Introduction to the Density Matrix Renormalization Group Method S. Ramasesha Solid State and Structural Chemistry Unit Indian Institute of Science, Bangalore , 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 Science Institute 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) (e 2 /r 12 )  k * (2)  l (2) d 3 r 1 d 3 r 2 This model requires further simplification to enable routine solvability. H Full = H o + ½ Σ [ij|kl] (E ij E kl –  jk E il ) ijkl E ij =  a † i,  a j,             o =  t ij (a i  a j  + H.c.) +  i n i † i      One-band tight binding model

Zero Differential Overlap (ZDO) Approximation [ij|kl] = [ij|kl]  ij  kl [ij|kl] =  i * (1)  j (1) (e 2 /r 12 )  k * (2)  l (2) d 3 r 1 d 3 r 2

 Hückel model + on-site repulsions [ii|jj] = 0 for i  j ; [ii|jj] = U i for i=j Hubbard Model (1964)  H Hub = H o + Σ U i n i (n i - 1)/2 i    In the large U/t limit, Hubbard model gives the Heisenberg Spin-1/2 model

Pariser-Parr-Pople (PPP) Model z i are local chemical potentials.  Ohno parametrization: V(r ij ) = { [ 2 / ( U i + U j ) ] 2 + r ij 2 } -1/2  Mataga-Nishimoto parametrization: V(r ij ) = { [ 2 / ( U i + U j ) ] + r ij } -1 [ii|jj] parametrized by V( r ij ) H PPP = H Hub + Σ V(r ij ) (n i - z i ) (n j - z j ) i>j   

Model Hamiltonian PPP Hamiltonian (1953)     H PPP = Σ t ij (a iσ a jσ + H.c.) + Σ(U i /2)n i (n i -1) + Σ V(r ij ) (n i - 1) (n j - 1) † σ i i>j   12 V 13

Exact Diagonalization (ED) Methods  Fock space of Fermion model scales as 4 n and of spin space as 2 n n is the number of sites.  These models conserve total S and M S.  Hilbert space factorized into definite total S and M S 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 N electrons = 14, N sites = 14, # of singlets = 2,760,615 N electrons = 16, N sites = 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  Diagonalize a small system of say 4 sites with two on the left and two on the right ● ● ● ● {|L>} = {| >, | >, | >, | >} 1 2 2’ 1’ H = S 1  S 2 + S 2  S 2’ + S 2’  S 1’ |  G > =  C LR |L>|R> L R The summations run over the Fock space dimension L  Renormalization procedure for left block Construct density matrix  L,L’ =  R  C LR C L’R Diagonalize density matrix  > =      Construct renormalization matrix O L = [      · · ·  m ] is L x m; m < L  Repeat this for the right block

 H L = O L H L O L † ; H L is the unrenormalized L x L left-block Hamiltonian matrix, and H L 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 O R  The system is augmented by adding two new sites ● ● ● ’ 2’ 1’  The Hamiltonian of the augmented system is given by

 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

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 (10 2  10 3 ) 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 of a 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 (1 1 A + g ); Lowest dipole excited state (1 1 B - u ); Lowest triplet state (1 3 B + u ); Lowest two-photon state (2 1 A + g ); mA + g state (large transition dipole to 1 1 B - 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:   a i † = b i ; ‘i’ on sublattice A a i † = - b i ; ‘i’ on sublattice B  

E-h symmetry divides N = N e space into two subspaces: one containing both ‘covalent’ and ‘ionic’ configurations, other containing only ionic configurations. Dipole operator connects the two spaces. N e = N E int. = 0, U, 2U,··· Covalent Space E int. = 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,S 2 ] = 0 ; [H,S z ] = 0  Exploiting invariance of the total S z is trivial, but of the total S 2 is hard.  When M S tot. = 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. M S = 0 S tot. = 0,1,2, ··· S tot. = 0,2,4, ··· S tot. =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, J i, has the property: J i |1> = |4> ; J i |2> =  |2> ; J i |3> =  |3> & J i |4> = - |1>  = +1 for ‘A’ sublattice and –1 for ‘B’ sublattice      The site parity operator, P i, has the property: P i |1> = |1> ; P i |2> = |3> ; P i |3> = |2> ; P i |4> = - |4>   

 The C 2 operation does not have a site representation  Matrix representation of system J and P    J of the system is given by J = J 1  J 2  J 3  ·····  J N       P of the system is, similarly, given by P = P 1  P 2  P 3  ·····  P N      The overall electron-hole symmetry and parity matrices can be obtained as direct products of the individual site matrices.  Polymer also has end-to-end interchange symmetry C 2 

Symmetrized DMRG Procedure  At every iteration, J and P matrices of sub-blocks are renormalized to obtain J L, J R, P L and P R.  From renormalized J L, J R, P L and P R, 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|  ’  ’> =  ’  J 1 |  ’> <  ’  J R  ’   Similarly, the matrix P is obtained.  

 C 2 |  ’,  ’> = (-1)  |  ’,  ’,  >;  = (n  ’ + n  ’ )(n  + n  ) and from this, we can construct the matrix for C 2.  C 2 Operation on the DMRG basis yields, J, P and C 2 form an Abelian group Irr. representations, e A +, e A -, o A +, o A -, e B +, e B -, o B +, o B - ; ‘e’ and ‘o’ imply even and odd under parity; ‘+’ and ‘-’ imply even and odd under e-h symmetry. Ground state lies in e A +, dipole allowed optical excitation in e B -, the lowest triplet in o B +.   

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 D   rows and M columns. M is dimensionality of full DMRG space.

 Symmetrized DMRG Hamiltonian matrix, H S, is given by, H S = SHS †  Symmetry operators J L, J R, P L, and P R 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, C 2 symmetry is used only at the end of each finite iteration.  H is Hamiltonian matrix in full DMRG space

Checks on SDMRG  Optical gap (E g ) in Hubbard model known analytically. In the limit of infinite chain length, for U/t = 4.0, E g exact = t ; U/t = 6.0 E g exact = t E g,N   = 1.278, U/t =4 E g,N   = 2.895, U/t =6 DMRG PRB, 54, 7598 (1996).

The spin gap in the limit U/t   should vanish for the Hubbard model. PRB, 54, 7598 (1996).