The DMRG and Matrix Product States

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The DMRG and Matrix Product States

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!

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!

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

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”

MPS for open boundary conditions
s1 s2 s3 s … sL

MPS for periodic boundary conditions
s1 s2 s3 s … sL

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

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

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

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:

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)

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

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

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

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…

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

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

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)

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