CFT project genome Slava Rychkov / Paris CERN ENS ERG Trieste 2016 ,
Critical state mum € 12 G vs y This talk D= 3
TIMELINE 1970 's period ancient Grillo Gatto Mack Polyakov Ferrara ; ; , , foundations formal period modern 2008 Vichi ] . Tonni [ Rattazzi Rychkov , , , results d=3 most impressive 2014-16 Collaboration the Simons on 2016 - Bootstrap Perturbation Non .
? approach Why new a Practical Reas= guy not Vsatisfactoz approaches Existing but qualitative great , RG * quantitative limitations a as has tool expensive Simulations , * inconclusive often
t÷ I Example in v r y al ] [ Lipa et 0.67094 2003 ) Exp v= : , 2006 ] [ Campos al trim et ( i ) MCTHT 0.6717 ' v= : , discrepancy 85
Cubic Heisenberg EXI anisotropy in YID.ge#EEI ? ? d) relevant play by . y - [ Calabrese Pelissetto <3 ( 4) , 2.991 D= , ] Vicari 2003 level @ 25 al et ] studies [ 2001 Tissier Other indicate 1>3
massless Ng with QED EXT } fermions 't NGZNF for ? CFT → 1608 ] [ Gukov
? approach Why new a conceptual universal behavior is Critical nomuniversal structure Reas=2 introduces RG But : intermediate steps the in "RG is a human-made thing. It's a smart way to calculate, but it does not have a breathtaking quality of, say, Dirac equation." [Polyakov] universal manifestly Need a ⇒ 'T ( F criticality to approach -
conformal ? Why invariance } Rotation invariance conformal + Scale → - invariance ) ^ + Locality generically e.g. , of interactions presence in counterexamples ( all Known ) gaussian are
What ? CFT is a local operators list of a • } A { }fe ) Azfu ) Ante ) . - . , , infinite generally correlates whose • > Aetna ) < . . . . conformal invariance have
Conformal transformations Fca ) x→ angles preserve dim ) z→f(z Zd x In - . dim d finite In >2 . . rotations Mµ translations pµ dilatation D stfmqd Koiiwersionofn . Kpn transf inversion .
Local Aik operators ) by characterized l ( vector tensor scalar 2- spin iii , , - i * (d) ) SO under dimension scaling - related critical exponents to Yndi Afp ~ k¥+1 ( Aik )7~ AJCY ) E.g. .
Poland [ Kos , , Ising: Scaling Dimensions Vichi Duffin Simmons - - Δ ϵ , , 1603.04436 ] •Ds←z Monte Carlo 1.4130 Aqq£f>V 1.41265 1.41264 1.4129 x 1.41263 1.41262 1.4128 1.41261 1.41260 0.518146 0.518148 0.518150 0.518152 1.4127 t 1.4126 Bootstrap 1.4125 Δ σ 0.51808 0.51810 0.51812 0.51814 0.51816 0.51818 ouserrorba= magnitude � ��✏ . The result is a new d - ( ∆ � , ∆ ✏ ) = (0 . 5181489(10) , 1 . 412625(10)), nations of the leading OPE coe ffi cients ( � rigor • [ Hasenbusk ] MC best better than factor 25-50 • best compatible -1000 2010 better than RG factor • Guida . Justin 98 ] Zinn ,
CFT magic 2- pt diagonal and is fixed imam Elie x-yl2=} Aicx ; )Ajly#E dependent spin Known (a) - Ie = structure tensor =L Io 2=:k2 fur In - = - . .
