Holographic Branching and Entanglement Renormalization

Transcription

1 KITP, December 7 th 2010 Holographic Branching and Entanglement Renormalization Glen Evenbly Guifre Vidal

2 Tensor Network Methods (DMRG, PEPS, TERG, MERA) Potentially offer general formalism to efficiently describe many-body wave functions ground states of large systems in D=1,2 (maybe 3) spatial dimensions strong or weak interactions, frustrated interactions etc different particle statistics (e.g. spins/bosons, fermions, or even anyons), Only limited by the amount of entanglement in the state! As Numerical Methods: Given Hamiltonian H what are properties x,y,z of the ground state? Conceptual Aspects: Framework for describing many-body systems - entanglement structure!

3 boundary law for entanglement entropy scaling: S D 1 branching MERA (Evenbly, Vidal, in preparation) generalization of the MERA to more complex holographic geometries in order to produce violations of the boundary law for entanglement entropy scaling - e.g. 2D branching MERA with entanglement: an ansatz for 2D systems with a 1D Fermi surface? we propose to use holographic geometry to classify entanglement in many-body ground states - geometric understanding of violations to the boundary law new notions of RG flow / RG fixed points S log - 2D fermions (with 1D fermi surface) as RG fixed points

4 Outline Entanglement and tensor network methods Scaling of entanglement entropy in ground states Scaling of entanglement entropy in tensor network ansatz - physical geometry vs holographic geometry Comparison of entropy scaling: - ground states vs tensor network ansatz The branching MERA Decoupling a many-body theory Holographic trees Scaling of entropy in the branching MERA Example: S =log entropy scaling in 2D fermions Example: S =(log ) 2 entropy scaling in 1D fermions Work in progress... Classification of many-body entanglement via holographic geometry Generalised RG flow / RG fixed points

5 Scaling of entanglement entropy for free fermions 1D S const. log( ) Gap. Crit. 1D Vidal, atorre, Rico, Kitaev, PR 2003 Srednicki, PR 1993 Callan, Wilczek, Phys ett B Fiola, Preskill, Strominger,Trivedi, PRD Holzhey, arsen,wilczek, Nucl.Phys.B Jin, Korepin, J. Stat. Phys Calabrese, Cardy, J. Stat. Mech D S Gap. Crit.I Crit.II log( ) 2D Wolf, PR Gioev, Klich, PR Barthel, Chung, Schollwock, PRA i, Ding, Yu, Haas, PRB Ding, Bray-Ali, Yu, Haas, PR Helling, eschke, Spitzer 2009, arxiv: Swingle 2009, arxiv: Can Tensor Network methods reproduce the appropriate entanglement entropy?

6 Entanglement Renormalization and the MERA isometry Coarse-graining transformation: Entanglement Renormalization disentangler MERA (multi-scale entanglement renormalization ansatz) length/energy scale log2 N layers MERA Holography Brian Swingle arxiv: Holographic geometry N sites Reproduce the pattern of entanglement in the ground state

7 AdS/CFT Correspondance t x t z x CFT 1+1 AdS 2+1 1D MERA x log

8 Computation of entanglement entropy D=1, MERA Geodesic curve Ω A Entanglement entropy as boundary in holographic geometry: A S( A) Ω A Holographic Entanglement Entropy, e.g. Nishioka, Ryu, Takayanagi, J. Phys. A 2009

9 Computation of entanglement entropy D=1, MERA Ω A Entanglement entropy as boundary in holographic geometry: A S( A) Ω A Holographic Entanglement Entropy, e.g. Nishioka, Ryu, Takayanagi, J. Phys. A 2009

10 MERA for D=1 spatial dimensions Entanglement entropy as boundary in holographic geometry: S( A) Ω A contributions to entropy const. 1 1 log Tensor network (MERA) based on holographic geometry can produce violations of boundary law, logarithmic violation: S ( ) log

11 MERA for D=2 spatial dimensions D=1 spatial dimensions Isometries Disentanglers D=2 spatial dimensions Disentanglers Isometries

12 MERA for D=2 spatial dimensions Entanglement entropy as boundary in holographic geometry: Ω A S( A) Ω A contributions to entropy const. /8 log /4 /2 A 1 1 S ( ) 2 4 boundary law for entropy scaling!

