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 O1(x), O2(x), ...
• Scaling dimensions Oi(x)Oi(y) = |x − y|−2∆i
• Operator Product Expansion (OPE)

Oi(x)Oj(0) =

• k

fijk x∆k−∆i−∆j (Ok(0) + . . .)

= ∑

i j k k

• Unitarity: ∆i bounded from below, fijk are real

Bootstrap Revival

• φ(x): a real scalar primary operator.
• It has the OPE

φ(x)φ(0) =

• O

fφφO x∆O−2∆φ (O(0) + . . .) Rattazzi, Rychkov, Tonni, Vichi ’08: Bootstrap constraints on φφφφ imply universal bounds on

• OPE coefficients fφφO
• Dimensions, spins ∆O, ℓO

Conformal Blocks & Crossing Symmetry

φ(x1)φ(x2)φ(x3)φ(x4) =

✁✁ ❆❆ ✁✁ ❆❆

• O

1 2 4 3 O

Crossing Symmetry

✁✁ ❆❆ ✁✁ ❆❆

• O

−

• 1

2 4 3 1 2 4 3 O O

=

✟ ✟❍ ❍ ✟ ✟ ❍ ❍

• O

f 2

φφO

• v∆φg∆,ℓ(u, v) − u∆φg∆,ℓ(v, u)
• F∆,ℓ(u, v)

= 0

Bounds from Crossing Symmetry

0 = F0,0(u, v) +

• O

f 2

φφOF∆,ℓ(u, v)

• Make an assumption about spectrum of ∆, ℓ’s.
• Try to find a linear functional α such that

α(F0,0) > 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]

3d Ising ?

78 comp. 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 Σ 1.0 1.2 1.4 1.6 1.8

Ε

• ǫ ≡ lowest dimension scalar in σ × σ
• Assumes only bootstrap constraints for σσσσ

3d O(N) Vector Models [Kos, Poland, DSD ’13]

∆φ ∆|φ|2

Ising O(10) O(20) O(2) O(4) O(6)

0.51 0.52 0.53 0.5 1 1.2 1.4 1.6 1.8 2 2.2 5

Fractional Spacetime Dimensions [El-Showk,

Paulos, Poland, Rychkov, DSD, Vichi ’13]

γǫ ≡ ∆ǫ − (d − 2) vs. γσ ≡ ∆σ − d−2

2

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.0 0.2 0.4 0.6 0.8 1.0 ΓΣ ΓΕ

2 3 2.25 2.5 2.75 3.25 3.5 3.7 3.8 4 3.9 ΓΕ 2ΓΣ

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 ∝ TµνTρσ

[El-Showk, Paulos, Poland, Rychkov, DSD, Vichi ’14]

0.50 0.52 0.54 0.56 0.58 0.60 Σ 0.95 1.00 1.05 1.10 1.15 1.20 1.25

CTCTfree

c at High Precision

0.5179 0.5180 0.5181 0.5182 0.5183 0.5184 0.5185

∆(σ)

0.9465 0.9466 0.9467 0.9468 0.9469 0.9470 0.9471 0.9472 0.9473

c/cfree c lower bound (153,190,231 comp.)

0.51815 0.51820 0.94655 0.94660

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: ∆σ = 1/2 + η/2, ∆ǫ = 3 − 1/ν, ∆ǫ′ = 3 + ω .

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 =

• O

fσσOfǫǫOg∆,ℓ(u, v) fσσOfǫǫO not necessarily positive.

Positivity for Mixed Correlators

• Consider σσσσ, σσǫǫ, ǫǫǫǫ together.

Crossing symmetry says:

• O

fσσO fǫǫO

• F (1,1)

∆,ℓ (u, v) F (1,2) ∆,ℓ (u, v)

F (2,1)

∆,ℓ (u, v) F (2,2) ∆,ℓ (u, v)

fσσO fǫǫO

• + · · · = 0
• Look for functionals α : F(u, v) → R such that
• α(F (1,1)

∆,ℓ ) α(F (1,2) ∆,ℓ )

α(F (2,1)

∆,ℓ ) α(F (2,2) ∆,ℓ )

• is positive semidefinite. Analog of α(F∆,ℓ) ≥ 0.

Mixed Correlator Bound for CFT3 w/ Z2

∆σ ∆ǫ 0.5181 0.5182 0.5183 1.4115 1.4125 1.4135 0.5 0.52 0.54 0.56 0.58 0.6 1 1.2 1.4 1.6

• 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