mera in 2d ( multi-scale entanglement renormalization ansatz ) Łukasz cincio ( poster ) jacek...

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MERA in 2D ( Multi-scale Entanglement Renormalization Ansatz ) Łukasz Cincio (POSTER) Jacek Dziarmaga Marek M. Rams (POSTER) @ Uniwersytet Jagielloński arXiv:0710.3829 accepted in Phys. Rev. Lett.

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Page 1: MERA in 2D ( Multi-scale Entanglement Renormalization Ansatz ) Łukasz Cincio ( POSTER ) Jacek Dziarmaga Marek M. Rams ( POSTER ) @ Uniwersytet Jagielloński

MERA in 2D( Multi-scale Entanglement Renormalization Ansatz )

Łukasz Cincio (POSTER)

Jacek Dziarmaga

Marek M. Rams (POSTER)

@

Uniwersytet Jagielloński

arXiv:0710.3829accepted in Phys. Rev. Lett.

Page 2: MERA in 2D ( Multi-scale Entanglement Renormalization Ansatz ) Łukasz Cincio ( POSTER ) Jacek Dziarmaga Marek M. Rams ( POSTER ) @ Uniwersytet Jagielloński

4321

14

14

32

32

2

43

1

21

21

kkkk

UU

WW

T

jjkk

jjkk

ijj

ijj

ii

MERA: periodic chain of 4 spins

)( :erdisentanglunitary -

.....1

)( : vectorslorthonorma -

1 : vectornormalized -

4231

21

43

21

21

212

21

1

21

2121

*

*

*

kkkkjjkk

jjkk

iiijj

ijj

i

iiii

UUU

di

WWW

TTT

G. Vidal, PRL 2007

Page 3: MERA in 2D ( Multi-scale Entanglement Renormalization Ansatz ) Łukasz Cincio ( POSTER ) Jacek Dziarmaga Marek M. Rams ( POSTER ) @ Uniwersytet Jagielloński

MERA: from N=4 to N=8 spins

amplitudes quantum2

encode tensorslog 2

N

N

)2 (

? s' are But...what Nd

d

Page 4: MERA in 2D ( Multi-scale Entanglement Renormalization Ansatz ) Łukasz Cincio ( POSTER ) Jacek Dziarmaga Marek M. Rams ( POSTER ) @ Uniwersytet Jagielloński

MERA representation of a quantum state

?

on 8 spins

? s' minimalith w

and , tensorsfind toHow

d

UWT

Page 5: MERA in 2D ( Multi-scale Entanglement Renormalization Ansatz ) Łukasz Cincio ( POSTER ) Jacek Dziarmaga Marek M. Rams ( POSTER ) @ Uniwersytet Jagielloński

MERA: entanglement renormalization (real space renormalization group on a quantum state)

U

4321

4

1

4

1

1

pppp

WWp

WWp

iid

ii

ii

iiU

1

11 maximize to

d

iipU

1st decimation

'

G. Vidal, PRL 2007

on 4 block spins

on 8 spins

U

U

U

Page 6: MERA in 2D ( Multi-scale Entanglement Renormalization Ansatz ) Łukasz Cincio ( POSTER ) Jacek Dziarmaga Marek M. Rams ( POSTER ) @ Uniwersytet Jagielloński

MERA: entanglement renormalization

2

12 maximize to

d

iipU

2nd decimation

'

''

on 4 block spins

on 2 block spins

U

iiddd

ii

iidd

iiU WWpWWp

11211

11

Page 7: MERA in 2D ( Multi-scale Entanglement Renormalization Ansatz ) Łukasz Cincio ( POSTER ) Jacek Dziarmaga Marek M. Rams ( POSTER ) @ Uniwersytet Jagielloński

MERA: entanglement renormalization

''T

3rd decimation

''

on 2 block spins

Page 8: MERA in 2D ( Multi-scale Entanglement Renormalization Ansatz ) Łukasz Cincio ( POSTER ) Jacek Dziarmaga Marek M. Rams ( POSTER ) @ Uniwersytet Jagielloński

MERA representation of a quantum state

U

d

ersdisentanglby small made arewhich

s' largely sufficientfor Accurate

Page 9: MERA in 2D ( Multi-scale Entanglement Renormalization Ansatz ) Łukasz Cincio ( POSTER ) Jacek Dziarmaga Marek M. Rams ( POSTER ) @ Uniwersytet Jagielloński

MERA: entanglement renormalization in critical quantum Ising chain G. Vidal, PRL 2007

1at ,

czj

ji

zi

i

xi ggH

ip

iii

ii WWp

Page 10: MERA in 2D ( Multi-scale Entanglement Renormalization Ansatz ) Łukasz Cincio ( POSTER ) Jacek Dziarmaga Marek M. Rams ( POSTER ) @ Uniwersytet Jagielloński

MERA in 2D

Page 11: MERA in 2D ( Multi-scale Entanglement Renormalization Ansatz ) Łukasz Cincio ( POSTER ) Jacek Dziarmaga Marek M. Rams ( POSTER ) @ Uniwersytet Jagielloński

MERA in 2D: quantum Ising model

04.3 ,,

czj

ji

zi

i

xi ggH

generatorshermitian imaginary

plaquette 44 aon states basis real

A

eU

e

ewW

etT

Aqi

abcdii

abcd

abcdabcd

parameters al variationreal ,, qwt i

Page 12: MERA in 2D ( Multi-scale Entanglement Renormalization Ansatz ) Łukasz Cincio ( POSTER ) Jacek Dziarmaga Marek M. Rams ( POSTER ) @ Uniwersytet Jagielloński

MERA in 2D: results

04.3 ,,

czj

ji

zi

i

xi ggH

Page 13: MERA in 2D ( Multi-scale Entanglement Renormalization Ansatz ) Łukasz Cincio ( POSTER ) Jacek Dziarmaga Marek M. Rams ( POSTER ) @ Uniwersytet Jagielloński

Conclusion

Work in progress

Symmetric MERA in 2D

&

First test in 2D with encouraging results

2D t-J model

Łukasz Cincio (POSTER ON MERA )

Marek M. Rams (POSTER ON ENTANGLEMENT IN

KIBBLE-ZUREK MECHANISM)

Page 14: MERA in 2D ( Multi-scale Entanglement Renormalization Ansatz ) Łukasz Cincio ( POSTER ) Jacek Dziarmaga Marek M. Rams ( POSTER ) @ Uniwersytet Jagielloński

MERA: expectation values

operations

) log ( 2NdO M

H H