Lawrence Berkeley National Laboratory Recent Work

Title Bootstrapping 3D fermions with global symmetries

Permalink https://escholarship.org/uc/item/5k0434j5

Journal Journal of High Energy Physics, 2018(1)

ISSN 1126-6708

Authors Iliesiu, Luca Kos, Filip Poland, David et al.

Publication Date 2018

DOI 10.1007/jhep01(2018)036

Peer reviewed

eScholarship.org Powered by the California Digital Library University of California JHEP01(2018)036 i ` e,f ψ and k ψ N j ) global ψ , i Springer N ψ ( h O January 9, 2018 : December 21, 2017 November 25, 2017 : : for which the Gross- N Published expansion at large Received Accepted /N [email protected] , and David Simmons-Duﬃn a Published for SISSA by ) symmetric Gross-Neveu-Yukawa ﬁxed . For values of https://doi.org/10.1007/JHEP01(2018)036 N N ( O Silviu S. Pufu transforms as a vector under an d i ψ [email protected] , David Poland, [email protected] , b,c . 3 1705.03484 The Authors. 1/N Expansion, Conformal and W Symmetry, Conformal Field Theory, Filip Kos, c

a We study the conformal bootstrap for 4-point functions of fermions , [email protected] Walter Burke Institute for1200 Theoretical Physics, E Caltech, California Blvd., Pasadena,E-mail: CA 91125, U.S.A. [email protected] Theoretical Physics Group, Lawrence Berkeley1 National Cyclotron Laboratory, Road, Berkeley, CADepartment 94720, of U.S.A. Physics, Yale University, 217 Prospect St, New Haven,School CT of 06520, Natural U.S.A. Sciences, Institute1 for Einstein Advanced Drive, Study, Princeton, NJ 08540, U.S.A. Joseph Henry Laboratories, Princeton University, Washington Road, Princeton, NJ 08544,Berkeley U.S.A. Center for Theoretical Physics,366 Department Le of Conte Physics, Hall University #7300, of Berkeley, California, CA 94720, U.S.A. b c e d a f Open Access Article funded by SCOAP Global Symmetries ArXiv ePrint: allow us to make nontrivialNeveu-Yukawa universality predictions classes at are small relevantour to results condensed-matter to systems, previous we analytic compare and numerical results. Keywords: Abstract: in parity-preserving 3dsymmetry. CFTs, We compute where bounds onin scaling our dimensions bounds and that central appear charges, topoints. finding coincide features Our with computations the are in perfect agreement with the 1 Luca Iliesiu, Bootstrapping 3D fermions with global symmetries JHEP01(2018)036 ψ 21 × ], the ψ 18 – 12 ] or of d-wave 21 , 8 20 ]. However, there are 15 11 , that belongs to a 3D CFT 10 ψ , 8 Majorana fermions [ 18 N 4 14 12 ] supersymmetric extensions of the plane showed features (kinks) that 26 16 } – σ 12 24 ∆ 3 , ], models of graphene [ ) parity-odd scalar appearing in the ψ 0 16 σ 19 ] has had striking success in constraining 3D ) vector models [ ∆ = 2 [ 3 { ) symmetry – N – 1 – ( 1 N N O of scaling dimension ∆ ( ψ O ] or ] and 13 9 – 4 = 1 [ ) current central charge N ) and the second ( N ( SDPB σ O ], 23 , 1 22 19 of the first ( 0 σ ] we initiated a study of the conformal bootstrap applied to the 4-point function of a single Majorana fermion 13 i In [ and ∆ 6.2 Bounds on the stress-energy tensor central charge 5.1 The lowest5.2 dimension parity-odd scalar The singlet lowest5.3 dimension parity-even scalar The singlet lowest dimension parity-odd symmetric traceless scalar 6.1 Bounds on the σ ψψψψ h with parity symmetry. We found∆ that by varying theOPE, gap the between the resulting scaling allowed dimensions region in the many CFTs of experimentalto interest be containing isolated fermionic via excitations,clude the the which 3D bootstrap are Gross-Neveu-Yukawa (GNY) by unlikely theories3D only with Hubbard considering model scalar on correlators.superconductors a honeycomb [ Such lattice theories [ in- Ising model, and more. In recent years theconformal conformal bootstrap field theories [ (CFTs)densed of matter relevance systems. to These critical studies focusedfunctions phenomena on of in systems scalar of statistical operators, bootstrap leading and equationsin to for con- models 4-point high such precision as determinations the of 3D Ising critical [ exponents 1 Introduction 7 Discussion A Implementation in 6 Universal bounds on central charges 3 Crossing and bootstrap with 4 Bootstrapping GNY models from5 scaling dimension bounds Universal bounds on scaling dimensions Contents 1 Introduction 2 Review of Gross-Neveu-Yukawa model JHEP01(2018)036 4 , a = 1 N ) and 0 N σ . From , where 0 i σ ` (∆ ψ σ k ψ corresponding j ψ ]. i N ψ h -expansion studies, 28 , ), though the story is 27 ∗ , we use the locations of OPE. As we will see, we N corresponds to the 0 j σ ψ × i ψ ) global symmetry. In particular, we we review the Gross-Neveu-Yukawa theories is worth further study. N 2 ( ∗ O = 1 supersymmetry [ N ], appearing close to the feature conjectured to – 2 – we study universal bounds on central charges. 13 . However, the precise value of , the family of kinks intersects the supersymmetry 6 N 0 σ . At small integer values of one can thus extract functional relations ∆ N has a possibility of becoming a unitary CFT. They are 0 ) global symmetry. In the present work, we will therefore σ N we study an expanded set of universal bounds on scaling N ( 5 ], where there is an additional fixed point besides the GNY O 28 ] to 4-point functions of Majorana fermions 13 = 1 supersymmetric Ising model. Our bootstrap results may support we discuss future directions. we discuss the crossing relations applicable to fermionic correlators with 2, and we suspect that it is this value of ∆ N 7 / 3 + 1 , again finding features that coincide with the GNY models. ) singlet operators do not make contact with the GNY models. (They do show, σ limit. At some value of ∆ . Next, in section N N ( N N = ∆ O ), which we expect should hold, conjecturally, even for the GNY theories away from 0 ψ σ ) symmetry and present the theoretical set-up for bootstrap applications. In section ) singlets and symmetric tensors appearing in the This paper is organized as follows. In section When making no assumptions about the spectrum, the bounds on the scaling dimen- In order to study the GNY theories more precisely, we should make use of the informa- (∆ transforms in the vector representation of an N N ψ ( ( i on scaling dimensions orIn by the placing bounds latter after case,values imposing of we further comment gaps ondimensions in a the of second spectrum. operators set andFinally, of in in kinks section section which solely appears for small currently unclear. The possible existence of GNY Model. In section O we specifically focus on the Gross-Neveu-Yukawa Model by either studying universal bounds e.g. recently discussed inmodel [ which for small valuesalso of similar to a secondcoincide kink with observed the in [ the existence of this second set of CFTs (which we refer to as GNY however, some evidence for thethe existence other of hand, new when “dead-end” we CFTs, doparity as make odd some we singlet, plausible will we assumptions discuss.) do aboutin find On the a gap the set above of GNY the kinks first models.second that set correspond of to Furthermore, expected kinks when scaling is dimensions visible. imposing This such is a reminiscent of gap structure for found in small values of compute universal bounds forfunction of the stress-energy tensor and current central charges as a sions of will use semidefinite programmingO methods to computeobserve universal a bounds sequence of on kinks theGNY in fixed leading the points space at of large allowed valuesthese theories of kinks which to match make precisely predictions with for the the critical exponents of these models. We additionally theory, which is expected to have enhanced tion that these theories have generalize the study ofψ [ to each of thesetracking kinks the could kinks be as inferred we∆ varied only ∆ approximately from thethe value large of ∆ line ∆ tracked the GNY fixed points at large JHEP01(2018)036 ] . ], i 1 ψ − 44 28 , (2.1) , ]. The 43 18 = 4 we will – 34 3 d 14 ) representa- ]. ], the Hubbard N ( 42 we also give the – O , with Lagrangian 33 , . 