The DMRG and Matrix Product States

Adrian Feiguin

The DMRG transformationWhen we add a site to the block we obtain the wave function for the larger block as:

11

,1,

llll

sllll ssA

Let’s change the notation…

llll ssAll 1, 1

1,

11lls

lllllLll ssU

We can repeat this transformation for each l, and recursively we find

}{1,21 ...|][...][][|

1211s

lll sssAsAsAll

Notice the single index. The matrix corresponding to the open end is actually a vector!

Some properties the A matricesRecall that the matrices A in our case come from the rotation matrices U

A= 2m

m

AtA= X =1

This is not necessarily the case for arbitrary MPS’s, and normalization is usually a big issue!

The DMRG wave-function in more detail…

}{1,2,11,21

,11

...|][...][][][...][][

||

32211211s

LLlllll

llll

sssBsBsBsAsAsALllllll

ll

}{,2, ...|][...][][|

321s

LlLlll sssBsBsBLllll

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:

}{1,2,21 ...|][][...][][

1211s

LL sssMsMsMsMLLL

MPS wave-function for open boundary conditions

Diagrammatic representation of MPS

The matrices can be represented diagrammatically as

][sA s

s

][sA

The dimension D of the left and right indices is called the “bond dimension”

And the contractions, as:

1 2 3s1 s2

MPS for open boundary conditions

}{1

1

}{121

}{1,2,21

...|][

...|][]...[][

...|][][...][][1211

sL

L

ll

sLL

sLL

sssM

sssMsMsM

sssMsMsMsMLLL

1 2 L

s1 s2 s3 s4 … sL

MPS for periodic boundary conditions

}{1

1

}{121

}{1,,2,21

...|][Tr

...|][]...[][Tr

...|][][...][][11211

sL

L

ll

sLL

sLL

sssM

sssMsMsM

sssMsMsMsMLLLL

1 2 3 L 1

s1 s2 s3 s4 … sL

Properties of Matrix Product States

Inner product:

1 2 Ls1 s2 s3 s4 … sL

1' 2' L'

Addition:

L

LL

sLL

sLL

sLL

MMMMMM

NNN

MM

N

ssNNN

ssMMMssMMM

~...~~...

...

~00

with

...|...

...|~...~~;...|...

21

2121

}{121

}{121

}{121

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

Operators

O

The operator acts on the spin index only

' elementsh matrix wit a is sOsO

Matrix product basis

}{

1,21 ...|][...][][|1211

slll sssAsAsA

ll

1 2 ls1 s2 s3 s4 sl

}{,2, ...|][...][][|

321s

LlLlll sssBsBsBLllll

1l 2llsl+1 sl+2 sl+3 sl+4 sL

L

ll '|As we saw before, in the dmrg basis we get:

1 2L

1' 2'L'

ll ',

The DMRG w.f. in diagrams

}{1,2,1,21

}{1,2,11,21

...|][...][][][...][][

...|][...][][][...][][

322111211

32211211

sLLlll

sLLlllll

sssBsBsBsAsAsA

sssBsBsBsAsAsA

Lllllllll

Lllllll

1 2 ls1 s2 s3 s4 sl

2l 3l1lsl+1 sl+2 sl+3 sL

Ll 1l

(It’s a just little more complicated if we add the two sites in the center)

The AKLT State 1 with

31 2

11 SSSSSHi

iiiiAKLT

We replace the spins S=1 by a pair of spins S=1/2 that are completely symmetrized

ibiai

ibiaibiai

ibiai

210

… and the spins on different sites are forming a singlet

biaibiai ,1,,1,2

1

a b

The AKLT as a MPS

The local projection operators onto the physical S=1 states are

1000

;0

21

210

;0001 0

ababab MMM

The mapping on the spin S=1 chain then reads

}{ ,

,,, },{}{...2

22

1

11s ba

sba

sba

sba basMMM L

LL

111

2

32

1

21

132

2

2221

1

11

,,,}{

,,,

}{,,,,,,

with }{...

}{...

ll

l

ll

l

ll

L

L

L

L

LL

abs

bas

aas

saa

saa

saaAKLT

sab

sbaab

sbaab

sbaAKLT

MAsAAA

sMMMP

Projecting the singlet wave-function we obtain

The singlet wave function with singlet on all bonds is

with },{...}{ ,

,,, 13221 s ba

ababab baL

02

12

10ab

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

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:

HA

min

DMRG does something very close to this…

MPS representation of the time-evolution

A MPS wave-function is written as

}{ }{212211

212211}{

,...,][...][][

,...,][]...[][Tr

14321

i j

N

i

sNNN

NNNs

ssssAsAsA

ssssAsAsA

The matrices can be represented diagramaticaly as

][sA s

And the contractions (coefficients), as:

1 2 3 N 1s1 s2 s3 s4 sN

s1 s2 s3 sN

MPS representation of the time-evolutionThe two-site time-evolution operator will act as:

1 2 3 N 1

U

s4 s5

Which translates as:

65

54

54

5454]'[]'[ 55

','

,','44 sAUsA

ss

ssss

s1 s2 s3

1 2 3

U

N 14 4 5 6 6

s4 s5

s6 sN

Swap gatesIn the MPS representation is easy to exchange the states of two sites by applying a “swap gate” si sj

s’i s’j

E.M Stoudenmire and S.R. White, NJP (2010)

],[ jiS

And we can apply the evolution operator between sites far apart as:

s1 s2 s3 sN

1 2 3 N 1

U

Matrix product basis

}{1,21 ...|][...][][|

1211s

lll sssAsAsAll

1 2 ls1 s2 s3 s4 sl

}{,2, ...|][...][][|

321s

LlLlll sssBsBsBLllll

1l 2llsl+1 sl+2 sl+3 sl+4 sL

L

(a)

(b)