13 Entanglement entropy in the MERA Vidal, quant-ph/ left out of PR 101, (2008)!!! Entanglement entropy as boundary in holographic geometry: S( A) Ω A 1D contributions to entropy const. 1 2D contributions to entropy const. /8 log 1 log /4 1 /2 S ( ) log log S ( ) 2 4

14 Entanglement entropy and Tensor Networks Can Tensor Network methods reproduce the proper entanglement entropy? Tensor networks in physical geometry: 1D 2D MPS: PEPS: S = S = const. 1D 2D S S Gap. Crit. const. log( ) Gap. Crit.I Crit.II log( ) First and second order phase transitions Frustrated Magnets Interacting Fermions All on large (or infinite) 2D lattices!

15 Entanglement entropy and Tensor Networks Can Tensor Network methods reproduce the proper entanglement entropy? Tensor networks in holographic geometry: 1D MERA: S = log 2D MERA: S = 1D 2D S S Gap. Crit. const. log( ) Gap. Crit.I Crit.II log( ) Natural description of scale-invariant 1D systems First and second order phase transitions Frustrated Magnets Interacting Fermions All on large (or infinite) 2D lattices!

16 Entanglement entropy and Tensor Networks PEPS MERA 2D S log( ) Gap. Crit.I Crit.II 2D S log( ) Gap. Crit.I Crit.II e.g. Fermi liquids Spin Bose Metals Certain types of critical 2D phases cannot be properly addressed (even in principle) with current tensor-network techniques The entanglement structure of these systems is not properly understood The Crit.II phases are not fixed points of the RG flow

17 Outline Entanglement and tensor network methods Scaling of entanglement entropy in ground states Scaling of entanglement entropy in tensor network ansatz - physical geometry vs holographic geometry Comparison of entropy scaling: - ground states vs tensor network ansatz The branching MERA Decoupling a many-body theory Holographic trees Scaling of entropy in the branching MERA Example: S =log entropy scaling in 2D fermions Example: S =(log ) 2 entropy scaling in 1D fermions Work in progress... Classification of many-body entanglement via holographic geometry Generalised RG flow / RG fixed points

18 Entanglement entropy and Tensor Networks Entanglement entropy in MERA as boundary in holographic geometry: S( A) Ω A By considering exotic holographic geometries we obtain a more general class of MERA that reproduces more entanglement entropy Branching MERA Evenbly, Vidal, in preparation 2D S log( ) Gap. Crit.I Crit.II Potentially a good ansatz for 2D systems with a 1D Fermi surface?

19 ow energy decoupling and holographic branching MOTIVATION: at low energies, sometimes sets of degrees of freedom decouple Examples: 1D system: spin-charge separation electrons low energy spinons holons 2D systems with 1D Fermi surface (or 1D Bose surface) free fermions Fermi liquids spin Bose metal e.g. Block, Sheng, Motrunich, Fisher, arxiv: Sheng, Motrunich, Fisher, arxiv: D system low energy many gapless modes

20 simplified diagrammatic representation for the MERA (3) (2) (3) (2) (1) (1) (0) (0) w ( τ + 1) u ( τ )

21 Entanglement renormalization in the presence of decoupling (3) (3) (3, A) branching MERA (3, A) (3, B) (3, B) (2) (2) (2, A) (2, A) (2, B) (2, B) (1) (1) (1) (1) (0) (0) (0) (0) ( τ + 1) ( τ + 1, A) ( τ + 1, B) ( τ ) ( τ )

22 Entanglement entropy? given by boundary in exotic holographic geometry: S( A) Ω A branching MERA (3, A) (3, B) (2, A) (2, B) (1) Ω A (0) A

23 Entanglement entropy? contributions to entropy log Ω A A Holographic branching can increase size of Ω A branching MERA can reproduce more entanglement entropy!