1 ). The dimensions φ 38 4 A 21 expansion [ , λφ also lists the values for 20 ) global symmetry with ) in addition to a parity − expansion — see table /N 3 2 N ( φ 2 − O m OPE by their ,...,N 1 2 ]. The Gross-Neveu-Yukawa de- . = 4) and for the semi-metallic to j 6 12 − ψ = 1 ), the two-index symmetric traceless N φ × i S µ i ψ φ∂ or in the 4 µ ∂ N 2 1 ], models of time-reversal symmetry-breaking − 19 and a parity-odd scalar field – 3 – i i ψ ψ ) ] and references therein). The models have also been gφ = 8). For convenience, table 33 + – N / ], models of 3-dimensional gapless semiconductors [ , this theory can be studied perturbatively in ∂ 28 ( i N -flux lattice [ 23 ψ , π 22 N =1 i X 1 2 − = 3, has been extensively studied using perturbative, analytic and nu- Majorana fermions = ] and numerically, using Monte Carlo simulations [ d N L ), or the two-index anti-symmetric representation ( 37 – T are coupling constants. The theory has an ]. Thus, we are optimistic that our results for this model can find future 35 λ 27 -expansion (see e.g. [ estimates for the central charges of the stress-energy tensor and of the conserved , while parity symmetry forbids a fermionic mass term. This critical point, thought and 2 g N m ) current, whose normalization we define in section In the context of this work, we consider a four-point function of fermionic operators Beyond serving as one of the most basic models for scalar-fermion interactions, the transforming in the fundamental representation ( N ( i representation ( of these operators canfor be such estimates estimated for at alarge- large few of the lowestO dimension operators. In table sition for spinless fermions oninsulator the transition honeycomb in lattice graphene ( ( the critical exponents obtained through several other methods. We can distinguish the operatorstion: appearing they in can the be in either the singlet representation ( and models that exhibitconductors emergent [ supersymmetry on theapplications boundary in a of number topologicalthose of super- interested different in experimentally-interesting the systems. applicationlist of the Specifically, bootstrap critical for exponents results for to the such systems, universality classes in relevant table to the metal-insulator tran- GNY-model and its variations frequentlytransitions appear in as universality condensed-matter classes systems foremployed as quantum with a phase model emergent for Lorentzmodel describing on symmetry. phase the transitions honeycomb in It and graphenein has [ d-wave superconductors been [ and in the recently studied from the perspectivecritical of exponents weakly-broken at higher this spinRG symmetry fixed methods in point [ [ have also been estimated through non-perturbative symmetry. For even valuesdimensions. of It has amass critical point that can beto achieved survive by down to appropriately tuningmerical the methods. scalar Specifically, the model has been studied in the 1 Here ψ While most of the bootstrap boundsfor presented the in space this of paper all will CFTs,the have as universal CFT implications mentioned data in the of Introduction,scription the our contains Gross-Neveu main model focus will at be criticality to [ study 2 Review of Gross-Neveu-Yukawa model JHEP01(2018)036 ]. 33 was – , the i ...... ψ 28 N models + + ∗ ) N ]. We list no ) symmetry 30 12 ... − − 13 ... N + ( N N ... + − − ) and thus it is O is decreased, the ) conserved current + - - 2.1 +6) ... N +36 +36 +6) +6) N +5 +6 3 N ( N N N N N -expansion) 2 N 8 N + O π 2( 6( 2( 9 N − theories become unitary than in GNY, such that , the dimensions of 2 − +132 +132 64 1 ∗ 2 2 π -expansion estimates for the ∗ N 3 2 9 1 + N N √ √ − ]. For large values of expansion, besides the critical N 28 ]. 2 + 3 + = = 1 45 )[ − ∗ J T ) anti-symmetric representation, as we , the GNY C C N ) ∆( ( N ...... N O + + + + + ) corrections are available for the dimensions , there are scaling dimensions in the theory ... 3 N N N N was computed in appendix B of [ ) symmetry N 2 2 2 2 N 4 2 2 3 . In fact, when computing scaling dimensions ) ( 32 32 64 32 π π π π j π 3 3 3 3 N O 8 + N ψ ( i – 4 – ( − ), respectively. At large coupling in the Lagrangian ( O ψ 3 + as determined in [ 1 + 1 2 + 2 + N -expansion for the GNY models can be found in [ λ ( N ) ∆ Central charge (large O , but irrelevant in GNY, and one could thus flow from ], where N ∗ ( ) ∆(large in the theory is lower in GNY 33 S A O N φ S S S T A V ( O , transforming in the fundamental vector of a global i 2 is non-unitary at large + + 2 . Prior work on the ψ Z ∗ + − + − − − Z φ denote the vector, singlet, rank-two traceless symmetric tensor, and rank-two ], while the dimension of A is relevant in GNY 16 representations and one-loop dimensions of low-lying operators in the 3D GNY , and – 4 Perturbative estimates for central charges in the GNY model T i φ φ µ 14 , ψ Perturbative estimates for lowest-lying scalars in the GNY model 2 J µν S ) ∂ Top: j , T µ i 2 3 representations and one-loop values for the central charges for the ψ . V ∂ φ i ψ φ φ in the UV to the GNY model in the IR. As we discuss below, the GNY ( 2 φ ψ ∗ µ As briefly mentioned in the Introduction, in the 4 J may appear in bootstrap bounds on the dimension of3 low-lying parity-odd operators. Crossing and bootstrapIn with this sectionMajorana we fermions, set up the conformal bootstrap constraints for 4-point functions of satisfied. Thus, it is possibleand that could at be small values detectedscaling of using dimension the of the conformalthe scalar bootstrap. operator Generically, oneGNY expects that the antisymmetric tensor representations of computed in [ additional fixed point hasexpected that a GNY negative at such a critical point, onethat finds become that for negative, large thus violatingscaling the unitarity dimensions bound. of However, as such operators grow and eventually the unitarity bound may be Table 1 models. Bottom: and for the stress-energy tensor at large GNY model one finds an additional fixed point (GNY corrections for dimension ofcould the not lowest operator find in any discussionscaling the of dimensions it are in availableassociated the in to literature. [ The most recent JHEP01(2018)036 , ] 4     I , ) t 13 2 if 2 if 3 , (3.2) (3.1) u,v 2 ↔ ( ↔ , and . In the . I 44 I a g     t ) r 2 ) = 1 )     ], where we I ) 4 O u,v λ 13 ( , . By explicitly ( odd ` u,v , of which two α I 44 ( s g O a O , ` 3 for I 44 2 − λ ) X g T 2 4 O ↔ ) λ O∈ 4 O ( )+ λ , ( odd even In addition, we can work a u,v operator r ( , ` , ` 1 ` a O − − I X 33 X S λ g A ( 2 I ) t O∈ a O∈ = 4 is symmetric under 1 3 O )+ ) global symmetry, the conformal ]. X 8 λ =1 a )+ ( I X 46 N even ( u,v = ( i ∝ u,v O , ` ) i ( I 33 ) 3 − X g 4 I 33 T 2 , s denotes whether the operators appearing g ) ,s 3 2 4 O∈ 3 O ) x )+ x λ ( 3 O ( ( ` ` odd λ 4) have odd parity. ( ψ u,v O , ) , ` ( – 5 – ) 3 2 − with independent coefficients X I ab even A ) singlet operators of even spin. Note that we work ,s 3 are anti-symmetric under the exchange 1 g , s = 3 3 a , , ` b O 2 N r x 2 O∈ a − λ ( X are symmetric under exchange 1 x ( )+ , ( S k a O I O 8. Additionally, when external dimensions are equal, 2 ψ t λ , ) u,v ψ O∈ 2 is