OPT ¥ fnak fixed if OD > ( ) A 3 ( ns ) Auln [ Art 1 Ae Dv¥= . . . = ¥+12 "!¥rmd ) # nmfu Arlo ) + / amnu you Arlo ) . symmetry OPE coefficient
- pt functions OPE n : tu functions pt t ( fixed ) - pit ' - functions 2 - li ) ( Ii all Cijk If }F'jaµdr*# know ⇒ we , , au compute can
of CFT view point From , fundamental just as are Cojk Di as . rarely RG they While are in compute ) to ( hard and discussed for 3d Ising Bootstrap : 1.0518537141 ) µ¥Eee C ogq = Ceee=k 53243549 ) f. ( I qqq [ Kos Poland affirm ... , , Simmons . D , ,
OPE :o) convergence one " ii Thm converges OPE 1 : . - Dade omitting from ops Error 2. [ Pappadopulo , Ryehbov , <~(Yf)D* is 2012 ] Ruttazti Espn , ,
OPE = convergence decoupling / low high energy energy . - low low D ←> energy D high high a energy
Not of Cftdata set every .li ) ( Di Cijk , CFT consistent define a
OPE associating . AICAJAK Ai Aj An ) ( ) = 4 WW data CFT condition on consistency
- fu for 4pt Bootstrap eqn . = -2¥ y 1 Al = Ay Zo¥Te{ • @ " " Crossing As Az 3 z diagrams Not Feynman NB : ) }k( [ . ) CMKCZ }yk( ? Gzkc Roughly " = . . . fns for all 4ft If imposed associatively OPE to equivalent
70 's formulated first Bootstrap in egns • universal - pertrbative manifestly Non , , ) . ( convergent mathematically def 'd well - unknowns for numerical a N eafns • 83 ] Zamolodohikov 1 [ Belavin : Polyakov • , , finitely many cases D= In 2 in some models ) solve minimal can variables ⇒ ( analytically d >z in progress 2008 • - techniques
do What can we solutions to approximate construct . ) bootstrap eqns the 10-100 arbitrary requested say accuracy , rigorously , numerically but • ) prove , solution does sometimes that not exist of CFT chunks rule out ⇒ parameter space
analysis CFT of Bootstrap 3d Ising ÷ global symmetry 2 C y ) 6 odd relevant zz - - one ( qy • Ee even - relevant zz - one fact experimental robust - tovctassify first makes it sense priori A , relevant of number ops small . with FT C ÷ ✓
( 44 l÷0 irrel -4,2 a . 1l=o ' ) 6×5=1 E { + + + . . . 't e=z +Tµ+ In ... l=9 6 + . . . , , different ( but ops { same E x = ) ueffs OPE . ( ys ) girrel ' 5 + 5 E + . . Ox . = l 3,9 + . . - , ,
weffs symmetric OPE are ; . ⇒ Core E 5×5 - " C J > 5 { ' x oer weffoftuv fixes OPE identity : Ward In 0×0 Coot > 1¥ Coot = charge central Ct
Unitarily positivity ) ( reflection Cijnek ⇒ ) Dmiulli Di 7 ) bounds unitarily ( d÷ beat E.g. ltd -2 De > > . models interesting most Applies to
Workflow ← dim of 's determine a al ; - lying operators low few coeffs OPE relative & their data ) ( low CFT ' ' for E E.g. 5 E 5 , , , 3d Isiy the
functions Ypt few / Pick a of interest . involving ops . ( exchanged ) internal Jemal or as look at 3d Ising for can E.g. > cross > } CEEEE rE{o> { > < cross or better the The more
Consider OPE expansion 2. functions of 4pt : these >€= high low ? + fixed data low CFT For , 3 operators . high find we can wefts ) so OPE ( their and for bootstrap conf egs that . ? satisfied fus Gpt selected are
Ising: Scaling Dimensions Δ ϵ Monte Carlo 1.4130 1.41265 1.41264 1.4129 1.41263 1.41262 1.4128 1.41261 1.41260 0.518146 0.518148 0.518150 0.518152 1.4127 1.4126 Bootstrap 1.4125 Δ σ 0.51808 0.51810 0.51812 0.51814 0.51816 0.51818 magnitude � ��✏ . The result is a new d ( ∆ � , ∆ ✏ ) = (0 . 5181489(10) , 1 . 412625(10)), nations of the leading OPE coe ffi cients ( � shaded outside for point every is there show that region we can , solution no
this IS ? surprising follow positivity from Exclusions . • violate low - ops crossing Once high ops " amount " by - positive a , undo this cannot " positive " identify algorithm Role of - directions Linear programming - definite Semi programming
that get surprising BUT : from constraints strong very of handful functions Gpt just a . open problem Why ? .
Ising: Scaling Dimensions just ;bbi°n Δ ϵ and Monte Carlo 1.4130 find it can we 1.41265 algorithm ) 1.41264 ( 1.4129 same 1.41263 1.41262 1.4128 1.41261 1.41260 0.518146 0.518148 0.518150 0.518152 1.4127 1.4126 Bootstrap 1.4125 Δ σ 0.51808 0.51810 0.51812 0.51814 0.51816 0.51818 magnitude � ��✏ . The result is a new d ( ∆ � , ∆ ✏ ) = (0 . 5181489(10) , 1 . 412625(10)), nations of the leading OPE coe ffi cients ( �
% ceeazdsiyeinmeenaaatt ¥ odd irrelevant Zz leading . weffs & dims OPE uncertainties on . allowed region from the varying in - Duffin 2016 ] Komargodski , Simmons [ of .
Precision improvement - results get precise more our WHY ? with time . physical add We . 1 instead . functions Ypt three E.g. morehtwnstraiuts etc. of one , function 4pt single , for a Even 2 funclionalequation . is bootstrap a conformal >cs ) satisfied for be xe ,xz,x3 any to ( has , which of we constraints many so ⇒ , subsets larger larger and use
good So shape Ising in -3 is , ? next What applied to method be Same can classes with universality other relevant number of small ops . a 01N ) a * Neveu Gross - ( - i already This has started work
OCN )ar=iphg Dopa Bi shot . Duffin 2015 ] Vichi Poland [ Kos Simmons , , , ,
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