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STABLE: generic site-reduced density matrices
Added a generic public subroutine to further reduce the cluster density matrix. ed_get_reduced_dm(rdm,Nsites,[doprint,optional]), in ED_IO module. It takes in the number of sites of the desired subsystem: - Nsites = 1 gives you the local density matrix (cfr. Walsh et al.) - Nsites = Nlat gives you the full-cluster density matrix (cdm) - 1 < Nsites < Nlat scales in between > NB] There is no choice on the "order" on which the sites are traced away. The other get_*_dm() procedures have been slightly touched to conform with the new one and improve the usability (i.e. no more need to allocate the output variables). > You can generally see how these density-matrix-related procedures are called and used looking at the HM-2DSQUARE driver. === >> TESTING << === Thorough testing has been successfully carried out on the 2d-square Hubbard model. For Nlat up to 5 (Nbath=Norb=1)... + We compare the local density matrix with the well-known analytical results in terms of the spin- and double-occupations > match to machine precision + We perform a site-by-site consistency test comparing the directly-reduced matrix (where we trace _at once_ Nlat-Nsites sites) and the iteratively-reduced one (where we call Nlat-Nsites times an ad hoc subroutine[*] that traces away _a unique site_ from a generic-size dm, feeding each time the result of the previous iteration). We obtain: 4-site REDUCED DENSITY MATRIX > match up to 0.0000000000000000 3-site REDUCED DENSITY MATRIX > match up to 6.9388939039072284E-018 2-site REDUCED DENSITY MATRIX > match up to 2.7755575615628914E-017 1-site REDUCED DENSITY MATRIX > match up to 1.6653345369377348E-016 where is evident that the discrepancy increases, iteration by iteration, due to the accumulation of the numerical error in the ad hoc recipe, but nevertheless stays quite low, proving the robustness of the algorithm. [*]subtrace() at the end of the hm-2dsquare driver; it will be deleted, for its purpose was just to carry on these tests. -------- More extensive testing, e.g. Norb going up to 3, has been performed with artificial random sectors (so to avoid the diagonalization step), building on the original test-code by @aamaricci, that has been extended to develop the subtracing algorithm. Everything is available, for record purposes, at < https://github.com/bellomia/TurboRDM-ED/commits/master >. -------- We should be ready for production... Merry X-MAS! 🎄
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,70 @@ | ||
| !+---------------------------------------------------------------------+ | ||
| !PURPOSE : further reduce the cdm by tracing out (Nlat-Nsites) sites | ||
| !+---------------------------------------------------------------------+ | ||
| subroutine ed_get_reduced_density_matrix_single(rdm,Nsites,doprint) | ||
| complex(8),dimension(4**Nimp,4**Nimp) :: cdm | ||
| complex(8),dimension(:,:),allocatable,intent(out) :: rdm | ||
| integer ,intent(in) :: Nsites | ||
| logical ,intent(in),optional :: doprint | ||
| logical :: doprint_ | ||
| integer :: i,j,io,jo,iUP,iDW,jUP,jDW | ||
| integer :: iIMPup,iIMPdw,jIMPup,jIMPdw | ||
| integer :: iREDup,iREDdw,jREDup,jREDdw | ||
| integer :: iTrUP,iTrDW,jTrUP,jTrDW,Nred | ||
| ! | ||
| if(Nsites>Nlat)then | ||
| stop "ERROR: cannot request a density matrix reduced to more sites then Nlat" | ||
| endif | ||
| ! | ||
| doprint_=.false.; if(present(doprint)) doprint_=doprint | ||
| ! | ||
| !Retrieve cdm from the global scope | ||
| if(.not.allocated(cluster_density_matrix))then | ||
| stop "ERROR: cluster_density_matrix is not allocated" | ||
| endif | ||
| cdm = cluster_density_matrix | ||
| ! | ||
| Nred=Norb*Nsites !Number of levels associated to the Nsites-reduced system | ||
| allocate(rdm(4**Nred,4**Nred)) | ||
| rdm=zero | ||
| ! | ||
| !Trace the cdm to the requested Nsites-rdm | ||
| do iUP = 1,2**Nimp | ||
| do iDW = 1,2**Nimp | ||
| i = iUP + (iDW-1)*2**Nimp | ||
| iIMPup = iup-1 | ||
| iIMPdw = idw-1 | ||
| iREDup = Ibits(iIMPup,0,Nred) | ||