odd. even = 1 7 ( 2 , ) , 1 ` I a ab , ` ,s ) singlets, two-index symmetric traceless tensors and two- 6 )+ =1 g 2 + , X , s x N T b O 1 is odd, while the structure a,b ( ( u,v λ x j ` ( ( O∈ O a O = 5 ψ 1 I ) ab     λ 1 g ψ odd 2 I , h b O kl ,s , ` δ 1 λ =1 + X x ij a O A ( δ a,b i λ denote ) global symmetry, we will thus generalize the analysis of [ 2 ψ O∈ N h N A     2 ( even − , ) O jk , ` jk +1 =1 , which are parity even δ δ + X ψ } and il S il a,b δ δ 2∆ 34 contributions vanish. For detailed definitions of the tensor structures T + O∈ x − 8 ): 2) have even parity and two ( jl , ,     even ]. jl , δ 4 +1 , S δ kl N ψ 3 ik δ ( 13 Similarly, 4-point functions of fermions have in general eight tensor structures Three 3-point functions between two fermions and a spin- , ` t ik δ These structures are simple to count using the formalism of [ ij δ = 1 2∆ 12 + 1 O δ is even and symmetric if is even and anti-symmetric if x S a +( × + and antisymmetric for the see [ Here index antisymmetric tensors. Theare superscript parity even or{ parity odd. So, forin example, a basis the such first that sum the in structures the second line runs over of ` ` case that the fermions transform asblock vectors decomposition under an of the 4-point function can be organized in terms of representations have four possible( tensor structures in a basis such that structures and contract the fermions with auxiliaryimposing commuting polarization the variables focused on the case of four identical Majorana fermions. of a parity-preserving 3d CFT. To keep our formulas compact, we follow the notation of [ JHEP01(2018)036 , = , .      7 7 T ,` , , (3.8) (3.9) (3.3) (3.4) (3.5) (3.6) (3.7) ` 2 6 2 6 ), are (3.10) ∆ , , , , , = I,R 44 , S u, v ~ = 1 = 5 = 1 = 5 V ( ` ) symmet- ·     I I I I I N ,` if if if if I,R ~α ( ,` ,` ij ∆ 2 ~ V ∆ ∆ O ) ,` ,` ,` ,` − R ∆ ∆ ∆ ∆ I A,ij, 4 O , , , , I I I I I A,ij, I B,ij, λ + − − + 44 44 44 44 ( A A . F F F F −A R 2 2 2 2 ` ) ) ) ) ) ∈     , are given by 4 O 4 O 4 O 4 O ,` λ λ X λ λ R v, u − = R . Here the sets of possible ( ( ( ( ` ( O A ,` odd odd odd odd ∆ I,A + . The vectors , , , ij or } , ` , ` , ` , ` ,` ~ V I ab, P P P P − − − − ∆ g , S S S S 1 2 , = 0 = 0 = 0 I,R 33 + O∈ O∈ O∈ O∈ ~ I B V I A I A     ψ + + + + S, T, · A ,` 3 can then be written as: A ∆ ,` ,` ,` ,` I ,` = u ∆ ∆ ∆ ∆ ∆ ~α , , , , I I I I + − A + ∆ 2 ↔ + − − + R ) 33 33 I 33 33 I I B A A I ) − F F F F R ,` B,ij, I all odd spins T T T 2 2 A,ij, 2 2 T 3 O ∆ { ) ) ) ) correspond to the contribution of the confor- T λ 3 O 3 O 3 O 3 O u, v ( ,l , respectively. , which sum the contributions of 2 = – 6 – ( 2 λ λ λ λ 2 N I N R N 2 A,ij, ( ( ( ( ∆ N ` ,` T A and their complements ∈ − ∆ − − ` I A/B I A/B − ,` even even even even 1 R X 1 1 I ab, A − R A 1 ` = , ` , ` , ` , ` g − − O I A/B,ij, P P P P 1 2 − − − − T S S S S + A + + ` and     ψ ,` + I A O∈ O∈ O∈ O∈ , and ∆ are defined as ∆ ) representations + + + + = S = I B v I ,` ,` ,` ,` I,R ,` S N A/B ab, I S , and A/B ( ∆ ∆ ∆ ∆ ~ T ≡ ` ∆ V I I I I I,T I ,l T ij · ), O + − − + ab, ab, ab, ab, ,` ~ ∆ , V ab, I F F F F ∆ ~α F I 3.5 b O b O b O b O and , ), + R ab, λ λ I λ λ A/B b O } F     )–( a O a O a O a O S I λ 3.9 A/B,ij, λ λ λ λ ,` ,` + T R 3.3 to ∆ ∆ , a O 2 2 2 2 ,l even even even even ,l λ 0 , , , , R ∆ 2 ∆ ` , ` , ` I , ` , ` I , A,ij, B,ij, =1 =1 =1 =1 P P P P ∈ I + + + + ij, S S S S S S =1 ,` a,b a,b a,b a,b F X + R     a,b O∈ O∈ O∈ O∈ I O A/B,ij, all even spins =                            S ). The functions { = = We can now exclude assumptions about the spectrum by applying a set of functionals The crossing equation under the exchange 1 I,S I,R 3.2 X = to equations ( I I ij B A ~ V S S I A 0= ~α where mal block where the sum isspins over all allowed in each representation, ` obtained from eq. ( in ( Similar definitions apply to ric tensors and antisymmetric tensors, following the pattern of quantum numbers appearing and where JHEP01(2018)036 . = O can i O with R λ (3.12) (3.13) (3.11) ∈ I O ) are pure i, j a O λ , , , . To make the R R R 2 ` ` ` ) and are related to , , , ∈ ∈ ∈ for all ( can R R R ` ` ` O i O ` ` ` j O /λ ∈ ∈ ∈ /λ i O ` ` ` with with with λ can contributes and the ) cannot be satisfied and a ) representation and parity j O R R R λ N O 3.5 ( with with with = O )–( , R R R 0 i O , for all for all for all 0 3.3 , that satisfies the inequalities: . Beyond unitarity, we can impose /λ I 11 ,` , we follow similar reasoning as above. I ~ V ~α O can for all for all for all , · , , , min i O R 0 0 0 I λ ∆ ~α = 1 ≥ ≥ , , , , ≥ 0 0 0 0 ,` ,` ,` O ) by ≤ − ,` – 7 – ∆ ∆ ∆ , , 2 > ≥ ≥ O O I,R I,R I,R ab, 33 44 3.9 , and since the operator is parity-even, we have ∆ ! 0 ,` ,` ,` ~ , ~ ~ V V V S I,R 0 ij, · ∆ ∆ ∆ · · , , can ~ i O V I,S I I I I,R I,R I,R ab, ) should hold for all values of the dimensions ∆ of = ab, 33 44 i O · λ ~α ~ ~α ~α V ~ ~ ~ V V V λ such that: I ) places an upper bound on ( · · ) representation · · O ~α

I α 3.11 I I I R N . ~α ~α ~α ~α ( can 3.12 a 1 b 1 j O O λ iδ λ a 1 λ = can 2 ) then imply the inequality: , a 1 i O , and λ λ =1 O . In this case, the canonically normalized OPE coefficients will be X obeying ( O 3.12 ` } a,b 2 I ∈I , ) − ~α 1 X , we will have i,j ( , spin 6 ) and ( O − } × { . The inequalities ( gives the set of structures to which the operator 3.9 2 , A O 1 I { or Eqs. ( Furthermore, in order to bound the OPE coefficient of a specific operator Concretely, we look for a vector of functionals = exists when imposing a gap, then the crossing eqs. ( I O ~α where the expression on theing r.h.s. a corresponds to functional the identity operator contribution. Find- For instance, if wedo want to in bound section theI OPE coefficient ofdetermined the by stress-energy the tensor, Ward identity as for we the stress-energy tensor. where are determined under some canonicalthe normalization OPE for coefficients the appearing operator in eq. ( imaginary, we have a negative signoperator in are the given first by line above. The OPE coefficientsdimension of ∆ the unit We now search for a functional places a lower boundfurther gaps on in the the scaling dimensionssector, dimensions ∆ of leading operators to in upper each each bounds sector if on a the suitable functional scaling can dimension be of found. the Specifically,CFT if lowest-lying with such operator such a scaling vector in of dimensions functionals cannot exist. Since in our conventions, all where we applyS, T, the functionalsoperators to present in blocks the spectrum. corresponding We always to assume the all theory representations is a unitary CFT, which JHEP01(2018)036 ], 7 [ ]. For 20), as (parity 28 cannot. , SDPB σ A . We focus 10 ψ , ) symmetric- ) that mini- 4 implementa- . , N ( 3 5 , T 3.12 O σ SDPB OPE by = 2 ) corrections [ j 4 OPE, but ψ N j ( /N × ) constraints by approx- ψ (1 i N for operators transforming × ψ O i 3.12 A ψ and ) or ( T 3.11 satisfying the relations ( I expansion up to ) antisymmetric tensors, respectively. We ~α N N ( = 8. O can all appear in – 8 – )-invariant scalars in T N N ( , , O A ]. A more detailed description of our OPE. Let us first focus on the most interesting bounds , σ 13 , while the resulting constraints on the space of CFTs are j T ψ A × σ, σ i ψ 10 and 20. The black symbols show the estimated values of the scaling obtained from the large- , that 4 T , σ 3 3 , for a unitary CFT containing fermions with scaling dimension ∆ . , T 0 σ , 0 , = 2 Kinks corresponding to the GNY model when bounding ∆ and ∆ I 11 . Due to the intensive computation required for each individual run, in this ψ ~ V N (parity even). As before, we use transcripts · 5 I . Upper bounds on the scaling dimension of the lowest-lying parity-odd ~α 20 we note strong agreement. This figure is a zoomed in version of figure ) symmetric traceless tensors and , − N ( We label the lowest dimension We search for functionals satisfying either the ( O = 10 well as some additional GNY results at odd) and as have seen in section In this section we startscalars off appearing by in presenting the severalin selected the context bounds of on GNY scalingfor models dimensions section and of leave a morework systematic we presentation will of present general bounds full bounds for a limited set of values of following the steps described intion [ is presented inpresented appendix below. 