1. The DMRG and Matrix Product States
Adrian Feiguin

2. The DMRG transformation
When we add a site to the block we obtain the wave function for the larger block as:
Let’s change the notation…
We can repeat this transformation for each l, and recursively we find
Notice the single index. The matrix corresponding to the open end is actually a vector!

3. Some properties the A matrices
Recall that the matrices A in our case come from the rotation matrices U 2m m A=
AtA= =1 X
This is not necessarily the case for arbitrary MPS’s, and normalization is usually a big issue!

4. The DMRG wave-function in more detail…
We can repeat the previous recursion from left to right…
At a given point we may have
Without loss of generality, we can rewrite it:
MPS wave-function for open boundary conditions

5. Diagrammatic representation of MPS
The matrices can be represented diagrammatically as s s
And the contractions, as:
s1 s2
The dimension D of the left and right indices is called the “bond dimension”

6. MPS for open boundary conditions
s1 s2 s3 s … sL

7. MPS for periodic boundary conditions
s1 s2 s3 s … sL

8. Properties of Matrix Product States
Inner product:
s1 s2 s3 s … sL
Addition:

9. Gauge Transformation X X-1 =
There are more than one way to write the same MPS.
This gives you a tool to othonormalize the MPS basis

10. The operator acts on the spin index only
Operators O
The operator acts on the spin index only

11. Matrix product basis s1 s2 s3 s4 sl sl+1 sl+2 sl+3 sl+4 sL
As we saw before, in the dmrg basis we get:

12. The DMRG w.f. in diagrams sl+1 sl+2 sl+3 sL s1 s2 s3 s4 sl
(It’s a just little more complicated if we add the two sites in the center)

13. The AKLT State
We replace the spins S=1 by a pair of spins S=1/2 that are completely symmetrized
… and the spins on different sites are forming a singlet a b

14. The AKLT as a MPS
The singlet wave function with singlet on all bonds is
The local projection operators onto the physical S=1 states are
The mapping on the spin S=1 chain then reads
Projecting the singlet wave-function we obtain

15. What are PEPS?
“Projected Entangled Pair States” are a generalization of MPS to “tensor networks” (also referred to as “tensor renormalization group”)

16. DMRG does something very close to this…
Variational MPS
We can postulate a variational principle, starting from the assumption that the MPS is a good way to represent a state. Each matrix A has DxD elements and we can consider each of them as a variational parameter. Thus, we have to minimize the energy with respect to these coefficients, leading to the following optimization problem:
DMRG does something very close to this…

17. MPS representation of the time-evolution
A MPS wave-function is written as
The matrices can be represented diagramaticaly as s
And the contractions (coefficients), as:
s1 s2 s3 s sN

18. MPS representation of the time-evolution
The two-site time-evolution operator will act as:
s4 s5 U
s1 s2 s sN
s4 s5
Which translates as:
s1 s2 s3 U
s sN

19. Swap gates si sj s’i s’j s1 s2 s3 sN U
In the MPS representation is easy to exchange the states of two sites by applying a “swap gate”
si sj
s’i s’j
And we can apply the evolution operator between sites far apart as: U
s1 s2 s sN
E.M Stoudenmire and S.R. White, NJP (2010)

21. Matrix product basis
(a)
s1 s2 s3 s sl
(b)
sl+1 sl sl sl sL