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mps_mpo_algebra_finite.f90
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mps_mpo_algebra_finite.f90
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MODULE mps_mpo_algebra_finite
USE utility
USE definitions_mps_mpo
USE mps_mpo_utility
IMPLICIT NONE
CONTAINS
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! MULT && SVD MPS-MPO SITES/BONDS !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!! Multiply sites of mps & mpo !!!
SUBROUTINE multiply_mps_and_mpo_site(mpsA, mpsB, mpoB, chi0, eps0)
TYPE(block_mps), INTENT(INOUT) :: mpsA, mpsB
TYPE(block_mpo), INTENT(IN) :: mpoB
INTEGER, INTENT(IN) :: chi0
REAL(KIND=DP), INTENT(IN) :: eps0
!! Dims
INTEGER :: chi
!! SVD matrices & temp tensor to store contraction results
COMPLEX(KIND=DP), ALLOCATABLE :: theta(:,:), U(:,:), Udag(:,:)
COMPLEX(KIND=DP), ALLOCATABLE :: tensO(:,:,:)
!! (1) Construct_theta
theta = RESHAPE_2D(mpsA % m, '12,3')
!! (2) Compute SVD
chi = chi0
CALL compute_lapack_svd(theta, U, chi, eps0)
!! (3) Copy back svd results to MPS --> copy U to MPS & create Udag
Udag = TensTRANSPOSE(U, 'HC')
CALL allocate_mps_site(mpsA, RESHAPE_3D(U, '12,3', (/mpsA % SNdim, mpsA % Wdim/)))
!! (4) Contract Udag -- theta -- {mpoB%m, mpsB%m} ---> write to MPS site
tensO = TENSMUL(TENSMUL(TENSMUL(mpoB % m, mpsB % m, MULT='21', FUSE='(32,43)'), theta, '22'), Udag, '22')
CALL allocate_mps_site(mpsB, tensO)
END SUBROUTINE multiply_mps_and_mpo_site
!!! Compute SVD of finite MPS bond !!!
SUBROUTINE calc_svd_of_mps_bond(mpsA, mpsB, chi0, eps0)
TYPE(block_mps), INTENT(INOUT) :: mpsA, mpsB
INTEGER, INTENT(IN) :: chi0
REAL(KIND=DP), INTENT(IN) :: eps0
!! Dims
INTEGER :: chi
!! SVD matrices & temp tensor to store contraction results
COMPLEX(KIND=DP), ALLOCATABLE :: theta(:,:), U(:,:), Udag(:,:)
COMPLEX(KIND=DP), ALLOCATABLE :: tensO(:,:,:)
!! (1) Construct_theta
theta = RESHAPE_2D(mpsA % m, '12,3')
!! (2) Compute SVD
chi = chi0
CALL compute_lapack_svd(theta, U, chi, eps0)
!! (3) Copy back svd results to MPS --> copy U to MPS & create Udag
Udag = TensTRANSPOSE(U, 'HC')
CALL allocate_mps_site(mpsA, RESHAPE_3D(U, '12,3', (/ mpsA % SNdim, mpsA % Wdim /)))
!! (4) Contract Udag -- theta -- mps(site+1)
tensO = TENSMUL(TENSMUL(mpsB % m, theta, '22'), Udag, '22')
CALL allocate_mps_site(mpsB, tensO)
END SUBROUTINE calc_svd_of_mps_bond
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! SWEEP && MULT THROUGH MPS-MPO BLOCK NETWORK !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!! Perform LEFT SWEEP through mps-mpo network !!!
SUBROUTINE left_sweep_mps_mpo_network(mps_block, mpo_block, is_multiplied, orth_centre, chi0, eps0)
TYPE(block_mps), ALLOCATABLE, INTENT(INOUT) :: mps_block(:)
TYPE(block_mpo), INTENT(IN) :: mpo_block(:)
LOGICAL, INTENT(INOUT) :: is_multiplied(:)
INTEGER, INTENT(IN) :: orth_centre
INTEGER, INTENT(IN) :: chi0
REAL(KIND=DP), INTENT(IN) :: eps0
!indices & dims
INTEGER :: site, N_sites
INTEGER :: SiteA, SiteB, SiteMpoB
N_sites = SIZE(mps_block)
!If orth_centre = start of block --> exit subroutine
IF(orth_centre .EQ. 1) RETURN
!Initialize block mult at site=1
IF(.NOT. is_multiplied(1)) CALL allocate_mps_site(mps_block(1), TENSMUL(mpo_block(1) % m, mps_block(1) % m, MULT='21', FUSE='(32,43)'))
DO site=2,orth_centre
IF(.NOT. is_multiplied(site)) THEN
!Multiply mps & mpo and perform SVD on each bond
SiteA = (site-1); SiteB = site; SiteMpoB = site
CALL multiply_mps_and_mpo_site(mps_block(SiteA), mps_block(SiteB), mpo_block(SiteMpoB), chi0, eps0)
is_multiplied(site) = .TRUE.