24 Outline Entanglement and tensor network methods Scaling of entanglement entropy in ground states Scaling of entanglement entropy in tensor network ansatz - physical geometry vs holographic geometry Comparison of entropy scaling: - ground states vs tensor network ansatz The branching MERA Decoupling a many-body theory Holographic trees Scaling of entropy in the branching MERA Example: S =log entropy scaling in 2D fermions Example: S =(log ) 2 entropy scaling in 1D fermions Work in progress... Classification of many-body entanglement via holographic geometry Generalised RG flow / RG fixed points

25 Holographic tree Collection of independent theories (3) (3) (2) (2) (1) (0) (1) (0) ow energies (3, A) (3, A) (3, B) (3, B) (2, A) (2, A) (2, B) (2, B) Original theory (1) (1) (0) (0)

26 Holographic Trees regular b-ary holographic trees: b=1 b=2 b=3 Branching Parameter, b

27 Outline Entanglement and tensor network methods Scaling of entanglement entropy in ground states Scaling of entanglement entropy in tensor network ansatz - physical geometry vs holographic geometry Comparison of entropy scaling: - ground states vs tensor network ansatz The branching MERA Decoupling a many-body theory Holographic trees Scaling of entropy in the branching MERA Example: S =log entropy scaling in 2D fermions Example: S =(log ) 2 entropy scaling in 1D fermions Work in progress... Classification of many-body entanglement via holographic geometry Generalised RG flow / RG fixed points

28 Holographic trees and entanglement entropy b=2 branching MERA in D=1 spatial dimensions Ω A A /2 contributions to entropy log2 1 8= 2 = /2 log 2 4= 2 1 2= 2 0 1= 2 = /4 = /8 = /16 S entropic bulk law (!)

29 Holographic tree and entanglement entropy b=2 branching MERA in D=2 spatial dimensions contributions to entropy 8 ( /8) log 4 ( /4) 2 ( /2) S log S log logarithmic violation (!)

30 Is the (b=2) branching MERA a good ansatz for S =log phase?? Example: free fermions in 2D xy, ( x y..) H = a a + hc Yes! critical model type II (1D Fermi surface) S log Branching MERA Evenbly, Vidal, in preparation 2D S log( ) Gap. Crit.I Crit.II

31 Scaling of entanglement: free fermions vs branching MERA Free Fermions: Dimension of Fermi Surface, Γ Spatial dimension Γ=0 Γ=1 Γ=2 1D log() 2D log() 3D log() Regular branching MERA: Spatial dimension Branching Parameter, b b=1 b=2 b=4 b=8 1D log() 2D log() 2 3D log() 3 Ω A A Proposed relation between dimensionality of Fermi surface and branching parameter: Γ b = 2

32 Corrections to the Boundary aw for Entanglement Entropy D 1 S = f ( ) Arbitrary Corrections! Boundary aw Correction f( ) = 1 Boundary aw b<2 D-1 f( ) = log ogarithmic Violation b=2 D-1 f( ) = Bulk aw b=2 D Sub-logarithmic e.g. f( ) = log ( log ) Super-logarithmic e.g. f( ) = ( log ) f( ) = α α Example: 1D branching MERA with S (log ) 2

33 Outline Entanglement and tensor network methods Scaling of entanglement entropy in ground states Scaling of entanglement entropy in tensor network ansatz - physical geometry vs holographic geometry Comparison of entropy scaling: - ground states vs tensor network ansatz The branching MERA Decoupling a many-body theory Holographic trees Scaling of entropy in the branching MERA Example: S =log entropy scaling in 2D fermions Example: S =(log ) 2 entropy scaling in 1D fermions Work in progress... Classification of many-body entanglement via holographic geometry Generalised RG flow / RG fixed points

34 Branching MERA beyond Regular Holographic Trees 1D branching MERA: log S (log ) 2 log Can we find a Hamiltonian that has this ground state entropy scaling? φ( d) H= a a + aa aa ( + + ) ˆ ˆ ˆ ˆ ˆ ˆ 2 r d r r r d µ r r r= d= d r d 0 ( d ) φ( d) cos log 2 Yes!

