Transcript Real Time Dynamics using DMRG Adrian Feiguin University of

On adaptive time-dependent DMRG based
on
Runge-Kutta methods*
Outline:
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Adrian Feiguin
University of California, Irvine
Review: DMRG
Targeting and DMRG
Time evolution using Suzuki-Trotter
An efficient targeting scheme for 2D / long-range interactions
Examples: Real-time Green's functions, Thermodynamic DMRG
Collaborator:
Steve R. White, UC Irvine
* A.E. Feiguin, S.R. White, Submitted to PRB (2005)
Density Matrix Renormalization Group
S.R. White, Phys. Rev. Lett. 69, 2863(1992), Phys. Rev. B 48, 10345 (1993)
Can we rotate our basis to one where the weights are more concentrated, to
minimize the error?
Cut here
|gs =∑ ai|xi , ∑ |a |2 =
i
1
Cut here
=> Error = 1-∑' |ai|2
The density matrix projection
superblock (universe)
system
environment
|i
|j
|y = ∑ijyij|i|j
We need to find the state |y' = ∑majaaj|ua |j
that minimizes the distance
S=||y' -|y|2
Solution: The optimal states are the eigenvectors of the
reduced density matrix with the largest eigenvalues wa
rii' = ∑jy*ijyi'j
; Tr r = 1
The Algorithm
How do we build the reduced basis of states?
We grow our basis systematically, adding sites to our
system at each step, and using the density matrix
projection to truncate
We grow the system by adding sites and applying the density
We
sweep
from
right
to left
left
to
matrix
projection
to truncate
the basis
until
the
We start
from a small
superblock
with
4 reaching
sites/blocks,
right m , small enough to be easily
desired
sizea dimension
each with
i
diagonalized
1
12
32
1113
121
23
22223
332
33
324
444423
23 43
… ans so on, until we converge…
4 4
Targeting states
If we target the ground state only, we cannot expect to have
a good representation of excited states (dynamics).
If the error is strictly controlled by the DMRG truncation error, we say that the algorithm
is “quasiexact”.
Non quasiexact algorithms seem to be the source of almost all DMRG “mistakes”. For
instance, the infinite system algorithm applied to finite systems is not quasiexact.
Time evolution: Suzuki-Trotter approach*
...
H=
H1
HA=
HB=
H1
+
H2
+
H3
H2
+
+
H3
+
H4
+
H5
H4
+
+
H5
+
H6
...
So the time-evolution operator is a product of individual link terms.
*G. Vidal, PRL (2004)
H6
Time dependent DMRG
S.R.White and A.E. Feiguin, PRL (2004), Daley et al, J. Stat. Mech.: Theor. Exp.
(2004)
WeWe
turnstart
off the
and start
applying
the evolution
operator
withdiagonalization
the finite system
algorithm
to obtain
the ground
state
1
12
e-iτHij
32
13
12
23
e-iτHij
23
23
e-iτHij
32
423
23 43
e-iτHij
4 4
e-iτHij
One sweep evolves one time step
Each link term only involves two-sites interactions => small matrix, easy to calculate!
Real-time dynamics using Runge-Kutta
We need to solve:
Time evolution and DMRG
Some history:
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Cazalilla and Marston, PRL 88, 256403 (2002). Use the infinite system
method to find the ground state, and evolved in time using this fixed
basis without sweeps. This is not quasiexact. However, they found that
works well for transport in chains for short to moderate time intervals.
t=0
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t= τ
t=2τ
t=3τ
t=4τ
Luo, Xiang and Wang, PRL 91, 049901 (2003) showed how to target
correctly for real-time dynamics. They target
ψ(t=0), ψ(t=τ) , ψ(t=2τ) , ψ(t=3τ)…
t=0
t= τ
t=2τ
t=3τ
t=4τ
This is quasiexact as τ→0 if you add sweeping.
The problem with this idea is that you keep track of all the history of the
time-evolution, requiring large number of states m. It becomes highly
inefficient.
Time-step targeting method
Feiguin and White, submitted to PRB, Rapid Comm.
To fix these problems, White and I have developed a new approach:
We target one time step accurately, then we move to the next step.
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The targeting principle is that of Luo et al. , but instead of keeping track of the whole
history, we keep track of intermediate points between t and t+τ
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t=0
t=τ
t=2τ
t=3τ
t=4τ
The time-evolution can be implemented in various ways:
1) Calculate Lanczos (tri-diagonal) matrix, and exponentiate. (time consuming)
2) Runge-Kutta.
Time-step targeting method (continued)
S=1 Heisenberg chain (L=32; t=8)
time targeting+RK
1st order S-T
4th order S-T
Fixed error, variable number of states
Time dependent correlation functions
(Example: S=1 Heisenberg chain)
S=1/2 Heisenberg ladder 2xL (L=32)
System coupled to a spin bath
V. Dobrovitski et al, PRL (2003), A. Melikidze et al PRB (2004)
|y(t=0)=||c0; |c0c0|=I
...
H=HS+ HB+ VSB
Comparing S-T and time step targeting
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S-T is fast and efficient for one-dimensional
geometries with nearest neighbor interactions
S-T error depends strongly on the Trotter error but it
can be reduced by using higher order expansions.
Time step targeting (RK) can be applied to ladders
and systems with long range interactions
It has no Trotter error, but is less efficient.
Evolution in imaginary time *,**
Thermo-field representation*:
O(β)|O(β)
|O(β)=e-βH/2 |I; Z(β)=O(β)|O(β)
where |I is the maximally mixed state for β=0 (T=∞) (thermal
vacuum)
Evolution in imaginary time is equivalent to evolving the
maximally mixed state in imaginary time. We can do so by
solving
-2
*Takahashi and Umezawa, Collect Phenom. 2, 55 (1975), ** Verstraete PRL 2004, Zwolak PRL 2004
Maximally mixed state for β=0 (T=∞)
CM: thermofield representation, QI: mixed state purification
|I =∑|n,ñ (auxiliary field ñ is called ancilla state)
with |n= |s1 s2 s3…sN 
2N states!!!
|I=|↑↑,↑↑+|↓↓,↓↓+|↑↓,↑↓+|↓↑,↓↑
each term can be re-written as a product of local “site-ancilla” states:
|I=|↑,↑|↑,↑+|↓,↓|↓,↓+|↑,↑|↓,↓+|↓,↓|↑,↑
after a “particle-hole” transformation on the ancilla we get
|I=|I0|I0 with |I0= |↑,↓+|↓,↑
→ only one product state!
and we can work in the subspace with Sz=0!!!
In DMRG language this looks like:
In this basis, left and right block have only one state!
As we evolve in time, the size of the basis will grow.
Thermodynamics of the spin-1/2 chain
L=64
Frustrated Heisenberg chain*
* TM-DMRG results from Wang and Xiang, PRB 97; Maisinger and Schollwoeck, PRL 98.
Conclusions
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If your DMRG program incorporates
wavefunction transformations, time-dependent
DMRG is easy to implement.
Time-targeting method allows to study 2D and
systems with long-range interactions.
Error is dominated by the DMRG truncation
error. Care must be taken in order to control it
by keeping more states.
Generalization to finite temperature (imaginary
time) is straightforward.