Introduction to the Density Matrix Renormalization Group Method
Solid State and Structural Chemistry UnitIndian Institute of Science, Bangalore 560 012, India
Collaborators: Anusooya Pati Swapan Pati
C. Raghu Manoranjan Kumar
Sukrit Mukhopadhyay Simil Thomas
Tirthankar Dutta Shaon Sahoo
H.R. Krishnamurthy Diptiman Sen
Funding: DST, India; CSIR, India; BRNS, India
K.S. Krishnan Discussion Meeting on Frontiers in Quantum ScienceInstitute for Mathematical Sciences, Chennai, March 19-21, 2012
Interacting One-Band Models
Explicit electron – electron interactions essential for realistic modeling
[ij|kl] = i*(1) j(1) (e2/r12) k
*(2) l(2) d3r1d3r2
This model requires further simplification to enable routine solvability.
HFull = Ho + ½ Σ [ij|kl] (EijEkl – jkEil)ijkl
Eij = a†i,aj,
o = tij (ai aj + H.c.) + i ni
†
<ij> i
One-band tight binding model
Zero Differential Overlap (ZDO) Approximation
[ij|kl] = [ij|kl]ij kl
[ij|kl] = i*(1) j(1) (e2/r12) k
*(2) l(2) d3r1d3r2
Hückel model + on-site repulsions
[ii|jj] = 0 for i j ;
[ii|jj] = Ui for i=j
Hubbard Model (1964)
HHub = Ho + Σ Ui ni (ni - 1)/2
i
In the large U/t limit, Hubbard model gives the Heisenberg Spin-1/2 model
Pariser-Parr-Pople (PPP) Model
zi are local chemical potentials.
Ohno parametrization:
V(rij) = { [ 2 / ( Ui + Uj ) ]2 + rij2 }-1/2
Mataga-Nishimoto parametrization:
V(rij) = { [ 2 / ( Ui + Uj ) ] + rij }-1
[ii|jj] parametrized by V( rij )
HPPP = HHub + Σ V(rij) (ni - zi) (nj - zj)i>j
Model Hamiltonian
PPP Hamiltonian (1953)
HPPP = Σ tij (aiσ
ajσ + H.c.) + Σ(Ui /2)ni(ni-1) + Σ V(rij) (ni - 1) (nj - 1)
†
<ij>σ i
i>j
12
V13
Exact Diagonalization (ED) Methods
Fock space of Fermion model scales as 4n and of spin space as 2n n is the number of sites.
These models conserve total S and MS.
Hilbert space factorized into definite total S and MS spaces using Rumer-Pauling VB basis.
Rumer-Pauling VB basis is nonorthogonal, but complete, and linearly independent.
Nonorthogonality of basis leads to nonsymmetric Hamiltonian matrix
Recent development allows exploiting full spatial symmetry of any point group.
Necessity and Drawbacks of ED Methods
ED methods are size consistent. Good for energy gap extrapolations to thermodynamic or polymer limit.
Hilbert space dimension explodes with increase in no. of orbitals
Nelectrons = 14, Nsites = 14, # of singlets = 2,760,615
Nelectrons = 16, Nsites = 16, # of singlets = 34,763,300
hence for polymers with large monomers (eg. PPV), ED methods limited to small oligomers. ED methods rely on extrapolations. In systems with large correlation lengths, ED methods may not be reliable.
ED methods provide excellent check on approximate methods.
Implementation of DMRG Method
Diag
on
alize a small system
of say 4 sites w
ith tw
o o
n th
e left an
d tw
o o
n th
e righ
t
● ● ●
● {|L>
} = {| >
, | >, | >
, | >
} 1 2 2’ 1’
H =
S1
S2 +
S2
S2
’ + S
2’
S1
’ |G>
= C
LR|L
>|R
> L
R
Th
e sum
matio
ns ru
n o
ver the F
ock sp
ace dim
ensio
n L
Ren
orm
alization
pro
cedu
re for left b
lock
Co
nstru
ct den
sity matrix
L,L
’ = RC
LR C
L’R
Diag
on
alize den
sity matrix
> =
Co
nstru
ct reno
rmalizatio
n m
atrix
OL =
[
· · · m] is L
x m; m
< L
Rep
eat this fo
r the rig
ht b
lock
HL = OL HLOL† ; HL is the unrenormalized L x L left-block
Hamiltonian matrix, and HL is renormalized m x m left-block Hamiltonian matrix.
Similarly, necessary matrices of site operator of left block are renormalized.
The process is repeated for right-block operators with OR
The system is augmented by adding two new sites
● ● ● ● ● ●1 2 3 3’ 2’ 1’
The Hamiltonian of the augmented system is given by
3 3 2 3 3 2tot L RH H H s s s s s s
Hamiltonian matrix elements in fixed DMEV Fock space basis
Obtain desired eigenstate of augmented system
New left and right density matrices of the new eigenstate
New renormalized left and right block matrices
Add two new sites and continue iteration
3 3 3 3
3 3 3 3
3 3
3 3
3 3 3 3
3 3 3 3 3 3
2 3 3 3
2 3 3 3
| | | |
| |
| | | |
| | | |
| | | |
tot L
R
H H
H
s s
s s
s s
Finite system DMRG
After reaching final system size, sweep through the system like a zipper
While sweeping in any direction (left/right), increase corresponding block length by 1-site, and reduce the other block length by 1-site
If the goal is a system of 2N sites, density matrices employed enroute were those of smaller systems.
The DMRG space constructed would not be the best
This can be remedied by resorting to finite DMRG technique
The DMRG Technique - Recap
DMRG method involves iteratively building a large system starting from a small system.
