Introduction to the Renormalization Group Hands-on course to the basics of the RG (based on: Introduction to the Functional Renormalization Group by P. Kopietz, L. Bartosch, and F. Schütz)
Andreas Kreisel Institut für Theoretische Physik Goethe Universität Frankfurt Germany
J. Phys. A: Math. Gen. 39 , 8205 (2006) J. Phys.: Condens. Matter 21 , 305602 (2009) Phys. Rev. B 80 , 104514 (2009) 1 SFB TRR 49 Outline
1. History of the RG
2. Phase Transitions and scaling
3. Mean-Field Theory
4. Wilsonian RG
5. Functional (exact) RG
6. Applications 2 Introduction
What is the renormalization group? All renormalization group studies have in common the idea of re- expressing the parameters which define a problem in terms of some other, perhaps simpler set, while keeping unchanged those physical aspects of a problem which are of interest. (John Cardy, 1996)
Meta-Theory about theories
Make the problem as simple as possible, but not simpler.
Describe general properties qualitatively, but not necessarily quantitatively. 3 Quantum Field Theory
Problem: Perturbation theory in quantum electrodynamics gives rise to infinite terms
Solution: all infinities can be absorbed in redefinition (=renormalization) of the parameters which have to be fixed by the experiment Renormalizable theories: finite number n of parameters sufficient, n experiments needed to fix them, predict all other experiments 4 Renormalization of QED
3 divergent diagramms 3 parameters 2 e Electron self energy §( P ) = (4m =P ) 8¼2² ¡ Zmmr electron mass m = renormalization Z2 e2 Photon self energy ¦¹º (P ) = (K ¹K º g¹º K 2) 6¼2² ¡ 2 er field Z3 = 1 2 renormalization ¡ 6¼ ² 2 Vertex correction e ¤( P;P + Q; Q)¹ = °¹ 8¼2² e2 2 charge er = e2 5 1 2 ln¹=¹ renormalization ¡ 6¼ 0 History
Kenneth Wilson (1971/1972) calculation of critical exponents which are universal for a class of models ¡® C(t) = t specific heat T Tc j j ¯ t = ¡ m(t) ( t) magnetization Tc » ¡ new formulation of the RG idea (Wilsonian RG) Nobel Prize in Physics 1982: ...for his theory of critical phenomena in connection with phase transitions...
6 Phase transitions and scaling hypothesis
Phase transitions: examples
Paramagnet-Ferromagnet Transition
m ordering parameter: magnetization
@f ¯ m = lim (Tc T ) ¡ h!0 @h / ¡
critical exponent: general for systems characterized by symmetry and dimensionality 0 c T
7 Phase transitions and scaling hypothesis
Liquid-Gas Transition order parameter: density n n n (T T )¯ ¡ c / ¡ c p1 same symmetry class as Ising model (classical spins in a magnetic field) same critical exponent 0 c T H = J s s h s ¡ i j ¡ i p2 ij i s = 1 X X p3 i § 8 Universality classes
Ising model, gas-liquid transition
H = J sisj Z : s s ¡ 2 i i ij !¡ X XY3 , Bose gas, (magnets in magnetic fields) H = J SxSx + SySy + (1 + ¸)Sz Sz ¡ i j i j i j ij X cos ' sin ' 0 £ O(2) : S~ S~0 = ¤sin ' cos ' 0 S~ ! 0 ¡ 0 0 1 1 @ A Heisenberg 0 H = J S~i S~j O(3) : S~ S~ ¡ ¢ ij ! 9 X Critical exponents
specific heat C(t) t ¡® / j j T Tc spontaneous ¯ t = ¡ m(t) ( t) T magnetization / ¡ c ¡° magnetic susceptibility Â(t) t / j j 1=± m(h) h sgn(h) t = 0 critical isotherm / j j e¡j rj=» correlation length G(~r) » t ¡º / »D¡3 r D¡1 / j j j j anomalous dimension p G(~k) ~k ¡2+´ T = T 10 » j j c Scaling Hypothesis
only two of six exponents are independent consider free energy density f(t; h) = fsing(t; h) + freg(t; h) singular part satisfies homogeneity relation
D=yt h fsing = t ©§( ) ©§(x) = fsing( 1; x) j j t yh =yt § j j critical exponents from derivatives 1 @2f C = t ¡® T @t2 / j j c ¯h=0 @f ¯ m = ¯ ( t)¯ ¡ @h ¯ / ¡ 11 ¯h=0 ¯ ¯ ¯ Scaling Hypothesis
relations between exponents 2 ® = 2¯ + ° = ¯(± + 1) ¡ scaling hypothesis for correlation function delivers two additional relations 2 ® = Dº ° = (2 ´)º ¡ ¡ relation between ® = 2 Dº º¡ thermodynamic ¯ = (D 2 + ´) exponents 2 ¡ ° = º(2 ´) and correlation ¡ D + 2 ´ function exponents ± = ¡ D 2 + ´ 12 ¡ Exercise 1: van der Waals Gas
equation of state N 2 p + a (V Nb) = NT V ¡ Ã µ ¶ !
