Bootstrapping the 3D Ising Model David Simmons-Duffin IAS Strings - PowerPoint PPT Presentation

Bootstrapping the 3D Ising Model David Simmons-Duffin IAS Strings 2014 with S. El-Showk, M. Paulos, F. Kos, D. Poland, S. Rychkov, A. Vichi

The Conformal Bootstrap Polyakov ’70: classify/solve CFTs using: • conformal symmetry • unitarity • associativity of the OPE Progress in d = 2 throughout 80’s and 90’s. Huge revival for d > 2 a few years ago...

CFT Review • Local operators O 1 ( x ) , O 2 ( x ) , ... • Scaling dimensions �O i ( x ) O i ( y ) � = | x − y | − 2∆ i • Operator Product Expansion (OPE) � f ijk x ∆ k − ∆ i − ∆ j ( O k (0) + . . . ) O i ( x ) O j (0) = k i = ∑ j k k • Unitarity: ∆ i bounded from below, f ijk are real

Bootstrap Revival • φ ( x ) : a real scalar primary operator. • It has the OPE � f φφ O x ∆ O − 2∆ φ ( O (0) + . . . ) φ ( x ) φ (0) = O Rattazzi, Rychkov, Tonni, Vichi ’08 : Bootstrap constraints on � φφφφ � imply universal bounds on • OPE coefficients f φφ O • Dimensions, spins ∆ O , ℓ O

Conformal Blocks & Crossing Symmetry 1 4 ❆❆ ✁✁ � O � φ ( x 1 ) φ ( x 2 ) φ ( x 3 ) φ ( x 4 ) � = ✁✁ ❆❆ O 2 3 Crossing Symmetry � � 1 4 1 4 ❍ ✟ ❍ ✟ ❆❆ ✁✁ � O − = 0 O ✁✁ ❆❆ ✟❍ ✟ ❍ O 2 3 2 3 � � � f 2 v ∆ φ g ∆ ,ℓ ( u, v ) − u ∆ φ g ∆ ,ℓ ( v, u ) = 0 φφ O O � �� � F ∆ ,ℓ ( u, v )

Bounds from Crossing Symmetry � f 2 0 = F 0 , 0 ( u, v ) + φφ O F ∆ ,ℓ ( u, v ) O • Make an assumption about spectrum of ∆ , ℓ ’s. • Try to find a linear functional α such that α ( F 0 , 0 ) > 0 α ( F ∆ ,ℓ ) ≥ 0 (convex optimization problem) • If α exists, assumption is ruled out.

Outline 1 Bounds in 3d CFTs 2 Mixed Correlators 3 Future Directions

Outline 1 Bounds in 3d CFTs 2 Mixed Correlators 3 Future Directions

Universal Bound in 3d CFTs [El-Showk, Paulos, Poland, Rychkov, DSD, Vichi ’12] � Ε 1.8 � 78 comp. � 3d Ising ? 1.6 1.4 1.2 0.64 � Σ 1.0 0.50 0.52 0.54 0.56 0.58 0.60 0.62 • ǫ ≡ lowest dimension scalar in σ × σ • Assumes only bootstrap constraints for � σσσσ �

3d O ( N ) Vector Models [Kos, Poland, DSD ’13] ∆ | φ | 2 O (20) 2 . 2 O (10) 2 O (6) 1 . 8 O (4) 1 . 6 O (2) Ising 1 . 4 1 . 2 ∆ φ 1 5 0 . 5 0 . 51 0 . 52 0 . 53

Fractional Spacetime Dimensions [El-Showk, Paulos, Poland, Rychkov, DSD, Vichi ’13] γ σ ≡ ∆ σ − d − 2 γ ǫ ≡ ∆ ǫ − ( d − 2) vs. 2 2 1.0 2.25 0.8 2.5 0.6 2.75 Γ Ε 3 0.4 3.25 3.5 0.2 Γ Ε � 2 Γ Σ 3.7 3.8 3.9 0.0 4 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Γ Σ

