Introduction to the Density Matrix Renormalization Group Method

S. Ramasesharamasesh@sscu.iisc.ernet.in

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).

The spin gap in the limit U/t should vanish for theHubbard model.

PRB, 54, 7598 (1996).