Download presentation

Presentation is loading. Please wait.

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

Similar presentations

Presentation is loading. Please wait....

OK

Quantum One: Lecture 17.

Quantum One: Lecture 17.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Download ppt on civil disobedience movement in war Ppt on power generation by coal Ppt on marie curie's discoveries Ppt on surface area and volume for class 9 free download Ppt on acid-base indicators are large organic molecules Ppt on ms project 2010 Inner ear anatomy and physiology ppt on cells Ppt on solar energy collectors Ppt on porter's five forces pdf Ppt on life and works of robert frost