ELSE
!If site has already been multiplied, we only need to do SVD & truncation
CALL calc_svd_of_mps_bond(mps_block(site-1), mps_block(site), chi0, eps0)
end IF
end DO
END SUBROUTINE left_sweep_mps_mpo_network
!!! Perform RIGHT SWEEP through mps-mpo network !!!
SUBROUTINE right_sweep_mps_mpo_network(mps_block, mpo_block, is_multiplied, orth_centre, chi0, eps0)
TYPE(block_mps), ALLOCATABLE, INTENT(INOUT) :: mps_block(:)
TYPE(block_mpo), INTENT(IN) :: mpo_block(:)
LOGICAL, ALLOCATABLE, INTENT(INOUT) :: is_multiplied(:)
INTEGER, INTENT(IN) :: orth_centre
INTEGER, INTENT(IN) :: chi0
REAL(KIND=DP), INTENT(IN) :: eps0
TYPE(block_mpo), ALLOCATABLE :: mpo_inverse(:)
!if orth_centre = start of block --> exit subroutine
IF(orth_centre .EQ. 1) RETURN
!Before R sweep --> invert mps_block & mpo_block
CALL invert_mps_block(mps_block)
CALL copy_mpo_block(mpo_inverse, mpo_block); CALL invert_mpo_block(mpo_inverse)
CALL invert_array(is_multiplied)
!Use L sweep routine, but with inverted (mps & mpo) network
CALL left_sweep_mps_mpo_network(mps_block, mpo_inverse, is_multiplied, orth_centre, chi0, eps0)
!After R sweep --> must revert mps_block & deallocate mpo_inverse
CALL invert_mps_block(mps_block); CALL deallocate_mpo_block(mpo_inverse)
CALL invert_array(is_multiplied)
END SUBROUTINE right_sweep_mps_mpo_network
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! SWEEP && SVD THROUGH MPS BLOCK !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!! Canonicalize (FINITE) MPS BLOCK !!!
SUBROUTINE canonicalize_mps_block(mps_block, orth_centre, chi, eps)
TYPE(block_mps), ALLOCATABLE, INTENT(INOUT) :: mps_block(:)
INTEGER, INTENT(IN) :: orth_centre
INTEGER, INTENT(IN) :: chi(2)
REAL(KIND=DP), INTENT(IN) :: eps(2)
INTEGER :: N_sites
!! Determine the length of mps
N_sites = SIZE(mps_block)
!! Left & Right sweeps
CALL left_sweep_mps_block(mps_block, orth_centre, chi(1), eps(1))
CALL right_sweep_mps_block(mps_block, N_sites - orth_centre + 1, chi(2), eps(2))
END SUBROUTINE canonicalize_mps_block
!!! Right SVD sweep through mps_block !!!
SUBROUTINE right_sweep_mps_block(mps_block, orth_centre, chi, eps)
TYPE(block_mps), ALLOCATABLE, INTENT(INOUT) :: mps_block(:)
INTEGER, INTENT(IN) :: orth_centre
INTEGER, INTENT(IN) :: chi
REAL(KIND=DP), INTENT(IN) :: eps
!if orth_centre = start of block --> exit subroutine
IF(orth_centre .EQ. 1) RETURN
!Before R sweep --> invert mps_block
CALL invert_mps_block(mps_block)
!Use L sweep routine, but with inverted mps_block
CALL left_sweep_mps_block(mps_block, orth_centre, chi, eps)
!After R sweep --> revert mps_block back
CALL invert_mps_block(mps_block)
END SUBROUTINE right_sweep_mps_block
!!! Left SVD sweep through mps_block !!!
SUBROUTINE left_sweep_mps_block(mps_block, orth_centre, chi, eps)
TYPE(block_mps), ALLOCATABLE, INTENT(INOUT) :: mps_block(:)
INTEGER, INTENT(IN) :: orth_centre
INTEGER, INTENT(IN) :: chi
REAL(KIND=DP), INTENT(IN) :: eps
!indices & dims
INTEGER :: site
!if orth_centre = start of block --> exit subroutine
IF(orth_centre .EQ. 1) RETURN
!sweep through all sites of mps_block performing SVD & truncating
DO site=2,orth_centre
CALL calc_svd_of_mps_bond(mps_block(site-1), mps_block(site), chi, eps)
end DO
END SUBROUTINE left_sweep_mps_block
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! MULT && SVD MPS-MPO BLOCKS !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!! Mult mps_block * binary_mpo_block !!!