35 Branching MERA beyond Regular Holographic Trees Holographic Tree: Hamiltonian: Gapped Ising 1 H = a a + a a + hc 2 ( ˆ ˆ ˆ ˆ r+ 1 r r r+ 1..) λ aa ˆ ˆ H= ( aˆ ˆ ˆ ˆ r+ 1ar + aa r r+ 1) r r r r Critical XX 1 2 r Dispersion: Ek ( ) π k π π k π

36 Branching MERA beyond Regular Holographic Trees Holographic Tree: Hamiltonian: H = 2 ( aˆ ˆ r+ dar+ hc..) r= d= d 0 φ( d) d r ˆ ˆ r r µ aa Dispersion: Ek ( ) k π Ek ( ) = sin cos π log 2 2 k π k π

37 Chemical Potential: µ = 0.6 µ = 0.2 µ = 0 0 µ = 0.6 µ = 0.2 µ = 0 0 0

38 Chemical Potential: µ = 0.6 µ = 0.2 µ = 0 Block entropy, S ( log ) 2 a + a log + a µ = 0 µ = 0.06 c = 5 µ = 0.2 c = 3 µ = 0.6 c =1 a 1 = / 6 a 2 = / 6 a 3 = S c log Blocksize,

39 Phase Transition in D=1 Free Fermions Chemical Potential: µ < 0 µ = 0 µ > 0 Crit: Crit.II: Crit: S 2 log S (log ) S log Holographic Geometry:

40 entanglement renormalization (real space RG) MERA holographic geometry (=spatial pattern of entanglement) new! decoupling branching MERA holographic tree Motivation: to find an ansatz that naturally decsribes 2D phases: S ~ log branching MERA: admits a holographic interpretation of entropy scaling (many different violations of boundary law) an efficient ansatz for critical phases beyond reach of MPS/PEPS (further on...) basis of an algorithm to simulate highly entangled critical phases of matter

41 entanglement renormalization (real space RG) MERA holographic geometry (=spatial pattern of entanglement) new! decoupling branching MERA holographic tree Work in progress... classification of entanglement in many-body ground states via holographic geometry systems with a surface of gapless modes as fixed points of a generalised RG flow.

42 Outline Entanglement and tensor network methods Scaling of entanglement entropy in ground states Scaling of entanglement entropy in tensor network ansatz - physical geometry vs holographic geometry Comparison of entropy scaling: - ground states vs tensor network ansatz The branching MERA Decoupling a many-body theory Holographic trees Scaling of entropy in the branching MERA Example: S =log entropy scaling in 2D fermions Example: S =(log ) 2 entropy scaling in 1D fermions Work in progress... Classification of many-body entanglement via holographic geometry Generalised RG flow / RG fixed points

43 Classification of entanglement in ground states 1D systems gapped crit. crit. crit.ii S const. S log S holographic geometry block entanglement entropy scaling Classification via holographic geometry: log S ( log ) 2 entanglement contributions from different length scales describes how different sets of degrees of freedom become decoupled at different length scales

44 Classification of entanglement in ground states 2D systems gapped crit.i crit.ii S S S log holographic geometry block entanglement entropy scaling Classification via holographic geometry: entanglement contributions from different length scales describes how different sets of degrees of freedom become decoupled at different length scales

45 Generalised RG flow Energy Momentum

46 Generalised RG flow: Toy example Ek ( ) 0 π k

47 Generalised Scale Invariance e.g. 1D free fermion toy model e.g. 2D free fermions (half filling) T T T H H H H H H H H H H H H H H H H Scale Invariance Generalised Scale Invariance finite number of gapless modes infinite (discrete) set of gapless modes 1D surface of gapless modes scale-invariant properties: scaling dimensions / scaling operators? correlators?

48 branching MERA (Evenbly, Vidal, in preparation) generalization of the MERA to more complex holographic geometries in order to produce violations of the boundary law for entanglement entropy scaling - e.g. 2D branching MERA with entanglement: an ansatz for 2D systems with a 1D Fermi surface? we propose to use holographic geometry to classify entanglement in many-body ground states - geometric understanding of violations to the boundary law new notions of RG flow / RG fixed points S log - 2D fermions (with 1D fermi surface) as RG fixed points