sketch of isotherms 6 T=9/8 T c T=T 4 c c T=9/7 T c p/p 2
0 0 1 2 3 13 V/V c Exercise 1: Critical properties
Thermodynamics: calculate free energy from
pressure V F (T;V ) = p(V 0)dV 0 + const.(T ) ¡ ZV0 h3 F (T;V )ideal = NkBT ln 3=2 + NkBT (2¼mkBT ) V
obtain quantities from derivatives of the free energy @2F C = T t ¡® specific heat v ¡ @T 2 / j j µ ¶V
14 Exercise 1: Critical exponents (continued)
2 NkB T Vc use equation of state p(V ) = 3pc V V =3 ¡ V 2 ¡ c susceptibility compressibility 1 @V 1 @p ¡1 · = = t¡° T ¡ V @p ¡ V @V / µ ¶T µ ¶T;V =Vc rewrite at the critical temperature
n = nc + ¢n p(¢n) = p + const.¢n± (n n ) (p p )1=± c ) ¡ c / ¡ c
15 Mean Field Theory
Example: Ising model H = J s s h s ¡ i j ¡ i ij i X X partition function Z(T; h) = e¡¯H fsi g X ¡¯H magnetization fsi g sie m = si = h i P Z simplify term in Hamiltonian s s = (m + ±s )(m + ±s ) = m2 + m(s + s ) + ±s + ±s i j i j ¡ i j j j free spins in field zJ H = N m2 (h + zJm)s MF 2 ¡ i i 16 X Mean Field Theory
calculate partition function and free energy 2 ¡¯NZJm =2 N ZMF (t; h) = e [2 cosh[¯(h + zJm)]]
magnetization m(t; h)
self consistency equation for magnetization ¡¯N L ZMF(t; h) = e @ L = 0 @m m0 = tanh[¯(h + zjm0)] free energy f(T; h) = (T; h; m ) L 0 17 Mean Field Theory: critical exponents (only correct at D>4)
@ Tc 3 minimum condition L = (T Tc)m0 + m h = 0 @m ¡ 3 0 ¡ T T magnetization m = c ¡ ( t)1=2 0 2T / ¡ ¯ = 1=2 r c @m 1 susceptibility  = 0 ° = 1 @h / T T ¯h=0 ¡ c critical isotherm ¯ m (h) h¯ 1=3 ± = 3 0 / ¯ @2f(T; h) specific heat C = T 2 @2(T ln 2) ¡ @T C Tc T > Tc ¼ @T 2 2 2 @ (T ln 2) 3(T Tc) ® = 0 18 C Tc 2 ¡ T > Tc ¼ @T ¡ 4Tc Wilsonian RG
Basic idea: take into account interactions iteratively in small steps Formulation in terms of functional integrals: example Ising model ('4 theory) 1 ~ 2 ~ ~ free action: particle is S¤0 ['] = [r0 + c0k ]'( k)'(k) characterized by the the 2 ~k ¡ values of two parameters n0 Z S1 = ±(~k1 + ::: + ~k4)'(~k1)'(~k2)'(~k3)'(~k4) 4! ~ ¢ ¢ ¢ ~ Zk1 Zk4 cutoff ~k < ¤ interaction between (scalar) particles: j j 0 characterized by the coupling constant only particles allowed up to a momentum for example due to a lattice in a condensed matter system
partition function = [']eS¤0 +S1 19 Z D Z Step 1: Mode elimination
integrate out degrees of freedom associated with fluctuations at high energies
