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 ´ Ecole Polytechnique & CEA Saclay Based on: arXiv:1203.6064 with M. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, A. Vichi ✄ � arXiv:1211.2810 with M. Paulos ✂ ✁ May 16, 2013 Crete Center for Theoretical Physics

New Skins for an Old Ceremony The Conformal Bootstrap and the Ising Model Sheer El-Showk ´ Ecole Polytechnique & CEA Saclay Based on: arXiv:1203.6064 with M. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, A. Vichi ✄ � arXiv:1211.2810 with M. Paulos ✂ ✁ May 16, 2013 Crete Center for Theoretical Physics

Motivation & Approach Why return to the bootstrap? Conformal symmetry very powerful tool that goes largely unused in D > 2. 1 Completely non-perturbative tool to study field theories 2 ◮ Does not require SUSY, large N , or weak coupling. In D = 2 conformal symmetry enhanced to Virasoro symmetry 3 ◮ Allows us to completely solve some CFTs ( c < 1). Long term hope: generalize this to D > 2? 4 Approach ◮ Use only “global” conformal group, valid in all D . ◮ Our previous result: ◮ Constrained “landscape of CFTs” in D = 2 , 3 using conformal bootstrap. ◮ Certain CFTs (e.g. Ising model) sit at boundary of solution space. ◮ New result: “solve” spectrum & OPE of CFTs (in any D) on boundary. ◮ Check against the D = 2 Ising model. ◮ The Future: 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. 1 Completely non-perturbative tool to study field theories 2 ◮ Does not require SUSY, large N , or weak coupling. In D = 2 conformal symmetry enhanced to Virasoro symmetry 3 ◮ Allows us to completely solve some CFTs ( c < 1). Long term hope: generalize this to D > 2? 4 Approach ◮ Use only “global” conformal group, valid in all D . ◮ Our previous result: ◮ Constrained “landscape of CFTs” in D = 2 , 3 using conformal bootstrap. ◮ Certain CFTs (e.g. Ising model) sit at boundary of solution space. ◮ New result: “solve” spectrum & OPE of CFTs (in any D) on boundary. ◮ Check against the D = 2 Ising model. ◮ The Future: 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. 1 Completely non-perturbative tool to study field theories 2 ◮ Does not require SUSY, large N , or weak coupling. In D = 2 conformal symmetry enhanced to Virasoro symmetry 3 ◮ Allows us to completely solve some CFTs ( c < 1). Long term hope: generalize this to D > 2? 4 Approach ◮ Use only “global” conformal group, valid in all D . ◮ Our previous result: ◮ Constrained “landscape of CFTs” in D = 2 , 3 using conformal bootstrap. ◮ Certain CFTs (e.g. Ising model) sit at boundary of solution space. ◮ New result: “solve” spectrum & OPE of CFTs (in any D) on boundary. ◮ Check against the D = 2 Ising model. ◮ The Future: 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. 1 Completely non-perturbative tool to study field theories 2 ◮ Does not require SUSY, large N , or weak coupling. In D = 2 conformal symmetry enhanced to Virasoro symmetry 3 ◮ Allows us to completely solve some CFTs ( c < 1). Long term hope: generalize this to D > 2? 4 Approach ◮ Use only “global” conformal group, valid in all D . ◮ Our previous result: ◮ Constrained “landscape of CFTs” in D = 2 , 3 using conformal bootstrap. ◮ Certain CFTs (e.g. Ising model) sit at boundary of solution space. ◮ New result: “solve” spectrum & OPE of CFTs (in any D) on boundary. ◮ Check against the D = 2 Ising model. ◮ The Future: Apply this to D = 3 Ising model? 2

Outline 1 Motivation 2 The Ising Model 3 CFT Refresher 4 The Bootstrap & the Extremal Functional Method 5 Results: the 2d Ising model 6 The (Near) Future 7 Conclusions/Comments 3