4 Bootstrapping GNY models from scaling dimension bounds bound as strong asmizes possible, we search for an imating the search as a semidefinite program and implementing it in the solver Figure 1 traceless tensor, ∆ on the cases dimensions ∆ N JHEP01(2018)036 . N , for , we σ T σ . In this 3 ψ , when > σ near 1, the 0 and ∆ 20. For lower ψ σ , ψ ) singlet, ∆ found for the GNY N ( ψ = 10 correspond to GNY O ]). They are reasonably 1 and 10. We notice that N , 33 4 , 3 , = 2 , when imposing a gap above this ψ . For a range of ∆ ]. While these second kinks are close N N 33 then gives a non-perturbative estimate 1 ] and [ = 10, but is inaccurate at smaller values of ) singlet parity-odd operator ∆ 28 . The lower kinks appear close to the three-loop N N as GNY models with ( . N = 10 the second kink does not exist at all. 1 O – 9 – N N expansion (black markers) for the scaling dimension of results for the scaling dimension ∆ N N , for N models (green hollow markers), obtained after performing a -expansion results for the dimensions of the GNY models after ∗ for different values of ψ ). 3. Once again, we focus on the cases approximation (see eqs. (11)–(13) and table II in [ ]. Using the large- 1] ,... > , 4 00 0 [2 in the GNY models at all σ , σ 0 we show the most general upper bounds on the dimension of the symmetric . Estimates from the large- T σ σ 1 1 , it is plausible to conjecture that the other kinks in figure as a function of ∆ N . Bound on the scaling dimension of the lowest-lying parity-odd T approximation, following the methods in [ imposing a gap above it in this sector up to the scaling dimension ∆ σ -expansion estimates for small and ∆ 2] , ψ [1 While it is satisfying to see that the GNY-model may saturate the universal bounds In figure also agree well with the position of the kink for Bounds on scaling dimension of lowest σ -expansion estimates for the GNY in the space of allowed scaling dimensions for this sector, we would want to determine the can identify the bottomvalues two of kinks in figure models as well. Usingof this ∆ as a conjecture, figure tensor bounds overlap and thensense, depart all from of the the bounds sharedkink seem to curve observed have in at the [ a feature of critical a value “kink,” of reminiscent of ∆ the Ising model Pade to the Higher dimension scalars inprimes a (e.g., given representation are labelled by increasing number of kinks in figure ∆ The red markers are theperforming three-loop a Pade close to the upper kinks in the bounds for small Figure 2 a unitary CFT containingoperator fermions up with to scaling ∆ when dimension imposing ∆ such a gap, we observe features that are close to the values of ∆ JHEP01(2018)036 σ = N are 0 σ σ = 10, 3. This theories , ∆ N ∗ > 1 -expansion 0 ) symmetry and ∆ σ models. N ψ ( ∗ ) as a proxy for O T . We aim here to 0 σ -expansion, shown σ for the GNY-model models. Specifically, : we plot the allowed 0 σ ∗ σ ]. . We take it as a working . 33 ) symmetry imposed, we N N N ( and ∆ were kindly provided by Grigory O too large. ] and [ ψ While this is encouraging, our ∗ 0 28 σ 2 estimated by the , for each value, we observe the existence ) when imposing the gap ∆ σ ) singlet parity-odd scalar sector of the σ with the dimension ∆ -expansion results for the GNY N , the position of this feature once again sits N 0 ( σ σ , ∆ O ]. There we did not impose ψ and ∆ 13 – 10 – expansion for the GNY theory: e.g. for ψ GNY models, where, as shown in table ], we can reconstruct the spectra of theories close to N ) representations and obtain tighter bounds, or, using N 48 N , ( O 47 . At small values of . In the present work, with the σ N is not larger than the correct value in the GNY model the kink . In the present work we will mostly focus on the former strategy, 3 is disfavored by the 0 1 σ -expansion also provides a possible explanation for the bottom set of > through a black square. We conjecture that for lower values of and ∆ 0 σ 2 ψ -expansion Padéestimates for GNY and GNY , but is not a priori justified for small values of : they may be alternative fixed-points, dubbed GNY ... 2 explicitly shows the consequence of imposing a gap on ∆ )+ 2 N 2 π ( / Interestingly, the For the set of kinks with higher values of ∆ Figure As an example, we impose gaps in the The three-loop 2 . For the each value of the gap, we observed a kink coinciding with the corresponding the values for theby the scaling green dimensions shapes, ∆ assumption are that close ∆ to the lower set of kinks. Tarnopolsky in private discussions, using the methods presented in [ at strong coupling.(shown This conjecture by is the further redsomewhat supported close shapes) by to whose results the top estimates from set the for of the kinks even scalingkinks for dimensions in small figure ∆ values of then discuss the connection between the bottom set ofnear kinks the and value the predicted GNY indicated by the in large- figure the kinks can be used to read off the scaling dimensions ∆ particular sector. By imposing the gaprevealing we new carve smoothed out the out allowed region “kinks”dimension below for on the each the free value theory of boundary ofof the two distinct allowed kinks region (besides for thatWe corresponding the will to scaling first the free comment theory, on in the the upper association left of corner). the top set of kinks with the GNY models, and region in the space ofassumption scaling certainly dimensions holds (∆ for3+64 large- hypothesis, as it has an appealing interpretation that there is only one relevant scalar in this N GNY values for ∆ have the possibility to findthe a assumed stable gap GNY on kinkcan ∆ for exist, a and range it of should gap disappear assumptions. only As once long we as choose ∆ though we will mention some preliminary results from studying extremal functionals below. theory. We will thus assumeand the assume existence a of gap an operatormake above in it a this until plot sector the analogous with operatorand to dimension used figure ∆ the 3 gap to of the ref. second [ parity odd scalar (which corresponds to more of the spectrumThere we are will two possibilities however in need thedimension to present of context: operators restrict in we the certain can either spacethe assume extremal of gaps functional CFTs for method the that [ scaling the we kinks study. seen in figure CFT data for operators in all the sectors of the low-lying spectrum. In order to learn about JHEP01(2018)036 ] 28 ] ] ] ] ] 40 41 41 41 40 obtained 5] , ] come from ]). [1 32 32 = 1 are taken N are obtained from ] for the anomalous ] 1.20(1) [ ] 1.30(5) [ /ν 33 ] 0.38(1) [ ] 0.62(1) [ ]- ]- ]- ]- ] 0.45(3) [ ]- ]- ]- 36 36 36 36 36 35 35 49 49 49 36 35 1), and correlation length directly. Other dimensions − 2 approximations obtained from ψ ]). The results shown from [ 1] , 33 and [2 ]). The results from [ 1 ] 0.994(2) [ ] 0.0276 [ ] 0.7765 [ = 2(∆ 28 28 28 and, respectively, the Padé ψ 28 η 