interface  ed_sector/map_allocate(h, n[, nsp])

Allocate the map(s) connecting the states in the symmetry sector to thos in the Fock space.

Parameters:

• h (various shapes) [sector_map]

• n (various shapes) [integer]

Options:

nsp [integer]

interface  ed_sector/get_sector(qn, n, isector, twojz)

Returns the index of isector of the symmetry sector given the quantum numbers qn

Parameters:

• qn (various shapes) [integer]

• n [integer]

• isector [integer]

• twojz [integer]

interface  ed_sector/get_quantumnumbers(isector, n, qn)

Returns the quantum numbers qn given the index of isector of the symmetry sector

Parameters:

• isector [integer]

• n [integer]

• qn (various shapes) [integer]

subroutine  ed_sector/build_sector(isector, self[, itrace])

This procedure build the sector sector \({\cal S}(\vec{Q})\) given the index isector which univocally determines the quantum numbers \(\vec{Q}\). All the components of the data structure sector are determined here. Moreover the map connecting the states of the sector to thos in the Fock space is constructed. To avoid integer overflow the loop over Fock space states is decomposed in spin up and down integer so that \(I = I_\uparrow + 2^{N}I_\downarrow\).

Parameters:

• isector [integer, in]

• self [sector]

Options:

itrace [logical]

subroutine  ed_sector/get_dim(isector, dim)

Returns the dimension of the symmetry sector with index isector

Parameters:

• isector [integer]

• dim [integer]