Introduction to Dynamic DMRG Methods

S. Ramasesha

Solid State and Structural Chemistry UnitIndian Institute of Science

Bangalore 560 012

Collaborators

Zoltan G. SoosSwapan Pati

Zhigang ShuaiTirthankar Dutta

H.R. Krishnamurthy

Institute for Mathematical SciencesChennai, March 19-21 2012.

Dynamic response to external perturbations

Response can be viewed as - a function of frequency or - a function of time. The two are related but, more accurate to compute them separately

Unperturbed Hamiltonian is an Interacting Hamiltonian

In Physics

– Hubbard Hamiltonian, Heisenberg Spin Hamiltonians and their many variants.

In Chemistry

– Long range interacting models like Pariser-Parr-Pople (PPP) Model or restricted Configuration Interaction (CI) matrices like single CI, singles and doubles CI etc.

To test the technique, we compare the rotationally averaged linear polarizability and THG coefficient

3 3

1 , 1

1 1; (2 )

3 15ii iijj ijjii i j

Computed at = 0.1t model exact values for a Hubbard chain of 12 sites at U/t=4 compared with DMRG computation with m=200

5.343 5.317 598.3 591.1exact DMRG exact DMRG

in 10-24 esu and in 10-36 esu in all casesThe dominant xx) is 14.83 (exact) and 14.81 (DMRG)and xxxx) 2873 (exact) and 2872 (DMRG).

THG coefficient in Hubbard models as a functionof chain length, L and dimerization :Superlinear behavior diminishes both with increase in U/t and increase in .

(a) (b)

gav. vs Chain Length and d in U-V Model

For U > 2V, (SDW regime) av. shows similar dependence on L as the Hubbard model, independent of d.

U=2V (SDW/CDW crossover point) Hubbard chains havelarger av. than the U-V chains

PRB, 59, 14827 (1999).

Time evolution operator: U(0,t) = exp[-iHt/ħ]

Discretized unitary form of time evolution is

U (t, t+t)

[1 - iH ]

[1 + iH ]

t2ħ

t2ħ

t2ħ

iH

Time evolution of (t) by t is given by

[1 + ] (t + t) = [1 - ] (t)t2ħ iH

Expressing (t) in an appropriate basis (eg.Slater Determinants), r.h.s. can be converted to a vector b, with (t + t) being expressed as an unknown x, the above equation can be converted to a set of linear inhomogeneous algebraic equations

Ax = b

19

Multistep Differencing (MSD)Techniques

MSD4:

)()ˆ

3

84(

ˆ5

2

22/ˆ2/ˆ2 tO

tHtHiee tHitHi

/ˆ/ˆ

2

22

2ˆ

tHitHi eetH

)0(

)](2[3

ˆ4 /ˆ/ˆ/ˆ2/ˆ2

onoperating

eeItHi

ee tHitHitHitHi

))]()((2)([3

ˆ4)2()2( ttttt

tHitttt

Fast - involves only one sparse matrix multiplication for time propagation. Time dependent quantities evaluatedas <O(t)> = <(t)|O|(t)>.

DMRG space of (0) (initial wave packet) 𝝍adapted to follow the time evolving wave packet | (t)>𝝍td-DMRG method:

Fundamental quantity in td-DMRG: Fundamental quantity in td-DMRG: weighted average reduced density weighted average reduced density matrixmatrix

/ /( ) | ( ) ( ) |L R R L j j jj

t Tr t t

𝝍(0)𝝍(tp) 𝝍(T)

Full Hilbert-spaceDMRG-space for 𝝍(0)DMRG-space for 𝝍(tp)DMRG-space for 𝝍(T)

𝝍(T)𝝍(tp)

𝝍(0)

“Sliding window” pace-Keeping (LXW) td-DMRG algorithm

Instead of retaining ALL time-dependent wave packets, retain ONLY ‘p’ of them (sliding time window) (each “sliding time window” has length t = p𝜟 𝜟τ)

Computational time reduces compared to parent LXW scheme

T. Dutta and SR ,Computing Letters, 3, 457 (2007).

Time Step Targeting (TST) td-DMRG algorithm (Phys. Rev. B, 72, 020404, 2005)

Combination of infinite and finite-system DMRG algorithms; accuracy < LXW; computational time ≈ parent LXW scheme

One or several finite-system ½-sweeps are required to update Hilbert space for time step t ; evolution time step = τ = t/p𝜟 𝜟 𝜟

Double Time Window Targeting (DTWT) td-DMRG algorithm (our development; Phys. Rev. B, 82, 035115, 2010 )

A hybrid of LXW and TST schemes, but at least twice as fast and more accurate than either

A completely generalized td-DMRG algorithm for any interacting one-dimensional system

a) Pace-Keeping or LXW algorithm(Liu, Xiang, Wang)

(PRL, 91, 049701, 2003)

b) “Sliding window” LXW algorithm (Dutta,

SR) (Comput. Lett., 3, 457,

2007 )

a) Time-step targeting (TST) algorithm (Feiguin, White)

(PRB, 72, 020404, 2005)

d) Double time window targeting (DTWT) technique (Dutta, SR) (PRB, 82, 035115, 2010)