c -Minimization • Perhaps � σσσσ � in 3d Ising lies on the boundary of the space of unitary, crossing-symmetric 4-pt functions. Natural conjecture: Ising minimizes c ∝ � T µν T ρσ � [El-Showk, Paulos, Poland, Rychkov, DSD, Vichi ’14] C T � C T free 1.25 1.20 1.15 1.10 1.05 1.00 0.95 0.60 � Σ 0.50 0.52 0.54 0.56 0.58

c at High Precision c lower bound (153,190,231 comp.) 0.9473 0.94660 0.9472 0.9471 0.94655 0.9470 c/c free 0.9469 0.9468 0.51815 0.51820 0.9467 0.9466 0.9465 0.5179 0.5180 0.5181 0.5182 0.5183 0.5184 0.5185 ∆( σ )

Spectrum from c -Minimization [El-Showk, Paulos, Poland, Rychkov, DSD, Vichi ’14] year Method ν η ω 1998 ǫ -exp 0.63050(250) 0.03650(500) 0.814(18) 1998 3D exp 0.63040(130) 0.03350(250) 0.799(11) 2002 HT 0.63012(16) 0.03639(15) 0.825(50) 2003 MC 0.63020(12) 0.03680(20) 0.821(5) 2010 MC 0.63002(10) 0.03627(10) 0.832(6) c -min 0.62999(5) 0.03631(3) 0.8303(18) Critical exponents: ∆ ǫ ′ = 3 + ω . ∆ σ = 1 / 2 + η/ 2 , ∆ ǫ = 3 − 1 /ν,

Outline 1 Bounds in 3d CFTs 2 Mixed Correlators 3 Future Directions

Mixed Correlators [Kos, Poland, DSD ’14] • So far, bootstrap studies have focused on 4-pt function of identical operators � φφφφ � . • Full bootstrap requires crossing-symmetry & unitarity for all 4-pt functions. • Mixed correlator: � σσǫǫ � in 3d Ising. • Consequences of unitarity are trickier: � � σσǫǫ � = f σσ O f ǫǫ O g ∆ ,ℓ ( u, v ) O f σσ O f ǫǫ O not necessarily positive.

Positivity for Mixed Correlators • Consider � σσσσ � , � σσǫǫ � , � ǫǫǫǫ � together. Crossing symmetry says: � � � F (1 , 1) ∆ ,ℓ ( u, v ) F (1 , 2) � ∆ ,ℓ ( u, v ) f σσ O � � f σσ O f ǫǫ O � F (2 , 1) ∆ ,ℓ ( u, v ) F (2 , 2) f ǫǫ O ∆ ,ℓ ( u, v ) O + · · · = 0 • Look for functionals α : F ( u, v ) → R such that � � α ( F (1 , 1) ∆ ,ℓ ) α ( F (1 , 2) ∆ ,ℓ ) � 0 α ( F (2 , 1) ∆ ,ℓ ) α ( F (2 , 2) ∆ ,ℓ ) is positive semidefinite. Analog of α ( F ∆ ,ℓ ) ≥ 0 .

Mixed Correlator Bound for CFT 3 w/ Z 2 ∆ ǫ 1 . 6 1 . 4135 1 . 4 1 . 4125 1 . 2 1 . 4115 0 . 5181 0 . 5182 0 . 5183 ∆ σ 1 0 . 5 0 . 52 0 . 54 0 . 56 0 . 58 0 . 6 • Monte-Carlo, c -min conjecture, rigorous bound • Assuming σ, ǫ are only relevant scalars.

Outline 1 Bounds in 3d CFTs 2 Mixed Correlators 3 Future Directions

Future Directions • Improve optimization algorithms/precision • Find more boundary-dwelling CFTs ( [3d, 5d: Nakayama, Ohtsuki] [4d N = 2 , 4 , 6d N = (2 , 0) : Beem, Lemos, Liendo, Peelaers, Rastelli, van Rees] [4d N = 4 Alday, Bissi] [3d N = 8 : Chester, Lee, Pufu, Yacoby] ) • Mixed correlators in other theories • Four-point functions of operators with spin (stress tensor, symmetry currents) • Nonlocal operators [Liendo, Rastelli, van Rees ’12] [Gaiotto, Mazac, Paulos ’13] • Analytic results, new consistency conditions