Bounds on the epsilon expansion Matthijs Hogervorst Ecole - - PowerPoint PPT Presentation

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!

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Multiscalar CFTs

Will look at system of N real scalars φi w/ quartic coupling in 2 d < 4 dimensions: L = 1 2(∂µφi)2 + 16π2Λ4−d 4! λijkl φiφjφkφl + counterterms. In MS in d = 4 − ε, ε ≪ 1, beta function reads β(λ)ijkl = −ελijkl + λijmnλklmn + λikmnλjlmn + λilmnλjkmn + higher loops. There exist perturbative solutions to β(λ) ≡ 0, of form λijkl = ελ(1)

ijkl + O(ε2).

Will look at one-loop term λ(1)

ijkl for rest of talk.

No need to discuss scheme dependencies, mass terms.

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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 β(λ)I = −ελI + CI

JKλJλK + . . .

e.g. could add Yukawas, φn-type interactions at diferent critical d. Small parameter can be something else (1/k in N = 2 Chern-Simons + matter).

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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? # of couplings grows as ∼

1 24N4.

N = 1: textbook. N = 2: solved in by [Osborn-Stergiou 2017], after 43 years! N = 3: 15 eqns, too many for Gr¨

• bner basis.

N 4: ? Different spirit: scan through theory space numerically? Difficult in its own right. [WIP with Z. Fisher]

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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) =

• i

bi(X) then ln B(X) (ln 3)(D(N) − 1) + ln 2 ∼

N≫1 0.0458 N4

where D(N) = N + 3 4

• is the dimension of the total space [Milnor-Oleinik-Thom].

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Discussion of (an)isotropy

Historically, often imposed isotropy: global symmetry group G has unique quadratic invariant δij. Many consequences in theory & practice: fields φi form irrep of G, φi(x)φj(0) = δij/|x|2∆φ can talk about critical exponent η unique mass operator O2 = δijφiφj, critical exponent ν typically O(few) number of quartic couplings I4 allowed, solvable tensor λijkl obeys various simplifying identities Without isotropy, RG picture is murky. ∼ 1

2N2 mass terms, ∼ 1 24N4 quartic

couplings.

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Famous solutions

Trivial theory λijkl = 0. Next: O(N) = N-vector = Heisenberg model λijkl = 1 N + 8 δijδkl + symm. which for N = 1, 2 is known as Ising/Wilson-Fisher resp. XY model. Zoo of other known solutions: cubic: O(N) deformed by

i φ4 i , discrete symmetry group |G| = 2N · N!

biconical-type solutions with symmetry O(N1) × O(N2) × . . .. “tensor” models built out of matrix fields Φab [SYK literature] . . . Note: solutions are additive.

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This talk

Complete classification every for N = O(few) seems too challenging. What will be done today: rule out parts of high-dimensional (∼

1 24N4) theory space

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.

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Orbits

Take sum of two Ising models λijklφiφjφkφl = 1 3(φ4

1 + φ4 2).

Rotate φ1,2: still solution — this is a 1d family of equivalent theories, S1 ⊂ R5. This is generic. Such sets of theories are called orbits: what you get when you act

• n λijkl with O(N).

In general dim(G) + dim(orbit) = dimO(N) = 1

2N(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 λ2

ijkl, λiijj, . . . , instead (cst. on orbits).

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Known results

Most basic invariant: norm λ2 = λ2

ijkl 0.

R-S recently showed that β(λ) = 0 ⇒ λ2     

1 36(3 + 4

√ 2) ≈ 0.240468 N = 2

1 12(1 + 2

√ 3) ≈ 0.372008 N = 3

1 8N

N 4 . Interpretation: fixed points can’t live in the whole space of dimension ∼ N4, they live inside a sphere of radius ∼ √ N. Proof: bound individual elements of λijkl using β = 0. For all but finite # of N, upper bound saturated. N = 3 mysterious.

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New: lower bound

Proceed in same spirit. Argue that any fixed point must satisfy: λijkl = 0

• r

λ 1 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 λijkl = λijmnλklmn + 2 terms. Now bound first term on RHS using Cauchy-Schwartz:

• mn

(λijmnλklmn)2

• mn

λ2

ijmn

• pq

λ2

klpq ⇒ RHS 3λ2.

But then λ 3λ2 ⇒ 3λ(λ − 1/3) 0.

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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 λijkl = RijSkl. By permutation symmetry of λijkl can argue that R = S and (S2)ij = tr(S)Sij. But then every eigenvalue ν of S must obey ν = 0

• r

ν = tr(S). Only possible if 1 non-zero eigenvalue. So ∃ ui s.t. Sij ∝ uiuj which implies bound saturated ⇔ λ = 1/3 ⇔ λijkl = 1

3uiujukul.

Conclusion: Ising model is the most weakly-coupled CFT, for any N!

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More bounds

To proceed, need to introduce more refined invariants, like λiijj =: a0 ∈ 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: a2 = 6 N + 4λ2

ijkk − 2(N + 2)

N + 4 a2 a4 = λ2 − 3 N(N + 2) a2

0 − a2.

Normalization of a2,4 given by projection operators. Naive bound 3 N(N + 2) a2

0 + a2 + a4 = λ2 1 8N

as follows from R-S.

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More bounds (2)

Start with simplest invariant, a0. Will argue that it lives inside a strip a0 ∈ [a−, a+] instead of R. Proof: apply O(N) projectors to the beta function equation, e.g. λiijj = λiimnλjjmn + 2 terms and decompose this into invariants. Messy but doable: 1 2N a0(N − a0) = λ2 + N + 4 12 a2. After tedious manipulation: N 2

• 1 −
• 1 − 8

9N

• < a0 N(N + 2)

N + 8 . Upper bound is the O(N) model. Any theories close to the lower bound a0 > 2 9 + O(1/N) = ?

