New Skins for an Old Ceremony

Transcription

1 New Skins for an Old Ceremony The Conformal Bootstrap and the Ising Model Sheer El-Showk École Polytechnique & CEA Saclay Based on: arxiv: with M. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, A. Vichi arxiv: with M. Paulos June 12, 2013 Twelfth Workshop on Non-Perturbative QCD, IAP

2 New Skins for an Old Ceremony The Conformal Bootstrap and the Ising Model Sheer El-Showk École Polytechnique & CEA Saclay Based on: arxiv: with M. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin, A. Vichi arxiv: with M. Paulos June 12, 2013 Twelfth Workshop on Non-Perturbative QCD, IAP

3 Motivation & Approach Why return to the bootstrap? 1 Conformal symmetry very powerful tool not fully exploited in D > 2. 2 Completely non-perturbative tool to study field theories Does not require SUSY, large N, or weak coupling. 3 In D = 2 conformal symmetry enhanced to Virasoro symmetry Allows us to completely solve some CFTs (c < 1). 4 Long term hope: generalize this to D > 2? 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?

4 Motivation & Approach Why return to the bootstrap? 1 Conformal symmetry very powerful tool not fully exploited in D > 2. 2 Completely non-perturbative tool to study field theories Does not require SUSY, large N, or weak coupling. 3 In D = 2 conformal symmetry enhanced to Virasoro symmetry Allows us to completely solve some CFTs (c < 1). 4 Long term hope: generalize this to D > 2? 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?

5 Motivation & Approach Why return to the bootstrap? 1 Conformal symmetry very powerful tool not fully exploited in D > 2. 2 Completely non-perturbative tool to study field theories Does not require SUSY, large N, or weak coupling. 3 In D = 2 conformal symmetry enhanced to Virasoro symmetry Allows us to completely solve some CFTs (c < 1). 4 Long term hope: generalize this to D > 2? 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?

6 Motivation & Approach Why return to the bootstrap? 1 Conformal symmetry very powerful tool not fully exploited in D > 2. 2 Completely non-perturbative tool to study field theories Does not require SUSY, large N, or weak coupling. 3 In D = 2 conformal symmetry enhanced to Virasoro symmetry Allows us to completely solve some CFTs (c < 1). 4 Long term hope: generalize this to D > 2? 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?

7 3 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

8 4 The Ising model

9 5 The Ising Model Original Formulation Basic Definition Lattice theory with nearest neighbor interactions H = J <i,j> s i s 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: 1 Ferromagnetism 2 Liquid-vapour transition 3...

10 6 The Ising Model A Field Theorist s Perspective Continuum Limit To study fixed point can take continuum limit (and σ(x) R) H = d D x [ ( σ(x)) 2 + t σ(x) 2 + a σ(x) 4] 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] = [ɛ] := [σ 2 ] = [ɛ ] := [σ 4 ] =

11 7 New Perspective At fixed point conformal symmetry emerges: Strongly constrains data of theory. Can we use symmetry to fix e.g. [σ], [ɛ], [ɛ ],...? Can we also fix interactions this way?

12 8 CFT Refresher

13 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) O i(x) O j(0) k C k ij D(x, x)o k (0) 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: O io j = δij x 2 i, OiOjO k C ijk Higher pt functions contain no new dynamical info: O 1(x 1)O 2(x 2) }{{} k Ck 12 D(x 12, x2 )O k (x 2 ) O 3(x 3)O 4(x 4) }{{} l Cl 34 D(x 34, x4 )(x 3 )O l (x 4 ) }{{} k,l Ck 12 Cl 34 D(x 12,x 34, x2, x4 ) O k (x 2 )O l (x 4 )

14 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) O i(x) O j(0) k C k ij D(x, x)o k (0) 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: O io j = δij x 2 i, OiOjO k C ijk Higher pt functions contain no new dynamical info: O 1(x 1)O 2(x 2) }{{} k Ck 12 D(x 12, x2 )O k (x 2 ) O 3(x 3)O 4(x 4) }{{} l Cl 34 D(x 34, x4 )(x 3 )O l (x 4 ) }{{} k,l Ck 12 Cl 34 D(x 12,x 34, x2, x4 ) O k (x 2 )O l (x 4 ) = k C k 12C k 34 G k,l k (u, v) }{{} conformal block

15 10 Crossing Symmettry CFT Background This procedure is not unique: φ 1φ 2φ 3φ 4 Consistency requires equivalence of two different contractions k C k 12C k 34 G 12;34 k,l k (u, v) = k C k 14C k 23 G 14;23 k,l k (u, v) 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 12x 34 x 13 x 24, v = x 14x 23 x 13 x 24 Crossing symmetry give non-perturbative constraints on ( k, C k ij).

16 11 How Strong is Crossing Symmetry?

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

18 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-ɛ Ising Blue = solution may exists. White = No solution exists. 13 Certain values of σ, ɛ inconsistent with crossing symmetry. Solutions to crossing: 1 white region 0 solutions. 2 blue region solutions. 3 boundary 1 solution (unique)! Ising model special in two ways: 1 On boundary of allowed region. 2 At kink in boundary curve.