2] , ] 0.180 [ ] 0.180 [ ] 1.075(4) [ ] 0.0645 [ ] 0.550 [ ] 1.408 [ [4 28 28 28 28 28 28 and ], 0.74 [ 1 2 ], 0.948 [ ], 0.042 [ ] 1.614 [ ] 0.112 [ ] 0.550 [ , Padé 2. The bootstrap results for 1 32 32 32 33 33 33 − 2] / , σ – 11 – [4 +1 ], 1.412 [ ], 0.176 [ ], 0.176 [ ], 0.852 [ ], 0.096 [ ], 0.506 [ 8. Conformal bootstrap results for ∆ -exp. Func. RG Monte Carlo σ , 33 33 33 33 33 33 4 , 2 ], 0.745 [ = 2 = ∆ ], 0.082 [ ], 0.931 [ , σ ψ 33 33 33 η expansion (see table 1 in [ can be obtained from figures = 1 T σ N − ] by intersecting the curve of kink found by imposing a gap on the 13 and 2 3 4 8 10 20 , for 3.49 3.47 3.45 3.29 3.24 3.14 3.02 3.09 3.32 3.52 3.59 3.39 2.14 2.17 2.25 2.12 2.09 2.03 and 4 σ 2.445 2.358 2.293 2.153 2.121 2.059 0.660 0.724 0.772 0.871 0.898 0.944 1.067 1.054 1.042 1.021 1.017 1.008 ] 0.162 [ ] 0.162 [ , ∆ ψ 13 13 − - 1.419 [ 0.86 1.493 [ 0.76 1.166 [ 0.88 1.048 [ 0.134 0.137 [ 0.320 0.282 [ 0.084 0.102 [ 0.544 0.463 [ 0.044 0.074 [ 0.742 0.672 [ = 3 with the SUSY line ∆ using the extremal functional method. The results shown from [ 0 0 0.164 [ 0.164 [ T T σ ψ σ σ Conf. boot. σ /ν N ∆ ∆ ∆ ∆ ∆ ∆ . Anomalous dimensions, . Dimensions of some of the low-dimensional operators in GNY models obtained in this -expansion at four loops with no Padéresummation performed (see table 2 in [ -expansion at three loops (see eqs. (11)–(13) and table II in [ σ σ σ σ ψ ψ ψ ψ = 1 = 2 = 4 = 8 /ν /ν /ν /ν η η η η η η η η 1 1 1 1 N N N N Table 2 work. Dimensions of were obtained from zeros of the extremal functionals and are thus given to less precision. Table 3 estimating ∆ dimensions and for the correlationthe length exponent are thefor Padé the same exponents are thefrom two-sided the Padé two-loop 2 + the exponent, 1 from our previous workdimension in ∆ [ JHEP01(2018)036 . 4 (4.1) OPE. j = 2 we ψ N × i ].) We have , ψ 9 1 ν ). We explore , 5 , the correlation N − ( 2), corresponding ψ than in section , η O -expansion Padéap- ψ − = 3 (corresponding to and It will be important to σ = 4 ) = (1 σ 3 σ η 9. ∆ ∆ . . For instance, at , 1 σ ψ ≈ 0 , the bound then increases mono- models in the future. σ , ∗ N ψ 2 η + , for any 3D parity-invariant CFT with a 2 ψ − . 3 2 2 – 12 – = ψ ∆ = 4, the estimate ∆ -expansion, the functional RG method, and Monte Carlo N ]. However, both our bootstrap results and the perturbative , as a function of ∆ 28 σ 2 σ η fermions. For each value of ] where no global symmetries were imposed. = 1 in order to extrapolate to three dimensions.) We note that + N 2 13 ]. To be effective, the extremal functional method requires a point , and the scaling dimensions computed in this paper are given by − 1 − 48 , ν = 1 5. , we give a comparison of critical exponents obtained from our bootstrap . 47 σ 2 , 3 ∆ ) symmetry. The bound starts at the point (∆ ≈ 9 , we plot universal upper bounds on the scaling dimension of the lowest-lying , 0 N 5 σ ( 3 O In table Thus, by imposing a minimal set of gaps, we find kinks corresponding to the GNY However, preliminary investigations of extremal functionals suggest the lower set of kinks may persist 3 tonically up to thea point marginal where it operator). intersectsafter At the which these horizontal the line intersection bounds ∆ points, plateau a at sharp much vertical higher discontinuity values occurs, down of to ∆ ∆ 5.1 The lowest dimensionIn parity-odd figure scalar singlet parity-odd scalar singlet ∆ global to the free theory with We consider scalars ofthe different bounds parities at in somewhatThe various results representations higher exhibit of values some of unusualfermion jumps bootstrap the reminiscent of fermion of [ dimension the features ∆ previously seen in the results differ significantly from the Monte Carlo simulations. 5 Universal bounds on scalingIn this dimensions section we present additional general bounds on low-lying scalars in the where one should set our results for theproximations anomalous performed dimensions in are [ closest to some of the study as well as resultssimulations. from the (The relation betweenlength the exponent anomalous dimensions on the boundary of(For the example, allowed region because of the CFTtional bounds data method that on gives is the good closeperformed Ising estimates to a preliminary model the for study theory several are of of operatorsresulting the so interest. extremal spectra in strong, functionals are that in the somewhat the theory extremaloperators, case noisy, [ of which but func- GNY we they models. have allow included The an in estimate table of low dimension scalar perform a bootstrap study better tailored to the GNY models that give informationever, we about can potentially some studymethod operator all [ dimensions sectors of in the those GNY model theories. using the How- extremal functional which suggest, for instance for JHEP01(2018)036 = 2. ], we σ 13 ) singlet, . 175 must σ . N 1 ( N O < ψ = 2 we conclude N with the scaling of the for which the CFTs must ψ increases with ψ 16. The intersection points ]. In figure 1 from [ σ 175. ≈ . 13 . Above, we focus on the cases 1 σ ψ 27 which would have very large . > ], the observed jump is similar to 1 ψ 13 ≈ ] when bounding the scaling dimension of ψ 13 OPE. For instance, for j ψ = 10 we find ∆ × – 13 – i (2) global symmetry that have ∆ N , while the jump in ∆ ψ O 7. When bounding dimensions in the parity-even . 1. As noted in [ N = 3. . 7 5 σ ≈ 1, the bound approaches the free theory value of ∆ ≈ are reminiscent of features previously encountered in σ 3 → 5, while for ψ . for larger 27, the bound for the dimension of the lowest-lying parity-odd 9 . 1 ψ 3 to ∆ ≈ ≈ σ ≈ ψ σ ]. By analogy with the existence of the 3D Ising model at the location 6 is precisely marginal, ∆ that we study. Once again, all jumps occur once the bound intersects the horizontal σ 10 and 20. As ∆ N , 4 . Upper bounds on the scaling dimension of the lowest-lying parity-odd , Upper bounds on scaling dimension of lowest-lying parity-odd singlet, ∆ 3 , The jumps shown in figure From these bounds, we can at least determine the values of ∆ = 2 , for a unitary CFT containing fermions with dimension ∆ σ lowest-lying parity-even scalar ∆ ones corresponding to themixed-correlators 3D [ Ising model, whenof studying the scalar jumps, dimension boundsCFT we without using therefore any relevant conjectured scalar that operators, there with ∆ exists a “dead-end” parity-invariant studies of fermions withoutobserved a that continuous when ∆ global symmetryscalar [ jumps from ∆ sector for such theories, one finds a kink at the same value of ∆ have a relevant parity-odd singlet inthat the 3D parity invariant CFTsalso with have an at least onehave relevant no parity-odd relevant singlet parity-odd scalar. singlet scalar Conversely, all must such have ∆ theories that each value of line on which find a plateau around ∆ occur at lower values of ∆ Figure 3 ∆ N Reminiscent of the jump noticedthe in lowest-lying our parity-odd previous operator work in [ a theory with one fermion, we observe a different jump for JHEP01(2018)036 , 7. 7. . . N 7 5 < < . The ψ σ ) singlet, ∆ = 2, in the N ( = 3. < O is increased, with ], the jump in the ψ 1 and 3 13 . 