Scale Invariance and the RG Scale invariance and a brief Introduction to the

- on this we 'll some the ( critical ) chapter , study of scaling the the properties of Ising modelusing theory description :

S={fdzdE[X+2X+ - X. JX .tiD×+X]

the the 'K that we derived in chapter . In we previous process ,

the the at : discuss of criticality ( A :o)

Scale and conformal invariance . We 'll also introduce some basic the to introduce of group scaling dimensions and critical exponents This analysis will also justify the of the and explain gradient expansion previous chapter , why critical the to the properties of Majonana Theory apply all systems with 22 ( in a ltld ⇐) classical Zd ) the case the going beyond exactly solvable of . =) Universality

& This chapter is meant to be a thorough intro . to NIT - Renormalizationduchbn the Renormalization Group ( RG ) . If not you're with idea RG Stat . condensed mat familiar the of in . mech and

a - will introduction and ta this chapter [email protected] , , give you vague I recommend that take P8l7 next basi you year ( which is - to the RG : dedicated 014 E . expansion cully entirely theory , took that coarse ( on 1 etc . . this chapter ) If you equivalent ,

be a should good reminder . Scale invariance Conformal invariance and correlation functions IO ,

D= 0 • Scale invariance Consider the To critical action ,

IX. + and s : . X+2x+) fade

→ dx It's invariant under rescaling : X X±t I "2X± Tt AT Z= dw

=i'2w ⇒ S = + . LOYNIYYLIN2. tfdwde It F+2wF. ) dt=×dw with X~± = Fl X±

Focus on : ) ) . X 61 < )=( Xlz , E) ) , =I'

. TXTEEEkgnskkon ,i¥,) - wnsknk c- unimportant =) ( X(z ) No ) ) = C- - - z Z= Xtit

X.tt/tD~tsJils.neJk)=thhEEysmam= x - t

course could this correlation : we 've computed function Note Of , : directly in the path integral formalism as a Gaussian integral L=EtX< t.kzt.ttX.

invariance . : In the action is invariant under Conformal fact ,

an large of called infinitely group symmetries conformal group ( preserve angles in Zd ) ) dude w=fCz 2w=¥*2 . =3 I sjeedzdz# = gtt ) # dropped Yn ¢ ×+iehf¥)kx¥, x. H=C¥t"ix. at

In general .

h ) ( ,I = conformal weights

ex :X :( ¥ " . a. El :(

. :C X+X 's , th .pe#ids:$(w.e1=ffFfhfffI)tifCz.E1- : h E) ) X+ ( , =p ,k called energy / thermal

operator . the Drives transition .

a scale Under transformation : w=ZA A

⇒ = with hth ¢( a. E) X. lot ,E) D=

Scaling Dimension

. h ) Rotation w=ei$z =) $(w,e ) . eilti loft ,E1

" "

- : Not s= h [ conformal spin . ( crucial for uss ) Fixed by Two point function : C- f$(z,E1 $6,01 ) = conformal EZE z2h invariance

For fields with s= o ( h = E) called sedan operator :

ZE = k$4 ) $6 ) ) = Say with D= 2h

Convince that a derivatives . if is field yourself $ primary , " " " like 2 not Those me called $ are primary fields . and " they still transform as dd under scale tnanspo oT= ¢ .

12-12=5descendantswith An . Dtn What is this formalism good for ?

to Let's we now a say want compute at finite we on T . temperature This should consider the theory a with ( conditions in time cylinder anti ) periodic boundary imaginary . In we could do this calculation discrete Sums over principle , using Matsubara but there's a much more elegant frequencies , way to proceed :

P

Conformal : HE ) mapping aye § f Z= W=£tYg2e YFW

Complex planeQHOI Cylinder " " " " to Finite T

w= Xtit

. z . lzleio ⇒ IH =e2%× TEG o=2pIt ,p )

F- 0 on cylinder result - kz - < X. (w.lxtw.DE#IwiTkfEwd) ¥ . 'E=R⇐e¥w, ¥, 4+4 C ) c 2T = e%( 2¥ " P ¥ .euTp% ⇒ Zsinhhtgcwixdf

inite Twnekton

spatial correlation : fx.KIX.to ) ) ~ 14 bon ×K{t=Ph W ,= X Thermal correlation Kao ~ e- ק+ for xD length { , ⇒ T cuts critical ) correlations finite off ( algebraic .

