SLIDE 1
CFT
genome
project
Slava Rychkov CERN / ENS Paris ERG 2016 , Trieste€
y This talk D= 3project genome Slava Rychkov / Paris CERN ENS ERG Trieste - - PowerPoint PPT Presentation
project genome Slava Rychkov / Paris CERN ENS ERG Trieste - - PowerPoint PPT Presentation
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
SLIDE 2 Critical
state
mum
12 G
vs
SLIDE 3 TIMELINE
1970 's
ancient
period
Polyakov
;
Ferrara
,
Gatto
,
Grillo
;
Mack
formal
foundations
2008
.
modern
period
[
Rattazzi
,
Rychkov ,
Tonni
,
Vichi ]
2014-16
most
impressive
d=3
results
2016
- Simons
- n
SLIDE 4 Why
a
new
approach
?
Reas=
Practical
guy
Existing
approaches
not Vsatisfactoz
*
RG
great
qualitative
, but has limitations as a quantitative tool * Simulations expensive ,- ften
SLIDE 5 Example
I
v
in
t÷
r y Exp : v= 0.67094 ) [ Lipa et al , 2003 ] MCTHT : v= 0.6717 ( i ) [Campos trim ' et al , 2006 ] 85 discrepancy SLIDE 6 EXI
Cubic
anisotropy
in
Heisenberg
YID.ge#EEI
? play by
. y d)- relevant
SLIDE 7 EXT
QED
}
with
Ng
massless
fermions
→
CFT
for
NGZNF
't
? [ Gukov 1608 ] SLIDE 8 "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]
Why
a
new
approach
?
Reas=2
conceptual Critical behavior is universal But : RG introduces nomuniversal structure in the intermediate steps ⇒ Need a manifestly universal approach to criticality- (
SLIDE 9 Why
conformal
?
Rotation
invariance
+
Scale
invariance }
→
conformal
- invariance
Locality
)
generically
, e.g. in presence- f
interactions
( all Known counterexamples are gaussian ) SLIDE 10 What
is
a
CFT
?
- a
- f
- perators
- }
- whose
SLIDE 11
Conformal
transformations x→ Fca ) preserve angles . In Zd z→f(z ) x- dim
SLIDE 12 Local
- perators
- spin
- scaling
*
related to critical exponentsAfp
~Yndi
E.g.( Aik
) AJCY)7~
k¥+1
. SLIDE 13 Monte Carlo Bootstrap 0.51808 0.51810 0.51812 0.51814 0.51816 0.51818 Δσ 1.4125 1.4126 1.4127 1.4128 1.4129 1.4130 Δϵ
Ising: Scaling Dimensions
0.518146 0.518148 0.518150 0.518152 1.41260 1.41261 1.41262 1.41263 1.41264 1.41265
magnitude ✏. The result is a new d (∆, ∆✏) = (0.5181489(10), 1.412625(10)), nations of the leading OPE coefficients (
[ Kos
,
Poland
,
Aqq£f>V
- Simmons
- Duffin
x
t- Ds←z
- rigor
- userrorba=
- factor
- factor
- 1000
SLIDE 14 CFT
magic
2- pt
is
diagonal
andfixed
imam
Aicx)Ajly#E
;Elie
x-yl2=}
Ie (a) = Known spin- dependent
- 2=:k2
SLIDE 15
OPT
( Ae 1OD
Auln ) A 3 ( ns ) . . .>
=
¥fnak
[Art
) + # nmfu Arlo )¥+12
.Dv¥=
amnu you Arlo )/
fixedif"!¥rmd
symmetry OPE coefficient SLIDE 16 OPE
:
n
- pt
tu
pt functions t 2- pit
- '
- ⇒
(
Ii , li ) , Cijk can compute au}F'jaµdr*#
SLIDE 17 From
CFT
point
- f
just
as fundamental as Di . While in RG they are rarely discussed ( and hard to compute ) Bootstrap for 3d Ising : C ogq = 1.0518537141 ) affirm , ... ,µ¥Eee
( qqq I f. 53243549 ) Ceee=k [ Kos , Poland , Simmons . D SLIDE 18 OPE
convergence
ii
"- ne
:o)
Thm : 1 . OPE converges- 2.
- mitting
- ps
<~(Yf)D*
[ Pappadopulo , Ryehbov, Espn , Ruttazti , 2012 ] SLIDE 19 OPE
convergence
=
low
- energy
/
high . energy decoupling low energy ←> low D high energy a high D SLIDE 20 Not
every
set- f
( Di
.li )
,Cijk
define
a consistent CFT SLIDE 21 OPE
associating
.