The Ising model 4

The Ising Model Original Formulation Basic Definition ◮ Lattice theory with nearest neighbor interactions � H = − J s i s j < i , j > with s i = ± 1 (this is O ( N ) model with N = 1). Relevance ◮ Historical: 2d Ising model solved exactly. [Onsager, 1944]. ◮ Relation to E -expansion. ◮ “Simplest” CFT (universality class) ◮ Describes: Ferromagnetism 1 Liquid-vapour transition 2 . . . 3 5

The Ising Model A Field Theorist’s Perspective Continuum Limit ◮ To study fixed point can take continuum limit (and σ ( x ) ∈ R ) � � ( ∇ σ ( x )) 2 + t σ ( x ) 2 + a σ ( x ) 4 � d D x H = ◮ In D < 4 coefficient a is relevant and theory flows to a fixed point. E -expansion Wilson-Fisher set D = 4 − E and study critical point perturbatively. Setting E = 1 can compute anomolous dimensions in D = 3: [ σ ] = 0 . 5 → 0 . 518 . . . [ ǫ ] := [ σ 2 ] = 1 → 1 . 41 . . . [ ǫ ′ ] := [ σ 4 ] = 2 → 3 . 8 . . . 6

New Perspective At fixed point conformal symmetry emerges: ◮ Strongly constrains data of theory. ◮ Can we use symmetry to fix e.g. [ σ ] , [ ǫ ] , [ ǫ ′ ] , . . . ? ◮ Can we also fix interactions this way? 7

CFT Refresher 8

Conformal Symmetry in D > 2 Primary Operators Conformal symmetry: SO ( 1 , D − 1 ) × R 1 , D − 1 + D (Dilatations) + K µ (Special conformal) � �� � Poincare Highest 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 . ◮ Example: Viraso descendents L − 2 O are primaries in our language. 9

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 fixes all correlatiors in the CFT: ◮ 2-pt & 3-pt fixed: δ ij �O i O j � = x 2 ∆ i , �O i O j O k � ∼ C ijk ◮ 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 l C l � 12 D ( x 12 ,∂ x 2 ) O k ( x 2 ) � 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 ) � 10

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 fixes all correlatiors in the CFT: ◮ 2-pt & 3-pt fixed: δ ij �O i O j � = x 2 ∆ i , �O i O j O k � ∼ C ijk ◮ Higher pt functions contain no new dynamical info: � C k 12 C k � O 1 ( x 1 ) O 2 ( x 2 ) O 3 ( x 3 ) O 4 ( x 4 ) � = 34 G ∆ k , l k ( u , v ) � �� � � �� � � �� � k k C k l C l conformal block � 12 D ( x 12 ,∂ x 2 ) O k ( x 2 ) � 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 ) � 10

Crossing Symmettry CFT Background � φ 1 φ 2 φ 3 φ 4 � This procedure is not unique: Consistency requires equivalence of two different contractions � � C k 12 C k 34 G 12 ; 34 C k 14 C k 23 G 14 ; 23 ∆ k , l k ( u , v ) = ∆ k , l k ( u , v ) k k Functions G ab ; cd ∆ k , l k are conformal blocks (of “small” conformal group): ◮ Each G ∆ k , l k corresponds to one operator O k in OPE. ◮ Entirely kinematical : all dynamical information is in C k ij . ◮ u , v are independent conformal cross-ratios: u = x 12 x 34 x 13 x 24 , v = x 14 x 23 x 13 x 24 ◮ Crossing symmetry give non-perturbative constraints on (∆ k , C k ij ) . 11

CFT Background Recap What have we learned so far: CFTs completely specified by primary operator spectrum and OPE. 1 { ∆ i , l i } , { C ijk } for all O i This data allows us to compute all correlators. Constrained by crossing symmetry 2 Crossing symmetry equations is sum over primary operators O k : 3 � � 34 G 12 ; 34 23 G 14 ; 23 C k 12 C k C k 14 C k ∆ k , l k ( u , v ) = ∆ k , l k ( u , v ) O k O k G ∆ k , l k ( u , v ) encode contribution of primary O k and its decendents. 4 12

How Strong is Crossing Symmetry? 13

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 14