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The DMRG and Matrix Product States Adrian Feiguin

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The DMRG transformation When we add a site to the block we obtain the wave function for the larger block as: Lets 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!

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

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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

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Diagrammatic representation of MPS The matrices can be represented diagrammatically as s s The dimension D of the left and right indices is called the bond dimension And the contractions, as: s 1 s 2

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MPS for open boundary conditions s 1 s 2 s 3 s 4 … s L

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MPS for periodic boundary conditions s 1 s 2 s 3 s 4 … s L

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Properties of Matrix Product States Inner product: s 1 s 2 s 3 s 4 … s L Addition:

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

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Operators O The operator acts on the spin index only

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Matrix product basis s 1 s 2 s 3 s 4 s l s l+1 s l+2 s l+3 s l+4 s L As we saw before, in the dmrg basis we get:

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The DMRG w.f. in diagrams s 1 s 2 s 3 s 4 s l s l+1 s l+2 s l+3 s L (Its a just little more complicated if we add the two sites in the center)

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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

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The AKLT as a MPS 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 The singlet wave function with singlet on all bonds is

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

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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…

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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: s 1 s 2 s 3 s 4 s N

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s 1 s 2 s 3 s N MPS representation of the time-evolution The two-site time-evolution operator will act as: U s 4 s 5 Which translates as: s 1 s 2 s 3 U s 4 s 5 s 6 s N

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Swap gates In the MPS representation is easy to exchange the states of two sites by applying a swap gate s i s j E.M Stoudenmire and S.R. White, NJP (2010) And we can apply the evolution operator between sites far apart as: s 1 s 2 s 3 s N U

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Matrix product basis s 1 s 2 s 3 s 4 s l s l+1 s l+2 s l+3 s l+4 s L (a) (b)

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