Bounds on the epsilon expansion Matthijs Hogervorst ´ Ecole polytechnique f´ ed´ erale de Lausanne, Fields and Strings Laboratory September 18, 2019 Joint ICTP/SISSA string seminar based on 1909. εεεεε with Chiara Toldo (´ Ecole Polytechnique) Matthijs Hogervorst (EPFL) 18/9/2019 1 / 26
Classifying CFTs Eric P. has hopefully talked about the conformal bootstrap . Grandiose ambition: (infinite) list of all relativistic, unitary CFTs in d � 2. Pretty difficult (seriously!). Bootstrap great for systems with “simple” low-energy spectrum/OPEs. Classic examples: 3d Ising/ O ( N ) CFTs. Have few “fundamental” DoFs or big symmetry groups. Not great when low-energy spectrum is a mess: too many eqns for numerics & analytics [at this point in time]. Yet expect that many (most?) CFTs are of this form. Today: look at class of systems where we can make progress analytically. At least get a feeling for complexity! Matthijs Hogervorst (EPFL) 18/9/2019 2 / 26
Multiscalar CFTs Will look at system of N real scalars φ i w/ quartic coupling in 2 � d < 4 dimensions: 2( ∂ µ φ i ) 2 + 16 π 2 Λ 4 − d L = 1 λ ijkl φ i φ j φ k φ l + counterterms . 4! In MS in d = 4 − ε , ε ≪ 1, beta function reads β ( λ ) ijkl = − ελ ijkl + λ ijmn λ klmn + λ ikmn λ jlmn + λ ilmn λ jkmn + higher loops . There exist perturbative solutions to β ( λ ) ≡ 0, of form λ ijkl = ελ (1) ijkl + O ( ε 2 ) . Will look at one-loop term λ (1) ijkl for rest of talk. No need to discuss scheme dependencies, mass terms. Matthijs Hogervorst (EPFL) 18/9/2019 3 / 26
Multiscalar CFTs (2) Well-explored since birth of RG [Wilson 1972] + many, many others . Very relevant for stat mech, after computing higher loops and setting ε → 1 , 2. Today: other POV. Many obvious open questions: Can we classify, or at least count, all solutions for a given N ? Geometry of set of solutions? Are there conformal manifolds? What does a typical solution look like? Global symmetry? What can we say about observables (e.g. critical exponents)? Although focus on multiscalar theories in d = 4 − ε , lot of reasoning applies to any beta function of the form JK λ J λ K + . . . β ( λ ) I = − ελ I + C I e.g. could add Yukawas, φ n -type interactions at diferent critical d . Small parameter can be something else (1 / k in N = 2 Chern-Simons + matter). Matthijs Hogervorst (EPFL) 18/9/2019 4 / 26
State of the art What has been done? Construct (families of) solutions with large global symmetry (few couplings) Prove structural theorems about cases with few couplings (stability etc.) Classify all isotropic CFTs for low N � 6 Extensive study of group-theoretical properties. Without imposing symmetry, classification? 1 24 N 4 . # of couplings grows as ∼ N = 1: textbook. N = 2: solved in by [Osborn-Stergiou 2017] , after 43 years! N = 3: 15 eqns, too many for Gr¨ obner basis. N � 4: ? Different spirit: scan through theory space numerically? Difficult in its own right. [WIP with Z. Fisher] Matthijs Hogervorst (EPFL) 18/9/2019 5 / 26
Let X be the space of solutions inside the set of all real couplings. We can actually bound the total number of solutions from above, using some tricks from algebraic geometry. If we define � B ( X ) = b i ( X ) i then N ≫ 1 0 . 0458 N 4 ln B ( X ) � (ln 3)( D ( N ) − 1) + ln 2 ∼ where � N + 3 � D ( N ) = 4 is the dimension of the total space [ Milnor-Oleinik-Thom] . Matthijs Hogervorst (EPFL) 18/9/2019 6 / 26
Discussion of (an)isotropy Historically, often imposed isotropy : global symmetry group G has unique quadratic invariant δ ij . Many consequences in theory & practice: fields φ i form irrep of G , � φ i ( x ) φ j (0) � = δ ij / | x | 2∆ φ can talk about critical exponent η unique mass operator O 2 = δ ij φ i φ j , critical exponent ν typically O (few) number of quartic couplings I 4 allowed, solvable tensor λ ijkl obeys various simplifying identities 2 N 2 mass terms, ∼ 24 N 4 quartic Without isotropy, RG picture is murky. ∼ 1 1 couplings. Matthijs Hogervorst (EPFL) 18/9/2019 7 / 26
Famous solutions Trivial theory λ ijkl = 0. Next: O ( N ) = N -vector = Heisenberg model 1 λ ijkl = N + 8 δ ij δ kl + symm. which for N = 1 , 2 is known as Ising/Wilson-Fisher resp. XY model. Zoo of other known solutions: i , discrete symmetry group | G | = 2 N · N ! i φ 4 cubic: O ( N ) deformed by � biconical-type solutions with symmetry O ( N 1 ) × O ( N 2 ) × . . . . “tensor” models built out of matrix fields Φ ab [SYK literature] . . . Note: solutions are additive. Matthijs Hogervorst (EPFL) 18/9/2019 8 / 26