L= Na

' d : k : in @ RG a nutshell 2En=2az± lattice Kenya ields = ¢(× ) Maximal Egneikxq, momentum = UV cutoff

"

: Main idea Coarse . / out "/ short grain integrate fast highenergy ,l scale degrees of freedom

+ "×&* ¢kl=E%eik×9. Fennel

momentum Shea ⇒ q : G. el= Itf #

out : . Integrate $ Is . (fD§Es , Z=§¢ §$< )

[ done . . easier said than . FED

⇒ New effective 5 [$<] with cutoff NIAEP they eff

→ ^ . Scale transformation : K KG so NE %-)

Now [ 01 ] and S[$] have the same cutoff but Seq , K RG : { } different couplings Kilt ; '{ =Re({ kit ) flow ' : A in → eg Ising A d etc . " " The idea of the RG is to iterate this transformation to rid of microscopic / of freedom and get high energy degrees , about wave . information coarse . gain grained , long length properties

. RG fixed : |Ki*f=Re({ki*f ) Theories invariant Under RG transformation

those will be CFT 's : us invariance . for , pointsconformal

: Once a • Perturbations we've we can identified fixed point ,

- - study its stability against small perturbations Ski - Ki KY .

that take us the are called Couplings away from fixed point Relevant back to are Perturbations that flow Ki* called irrelevant . that perturbations don't flow are called marginal

# Universality of the Ising transition

the in Ve 'll see more explicit applications of RG later on this " "

course For 'll a to do . now we and , adopt quick dirty way A6 to order the to leading , by essentially reducing problem " "

= level RG dimensional analysis ( tree ) .

Let's back to the action in the Dirac go Ising fermin formulation ;

Sfdtdx 2+4 ]td44]+ @E[ 43×4-42×4 " - " Freund " critical the part of action = So

invariant under × → @ X " perturbation 4 → 24 T J Dnftz-5ftf-scaling dimension of 4 Under this transformation we have : . ,

dxdt 54 → Obviously d) fdxdtyy ,not } invariant . new @coupling g/

RG , DI bd > D : coupling grows under relevant perturbation

if we write b= el ⇒ = dyed ly;

valid FM only D= o PM for small < < • > > A s

.tt#=tdt=Majoran . CFT

b at D . : has Correlation length Length scale } = a which

to be : evoked 04 ) d( b) = D= 041 a

' " =) D with { ~ bB=%d=l

to a : we 'd . Universality Say like study modified Ising

chain :

'

H= - .× ) + J (oitoint . JE *go , ? 9×0.1 " " ' small perturbations t Jz g. tout t ... the ? preserving Ischg symmetry

E- . t.to;×

This model also has a transition between a PM and a FM '

at .

phase CJ . g- g. , , Ji ) 41 this Unlike the model model isn't solvable as it Ising , exactly

onto ( non a Jordan maps interacting quadratic ) fermions after Wigna ! transformation Check this ) . However the universal ( , properties the of transition like the D= I are unchanged !

' ' ' all J - * ( ... for , ,Ji Sal , ) 948 uneven

Teti non , ? universal Why Field theory of the modified model :

~ ~ L= ' 43+4 + E (4-2×5.42×4) tD44 + X

also ) t . . . ,YT2I5tdd42gHpresent for ) o IEJE . : higher order interaction term in gradient expansion (Tty )2=o - . Dx Now : c and ... to # 2J ( g- GDJIJI )) t leading - order

D= o Taylor expand for g= go Scale transformation : d. fdxdt 4-2×25 → Xiqfkdtysx 't ' " d 52×5) → ) , fKdt( Xy¢|d×dtk2×4

⇒ RG =) = - d irrelevant : decrease under day , couplings

hoarse . graining = - 2h ¥¥ ⇒ can be ignored .

a our . This justifies posteriori gradient expansion : 4214

irrelevant .

. This also explain universality : interactions in Majonana theory " (4-2×4) are irrelevant ! N irrelevant . couplings . , × , } ✓ D~lg-g.ly#,.oo..PIew.irdinei.n

^ €

order RG : . General leading

d +1=2 ( D= for on case )

So = Lo and 1061061 ) ~

do : xtzjo Scaling dimension of field O

O → AOO under Scale transformation : f-

we now So with 0 : 5=5 OC× ) If perturb ,

→

@ = . renormalized coupling el deisdofdxokl* ,

⇒ - = D relevant tdfdxddfpxokl0¥ ( d.) d =) ie Dokd .

Examples : Ising CFT : Duty =L : relevant fdbx= n : irrelevant n 1 Dyzny It for >

= dimensional D(yzxnygmm ( Itn) analysis [4]=k [ 2×7=1