Entanglement Renormalization, critical phenomena and CFT

Transcription

1 NCKU 20 th Dec 2009 Wrkshp n Quantum Infrmatin Science and Many-Bdy Physics Entanglement Renrmalizatin, critical phenmena and CFT Guifre Vidal

2 Outline MERA = Multi-scale Entanglement Renrmalizatin Ansatz Entanglement Renrmalizatin/MERA Renrmalizatin Grup (RG) transfrmatin Quantum Cmputatin Scale invariant MERA RG fixed pint ( nn-critical fixed pint tplgical rder TQFT ) critical fixed pint cntinuus quantum phase transitin CFT bulk (lcal/nn-lcal) bundary defect cllabratin ith Glen Evenbly, R. Pfeifer, P. Crbz, L. Tagliaczz, I.P. McCullch V. Pic (U. Barcelna) S. Iblisdir (U. Barcelna)

3 MERA (multi-scale entanglement renrmalizatin ansatz) Lattice ith N sites Vidal, Phys. Rev. Lett. 99, (2007); ibid 101, (2008) N d N d d d GS i12 i... in 1 2 N i 1 i 1 i N c i i... i i 1 i 2 i 17 i i 1 i 2 i 3 i 4 i 5 i 6 i 7 i 8 i 9 i 10 i 11 i 12 i 13 i 14 i 15 i 16 i 17 i 18

4 MERA (multi-scale entanglement renrmalizatin ansatz) k=1,2,,r 1,2,, ismetry u disentangler u u I I disentangler ismetry

5 MERA as a quantum circuit ismetry 0 0 unitary N (2) (1) (0)

6 Many pssibilities examples: binary 1D MERA k=1,2,,r ternary 1D MERA k=1,2,,r u u i 1 i 2 i 3 i 4 i 5 i 6 i 7 i 8 i 9 i 10 i 11 i 12 i 13 i 14 i 15 i 16 Defining prperties: 1) ismetric tensrs i 1 i 2 i 3 i 4 i 5 i 6 i 7 i 8 i 9 i 10 i 11 i 12 i 13 i 14 i 15 i 16 i 17 i 18 2) past causal cne ith bunded idth g g I

7 What is the MERA useful fr? grund states/l energy subspaces in 1D, 2D lattices at criticality: critical expnents/cft Simulatin csts: ON ( ) p N inhmgeneus O(lg( N)) p lg N translatin invariance O(1) p translatin & scale invariance N p + (scale invariant) bundary/defect/ interface (n translatin invariance!!! )

8 RG transfrmatin Vidal, Phys. Rev. Lett. 99, (2007); ibid 101, (2008) The MERA defines a carse-graining transfrmatin L 3 3 L 2 2 L 1 1 L 0 0 Entanglement renrmalizatin ismetry disentangler 1 ' L τ+1 L τ

9 RG transfrmatin The MERA defines a carse-graining transfrmatin L 3 L 2 L 1 L 0 transfrmatin f lcal peratrs 1 L τ+1 L τ

10 RG transfrmatin The MERA defines a carse-graining transfrmatin L τ L τ+1 τ τ+1 τ disentangler u u I τ ismetry τ+1 τ τ I Ascending superperatr

11 L 3 3 L 2 2 L 1 1 L 0 0 ascending superperatr ( ) L τ+1 L τ past causal cne ith bunded idth Lcal peratrs remain lcal 1 L( ) 1 C( ) 1 R( )

12 L 3 3 L 2 2 L 1 1 L 0 0 descending superperatr ( 1) L τ+1 L τ ' ' ' L ( ') C ( ') R( ')

13 Descriptin f the system at different length scales Ascending and descending super-peratrs: change f length scale (r time in a quantum cmputatin) L 3 L 2 L 1 L 0

