Presentation on theme: "The DMRG and Matrix Product States"— Presentation transcript:
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 findNotice 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 U2mmA=AtA==1XThis 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 haveWithout 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 asssAnd the contractions, as:s1 s2The 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 … sLAddition:
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 OperatorsOThe 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 StateWe 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 singletab
14 The AKLT as a MPSThe singlet wave function with singlet on all bonds isThe local projection operators onto the physical S=1 states areThe mapping on the spin S=1 chain then readsProjecting 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 MPSWe 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 asThe matrices can be represented diagramaticaly assAnd 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 s5Us1 s2 s sNs4 s5Which translates as:s1 s2 s3Us 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 sjs’i s’jAnd we can apply the evolution operator between sites far apart as:Us1 s2 s sNE.M Stoudenmire and S.R. White, NJP (2010)
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