New Skins for an Old Ceremony The Conformal Bootstrap and the Ising - PowerPoint PPT Presentation

New Skins for an Old Ceremony The Conformal Bootstrap and the Ising Model Sheer El-Showk CEA Saclay Based on: arXiv:1203.6064 with M. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, A. Vichi arXiv:1211.XXXX with M. Paulos November 8, 2012 DAMTP, Cambridge

Motivation & Approach Why return to the bootstrap? ◮ Conformal symmetry very powerful tool that goes largely unused in D > 2. ◮ Completely non-perturbative tool to study field theories Does not require SUSY, large N , or weak coupling. 1 ◮ In D = 2 conformal symmetry enhanced to Virasoro symmetry Allows us to completely solve some CFTs ( c < 1). 1 ◮ How far could we get in D = 2 with only “global” conformal group? Approach ◮ Use only “global” conformal group, valid in all D . ◮ Constrain “landscape of CFTs” in D = 2 , 3 using conformal bootstrap. ◮ Result: CFTs (e.g. Ising model) sit at boundary of solution space. 1 ◮ Use to solve (partially) spectrum & OPE of 2D Ising without Virasoro. ◮ The Hope: Apply this to D = 3 Ising model? 2

Motivation & Approach Why return to the bootstrap? ◮ Conformal symmetry very powerful tool that goes largely unused in D > 2. ◮ Completely non-perturbative tool to study field theories Does not require SUSY, large N , or weak coupling. 1 ◮ In D = 2 conformal symmetry enhanced to Virasoro symmetry Allows us to completely solve some CFTs ( c < 1). 1 ◮ How far could we get in D = 2 with only “global” conformal group? Approach ◮ Use only “global” conformal group, valid in all D . ◮ Constrain “landscape of CFTs” in D = 2 , 3 using conformal bootstrap. ◮ Result: CFTs (e.g. Ising model) sit at boundary of solution space. 1 ◮ Use to solve (partially) spectrum & OPE of 2D Ising without Virasoro. ◮ The Hope: Apply this to D = 3 Ising model? 2

Motivation & Approach Why return to the bootstrap? ◮ Conformal symmetry very powerful tool that goes largely unused in D > 2. ◮ Completely non-perturbative tool to study field theories Does not require SUSY, large N , or weak coupling. 1 ◮ In D = 2 conformal symmetry enhanced to Virasoro symmetry Allows us to completely solve some CFTs ( c < 1). 1 ◮ How far could we get in D = 2 with only “global” conformal group? Approach ◮ Use only “global” conformal group, valid in all D . ◮ Constrain “landscape of CFTs” in D = 2 , 3 using conformal bootstrap. ◮ Result: CFTs (e.g. Ising model) sit at boundary of solution space. 1 ◮ Use to solve (partially) spectrum & OPE of 2D Ising without Virasoro. ◮ The Hope: Apply this to D = 3 Ising model? 2

Outline 1 Motivation 2 CFT Refresher ◮ What is crossing symmetry (and why is it useful)? 3 Results ◮ Using our techniques to “solve” 2d Ising. 4 Methodology ◮ The bootstrap in (gory) detail. 5 The (Near) Future ◮ Solving (??) the 3D Ising Model. 6 Conclusions/Comments 3

CFT Refresher 4

Conformal Symmetry in D > 2 Primary Operators Conformal symmetry: SO ( 1 , D − 1 ) × R 1 , D − 1 + D (Dilatations) + K µ (Special conformal) � �� � Poincare Higher weight representation built on primary operators O : Primary operators: K µ O ( 0 ) = 0 Descendents: P µ 1 . . . P µ n O ( 0 ) All dynamics of descendants fixed by those of primaries. Clarifications vs 2D ◮ Primaries O called quasi-primaries in D = 2. ◮ Descendents are with respect to “small” conformal group: L 0 , L ± 1 . ◮ Viraso descendents L − 2 O are primaries in our language. ◮ In this talk we always mean small conformal group (i.e. for descendants, conformal blocks, primaries, . . . ). 5

Spectrum and OPE CFT Background CFT defined by specifying: ◮ Spectrum S = {O i } of primary operators dimensions, spins: (∆ i , l i ) ◮ Operator Product Expansion (OPE) � C k O i ( x ) · O j ( 0 ) ∼ ij D ( x , ∂ x ) O k ( 0 ) k O i are primaries. Diff operator D ( x , ∂ x ) encodes descendent contributions. This data fixed all correlatiors in the CFT: ◮ 2-pt & 3-pt fixed: δ ij �O i O j � = �O i O j O k � ∼ C ijk x 2 ∆ i , ◮ Higher pt functions contain no new dynamical info: � O 1 ( x 1 ) O 2 ( x 2 ) O 3 ( x 3 ) O 4 ( x 4 ) � � �� � � �� � � k C k 12 D ( x 12 ,∂ x 2 ) O k ( x 2 ) � l C l 34 D ( x 34 ,∂ x 4 )( x 3 ) O l ( x 4 ) � �� � k , l C k 12 C l � 34 D ( x 12 , x 34 ,∂ x 2 ,∂ x 4 ) �O k ( x 2 ) O l ( x 4 ) � 6

Crossing Symmettry CFT Background � φ 1 φ 2 φ 3 φ 4 � This procedure is not unique: Consistency requires equivalence of two different contractions � � 34 G 12 ; 34 23 G 14 ; 23 C k 12 C k C k 14 C k ∆ k , l k ( x 1 , x 2 , x 3 , x 4 ) = ∆ k , l k ( x 1 , x 2 , x 3 , x 4 ) k k Functions G ab ; cd ∆ k , l k are conformal blocks (of “small” conformal group): ◮ Encode contribution of operator O k to double OPE contraction. ◮ Entirely kinematical : all dynamical information is in C k ij . ◮ Crossing symmetry give non-perturbative constraints on (∆ k , C k ij ) . 7