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More bounds (3)

Next invariant a2 is measure of anisotropy. Appears in refinement of R-S: 1 9 < λ2 + 1 12(N + 4) a2 3N(N+2)

(N+8)2

N = 2, 3

1 8N

N 4 (notice that by construction a2 0). For isotropic theories a2 = 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.

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More bounds (4)

Finally can obtain 1 12(N + 16)a2 + a4 N(N + 2) 8(N + 8) . For N = 3 this is almost saturated (Ising + Ising + trivial CFT):

0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15

In particular can show that the anisotropy a2 is at most of order unity: a2 3/2 + O(1/N) even though a priori it could be of order N.

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Bounds on anomalous dimensions

Composite operators Oa have an anomalous dimension at one loop: ∆[Oa] = ∆classical[Oa] + γaε + O(ε2) which can be determined through eigenvalue problem (operator mixing!): V(x)Oa(0) =

• b

1 |x|4 C b

a (λ)Ob(0),

V := λijklφiφjφkφl Focus on operators of form O ∼ φr: Oa = Ta|i1···ir φi1 · · · φir for some tensor Ta of rank r 2. There are D(N, r) ∼ 1 r!Nr tensors: huge mixing matrix that needs to be diagonalized.

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Anomalous dimensions (3)

First, get bound: for operators with r copies of φ, we find Oa ∼ φr : |γa| r(r − 1) 2 λ. Second: bound saturated ↔ λ = Ising model and O = (u · φ)4. Third, get sum rules which are of the form: 1 D(N, r)

• a

γa = r(r − 1) N(N + 1)a0, 1 D(N, r)

• a

γ2

a = p0a2 0 + p1a2 + p2λ2

for some rational functions pi(N, r).

[covariance, r=0,1,3,4]

Note: for r = 2 + isotropic, bound on ν appeared in [Br´

ezin et al. 1973].

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Anomalous dimensions (3)

In particular, find that “typical anom. dim.” :=

• 1

D(N, r)

• a

γ2

a r≫1

r 2 3N + O(1/N). This has the same scaling as a bound previously found by [Kehrein-Wegner-Pismak], valid for the O(N) model: O(N) model, any operator with r fields : 0 γ

r≫1

3r 2 2(N + 8) (Evanescents.) Lower bound in general case? Can we repeat this for other operators? For example double-trace operators φi∂ℓ(∂2)nφj + (i ↔ j) currents φi∂ℓφj − (i ↔ j) . . . in complete generality?

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Anomalous dimensions of φi at NLO

The anomalous dimensions of operators φi are generated at two loops: ∆[Φa] = 1

2(d − 2) + γaε2 + O(ε3),

Φa =: Ta|iφi so slightly different framework. Now get

1 12λipqrλjpqrTa|j = γaTa|i.

The sum rule/bound reads 1 N

• a

γa = 1 12N λ2, 0 γa 1 12λ2. Again it can be shown that γ = 1 12λ2 ↔ λ = Ising and Φ = u · φ and if there are k zero eigenvalues ⇒ interaction involves at most N − k fields. Note: isotropic case already in [Br´

ezin et al. 1973].

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Complex/gauged case

Can instead look at N complex scalars, imposing overall U(1) + C — want to gauge later: L = |∂µφi|2 + 24π2 6 gijklφiφj(φ∗)k(φ∗)l. At fixed points, reality condition gijkl ∈ R sufficient for unitarity. Can be embedded in action with 2N real fields. No new interesting bounds (surprisingly?), except Ising is replaced by O(2) = XY model. More interesting: gauge the U(1) that rotates φi = (multi)scalar QED: L ′ = 1

4F 2 µν + |Dµφi|2 + quartic interaction.

Beta function for coupling e reads β(e) = −ε 2e + N 48π2 e3 + . . . so at any fixed point e∗ = 0

• r

e2

∗ ∼ ε

N .

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Known cases

Well-known solution: PSU(N) = SU(N)/ZN global symmetry, having an interaction V (φ) ∝ (

• i

|φi|2)2 which exists only for N 183 (famous result!). For even N, can split fields into two groups and get other fixed points with big global symmetries G ∼ SU(N/2)2 × Abelian factors.. These two only exist for N 198 — see [Benvenuti-Khachatryan 2019] × 2. The 3d physics of scalar QED (= Abelian Higgs) at N = O(few) is of serious experimental interest. Not in this talk. . .

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Gauged beta functions

The beta functions (omitting tensor structures and O(1) coefficients) are of the form β(g)ijkl = −εg + g 2 − e2g + e41 + 2 loops so right scaling with ε. Since e2

∗ ∼ 1/N, the beta functions differ from the ungauged case only by terms

• f order 1/N and 1/N2.

Phenomenology should not be too different from the real case for N O(few). Introduce invariants as before, only PSU(N) instead of O(N).

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Bounds on gauged case

Allowed region in theory space (PSU(N) dotted):

200 400 600 800 1000 1200 0.2 0.4 0.6 0.8 1.0

No solutions at all for N < N∗ := 90 + 24 √ 15 ≈ 182.9516. Bound for PSU(N) theory applies to all possible fixed points!

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Discussion

Like to think that Wilson-Fisher/Ising is a completely generic CFT. But in many ways, it’s a special point in theory space. To a lesser extent O(N). Useful to consider higher invariants? N = 3 case special. Can we show that there’s nothing new? Lemma about discrete symmetries. Representation as sum of quadratic forms (Hilbert)? For 4 N O(10), brute-force scan through landscape and compare with known results. Can we compare # of solutions to algebraic geometry bound? Explore bosonic QED more. Impossible (?) to explore numerically. Other theories.

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