On adaptive timedependent DMRG based on RungeKutta methods Adrian Feiguin University of California,

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On adaptive time-dependent DMRG based on
Runge-Kutta methodsAdrian Feiguin University
of California, Irvine
Outline

• 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)
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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?
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The density matrix projection
y? ?ijyiji?j?
We need to find the state y'? ?majaajua? j?
that minimizes the distance
Sy'? -y?2
Solution The optimal states are the eigenvectors
of the reduced density matrix with the largest
eigenvalues wa
rii' ?jyijyi'j Tr r 1
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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 matrix projection to truncate the
basis until reaching the desired size
We start from a small superblock with 4
sites/blocks, each with a dimension mi , small
enough to be easily diagonalized
ans so on, until we converge
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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.
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Time evolution Suzuki-Trotter approach
HA H2
H4
H6
HB H1
H3 H5
So the time-evolution operator is a product of
individual link terms.
G. Vidal, PRL (2004)
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Time dependent DMRGS.R.White and A.E. Feiguin,
PRL (2004), Daley et al, J. Stat. Mech. Theor.
Exp. (2004)
We turn off the diagonalization and start
applying the evolution operator
We start with the finite system algorithm to
obtain the ground state
e-itHij
e-itHij
e-itHij
e-itHij
e-itHij
One sweep evolves one time step
Each link term only involves two-sites
interactions gt small matrix, easy to calculate!
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Real-time dynamics using Runge-Kutta
We need to solve
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Time evolution and DMRG
Some history

• 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.

t0
t t
t2t
t3t
t4t

• Luo, Xiang and Wang, PRL 91, 049901 (2003) showed
  how to target correctly for real-time dynamics.
  They target
• ?(t0), ?(tt) , ?(t2t) , ?(t3t)

t0
t t
t2t
t3t
t4t
This is quasiexact as t?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.
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Time-step targeting methodFeiguin 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.
• 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 tt

t4t
t0
tt
t2t
t3t
The time-evolution can be implemented in various
ways 1) Calculate Lanczos (tri-diagonal)
matrix, and exponentiate. (time consuming) 2)
Runge-Kutta.
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Time-step targeting method (continued)
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S1 Heisenberg chain (L32 t8)
time targetingRK
1st order S-T
4th order S-T
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Fixed error, variable number of states
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Time dependent correlation functions(Example
S1 Heisenberg chain)
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S1/2 Heisenberg ladder 2xL (L32)
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System coupled to a spin bathV. Dobrovitski et
al, PRL (2003), A. Melikidze et al PRB (2004)
y(t0)????c0? c0??c0I
HHS HB VSB
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Comparing S-T and time step targeting

• 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.

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Evolution in imaginary time ,
Thermo-field representation
?O(ß)O(ß)?
O(ß)?e-ßH/2 I? Z(ß)?O(ß)O(ß)?
where I? is the maximally mixed state for ß0
(T8) (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
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Maximally mixed state for ß0 (T8)
CM thermofield representation, QI mixed state
purification
I? ?n,ñ?
(auxiliary field ñ is called ancilla state)
with n? s1 s2 s3sN ?
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
Sz0!!!
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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.
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Thermodynamics of the spin-1/2 chainL64
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Frustrated Heisenberg chain
TM-DMRG results from Wang and Xiang, PRB 97
Maisinger and Schollwoeck, PRL 98.
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Conclusions

• 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.