This talk Complete classification every for N = O (few) seems too challenging. What will be done today: 24 N 4 ) theory space 1 rule out parts of high-dimensional ( ∼ comments about (non)existence of conformal manifolds bounds on one-loop anomalous dimensions as well as generalization to N gauged complex scalars. First hard bounds found in [Br´ ezin–Le Guillou–Zinn-Justin 1973] for simple = isotropic systems. Recent revival of this strategy by [Osborn-Stergiou 2017] and [Rychkov-Stergiou 2018] . Will build on this work. Matthijs Hogervorst (EPFL) 18/9/2019 9 / 26
Orbits Take sum of two Ising models λ ijkl φ i φ j φ k φ l = 1 3( φ 4 1 + φ 4 2 ) . Rotate φ 1 , 2 : still solution — this is a 1d family of equivalent theories, S 1 ⊂ R 5 . This is generic. Such sets of theories are called orbits : what you get when you act on λ ijkl with O ( N ). In general dim( G ) + dim(orbit) = dim O ( N ) = 1 2 N ( N − 1) so the O ( N ) fixed point is a single point, but less symmetric theories are manifolds in the landscape of couplings. Easily observed in numerics! Consequence: instead of talking about coordinates in theory space, should discuss invariants λ 2 ijkl , λ iijj , . . . , instead (cst. on orbits). Matthijs Hogervorst (EPFL) 18/9/2019 10 / 26
Known results Most basic invariant: norm � λ � 2 = λ 2 ijkl � 0 . R-S recently showed that √ 1 36 (3 + 4 2) ≈ 0 . 240468 N = 2 √ β ( λ ) = 0 ⇒ � λ � 2 � 1 12 (1 + 2 3) ≈ 0 . 372008 N = 3 . 1 8 N N � 4 Interpretation: fixed points can’t live in the whole space of dimension ∼ N 4 , they √ live inside a sphere of radius ∼ N . Proof: bound individual elements of λ ijkl using β = 0. For all but finite # of N , upper bound saturated. N = 3 mysterious. Matthijs Hogervorst (EPFL) 18/9/2019 11 / 26
New: lower bound Proceed in same spirit. Argue that any fixed point must satisfy: � λ � � 1 λ ijkl = 0 or 3 . Interpretation: fixed points don’t live inside a disk, but inside an annulus. Or: can’t have arbitrarily weak CFTs. Proof: fixed point obeys λ ijkl = λ ijmn λ klmn + 2 terms . Now bound first term on RHS using Cauchy-Schwartz: ( λ ijmn λ klmn ) 2 � � � � λ 2 λ 2 klpq ⇒ � RHS � � 3 � λ � 2 . ijmn mn mn pq But then � λ � � 3 � λ � 2 ⇒ 3 � λ � ( � λ � − 1 / 3) � 0 . Matthijs Hogervorst (EPFL) 18/9/2019 12 / 26
Lower bound (2) C-S argument gives info about limiting cases. Here: learn that the bound is saturated if there exist matrices R , S such that λ ijkl = R ij S kl . By permutation symmetry of λ ijkl can argue that R = S and ( S 2 ) ij = tr( S ) S ij . But then every eigenvalue ν of S must obey ν = 0 or ν = tr( S ) . Only possible if � 1 non-zero eigenvalue. So ∃ u i s.t. S ij ∝ u i u j which implies λ ijkl = 1 bound saturated ⇔ � λ � = 1 / 3 ⇔ 3 u i u j u k u l . Conclusion: Ising model is the most weakly-coupled CFT, for any N ! Matthijs Hogervorst (EPFL) 18/9/2019 13 / 26
More bounds To proceed, need to introduce more refined invariants, like λ iijj =: a 0 ∈ R . Use that space of rank-4 tensors splits in irreps as vector space of all couplings = spin-0 ⊕ spin-2 ⊕ spin-4 with associated projection operators. Gives rise to invariants: 6 ijkk − 2( N + 2) N + 4 λ 2 a 2 a 2 = 0 N + 4 3 a 4 = � λ � 2 − N ( N + 2) a 2 0 − a 2 . Normalization of a 2 , 4 given by projection operators. Naive bound 3 0 + a 2 + a 4 = � λ � 2 � 1 N ( N + 2) a 2 0 � 8 N as follows from R-S. Matthijs Hogervorst (EPFL) 18/9/2019 14 / 26
More bounds (2) Start with simplest invariant, a 0 . Will argue that it lives inside a strip a 0 ∈ [ a − , a + ] instead of R . Proof: apply O ( N ) projectors to the beta function equation, e.g. λ iijj = λ iimn λ jjmn + 2 terms and decompose this into invariants. Messy but doable: 2 N a 0 ( N − a 0 ) = � λ � 2 + N + 4 1 a 2 . 12 After tedious manipulation: � � � N 1 − 8 < a 0 � N ( N + 2) 1 − . 2 9 N N + 8 Upper bound is the O ( N ) model. Any theories close to the lower bound a 0 > 2 9 + O (1 / N ) = ? Matthijs Hogervorst (EPFL) 18/9/2019 15 / 26
More bounds (3) Next invariant a 2 is measure of anisotropy. Appears in refinement of R-S: � 3 N ( N +2) N = 2 , 3 1 9 < � λ � 2 + 1 ( N +8) 2 12( N + 4) a 2 � 1 8 N N � 4 (notice that by construction a 2 � 0). For isotropic theories a 2 = 0, so for those theories it reduces to R-S for N � 4. For N = 2 there is a complete classification, skip this case. For N = 3 it shows that the O (3) model is the most strongly coupled CFT. Refinement only tiny: 0 . 371901 < 0 . 372008 despite the fact that proof is very different. Matthijs Hogervorst (EPFL) 18/9/2019 16 / 26
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