5 ≈ . Once again, we focus on in the parity even sector. . Thus, if the conjectured ψ ψ 4 ] and conjecture the existence 6 shows an upper bound on the 4 (2) symmetry should have ∆ is increased. In [ O N ) global symmetry. For each value of N ( 1 as O were to exist, they should lie within the allowed → – 14 – 1, the bound approaches the value of ∆ ψ 5.1 → and increases monotonically as ∆ ψ ) is the upper bound of each jump which is increasing 4 N ( 3), , max σ ) = (1 . In the future it will be interesting, for instance, to study the ∆ N , 10 and 20. As ∆ ψ , ) where ∆ 4 , N ( 3 , max σ = 2 ∆ . Upper bounds on the scaling dimension of the lowest-lying parity-even N < Upper bounds on scaling dimension of lowest-lying parity-even singlet, ∆ σ )-symmetric “dead-end” CFTs. ∆ This places the free theory, for which the lowest-lying parity-even singlet scalar has ∆ N 4 , for a unitary CFT containing fermions with scaling dimension ∆ < ( However, this is not thefamily case of for “dead-end” the CFTs from bounds section presentedregion, in for figure instance implying that the theories with allowed region. We now focus onscaling the dimension parity-even singlet of sector. thebound starts lowest-lying Figure at operator (∆ inan inflection this point sector that asparity gets odd closer a sector to function coincided ∆ with of a ∆ kink at the same value of ∆ extremal spectra of theseO jumps and try to find further evidence for5.2 the existence of new The lowest dimension parity-even scalar singlet anomalous dimensions in both the parity-evenIt and is odd therefore sectors, tempting ∆ to extendof the a line of family reasoning of fromthe [ “dead-end” dimension theories of with the an 3 scalar singlet in themonotonically parity-odd scalar with in such a theory would satisfy Figure 4 ∆ the cases JHEP01(2018)036 N . Once = 2, to ψ N , the value 2). For all ) symmetric- , N 1 ( O 45) for . 2 ) = (1 , T . σ 4 ∆ 069 . , . Once again, the bound ψ ψ (1 ≈ , encountering a first kink at 1, the bound approaches the free ) ψ T → σ , which is identified with the scaling ψ ∆ , ψ T σ , as a function of ∆ T – 15 – σ 10 and 20. As ∆ operator, ∆ , 4 at which we find the inflection point for the boundary ranging from (∆ , 3 ψ T , = 20. The location of these kinks can be seen more clearly σ N = 2 N and ∆ ψ 06) for . 2 coincidentally seems to match the value predicted by the large- = 3. There are two sets of kinks. The first set of kinks correspond to the , for a unitary CFT containing fermions with scaling dimension ∆ , T 4 T σ σ 008 , is well within the bootstrap-determined universal bounds. . 2 φ (1 , each bound increases monotonically with ∆ ≈ we show an upper bound for the scaling dimension of the lowest-lying parity- N ) symmetric traceless scalar, ∆ ) and discuss the meaning of the kinks in more detail in section . Upper bounds on the scaling dimension of the lowest-lying parity-odd 5 T 1 N σ ( ∆ O Thus, the connection between these features and the GNY models is uncertain. How- Interestingly, the values of ∆ , Upper bounds on scaling dimension of lowest-lying parity-odd traceless symmetric ψ odd starts at thevalues point of corresponding topoints the with free values of theory(∆ ∆ with (∆ sion of the parity-oddboundary symmetric becomes traceless apparent. scalar, the presence of the5.3 GNY model on The the lowest dimensionIn parity-odd figure symmetric traceless scalar expansion for thepredicted GNY-model. for the dimension of However, thedimension as parity-even of scalar can ∆ be seen in topever, as of discussed in table the next section, when placing universal bounds on the scaling dimen- GNY-models and are locatedin in figure the square on the lower left. We zoom onto this lower left square curve from figure Figure 5 traceless tensor, ∆ again, we focus ontheory the value cases of ∆ JHEP01(2018)036 , , at N = 1 = 2 ψ (6.3) (6.1) (6.2) ρσ is the η N N ) i µν µν , ) can η 2 allows us expansion T 21. 1 3 x . 5 ( 1 l 21 for N is the spin-1, . − ψ ) = 0 1 > ) i )) ) 1 µ can ψ 2 ≈ x 12 J ( x decreases with i ψ x ( = 20) that we have ) conserved current = 3. i ( T ψ . For large values of h ψ σ N T νρ N ) ( 5 ( I 1 jk ) O δ x ( N 12 as well as in our previous k + , in GNY-theories. x that occurs precisely where i ψ T 21 then the theory needs at ( . ) h σ 2 1 T 5.1 jl µσ σ x δ I ( < l + = 2, while the operator ψ . These quantities are defined as ψ , ) i denote the canonically normalized ) + and ∆ T , ranging from ∆ J ) 1 ) 2 12 C x ψ ) global symmetry. N ( x 12 x µν can ( j ( N x j 4 12 T ( ( ψ is precisely marginal. The value of ∆ x νσ h ψ O ) we find yet another discontinuity: for all I µν T ik 1 ) I there must be at least one relevant operator σ δ x ψ and 2 12 ( − ψ ) = 1. The conserved current k x J ( π )( ψ – 16 – C µ 1 can h (4 µρ x J free il T ) I δ ( C − jk − 1 2 δ x 2. )( il ( . = 2 we find that if ∆ 2 δ δ 7 x )-transformations, 6 12 1 6, while for the highest value of . Here − + N . x ≤ 2 N − appears in the conformal block decomposition. We start by 8 i ( jl 2 ) = 2 and T ) current central charge ) x δ J /x 2 T σ O ( ≤ π ν ik x C δ N C ( δ x T ( free l J (4 σ µ ) singlet operator with scaling dimension ∆ + ψ C O x ) = ( = N 2 1 ( i i x = 20. The range of the jump in the value of ∆ ) ) O ( − 2 2 k x x N ψ ( ( µν ) η x ,kl ρσ can ( T ≡ ν we once again find a jump in the bound on ∆ can ,ij ) 28 for 1 J ) . , where we zoom in on the region of interest from figure ) x µ x can N 1 1 = 2 one finds ∆ ( ( ], we are led to consider the possibility that the set of jumps follows a second 1 J x h ( ≈ µν µν 13 N can µ I ,ij T ) conserved current and stress-energy tensor, defined such that a theory of free ψ ∂ While we cannot yet check the existence of these conjectured CFTs, figure Once again following the reasoning presented in section As we move towards larger values of ∆ h ∂x we find a strong agreement between the position of the kinks and large- , and on the stress-energy tensor central charge N µ can ( J J h Let us first determine how using the Ward identity for Majorana fermions has parity-even, antisymmetric operator with dimensionspin-2, ∆ parity-even, 6.1 Bounds on the with O In this section we placeC lower bounds on thecoefficients central charge in of the the two-point function of the corresponding currents, least one relevant operator inwant the to symmetric study traceless CFTs representation, with and no conversely, if relevant we operators in6 this sector, we need Universal ∆ bounds on central charges family of theories, namelyuniversal a bounds second on extension fermionic of theories with the no “dead-end” CFTto seen conservatively when claim for placing whatin values this of sector: ∆ for instance, for which the jump occurs increasesto monotonically with ∆ as for used numerics for we find ∆ work [ N estimates for the values of the scaling dimensions, ∆ values of the bound intersects the horizontal line where in figure JHEP01(2018)036 : a µ , (6.8) (6.9) (6.4) (6.5) (6.6) (6.7) OPE ,` ) irre- j . J ∆ , ψ J . At the C I,R 44 ψ C × π ~ V 2 i 4 in the units 2 ) ψ √ 1 J , − R 2 , C 4 O can 2 I,A , corresponding to λ 22 . For large values 1 J, ( ˆ ~ V ), antisymmetric in R µ ψ N 2 ` to indicate that we ∈ to put a lower bound = 6.1 2 J , ` . 3 ~ (equivalently, 2 V X µ , − R ` 1 J is fixed by Ward identity, 1 J 2 O ) µ − 2 − . s 0 can ` , ) , µ . ,` 2 0 1 J, 2 λ ∆ +2 J I,S π , x µ i ab, . In other words, the bound is O . 2 I,R C π ~ 33 V ’s. We now search for functionals − 2 4 + 2 b 1 ~ a V − 2∆ 12 free 1 J 2 λ 1 µ √ x ) = 0 x a 1 = C OPE: λ ( − R λ i 1 j can 2 1 3 O 2 , ≡ s , , = ψ 1 J, can ( 2 λ , 2 ` ( λ 2 J, =1 J I,A 2 × R 21 X λ i ` i C ˆ ~ = V a,b ∈ ψ − = 2 , , ` + − X 1 J i − R 1 ) – 17 – , µ λ , can = O 2 2 2 , 1 J, i λ I,A , s − λ 12 1 , 2 ˆ 2 ~ V 2 ,` − x , h ( ∆ 1 I,A ≡ ` · 21 λ ˆ I,R ~ V I ab, O 2 ) ~ ~α V can 1 + -dependence and used the hat on + ≡ 1 J, free Majorana fermions for all values of 1 , , s O b ). Notice, however, that only 1 , and to avoid the costly scanning over all possible values of µ 2 1 , λ 2 J appearing in the µ N u, v x I,A + µ 12 ( ) gives: and is otherwise independent of the details of the CFT we study. ` ˆ O a 3.12 ~ ,ij V = 2, as a function of the fermion scaling dimension ∆ ` h λ 6.1 hO µν can 2 J and consequently, = 1 the bound goes to J µ free J even 2 1 J , ψ . From this Ward identity, we find a condition on the OPE coefficients C R µ j , ` =1 ∈ X + µ can , ` a,b 1 , J + R denotes the canonically normalized conserved current ( and 2 shows universal lower bounds for the current central charge is at this stage arbitrary. To place the bound on , J O i I,A 6 ,ij 11 C π − ˆ ~ 2 V 4 can 2 µ can 2 J, √ J of the current µ . 