| iREDdw = Ibits(iIMPdw,0,Nred) | ||
| iTrUP = Ibits(iIMPup,Nred,Nimp) | ||
| iTrDW = Ibits(iIMPdw,Nred,Nimp) | ||
| do jUP = 1,2**Nimp | ||
| do jDW = 1,2**Nimp | ||
| j = jUP + (jDW-1)*2**Nimp | ||
| jIMPup = jup-1 | ||
| jIMPdw = jdw-1 | ||
| jREDup = Ibits(jIMPup,0,Nred) | ||
| jREDdw = Ibits(jIMPdw,0,Nred) | ||
| jTrUP = Ibits(jIMPup,Nred,Nimp) | ||
| jTrDW = Ibits(jIMPdw,Nred,Nimp) | ||
| if(jTrUP/=iTrUP.or.jTrDW/=iTrDW)cycle | ||
| io = (iREDup+1) + iREDdw*2**Nred | ||
| jo = (jREDup+1) + jREDdw*2**Nred | ||
| rdm(io,jo) = rdm(io,jo) + cdm(i,j) | ||
| enddo | ||
| enddo | ||
| enddo | ||
| enddo | ||
| ! | ||
| !Print to file (if requested) | ||
| if(doprint_)then | ||
| call ed_print_dm(rdm,4**Nred) | ||
| endif | ||
| ! | ||
| end subroutine ed_get_reduced_density_matrix_single | ||
|
|
||
| ! NB: the spin-factorization of the loops is paramount, | ||
| ! since we build in such way the original cdm and the | ||
| ! ordering of the basis states has to be preserved | ||
| ! in order to trace along the proper matrix elements | ||
|
|
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| Original file line number | Diff line number | Diff line change |
|---|---|---|
| @@ -0,0 +1,80 @@ | ||
| !+---------------------------------------------------------------------+ | ||
| !PURPOSE : further reduce the cdms by tracing out (Nlat-Nsites) sites | ||
| !+---------------------------------------------------------------------+ | ||
| subroutine ed_get_reduced_density_matrix_lattice(rdm,Nsites,doprint) | ||
| complex(8),allocatable :: cdm(:,:,:) | ||
| complex(8),allocatable,intent(out) :: rdm(:,:,:) | ||
| integer ,intent(in) :: Nsites | ||
| logical ,intent(in),optional :: doprint | ||
| logical :: doprint_ | ||
| integer :: ii,Nineq | ||
| integer :: i,j,io,jo,iUP,iDW,jUP,jDW | ||
| integer :: iIMPup,iIMPdw,jIMPup,jIMPdw | ||
| integer :: iREDup,iREDdw,jREDup,jREDdw | ||
| integer :: iTrUP,iTrDW,jTrUP,jTrDW,Nred | ||
| ! | ||
| if(Nsites>Nlat)then | ||
| stop "ERROR: cannot request a density matrix reduced to more sites then Nlat" | ||
| endif | ||
| ! | ||
| doprint_=.false.; if(present(doprint)) doprint_=doprint | ||
| ! | ||
| if(.not.allocated(cluster_density_matrix_ii))then | ||
| stop "ERROR: cluster_density_matrix_ii is not allocated" | ||
| endif | ||
| ! | ||
| Nineq=size(cluster_density_matrix_ii,1) | ||
| allocate(cdm(Nineq,4**Nimp,4**Nimp)); cdm=zero | ||
| Nred=Norb*Nsites | ||
| allocate(rdm(Nineq,4**Nred,4**Nred)); rdm=zero | ||
| ! | ||
| do ii=1,Nineq | ||
| ! | ||
| cdm(ii,:,:) = cluster_density_matrix_ii(ii,:,:) | ||
| ! | ||
| !Trace the cdm(ii,:,:) to the requested Nsites-rdm(ii,:,:) | ||
| do iUP = 1,2**Nimp | ||
| do iDW = 1,2**Nimp | ||
| i = iUP + (iDW-1)*2**Nimp | ||
| iIMPup = iup-1 | ||
| iIMPdw = idw-1 | ||
| iREDup = Ibits(iIMPup,0,Nred) | ||
| iREDdw = Ibits(iIMPdw,0,Nred) | ||
| iTrUP = Ibits(iIMPup,Nred,Nimp) | ||
| iTrDW = Ibits(iIMPdw,Nred,Nimp) | ||
| do jUP = 1,2**Nimp | ||
| do jDW = 1,2**Nimp | ||
| j = jUP + (jDW-1)*2**Nimp | ||
| jIMPup = jup-1 | ||
| jIMPdw = jdw-1 | ||
| jREDup = Ibits(jIMPup,0,Nred) | ||
| jREDdw = Ibits(jIMPdw,0,Nred) | ||
| jTrUP = Ibits(jIMPup,Nred,Nimp) | ||
| jTrDW = Ibits(jIMPdw,Nred,Nimp) | ||
| if(jTrUP/=iTrUP.or.jTrDW/=iTrDW)cycle | ||
| io = (iREDup+1) + iREDdw*2**Nred | ||
| jo = (jREDup+1) + jREDdw*2**Nred | ||
| rdm(ii,io,jo) = rdm(ii,io,jo) + cdm(ii,i,j) | ||
| enddo | ||
| enddo | ||
| enddo | ||
| enddo | ||
| ! | ||
| !Print to file (if requested) | ||
| if(doprint_)then | ||
| call ed_print_dm(rdm(ii,:,:),4**Nred,ineq=ii) | ||
| endif | ||
| ! | ||
| enddo | ||
| ! | ||
| deallocate(cdm) | ||
| ! | ||
| end subroutine ed_get_reduced_density_matrix_lattice | ||
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| ! NB: the spin-factorization of the loops is paramount, | ||
| ! since we build in such way the original cdm and the | ||
| ! ordering of the basis states has to be preserved | ||
| ! in order to trace along the proper matrix elements | ||
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