Introduction to Dynamic DMRG Methods
S. Ramasesha
Solid State and Structural Chemistry Unit
Indian Institute of Science
Bangalore 560 012
Collaborators
Zoltan G. Soos
Swapan Pati
Zhigang Shuai
Tirthankar Dutta
H.R. Krishnamurthy
Institute for Mathematical Sciences
Chennai, 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.
test the technique, we compare the rotationally
raged linear polarizability
and THG coefficient
γ
α
1 3
α= ∑
αii ;
3 i =1
1 3
γ=
(2γ
∑
15 i , j =1
puted at ω = 0.1t model exact values for a Hubbard chain of
ites at U/t=4 compared with DMRG computation with m=200
γ γ
αα
5.343
exact DMRG
exact DMRG
5.317
598.3
591.1
e dominant α (α xx) is 14.83 (exact) and 14.81 (DMRG)
in 10-24 esu and γ in 10-36 esu in all cas
d γ (γ xxxx) 2873 (exact) and 2872 α
(DMRG).
THG coefficient in Hubbard models as a function
of chain length, L and dimerization δ:
Superlinear behavior diminishes both with
increase in U/t and increase in δ.
(a)
(b)
gav. vs Chain Length and d in U-V Model
For U > 2V, (SDW regime) γ av. shows similar dependence
on L as the Hubbard model, independent of d.
U=2V (SDW/CDW crossover point) Hubbard chains have
larger γ 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
[1 - iH
U (t, t+∆ t) ≈
∆t
]
2ħ
∆t
[1 + iH 2ħ ]
 Time evolution of Ψ (t) by ∆ t is given by
[1 + iH
∆t
∆t
] Ψ (t + ∆ t) = [1 -iH
] Ψ (t)
2ħ
2ħ
 Expressing Ψ (t) in an appropriate basis (eg.Slater Determinants), r.h.s.
can be converted to a vector b, with Ψ (t + ∆ t) being expressed as an
unknown x, the above equation can be converted to a set of linear
inhomogeneous algebraic equations
Ax = b
Multistep Differencing (MSD)Techniques
MSD4:
e
i 2 Hˆ ∆ t / 
−e
− i 2 Hˆ ∆ t / 
iHˆ ∆ t
8 Hˆ 2 ∆ t 2
5
=
(− 4 +
)
+
O
(
∆
t
)
2

3 
2 2
ˆ
H ∆t
iHˆ ∆ t / 
− iHˆ ∆ t / 
= 2− e
−e
2

ˆ
4
i
H
∆t
ˆ
ˆ
− i 2 H∆ t / 
i 2 H∆ t / 
iHˆ ∆ t / 
− iHˆ ∆ t / 
e
≈e
−
[ I + 2(e
+e
)]
3
operating on ψ (0)
4iHˆ ∆ t
Ψ (t + 2∆ t ) ≈ Ψ (t − 2∆ t ) −
[Ψ (t ) + 2(Ψ (t − ∆ t ) + Ψ (t + ∆ t ))]
3
Fast - involves only one sparse matrix multiplication
for time propagation. Time dependent quantities evaluated
as <O(t)> = <ψ(t)|O|ψ(t)>.
19
td-DMRG
method:
DMRG space of �(0) (initial wave packet)
adapted to follow the time evolving wave
packet |�(t)>
Full Hilbertspace
�(0)
�(0)
�(tp
)
�(T)
DMRG-space for
�(0)
DMRG-space for
�(tp)
�(tp
)
�(T)
DMRG-space for
�(T)
Fundamental quantity in


ρ L / R (t ) = TrR / L  ∑ ω j | ψ j (t ) >< ψ j (t ) | ÷
td-DMRG: weighted
 j

average reduced density
“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 = p �τ)
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/p
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 )
c) Time-step targeting
(TST) algorithm
(Feiguin, White)
(PRB, 72, 020404, 2005)
d) Double time window
targeting (DTWT)
technique (Dutta, SR)
(PRB, 82, 035115, 2010)