14 Scale invariant MERA (3) (2) (1) (0) critical systems (1D) critical expnents OPE, CFT Vidal, Phys. Rev. Lett. 99, (2007) Vidal, Phys. Rev. Lett. 101, (2008) Evenbly, Vidal, arxiv: Evenbly, Vidal, arxiv: Givannetti, Mntanger, Fazi, Phys. Rev. Lett. 101, (2008) Pfeifer, Evenbly, Vidal, Phys. Rev. A 79(4), (R) (2009) Mntanger, Rizzi, Givannetti, Fazi, Phys. Rev. B 80, (2009) Givannetti, Mntanger, Rizzi, Fazi, Phys. Rev. A 79, (2009) bundary & defects nn-lcal peratrs tplgically rdered systems (2D) Evenbly, Pfeifer, Pic, Iblisdir, Tagliaczz, McCullch,Vidal,arXiv: Evenbly, Crbz, Vidal, arxiv: Silvi, Givannetti, Calabrese, Santr, Fazi, arxiv: Aguad, Vidal, Phys. Rev. Lett. 100, (2008) Kenig, Reichardt, Vidal, Phys. Rev. B 79, (2009)

15 N Scale invariant MERA (3) (2) (1) (0) Vidal, Phys. Rev. Lett. 99, (2007) Vidal, Phys. Rev. Lett. 101, (2008) Evenbly, Vidal, arxiv: Evenbly, Vidal, arxiv: MERA is a gd ansatz fr grund state at quantum critical pint it recvers scale invariance! same state and Hamiltnian at each level [shn fr Ising mdel, free fermins, free bsns] prper scaling f entanglement entrpy SN /2 lg( N) lg( N)

16 Scale invariant MERA (3) (2) (1) (0) Vidal, Phys. Rev. Lett. 99, (2007) Vidal, Phys. Rev. Lett. 101, (2008) Evenbly, Vidal, arxiv: Evenbly, Vidal, arxiv: MERA is a gd ansatz fr grund state at quantum critical pint it recvers scale invariance! same state and Hamiltnian at each level [shn fr Ising mdel, free fermins, free bsns] prper scaling f entanglement entrpy plynmial decay f crrelatins cnstant ascending superperatr ' ( ) ' ( ) '' scaling superperatr

17 Scale invariant MERA plynmial decay f crrelatins Vidal, Phys. Rev. Lett. 101, (2008) 2 z 1 2 z 1 2 lg( r) r s s 1 2 lg( r) lg( z) q C ( s, s ) z r r, q lg( z) ( 1) z1 ( 2) z2 Eigenvalue decmpsitin Givannetti, Mntanger, Fazi, Phys. Rev. Lett. 101, (2008) ( ) scaling peratr 3 lg scaling dimensin 3

18 Eigenvalue decmpsitin Givannetti, Mntanger, Fazi, Phys. Rev. Lett. 101, (2008) ( ) scaling peratr 3 lg scaling dimensin 3 Critical expnents can be extracted frm the scaling superperatr (= qumera channel, MERA transfer map) Cnnectin t Cnfrmal Field Thery Pfeifer, Evenbly, Vidal, Phys. Rev. A 79(4), (R) (2009) spin lattice at quantum critical pint CFT central charge c primary fields p p cnfrmal dimensins ( h, h ) p p p h h peratr prduct expansin OPE p p p p C

19 Pfeifer, Evenbly, Vidal, Phys. Rev. A 79(4), (R) (2009) peratr prduct expansin OPE frm three pint crrelatrs p p p C L 3 L 2 L 1 ( x) ( y) ( z) ( x ) ( y ) ( z) x y y z z x C L 0

20 Scale invariant MERA (bulk) Example: Ising mdel Pfeifer, Evenbly, Vidal, Phys. Rev. A 79(4), (R) (2009) =36 =20 scaling peratrs/dimensins: identity spin energy Operatr prduct expansin (OPE): C C C C C ( 5 10 ) I scaling dimensin (exact ) scaling dimensin (MERA) errr % % % % % % % % fusin rules I I+

21 Example: ther mdels Energy Errr, E Ising 3-state Ptts XX Heisenberg Refinement Parameter,

22 Recent develpments: nn-lcal scaling peratrs (bulk) bundary critical phenmena defects (in bulk)

23 Nn-lcal scaling peratrs Evenbly, Crbz, Vidal, arxiv: N glbal symmetry G V H V H g G g [ N N example H Z H Z H Ising X i X i 1 Z ] i Ising Ising g N u G-symmetric MERA u u V g u u V g V g V g

24 the symmetry cmmutes ith the carse-graining V g V g nn-lcal peratrs f the frm can be lcally carse-grained Vg Vg