The eigenstate of superblock consisting of system and surroundings is used to build density matrix of system.
Dominant eigenstates (102 103) of the density matrix are used to span the Fock space of the system.
The superblock size is increased by adding new sites.
Very accurate for one and quasi-one dimensional systems such as Hubbard, Heisenberg spin chains and polymers
S.R. White (1992)
Exact entanglement entropy of Hubbard (U/t=4) and PPP eigenstates ofa chain of 16 sites
DMRG technique is accurate for long-range interacting models with diagonal density-density interactions
Important states in conjugated polymers:
Ground state (11A+g);
Lowest dipole excited state (11B-u);
Lowest triplet state (13B+u);
Lowest two-photon state (21A+g);
mA+g state (large transition dipole to 11B-
u);
nB-u state (large transition dipole to mA+
g)
In unsymmetrized methods, too many intruder states
between desired eigenstates.
In large correlated systems, only a few low-lying states can be targeted; important states may be missed altogether.
Symmetrized DMRG Method
Why do we need to exploit symmetries?
Symmetries in the PPP and Hubbard Models
When all sites are equivalent, for a bipartite system,electron-hole or charge conjugation or alternancy symmetry exists, at half-filling.
At half-filling the Hamiltonian is invariant under the transformation
Electron-hole symmetry:
ai † = bi ; ‘i’ on sublattice A
ai † = - bi ; ‘i’ on sublattice B
E-h symmetry divides N = Ne space into two subspaces: one containing both ‘covalent’ and ‘ionic’ configurations,other containing only ionic configurations. Dipole operator connects the two spaces.
Ne = NEint. = 0, U, 2U,···
Eint. = 0, U, 2U,···
Covalent Space
Eint. = U, 2U,···
Even e-h space
Odd e-h space
Dipole operator
Includes covalent states
Excludes covalent states
Full Space
Ionic Space
Spin symmetries
Hamiltonian conserves total spin and z – component of total spin.
[H,S2] = 0 ; [H,Sz] = 0
Exploiting invariance of the total Sz is trivial, but of
the total S2 is hard.
When MStot. = 0, H is invariant when all the spins are
rotated about the y-axis by This operation flips all the spins of a state and is called spin inversion.
Spin inversion divides the total spin space into spaces of even total spin and odd total spin.
MS = 0
Stot. = 0,1,2, ···
Stot. = 0,2,4, ···
Stot. =1,3,5, ···
Even space
Odd space
Full Space
Matrix Representation of Site e-h and Site Parity Operators in DMRG
Fock space of single site:
|1> = |0>; |2> = |>; |3> = |>; & |4> = |>
The site e-h operator, Ji, has the property:
Ji |1> = |4> ; Ji |2> = |2> ; Ji |3> = |3> & Ji |4> = - |1>
= +1 for ‘A’ sublattice and –1 for ‘B’ sublattice
The site parity operator, Pi, has the property:
Pi |1> = |1> ; Pi |2> = |3> ; Pi |3> = |2> ; Pi |4> = - |4>
The C2 operation does not have a site representation
Matrix representation of system J and P
J of the system is given by
J = J1 J2 J3 ····· JN
P of the system is, similarly, given by
P = P1 P2 P3 ····· PN
The overall electron-hole symmetry and paritymatrices can be obtained as direct productsof the individual site matrices.
Polymer also has end-to-end interchange symmetry C2
Symmetrized DMRG Procedure
At every iteration, J and P matrices of sub-blocks are renormalized to obtain JL, JR, PL and PR.
From renormalized JL, JR, PL and PR, the super block matrices, J and P are constructed.
Given DMRG basis states ’, ’>
( |’> are eigenvectors of density matrices, L & R
|’> are Fock states of the two new sites)
super-block matrix J is given by
J ’ ’ ’ ’ = ’, ’|J| ’ ’>
= < JL J1|> ’J1|’> < ’JR’
Similarly, the matrix P is obtained.
C2| ’,’> = (-1) | ’,’, >;
= (n’ + n’)(n+ n)
and from this, we can construct the matrix for C2.
C2 Operation on the DMRG basis yields,
J, P and C2 form an Abelian group
Irr. representations,eA+, eA-, oA+, oA-, eB+, eB-, oB+, oB-;
‘e’ and ‘o’ imply even and odd under parity;
‘+’ and ‘-’ imply even and odd under e-h symmetry.
Ground state lies in eA+,
dipole allowed optical excitation in eB-,
the lowest triplet in oB+.
Projection operator for an irreducible representation,, P is
P = R) R
R
1/h
The dimensionality of the space is given by,
D= 1/h R) red.R)
R
Eliminate linearly dependent rows in the matrix of P
Projection matrix S with Drows and M columns.
M is dimensionality of full DMRG space.
Symmetrized DMRG Hamiltonian matrix, HS , is given by,
HS = SHS†
Symmetry operators JL, JR, PL, and PR for the augmented sub-blocks can be constructed and renormalized just as the other operators.
To compute properties, one could unsymmetrize the eigenstates and proceed as usual.
To implement finite DMRG scheme, C2 symmetry is used only at the end of each finite iteration.
H is Hamiltonian matrix in full DMRG space
Checks on SDMRG
Optical gap (Eg) in Hubbard model known analytically. In the limit of infinite chain length, for
U/t = 4.0, Egexact
= 1.2867 t ; U/t = 6.0 Egexact = 2. 8926 t
Eg,N = 1.278, U/t =4
Eg,N = 2.895, U/t =6
DMRG
DMRG
PRB, 54, 7598 (1996).