How Strong is Crossing Symmetry? 8

The “Landscape” of CFTs Constraints from Crossing Symmetry Constraining the spectrum ◮ Unitarity implies: Figure: A Putative Spectrum in D = 3 ∆ ≥ D − 2 ( l = 0 ) , � 2 6 ∆ ≥ l + D − 2 ( l ≥ 0 ) 5 ◮ “Carve” landscape of CFTs by imposing gap in scalar 4 sector. ◮ Fix lightest scalar: σ . 3 ◮ Vary next scalar: ǫ . Unitarity Bound 2 ◮ Spectrum otherwise Ε unconstrained : allow any Gap 1 Σ other operators. 0 L 0 2 4 9

Constraining Spectrum using Crossing Symmetry Is crossing symmetry consistent with a gap? σ four-point function: � σ 1 σ 2 σ 3 σ 4 � Crossing symmetric values of σ - ǫ � Ε ◮ Only certain values of σ, ǫ consistent with crossing symmetry. 1.8 Ising ◮ Solutions to crossing: 1.6 white region ⇒ 0 solutions. 1 blue region ⇒ ∞ solutions. 2 1.4 boundary ⇒ 1 solution (unique)! 3 ◮ Ising model special in two ways: 1.2 On boundary of allowed region. 1 0.80 � Σ 1.0 At kink in boundary curve. 2 0.50 0.55 0.60 0.65 0.70 0.75 Blue = solution exists. White = No solution exists. 10

Crossing Symmetry on the Boundary (the extremal functional method) 11

How Powerful is Crossing Symmetry? To check our technique lets apply to 2d Ising model. ◮ Get exacty same plot as above in 2d with “kink” at Ising point. ◮ Completely solvable theory. ◮ Using Virasoro symmetry can compute full spectrum & OPE. Can we reproduce using crossing symmetry & only “global” conformal group? Approach: Use same methods (even code) that works in 3d. 1 Apply to 2d Ising. 2 Look at unique solution on boundary. 3 Fix σ using value at “kink” or by hand. 4 Compute spectrum & OPE from unique solution on boundary. 5 Compare to known exact results. 6 12

Crossing Symmetry vs. Exact Results Exact (Virasoro) results compared to unique solution at “kink” on boundary: Spin 0 L Bootstrap Virasoro ∆ Error Bootstrap Virasoro OPE Error ∆ ∆ (in %) OPE OPE (in %) 0 1. 1 0.0000106812 0.500001 0.5 0.000140121 0 4.00145 4 0.03625 0.0156159 0.015625 0.0582036 0 8.035 8 0.4375 0.00019183 0.000219727 12.6962 3 . 99524 × 10 − 6 6 . 81196 × 10 − 6 0 12.175 12 1.45833 41.3496 But not just low spin: Spin 8 L Bootstrap Virasoro ∆ Error Bootstrap Virasoro OPE Error ∆ ∆ (in %) OPE OPE (in %) 1 . 25 × 10 − 9 8 8. 8 0.000731575 0.000734387 0.382872 8 9.01 9 0.111111 0.000283447 0.000273566 3.61199 8 12.02 12 0.166667 0.0000205853 0.0000193865 6.18355 13

How Powerful is Crossing Symmetry? Mileage from Crossing Symmetry? ◮ 12 OPE coefficients to < 1 % error. ◮ Spectrum better: In 2d Ising expect operators at L , L + 1 , L + 4. 1 We find this structure up to L = 20 2 ∼ 38 operator dimensions < 1 % error! Spec � � � Error � � � OPE Error � � � 4 15 3 10 2 5 1 0.001 0.01 0.1 1. 10. 100. 1. � 10 � 8 1. � 10 � 6 0.0001 0.01 1. 14

The Bootstrap in Detail 15

Crossing Symmetry Nuts and Bolts Bootstrap So how do we enforce crossing symmetry in practice? � φ ( x 1 ) φ ( x 2 ) φ ( x 3 ) φ ( x 4 ) � dim ( φ ) = ∆ φ Consider four identical scalars: Recall crossing symmetry constraint: � � φφ ) 2 G 12 ; 34 φφ ) 2 G 14 ; 23 ( C k ( C k ∆ k , l k ( x i ) = ∆ k , l k ( x i ) k k 16

Crossing Symmetry Nuts and Bolts Bootstrap So how do we enforce crossing symmetry in practice? � φ ( x 1 ) φ ( x 2 ) φ ( x 3 ) φ ( x 4 ) � dim ( φ ) = ∆ φ Consider four identical scalars: Move everything to LHS: � � φφ ) 2 G 12 ; 34 φφ ) 2 G 14 ; 23 ( C k ( C k ∆ k , l k ( x i ) − ∆ k , l k ( x i ) = 0 k k 16

Crossing Symmetry Nuts and Bolts Bootstrap So how do we enforce crossing symmetry in practice? Consider four identical scalars: � φ ( x 1 ) φ ( x 2 ) φ ( x 3 ) φ ( x 4 ) � dim ( φ ) = ∆ φ Express as sum of functions with positive coefficients: � φφ ) 2 [ G 12 ; 34 ∆ k , l k ( x i ) − G 14 ; 23 ( C k ∆ k , l l ( x i )] = 0 � �� � � �� � k p k F k ( x i ) 16