25 lcal scaling peratrs u ' ( ) scaling superperatr 1 3 ( ) scaling peratr scaling dimensin 3 lg 3 nn-lcal scaling peratrs ' ( ) g mdified scaling superperatr 1 3 u Vg g ( g, ) g, g, g, g, nn-lcal scaling peratr, 3 g g, g, lg scaling dimensin g, 3 g,

26 Nn-lcal scaling peratrs Scale invariant MERA (bulk) Example: Ising mdel lcal nn-lcal I identity spin energy disrder fermins

27 Nn-lcal scaling peratrs Scale invariant MERA (bulk) Example: Ising mdel disrder fermins scaling dimensin (exact ) scaling dimensin (MERA) lcal errr 1/ % 1/2 0.5 <10 8% 1/2 0.5 <10 8% 1+1/ % 1+1/ < 10 5% 1+1/ < 10 5% 2+1/ % 2+1/ % 2+1/ % nn-lcal

28 Nn-lcal scaling peratrs Scale invariant MERA (bulk) Example: Ising mdel OPE fr lcal & nn-lcal primary fields fusin rules C C 1/ 2 1/ 2 C C i i I I+ C C e e i i /4 / 2 /4 / 2 4 ( 6 10 ) I I I {I,,,,, } lcal and semi-lcal subalgebras {I, } {I,, } {I,, } {I,,, }...

29 Example: quantum XX mdel H ( X X YY ) XX i i 1 i i 1 G = U(1) symmetry =54 =32 (expliting U(1) symmetry)

30 Bundary critical phenmena infinite chain (bulk) Evenbly, Pfeifer, Pic, Iblisdir, Tagliaczz, McCullch,Vidal, arxiv: see als Silvi, Givannetti, Calabrese, Santr, Fazi, arxiv: u bulk MERA semi-infinite chain (bundary) u u MERA ith bundary

31 < z > Example: Ising mdel Free bundary cnditins: lcal magnetizatin exact MERA bulk Distance frm bundary

32 < z > MERA - 2/ < z > exact - 2/ z < Bulk > MERA -2/ < z > - < z > exact MERA Bundary effects still nticeable far aay frm the bundary MERA gets crrect magnetizatin everyhere (apprx. same accuracy as ith bulk MERA ithut bundary) distance frm bundary, d

33 Bulk expectatin values in the presence f a bundary bulk ( r) 0 ( r) ( r') 1 r r' 2 r r () r ( r ) bundary () r 1 r r () r

34 Bundary scaling peratrs/dimensins u L L L bundary scaling superperatr ( ) L ( ) bundary scaling peratr bundary scaling dimensin 3 lg 3

35 Bundary scaling peratrs/dimensins Example: Ising mdel Bulk Free BC Fixed BC 4 2 1/ / / / 2 1/ / 2 0

36 Bundary scaling peratrs/dimensins Free BC identity spin I Fixed BC identity I scaling dimensin (exact ) 2 1/ % / % 1 1/ 8 1/ 8 0 scaling dimensin (MERA) errr Bulk 1/ % 1+1/ % scaling dimensin (exact ) scaling dimensin (MERA) 1 errr % Free BC 2 1/ 2 1 1/ 2 1/ 2 Fixed BC % % %

37 Finite system ith t bundaries finite MERA ith t bundaries L T R Energy spectrum

38 defect / interface Evenbly, Pfeifer, Pic, Iblisdir, Tagliaczz, McCullch,Vidal,arXiv: u DL, u D DR, u defect u I ul L IL, IR, R ur left regin interface right regin

39 Lattice Defects: Hamiltnian: -XX -XX -XX -αxx -XX -XX -XX -XX Z Z Z Z Z Z Z Z Z

40 < z > = 1.5 = 1 = 0.8 = 0 = 2 decupled semi-infinite chains ith Free BC cupled semi-infinite chains ith Fixed BC Lattice Site

41 Bulk Summary: scale invariant MERA, critical phenmena, and CFT (lcal & nn-lcal) scaling peratrs/dimensins CFT: primary fields and OPE Bundary bundary scaling peratrs BCFT: primary fields and OPE finite system ith t bundaries Defect defect scaling peratrs/dimensins interface