19 14 The Extremal Functional Method

20 Extremal Functional Method Solving CFTs on the boundary via Crossing 1 Consider four identical scalars: σ(x 1 )σ(x 2 )σ(x 3 )σ(x 4 ) dim(σ) = σ 2 Crossing equations simplify: sum of functions with positive coefficients. (Cσσ) k 2 }{{} O k 12;34 [G p k k,l k (x) G 14;23 k,l l (x)] = 0 }{{} F k (x) 3 Convert to geometric cone problem by expanding in derivatives: p 1 (F 1, F 1, F 1,... ) }{{} v 1 +p 2 (F 2, F 2, F 2,... ) }{{} v 2 +p 3 (F 3, F 3, F 3,... ) + = }{{} 0 v Each vector v k represents the contribution of an operator O k. 5 If { v 1, v 2,... } span a positive cone there is no solution. 6 Efficient numerical methods to check if vectors v k span a cone. 7 When cone unfolds solution is unique!

21 16 Spectrum and OPE from EFM? Checking the extremal functional method

22 7 How Powerful is Crossing Symmetry? To check our technique lets apply to 2d Ising model. Same plot in 2d. Completely solvable theory. Using Virasoro symmetry can compute full spectrum & OPE. 2.0 M 5, d Ising Can we reproduce using crossing symmetry & only global conformal group?

23 18 Spectrum from Extremal Functional Method

24 19 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 %) Mileage from Crossing Symmetry? 12 OPE coefficients to < 1% error. Spectrum better: 1 In 2d Ising expect operators at L, L + 1, L We find this structure up to L = operator dimensions < 1% error!

25 20 What about 3d Ising Model?

26 21 Current State-of-the-Art 3d Ising model Using E-expansion, Monte Carlo and other techniques find partial spectrum: Field: σ ɛ ɛ T µν C µνρλ Dim ( ): (3) 1.413(1) 3.84(4) (12) Spin (l): Only 5 operators and no OPE coefficients known for 3d Ising! Lots of room for improvement! Our Goal Compute these anomolous dimensions (and many more) and OPE coefficients using the bootstrap applied along the boundary curve.

27 Spectrum of the 3d Ising Model Computing 3d Spectrum from Boundary Functional? A first problem: what point on the boundary? what is correct value of σ? Ising Ising In D = 2 we know σ by other means. 2 Kink is not so sharp when we zoom in. 3 Gets sharper as we increase number of constraints should taylor expand to higher order!

28 3 Origin of the Kink Re-arrangment of spectrum? Spectrum near the kink undergoes rapid re-arrangement. Plots for next Scalar and Spin 2 Field Ising Ising Kink in (ɛ, σ) plot due to rapid rearrangement of higher dim spectrum. 2 Important to determine σ to high precision. 3 Does this hint at some analytic structure we can use?

29 The Future What s left to do? Honing in on the Ising model? Fix dimension of σ in 3d Ising using kink or other features. Use boundary functional to compute spectrum, OPE for 3d Ising. Compare with lattice or experiment! Additional constraints: add another correlator σσɛɛ. Study spectrum, OPE as a function of spacetime dimension. Exploring the technology How specific is this structure to Ising model? Can we impose more constraints and find new kinks for other CFTs? Can any CFT be solved by imposing a few constraints (gaps) and then solving crossing symmetry? What about SCFTs? Need to know structure of supersymmetric conformal blocks.

30 25 Thanks

31 26 Implementing Crossing Symmetry

32 27 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 Recall crossing symmetry constraint: O k (C k φφ) 2 G 12;34 k,l k (x) = O k (C k φφ) 2 G 14;23 k,l k (x)

33 27 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 Move everything to LHS: O k (C k φφ) 2 G 12;34 k,l k (x) O k (C k φφ) 2 G 14;23 k,l k (x) = 0

34 27 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: (Cφφ) k 2 }{{} O k 12;34 [G p k k,l k (x) G 14;23 k,l l (x)] = 0 }{{} F k (x)

35 27 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 (Cφφ) k 2 }{{} O k 12;34 [G p k k,l k (x) G 14;23 k,l l (x)] = 0 }{{} F k (x) Functions F k (x) are formally infinite dimensional vectors. p 1 (F 1, F 1, F 1,... ) }{{} v 1 +p 2 (F 2, F 2, F 2,... ) }{{} v 2 +p 3 (F 3, F 1 Each vector v k represents the contribution of an operator O k. 2 If { v 1, v 2,... } span a positive cone there is no solution. 3, F 3,... ) + = }{{} 0 v 3 3 Efficient numerical methods to check if vectors v k span a cone. 4 When cone unfolds solution is unique!

36 27 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 (Cφφ) k 2 }{{} O k 12;34 [G p k k,l k (x) G 14;23 k,l l (x)] = 0 }{{} F k (x) Functions F k (x) are formally infinite dimensional vectors. p 1 (F 1, F 1, F 1,... ) }{{} v 1 +p 2 (F 2, F 2, F 2,... ) }{{} v 2 +p 3 (F 3, F 1 Each vector v k represents the contribution of an operator O k. 2 If { v 1, v 2,... } span a positive cone there is no solution. 3, F 3,... ) + = }{{} 0 v 3 3 Efficient numerical methods to check if vectors v k span a cone. 4 When cone unfolds solution is unique!