1 J the bound has a sharp local minimum at scaling dimensions ∆ J = µ Figure In our bootstrap setup, the normalization of operators appearing in the C satisfying conditions ( µ N can , we look for the functionals which satisfy the additional condition: i J 2 J a J, ~α of free field value unitarity bound ∆ saturated by the theory of of µ We can now proceed in exactlyon the same way as described in section where we suppressedswitched the to a 3-point function basis corresponding to while spectively of the value of The comparison with ( We can now rearrange the terms in the crossing equation as follows: Thus, we need to relatesetup, the which canonically have normalized normalization current fixed to by the operators we use in our depends only on ∆ and the indices λ Here we introduced a new basis for 3-point functions and corresponding OPE coefficients where JHEP01(2018)036 J N C (6.10) ]. This ] where 45 10 , 1, we set the = 0 . Note that we → estimates of i 4 ) ] where the Ising ψ as a function of the n N J 5 x J , ( C 4 C n O to extract the spectrum have no similar features. J ... in section N ) C 1 T σ x ( such that as ∆ 1 J bound, and similarly in [ hO C ) corrections (black markers) [ ν i T 2 ∂ C ∂x /N ) i at least, we can see that the GNY models (1 x O − N ) current central charge x ( N – 18 – δ ( O n =1 i X + ) conserved current central charges i . For large ) N at the minimum agree with the large- ( n N x O J ) would improve the bounds significantly and result in local ( we do not observe any local minima or even sharp changes C 20 the bound has a local minimum that lies close to the large- n ) vector models under certain assumptions. , A O N N ( ... O = 10 ) 1 N x . We normalize the central charges ( ψ 1 Lower bounds on the O ) in GNY models obtained up to ) J x ( 20. This is analogous to the phenomena observed in [ ,C , = 1. For µν ψ can T J . Lower bounds on the h C µ = 10 ∂ ∂x For smaller values of N 6.2 Bounds onUsing the the stress-energy Ward tensor identity central for translations, charge saturate the bootstrapbound, bound similarly and to furthermore thecould lie bounds try at on to the a use scalingof special the dimension the point ∆ functional (minimum) GNY obtainedfunctionals of model in obtained the in minimization from scaling of question. dimension bounds In seem to our give preliminary more precise investigations results. we have found that the same was found for in the slopecomputations of (see the appendix bound.minima even It for is small values possible of that increasing the derivative cutoff Λ in our corresponding to 3D Ising model. The bounds for smaller valuesGNY of models. The valuesfor of model was found lying in the local minimum of the Figure 6 scaling dimension ∆ value of values of (∆ is reminiscent of the central charge bound in the scalar bootstrap which had a local minimum JHEP01(2018)036 N (6.14) (6.11) (6.12) (6.13) 465, at . and can . We do . 1 ) 2 , , N 1 T ) ≈ λ ρσ u, v η ψ . ( u, v ,` ( µν T ∆ ,` η π C ,R 2 ∆ I 1 3 . For all values of , ab, √ ψ ~ I V 44 − ] until ∆ + ~ V can 2 2 b O , )) ψ 13 ) 1 T, λ 12 − R . λ ) symmetry. We observed + x 4 O i a O ( π N = λ 3 ( λ 4 ( νρ 2 , R O I − 1 T ` ) ∈ λ ) are satisfied and, at the same . At larger values of the fermion even 2 , = 12 , ` R X N x ` − R =1 ( ∈ X O can 3.12 is in the allowed region. , ` a,b µσ 2 T, ), which is equivalent to imposing that as a function of ∆ + − R I T ) O T C 6.6 − C ) + , it is important to check that the recently of the canonically normalized stress-energy u, v ) ( 12 7 , λ x ,` by bounding the OPE operators ( ), can ,R ) Gross-Neveu fixed points at large ∆ u, v 1) , T ( – 19 – ), which is saturated by the free theory with a T, νσ I N 0 33 , 3.5 λ C − ( I ,S 0 ~ V ) π ,N ψ O 2 I OPE. Specifically, one finds [ 8 )–( ) ab, 12 ~ j x V − R and consequently, (∆ ( b 1 i ψ 3.3 3 O λ 3 λ µρ a 1 × ) = (1 ( I λ i R ( = T )]. ` 2 ψ consistent with unitarity become allowed. While the GNY- , , 1 2 ∈ ,C , ` can T =1 such that the conditions ( u, v ψ X µν can − R X ( 1 T, C i 6 12 T 0 a,b O , λ 1 ~α x 0 T , 4 − − π I C 11 2 ~ V = √ [ i ) = i estimate for the central charge ) ~α = 2 − x u, v N µν ( ( 2 , T shows universal lower bounds on ,S ρσ 3 , appearing in the T 7 I ) ab, ) and the equation above, one finds 1 ~ V µν can x b T 6.2 T ( λ µν a T Figure We can now put a lower bound on Once again we must relate the canonically normalized stress-energy tensor to the one λ T , the bound starts at (∆ h parity-invariant CFTs containing fermions charged under an a sequence of “kinks”potentially that be match used the to learndimension, about we the also theories at observedwhen small a some sequence scalar of operator dimension discontinuous passes jumps through in marginality. the It bounds will occurring be important in model does not saturateobtained the large- bounds in figure 7 Discussion In this work we computed universal bounds on scaling dimensions and central charges of 3d time, minimize N Majorana fermions. Thewhich bound point then all decreases values as of a function of ∆ where the summation inenergy the tensor second and the term identity onsearch operator, the for whose contributions right-hand the we side function wrote now separately. We excludes now the stress singlet sector in the crossing eqs. ( this by isolating the contribution of the parity-even spin-2 operator with ∆ = 3 from the the 2-point function of the stress-energy tensor is normalized as From ( tensor we use in our bootstrap program, written in eq. ( one can determine the OPE coefficients JHEP01(2018)036 ) N = 1 ( O N T C as a function of the scaling T to be equal to the number of C T C ]) or additional supersymmetry. 51 estimates for GNY models obtained up to ) fundamental representation. We normalize N N ( O – 20 – 1, we set the value of → ]. This will likely require extending the bootstrap to ψ 11 , 8 ]. An important question is whether the GNY* fixed points 50 ]. such that as ∆ 45 T C Lower bounds on the stress-energy tensor central charge of the fermion transforming in the ψ . Lower bounds on the stress energy tensor central charge ) corrections [ 2 Finally, it could be interesting to relax the assumption of parity and probe 3d fermionic We would also like to understand how to isolate the GNY (and possibly GNY*) fixed /N (1 of the GNY models with multipleconnected scalar to the order Hubbard parameters model (e.g., on the a variant honeycomb with lattice 3 [ scalars is CFTs with parity violation.fermion It bootstrap should in also 4d, be making possible contact to with perform a BSM similar ideas study related of to the partial composite- are fully unitarykinks in could 3d be or turnedpush whether into the they bootstrap closed to islands are higher thenAlong derivative only order these a and approximately lines way check unitary. it if to thesupersymmetric will probe islands extension also If eventually this of be disappear. the the question interesting Ising would GNY* to model, be and isolate to to and find learn ways to more probe about important the variants 3d points as closed islands in thevector allowed models space were of scaling isolated dimensions,mixed in similar correlators [ to how containing the bothblocks fermions were worked and out scalars, in for [ which the needed conformal future work to clarify two questions:tor becoming 1) Why marginal do and these can this jumpswith be always new coincide understood with physical analytically? an “dead-end” 2) opera- CFTs? Dois these to One jumps more coincide avenue carefully for study making extremal progress spectra on as these one questions passes through these discontinuities. Figure 7 dimension ∆ the central charges fermions in the theory.O The black markers are large- JHEP01(2018)036 (A.1) model at the ∗ ~ f )-symmetric N ( ) derivatives of z O . The derivatives z, I ~α ] for the GNY 33 , 1 2 satisfying constraints such = z i = ~α z