(
Ai Aj
)An
=AICAJAK
) WW 4 consistency condition- n
SLIDE 22 Bootstrap
eqn
for
4pt
- fu
- @
Zo¥Te{
= -2¥ " Crossing "=
Az As z 3 NB : Not Feynman diagrams Roughly?
Gzkc }yk( . . . ) =[
CMKCZ }k( . " ) If imposed for all 4ft fns equivalent to OPE associatively SLIDE 23
- Bootstrap
- pertrbative
- def 'd
- N
- [ Belavin
- 2008
- progress
numerical
techniques SLIDE 24 What
we
can
do
.)
construct
approximate
solutions
to
the
bootstrap
eqns
arbitrary
requested
accuracy
,
say
10-100
- )
- ut
- f
SLIDE 25 Bootstrap
analysis
- f
÷
2 global symmetry- ne
- dd
- 6
- ne
- even
- Ee
- robust
tovctassify
C FT with small number- f
- ps
÷
✓ SLIDE 26 a
irrel
.
( 44
6×5=1
+
{
+
E
' +
. .
.
1l=o
)
+Tµ+
In 't ... e=z + l=9 , 6 , . . . E x { = same- ps
l÷0
+ l- 4,2
- .
SLIDE 27 OPE
weffs
.
are
symmetric
;
5×5
⇒
Core
- E
- er
identity
fixes OPE weffoftuv : 0×0 > CootIn
Coot
=1¥
Ct central charge SLIDE 28
Unitarily
(
reflection positivity ) ⇒Cijnek
Di 7Dmiulli
)
( unitarily bounds ) E.g. beatd÷
De > . > ltd- 2
SLIDE 29
Workflow
←
al ; determine dim 's- f
- lying
- perators
Isiy
SLIDE 30 /
.
Pick
a
few
Ypt
functions
involving
- ps
- f
- r
- r
SLIDE 31 2.
Consider
OPE expansion
- f
? >€=
low + high 3 . For low CFT data fixed , can we find high- perators
SLIDE 32 Monte Carlo Bootstrap 0.51808 0.51810 0.51812 0.51814 0.51816 0.51818 Δσ 1.4125 1.4126 1.4127 1.4128 1.4129 1.4130 Δϵ
Ising: Scaling Dimensions
0.518146 0.518148 0.518150 0.518152 1.41260 1.41261 1.41262 1.41263 1.41264 1.41265
magnitude ✏. The result is a new d (∆, ∆✏) = (0.5181489(10), 1.412625(10)), nations of the leading OPE coefficients (
for every
point
- utside
SLIDE 33 IS
this
surprising
?- Exclusions
- ops
- ps
- f
- identify
- definite
SLIDE 34 BUT
:
surprising
that
get very strong
constraints
from
just
a
handful
- f
- pen
SLIDE 35 Monte Carlo Bootstrap 0.51808 0.51810 0.51812 0.51814 0.51816 0.51818 Δσ 1.4125 1.4126 1.4127 1.4128 1.4129 1.4130 Δϵ
Ising: Scaling Dimensions
0.518146 0.518148 0.518150 0.518152 1.41260 1.41261 1.41262 1.41263 1.41264 1.41265
magnitude ✏. The result is a new d (∆, ∆✏) = (0.5181489(10), 1.412625(10)), nations of the leading OPE coefficients (
and
just
;bbi°n
we can find it ( samealgorithm)
SLIDE 36
%
ceeazdsiyeinmeenaaatt¥
leading irrelevant Zz .- dd
- n
- Duffin
SLIDE 37 Precision
improvement
- ur
morehtwnstraiuts
. E.g. three Ypt functions instead- f
- ne
- f
SLIDE 38 So
,
Ising
- 3
applied
to- ther
- f
- ps
- Neveu
- i
SLIDE 39 Dopa
OCN)ar=iphg
Bi
shot [ Kos , Poland , Simmons . Duffin , Vichi , 2015 ] SLIDE 40 0 ( 2)
:
86
V
discrepancy
- ¢2←>VBtd
eaD¢
rigorous bootstrap islands [ Kos , Poland , Simmons- Duffin
SLIDE 41
CFTgenomef.ro#
Determine CFT data- f
- D
- perators
- f
- f
- ps