37 28 Cones in Derivative Space 0.5 L 2 L 4 L 6 L 8 L 10 Two-derivative truncation Consider σσσσ with (σ) = We plot e.g. (F (1,1), F (3,0) ). Consider putative spectrum { k, l k } = unitarity l = 0 to 10 Vectors represent operators. 0.5 All vectors lie inside cone Inconsistent spectrum! 1.0 L 0

38 28 Cones in Derivative Space Derivatives Putative Spectrum L 2 L 4 L 6 L 8 L Unitarity Bound L 0

39 28 Cones in Derivative Space 0.5 L 2 L 4 L 6 L 8 L 10 Allow even more operators in putative spectrum. Scalar channel plays essential role. vectors span plane. In particular can find p k 0 p k F k,l k = 0 k crossing sym. can be satisfied Why does this work? Cone boundary defined by low-lying operators. Higher, l operators less important. Follows from convergence of CB expansion. 1.0 L 0

40 28 Cones in Derivative Space Derivatives Putative Spectrum L 2 L 4 L 6 L 8 L 10 L L 14 L 16 L 18 L 20 L 22 L 24 L 26 L 28 L L 0 0 Unitarity Bound

41 28 Cones in Derivative Space L 2 L 4 L 6 L 8 L 10 L L 14 L 16 L 18 L 20 L 22 L 24 L 26 L 28 L 30 Carving the Landscape of CFTs 1 Plot imposes necessary conditions. 2 Carve landscape via exclusion Any CFT in D = 3 with dim(σ) = must have another scalar with L 0

42 9 The Extremal Functional Method 0.5 L 2 L 4 L 6 L 8 L 10 Uniqueness of Boundary Functional Consider 0 < 0.76 No combination of vecs give a zero. p i F i 0 for p i > 0 i Consider 0 > 0.76 Many ways to for vects to give zero Families of possible {p i }. Neither spectrum nor OPE fixed. Consider 0 = 0.76 Only one way to form zero. 0.5 Non-zero p i fixed unique spectrum. 1.0 L 0 Value of p i := (C k ii )2 fixed unique OPE. Non-zero p i : 0.76, L = 0 = 2, L = 2 NOTE: Num operators = num components of F i

43 29 The Extremal Functional Method Uniqueness of Boundary Functional 0.5 L 2 L 4 L 6 L 8 L 10 Consider 0 < 0.76 No combination of vecs give a zero. p i F i 0 for p i > 0 i Consider 0 > 0.76 Many ways to for vects to give zero. Families of possible {p i } Neither spectrum nor OPE fixed. Consider 0 = 0.76 Only one way to form zero. 0.5 Non-zero p i fixed unique spectrum. Value of p i := (C k ii )2 fixed unique OPE. Non-zero p i : 0.76, L = 0 = 2, L = L 0 NOTE: Num operators = num components of F i

44 29 The Extremal Functional Method Uniqueness of Boundary Functional 0.5 L 2 L 4 L 6 L 8 L 10 Consider 0 < 0.76 No combination of vecs give a zero. p i F i 0 for p i > 0 i Consider 0 > 0.76 Many ways to for vects to give zero. Families of possible {p i } Neither spectrum nor OPE fixed. Consider 0 = 0.76 Only one way to form zero. 0.5 Non-zero p i fixed unique spectrum. Value of p i := (C k ii )2 fixed unique OPE. Non-zero p i : 0.76, L = 0 = 2, L = L 0 NOTE: Num operators = num components of F i

45 29 The Extremal Functional Method Uniqueness of Boundary Functional 0.5 L 2 L 4 L 6 L 8 L 10 Consider 0 < 0.76 No combination of vecs give a zero. p i F i 0 for p i > 0 i Consider 0 > 0.76 Many ways to for vects to give zero. Families of possible {p i } Neither spectrum nor OPE fixed. Consider 0 = 0.76 Only one way to form zero. 0.5 Non-zero p i fixed unique spectrum. Value of p i := (C k ii )2 fixed unique OPE. Non-zero p i : 0.76, L = 0 = 2, L = L 0 NOTE: Num operators = num components of F i

46 30 Cones in Derivative Space Vectors form cone no solution. No Solutions to Crossing Axes are derivatives: 0.5 F,l(x), F,l(x), F,l(x) 2 Vectors represents operators 3 All operators lie inside half-space. 4 0 not in positive cone

47 30 Cones in Derivative Space Cone unfolds giving unique solution. Unique Solution to Crossing Axes are derivatives: F,l(x), F,l(x), F,l(x) 2 Vectors represents operators 3 Boundary of cone (red) spans a plane. 4 0 in span of red vectors

48 30 Cones in Derivative Space As more operators added solutions no longer unique. Many Solutions to Crossing Axes are derivatives: F,l(x), F,l(x), F,l(x) 2 Vectors represents operators 3 Vectors span full space. 4 Many ways to form