). As noticed in our previous , which we will avoid by multiply- ) z z 3.10 ( z, → ( ~ ,R z f n z I ∂ ~ V m z ∂ -expansion results of [ – 21 – I mn ~a before applying the functional Λ 2. ≤ 5 m, ) and have evaluated the vector of functions / ) n z ≤ z X + to search for functionals that take the form: n m − = 1 − I z z α SDPB )(1 ] = SDPB for the fermionic four point functions were determined using a = ~ z f [ z − I I g ~α ]. In order to implement a semi-definite program we limit our search appearing in the vectors = (1 7 v ,R )[ I g script previously used when computing fermionic bounds without imposing and 3.12 z ]. We hope that our study helps to illustrate that the spinning bootstrap, even z 54 ], these functions will have singularities as ) or ( – = 13 52 u Applying these functionals to our crossing equations involves the ( 3.11 the functions work [ ing the crossing equationof by the ( conformal blocks Mathematica with crossing symmetric point fermionic bootstrap, using as ( over the space of functionals Computing Staff at the Institute for Advanced Study. A Implementation in We now give a description of the numerical implementation of the and Della clusters supportedclusters supported by by Princeton the University, facilitiesences the and High staff Omega Performance of Computing and the Center, Yale Grace Savioby University computational computing Faculty the of cluster resource Arts Berkeley provided and Researchas Sci- Computing well program as at the the Hyperion University computing of cluster California supported Berkeley, by the School of Natural Sciences and for numerous other usefulNSF under discussions. grant No. LVI PHY-1418069 and andsupported SSP by the by are Simons NSF supported Foundation grant grant in PHY-1350180 No.ported part 488653. and by by Simons DP DOE Foundation the is grant grant US 488651. DE-SC0009988,for a DSD Advanced William is Study, D. sup- and Loughlin SimonsNon-perturbative Membership at Bootstrap). Foundation grant the The Institute 488657 computations (Simons in Collaboration this on paper the were run on the Feynman Acknowledgments We thank Simone Giombidiscussions and and Igor collaboration Klebanov in for theory early Tarnopolsky discussions, for stages and adapting of the Ran this three-loop Yacoby work. for Special many thanks also to Grig- with additional global symmetry structure, isial currently results viable may and be that many attainedin more by nontriv- other studying global other symmetry external representations,and spinning global operators more. symmetry currents, such stress-energy as tensors, fermions ness [ JHEP01(2018)036 0 O (A.4) (A.5) (A.2) (A.3) , some 5 ) z , − z odd ` using the built-in ]. Similarly, we can script to manipulate , 13 , . 4), are polynomials in ∆ (∆) (∆) , ,R SDPB ) in the form of a polynomial ,I ) ,I ) , ) = (4 Mathematica m,n 3.12 ( m,n ab,` Mathematica ( ab,` P ,I a, b ~ ) P (∆) m,n ) and ( ` 3), ( (∆) ( ab,` ` , χ P χ ≈ 3.11 I mn ≈ 2 a / 2 4). In our applications, we take the operator ) = (3 / , =1 – 22 – z =1 . This can be done numerically. We can view the a, b z , , , X = z = ]. We have used a ], m,n,I 0 for all parity-odd operators with 0 0 | z 7 7 ) | [ . We then only need to find the linearly independent or ( ) = (4 ) = z SDPB ≥ ≥ z } ) = 1 z, I mn 2 a, b ( 0 z, a ab,` , ( ,` ). SDPB Y (∆) (∆) (∆) 1 ,` ∆ (∆ ,` ,` ,R ∆ 0 3), ( manual [ 0 R R R I using the set of differential operators that relates the fermionic ab,` 33 44 , ab, 2, O ∈ { I 3.10 ab, Y / Y Y F R ab,` ~ V n ) are identically zero, or their linear combinations are identically z such that: Y n z ∂ a, b 0 SDPB implementation. For Λ = 19 generating the input file required by ∂ = 1 m b O z ) = (3 ) as a matrix with rows labeling the constraints and columns labeling A.4 m z z I mn ∂ λ ∂ 0 ~a = a, b a O A.4 SDPB (∆) are now vectors of polynomials whose elements are simply related to λ z 2 (∆) for are polynomials defined as ,R , or ( Find Mathematica } =1 ,I ,I 2 ) X ) ab,` function. Notice that this step needs to be done only once for a given Λ. , a,b Y 1 in our m,n (∆) by using eq. ( m,n − ( ( ab,` ab,` )-symmetry explicitly. Using this script one can consequently write, at the crossing 4 ∈ { ~ P P ,I N takes about 30 minutes (on a single core), while solving each semi-definite program ) ( , making it run indefinitely. We want to remove such “flat directions” and give only can be found in the In order to obtain numerically accurate results we have used the parameters presented The full description of implementing the polynomial matrix program required to find We can approximate the set of constrains ( a, b O m,n ( ab,` I mn the fermionic conformal blocks to obtain the matrix inputin for table SDPB takes about 1 hour (allowed points) or 10 hours (disallowed points) on an 16 core machine. set of constraints ( the components of arows functional, of theRowReduce matrix. That can be done fora example in energy tensor. Note that becauseof of the the multiplication constraints of in crossing equation ( zero, by ( i.e. the setSDPB of constraints is notlinearly linearly independent independent. constraints This to can cause instabilities in for used for the normalization of the functionals to be either the identity operator or the stress- where the matrix program solvable using blocks to the rational approximationthen of write the scalar conformal blocks [ where P symmetric point where determined in an JHEP01(2018)036 ]. Ising 06 Solving Solving D 3 ]. SPIRE IN JHEP [ , Zh. Eksp. Teor. , SPIRE IN . [ } 40 35 35 10 ]. 40 40 50 19 130 − − − − 22 , S (1973) 161 896 (2012) 025022 10 10 10 10 parameters that affect the ) are included. The set of 49 , 76 SPIRE ) 0.3 46 (1977) 445 IN , D 86 SDPB ][ 45 ]. ) 0.1 , ]. 42 , infeasible B 118 maxRuntime β 41 ( feasible SPIRE ) 10 , ) 10 SPIRE θ β IN Annals Phys. P ( ( 38 and IN Phys. Rev. ) 10 , , , D ][ (Ω 37 , (Ω Bootstrapping Mixed Correlators in the ) 0.7 34 Nucl. Phys. , arXiv:1403.4545 γ ]. , ( [ – 23 – (1974) 9] [ 33 , Tensor representations of conformal algebra and 39 maxThreads 30 , SPIRE 29 IN ][ (2014) 869 ), which permits any use, distribution and reproduction in arXiv:1406.4858 ]. } ∪ { [ 25 157 SPIRE ,..., IN A semidefinite program solver for the conformal bootstrap 2 , Sov. Phys. JETP ][ CC-BY 4.0 1 (2014) 109 , Nonhamiltonian approach to conformal quantum field theory 0 maxComplementarity choleskyStabilizeThreshold stepLengthReduction infeasibleCenteringParameter feasibleCenteringParameter initialMatrixScaleDual initialMatrixScalePrimal dualErrorThreshold primalErrorThreshold dualityGapThreshold precision spins κ Λ 19 This article is distributed under the terms of the Creative Commons { 11 = arXiv:1502.02033 J. Stat. Phys. 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