SUBROUTINE multiply_mps_bimpo(mps_block, mpo_block, chi, eps)
TYPE(block_mps), ALLOCATABLE, INTENT(INOUT) :: mps_block(:) !MPS
TYPE(block_mpo), INTENT(IN) :: mpo_block(:,:) !binary MPO
INTEGER, INTENT(IN) :: chi(2) !svd params
REAL(KIND=DP), INTENT(IN) :: eps(2) !svd params
CALL multiply_mps_mpo(mps_block, mpo_block(2,:), chi, eps)
CALL multiply_mps_mpo(mps_block, mpo_block(1,:), chi, eps)
END SUBROUTINE multiply_mps_bimpo
!!! Multiply MPS_BLOCK * MPO_BLOCK (full multiplication with SVD) !!!
SUBROUTINE multiply_mps_mpo(mps_block, mpo_block, chi, eps, excludeSites)
TYPE(block_mps), ALLOCATABLE, INTENT(INOUT) :: mps_block(:) !MPS
TYPE(block_mpo), INTENT(IN) :: mpo_block(:) !MPO
INTEGER, INTENT(IN) :: chi(2) !svd params
REAL(KIND=DP), INTENT(IN) :: eps(2) !svd params
INTEGER, INTENT(IN), OPTIONAL :: excludeSites(:) !optional exclude MPS sites from multiplication
INTEGER :: orth_centre, N_sites, i
LOGICAL, ALLOCATABLE :: is_multiplied(:)
!! Size of mps-mpo network && pos of orth_centre (mixed canonical: CEILING(0.5*N_sites))
N_sites = SIZE(mps_block)
orth_centre = N_sites
!! Alloc & init [is_multiplied] array to track which mps-mpo sites have been multiplied
ALLOCATE(is_multiplied(N_sites)); is_multiplied(1:N_sites) = .FALSE.
!! If any sites have been multiplied during an earlier process
IF(PRESENT(excludeSites)) THEN
DO i=1,SIZE(excludeSites)
is_multiplied(excludeSites(i)) = .TRUE.
end DO
end IF
!! (1) Sweep Left--to--Right through MPS-MPO network (the default choice)
CALL left_sweep_mps_mpo_network(mps_block, mpo_block, is_multiplied, orth_centre, chi(1), eps(1))
!CALL right_sweep_mps_mpo_network(mps_block, mpo_block, is_multiplied, N_sites - orth_centre + 1, chi(1), eps(1))
!! (2) Backward sweep Left--Right:
!! canonicalize post-mult MPS --> [Oc on backward sweep] = opposite to [initial val of Oc]
CALL canonicalize_mps_block(mps_block, N_sites - orth_centre + 1, (/chi(2), chi(2)/), (/eps(2), eps(2)/))
END SUBROUTINE multiply_mps_mpo
!!! Simple multiplication: MPO * MPS blocks (no SVD) !!!
SUBROUTINE simple_mult_mps_mpo(mps_block, mpo_block)
TYPE(block_mps), ALLOCATABLE, INTENT(INOUT) :: mps_block(:)
TYPE(block_mpo), INTENT(IN) :: mpo_block(:)
!! Indices
INTEGER :: site, N_sites
N_sites = SIZE(mpo_block)
!! Multiply mpo_block * mps_block
DO site=1,N_sites
CALL allocate_mps_site(mps_block(site), TENSMUL(mpo_block(site) % m, mps_block(site) % m, MULT='21', FUSE='(32,43)'))
end DO
END SUBROUTINE simple_mult_mps_mpo
!!! Combine two MPS vectors into a single MPO matrix !!!
SUBROUTINE kronecker_mult_mps_mps(mpoCombo, mpsR, mpsL)
TYPE(block_mpo), ALLOCATABLE, INTENT(INOUT) :: mpoCombo(:)
TYPE(block_mps), INTENT(IN) :: mpsR(:), mpsL(:)
INTEGER :: site, N_sites
N_sites = SIZE(mpsR)
CALL allocate_empty_mpo_block(mpoCombo, N_sites)
DO site=1,N_sites
CALL allocate_mpo_site(mpoCombo(site), TensKRON(mpsR(site) % m, mpsL(site) % m))
end DO
END SUBROUTINE kronecker_mult_mps_mps
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! MULT && SVD MPO-MPO BLOCKS !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!! Simple multiplication: MPO * MPO blocks (no SVD) !!!
