1. Generalized DMRG with Tree Tensor Network
Sujay Ray (Int. PhD)
Department of Physics, IISc
Quantum Condensed Matter Journal Club
11/12/2018
Date:
11/12/2018

2. Outline Brief introduction of DMRG. Matrix Product State.
Tensor network and Tree network.
Generalized DMRG with TTNS.
Application to problems.
11/12/2018
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

3. Outline Brief introduction of DMRG. Matrix Product State.
Tensor network and Tree network.
Generalized DMRG with TTNS.
Application to problems.
11/12/2018
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

4. Brief introduction of DMRG
Density Matrix Renormalization Group (DMRG) was introduced by
Steve White in 1992.
One of the dominating numerical techniques to simulate strongly
correlated systems.
It relies on exact diagonalization and numerical renormalization
group ideas.
It is widely used to study Fermionic spin chain problems in 1D
Higher dimensional problems can be mapped to 1D chain and
DMRG can be used.
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

5. Brief introduction of DMRG
Block growth
2 sites basis with dim 4:
For 3 sites:
For i sites:
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

6. Brief introduction of DMRG
Some basic concepts
Reduced Density Matrix: U S V
Singular Value Decomposition:
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

7. Brief introduction of DMRG
DMRG process
Hilbert space for single site has
dimension d
Block A
Block B
Keep creating Hamiltonian until
blocks become larger than d x m.
Obtain full spectrum and eigen-
vectors of reduced density matrices.
Truncate basis keeping m eigen-
Vectors with largest eigenvalues.
Rotate Hamiltonian and operators
to new basis and reiterate the process.
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

8. Brief introduction of DMRG
DMRG process
Schmidt decomposition:
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

9. Outline Brief introduction of DMRG. Matrix Product State.
Tensor network and Tree network.
Generalized DMRG with TTNS.
Application to problems.
11/12/2018
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

10. Matrix product state Introduction The quantum state is described by:
ith site
L th site
Local basis of dimension d
The quantum state is described by:
where
Through a sequence of singular value decomposition we get:
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

11. Matrix product state Introduction
Then we have the diagrammatic representation:
Physical index
Dummy index
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

12. Matrix product state Operations Some other forms:
Dot product as sum over physical indices:
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

13. Matrix product state Matrix product Operators
Most general MPO
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

14. Matrix product state DMRG revisited DMRG as variational problem:
Minimize using Lagrange multiplier
DMRG as variational problem:
Considering in MPO language and after minimization:
dimension:
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

15. Matrix product state DMRG revisited Finite size DMRG:
Sweeping is important
during sweeping we add one
extra site to one block and
shrink one site from other
block.
Right to left sweep
MPS site by site optimization
of M matrices are same.
Both lead to the same algorithm.
Left to right sweep
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

16. Outline Brief introduction of DMRG. Matrix Product State.
Tensor network and Tree network.
Generalized DMRG with TTNS.
Application to problems.
11/12/2018
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

17. Tensor network and Tree network
Introduction
What is a tree network?
A network of tensors that has a tree structure.
Bipartition
Weight of the index
Edges represent qudits.
Canonical form of bipartition: When weight is set of Schmidt coefficient
and bases are Schmidt bases.
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

18. Tensor network and Tree network
Introduction :
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

19. Tensor network and Tree network
Introduction
Two random qudits are connected through
O(log(n)) tensors in binary tree and through
O(n) in MPS.
Time required is a factor log(n)/n lower when
Tree TN is used.
Red dots represent atoms and red
dots are interaction pattern.
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

20. Outline Brief introduction of DMRG. Matrix Product State.
Tensor network and Tree network.
Generalized DMRG with TTNS.
Application to problems.
11/12/2018
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

21. Generalized DMRG with TTNS
Introduction
How tree network can be related to generalized DMRG
with z blocks?
In the approach we are considering all the sites represent physical
sites and entanglement is transferred through virtual bonds .
Physical bond
Virtual bond
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

22. Generalized DMRG with TTNS
Long range correlation
where
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

23. Generalized DMRG with TTNS
Long range correlation
Logarithmic scaling counteracts
this exponential decay.
Correlation scales polynomially
with number of sites M.
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

24. Generalized DMRG with TTNS
Separation of blocks
At each site we can associate with z +1 indices.
D: virtual dimension.
d: physical dimension.
Separation of states into z blocks:
Basis of environment block i.
State of site m.
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

25. Generalized DMRG with TTNS
Network optimization
Orthonormality:
Network optimization:
Minimise -
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

26. Generalized DMRG with TTNS
Interaction
Suppose two body interaction acting on site 7 and 15 :
Network is contracted from leaves
and working in the inward direction.
Numerical effort:
number of sites x
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

27. Generalized DMRG with TTNS
Fermions
In case of fermions:
Interactions with local support turned into interactions with non
local supports via Jordan –Wigner transformation.
Suppose for two particle
interaction:
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

28. Generalized DMRG with TTNS
Fermions
In case of fermions:
Introducing an index pair
tensor is
made block diagonal.
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

29. Generalized DMRG with TTNS
Fermions
Fermionic number parity:
Z can be moved to virtual bond:
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

30. Generalized DMRG with TTNS
Fermions
The calculation of effective Hamiltonian with supports on a few
sites simplifies.
Z is moved to virtual bond and only one Z remains.
So it is sufficient for two site interaction to take only the path
connecting the sites.
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

31. Outline Brief introduction of DMRG. Matrix Product State.
Tensor network and Tree network.
Generalized DMRG with TTNS.
Application to problems.
11/12/2018
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

32. Application to problems
2D Heisenberg model
4 x 4
4 x 4
6 x 6
Z =3
Z =4
Z =3
6 x 6
Z =4
Virtual bond dimension was kept fixed D=4
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

33. Application to problems
2D interacting spinless fermions
4 x 4
6 x 6
J = 1
U = 0.5
N = 3
Periodic boundary
condition assumed.
Virtual bond dimension was kept fixed D=4
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

34. Conclusion and outlook
Tree tensor network state method is used to treat strongly correlated
systems with long range interactions with an arbitrary coordination
number z.
This tree tensor network method has low computational cost compared to
DMRG-based method.
The long range correlation deviates from mean field value polynomially
with distance in contrast to MPS states which deviates exponentially.
Calculations done in numerical problems with fixed coordination number
z but it can be varied from site to site and may give better results.
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

35. Thanks to Tanmoy Das, Kausik Ghosh, Adhip Agarwala, Abhas Mallick
for valuable discussions.
References:
V. Murg and F. Verstraete, Ö. Legeza and R. M. Noack, PHYSICAL REVIEW B 82, (2010)
Y.-Y. Shi, L.-M. Duan, and G. Vidal, PHYSICAL REVIEW A 74, (2006)
Ö. Legeza and J. Solyom, Phys. Rev. B 68, (2003).
F. Verstraete, J. I. Cirac, and V. Murg, Adv. Phys. 57, 143 (2008).
F. Verstraete and J. Cirac, Phys. Rev. B 73, (2006).
N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 100, (2008).
A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States, Roman Orus.
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

36. Thank you
Generalized DMRG with Tree Tensor Network
Date:
11/12/2018

37. MPS MPS operations: Order of contraction is important: MPS Date:
11/12/2018
MPS
Date:
11/12/2018