Scale Invariance and the RG Scale invariance and a brief Introduction to the Renormalization Group - on this we 'll some the ( critical ) chapter , study of scaling the the properties of Ising modelusing field 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 symmetries of action 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 spin systems with 22 symmetry ( in a ltld ⇐) classical Zd ) the case the going beyond exactly solvable of Ising model . =) 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 correlation function at finite we on T . temperature This means 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 critical exponent 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.