SUBROUTINE simple_mult_MPO_MPO(mpoSELF, mpoOTHER, MULT)
TYPE(block_mpo), ALLOCATABLE, INTENT(INOUT) :: mpoSELF(:)
TYPE(block_mpo), INTENT(IN) :: mpoOTHER(:)
CHARACTER(LEN=*), INTENT(IN) :: MULT
!! Indices
INTEGER :: site, N_sites
!! Get MPO size
N_sites = SIZE(mpoSELF)
!! Multiply mpoSELF * mpoOTHER
SELECT CASE(MULT)
CASE('R')
DO site=1,N_sites
CALL allocate_mpo_site(mpoSELF(site), TENSMUL(mpoOTHER(site) % m, mpoSELF(site) % m, MULT='21', FUSE='(33,44)'))
end DO
CASE('L')
DO site=1,N_sites
CALL allocate_mpo_site(mpoSELF(site), TENSMUL(mpoSELF(site) % m, mpoOTHER(site) % m, MULT='21', FUSE='(33,44)'))
end DO
CASE DEFAULT
CALL invalid_flag("simple_mult_mpo_mpo -- invalid MULT ", MULT)
end SELECT
END SUBROUTINE simple_mult_MPO_MPO
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! CONTRACT MPS--MPO--MPS, MPS--biMPO--MPS, and MPS--MPS networks !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!! CONTRACT MPS-MPS NETWORK, FIND EXPVAL !!!
SUBROUTINE contract_mps_mps(expval, mps1, mps2)
COMPLEX (KIND=DP), INTENT(OUT) :: expval
TYPE(block_mps), INTENT(IN) :: mps1(:), mps2(:)
!! Overlap variable
COMPLEX(KIND=DP), ALLOCATABLE :: mps_overlap(:,:)
!! Indices
INTEGER :: N_sites, site
N_sites = SIZE(mps1)
!! Initialize mps_overlap at site=1 of mps-mps network
mps_overlap = TENSMUL(mps1(1) % m, mps2(1) % m, MULT='11', FUSE='(22,33)')
!! Sweep through mps-mps network, zip the two MPS site-by-site & contract with mps_overlap
!! the whole mps-mps network is contracted into a single tensor
DO site=2,N_sites
mps_overlap = TENSMUL(mps_overlap, TENSMUL(mps1(site) % m, mps2(site) % m, MULT='11', FUSE='(22,33)'))
end DO
!! Check if mps_overlap indeed has singleton dim
IF((SIZE(mps_overlap,1) .NE. 1) .OR. (SIZE(mps_overlap,2) .NE. 1)) THEN
WRITE(*,*) "Error in contract_two_mps: mps_overlap has dim > 1: ", shape(mps_overlap)
STOP
end IF
!! Return output
expval = mps_overlap(1,1)
DEALLOCATE(mps_overlap)
END SUBROUTINE contract_mps_mps
!!! Contract mps-mpo-mps network, find expval !!!
SUBROUTINE contract_mps_mpo_mps(expval, mps1, mpo, bimpo, mps2, chi, eps)
COMPLEX(KIND=DP), INTENT(OUT) :: expval
TYPE(block_mps), INTENT(IN) :: mps1(:)
TYPE(block_mps), INTENT(IN) :: mps2(:)
TYPE(block_mpo), INTENT(IN), OPTIONAL :: mpo(:)
TYPE(block_mpo), INTENT(IN), OPTIONAL :: bimpo(:,:)
INTEGER, INTENT(IN) :: chi(2)
REAL(KIND=DP), INTENT(IN) :: eps(2)
!! Local copy of mps1
TYPE(block_mps), ALLOCATABLE :: mpsC1(:)
!! Create a local copy of mps1
CALL copy_mps_block(mpsC1, mps1)
!! Mult MPS-1 with MPO or biMPO
IF(PRESENT(mpo)) THEN
CALL multiply_mps_mpo(mpsC1, mpo, chi, eps)
ELSEIF(PRESENT(bimpo)) THEN
CALL multiply_mps_bimpo(mpsC1, bimpo, chi, eps)
end IF
!! Calc normalized expval
CALL contract_mps_mps(expval, mpsC1, mps2)
END SUBROUTINE contract_mps_mpo_mps
!!! Normalize MPS !!!
SUBROUTINE normalize_mps(mps_block, C_mps)
TYPE(block_mps), ALLOCATABLE, INTENT(INOUT) :: mps_block(:)
COMPLEX(KIND=DP), INTENT(OUT), OPTIONAL :: C_mps
!HC of mps_block
TYPE(block_mps), ALLOCATABLE :: mps_hc(:)
!norm const
COMPLEX(KIND=DP) :: C_norm
!dims & indices
INTEGER :: site
!Obtain norm constant C_norm
CALL copy_mps_block(mps_hc, mps_block, 'HC')
CALL contract_mps_mps(C_norm, mps_block, mps_hc)
!Normalize mps_block by C_norm
CALL mult_mps_by_value(mps_block, C_norm**(-0.5))
!Output C_norm if present
IF(PRESENT(C_mps)) C_mps = C_norm
END SUBROUTINE normalize_mps
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
END MODULE mps_mpo_algebra_finite