= [']e¡S¤0 ¡S1 = ['<] ['>]e¡S¤0 ¡S1 Z D D D Z Z Z e¡S¤0 +S10 = ['>]e¡S¤0 ¡S1 D end up: Ztheory with modified couplings due to interactions 0 1 < <~ 2 ~ ~ S¤['] = [r + c k ]'( k)'(k) 2 ~ ¡ Zk < 0 n ~ ~ ~ ~ ~ ~ S1 = ±(k1 + ::: + k4)'(k1)'(k2)'(k3)'(k4) 4! ~ ¢ ¢ ¢ ~ Zk1 Zk4 20 Step 2: Rescaling
Fields: defined on reduced space
blow up again the momentum space rescale wave vectors to get action with the same form as before (free and interaction part) 0 ~k = ¤0=¤~k 0 ¡1 < ' = ³b ' 1+D=2 < ³b = b c0=c b = ¤0=¤ p 21 Step 3: Iterative Procedure
get relations for mode elimination and rescaling (semi-group) ¤0 D 0 2 n0 d k 1 r (r0; n0) = b Zb r0 + D 2 ¤ (2¼) r + c ~k2 " Z 0 0 # 2 ¤0 D 0 4¡D 2 3n0 d k 1 n (r0; n0) = b Zb n0 D ¡ 2 ¤ (2¼) (r + c ~k2)2 " Z 0 0 # iteration in infinitesimal steps (differential equations: flow equations) ¤ = ¤ e¡±l ¤ (1 + ±l) 0 ¼ 0 1 n(l) @ r(l) = 2r(l) + l 2 1 + r(l) 3 n(l)2 @ln(l) = (4 D)n(l) + 22 ¡ 2 (1 + r(l))2 Flow diagrams
solve coupled differential equations for different initial conditions (parameters of real system) critical surface: critical systems determined by the values of the coupling constants
two fixed points: Gaussian, one fixed Wilson-Fisher point: Gaussian critical surface fixed point (mean field) 23 RG fixed points and critical exponents
RG fixed points: describe scale invariant system critical fixed points relevant / irrelevant directions correlation length: infinite critical manifold (surface describes system at critical point) critical exponents eigenvalues of linearised flow equations near to fixed point 24 Functional renormalization group
basic idea: Express Wilsonian mode elimination in terms of formally exact functional differential equations generating functional of Green functions ¡S[©] (n) [©] e ©®1 ©®n derive RG G®1 :::®n = D ¡S[©]¢ ¢ ¢ equations for [©] e generating R D functional ¡S[©]+(J;©) Green functions [©] e R ©®1 ©®n [J] = D ¢ ¢ ¢ then given as G [©] e¡S[©] derivatives R n D (n) ± [J] example: two (2) ~ 1 G®1 ¢¢¢ ®n = RG point function G (k) / ~ 2¡´ ±J®1 ±J®n of Ising model k ¢ ¢ ¢ j j 25 Exact renormalization group
introduce cutoff: modify Gaussian propagator: allow only propagation of modes up to a certain momentum 1 1 S = (r + c ~k2)'( ~k)'(~k) = G¡1(~k)'( ~k)'(~k) 0 2 0 0 ¡ 2 0 ¡ Z Z 1 1 £c( ~k ¤) G0(~k) = ¡ j j ¡ 2 2 r0 + c0~k ! r0 + c0~k
take derivative of generating functional with respect to cutoff FRG flow equation 26 Exact FRG flow equation
Wetterich equation cutoff function
1 @¤R¤ @¤¡¤[©] = Tr 2 2 @ ¡¤[©] " @©2 + R¤ # flow equations for vertex functions (coupling constants)
27 Applications
BCS-BEC crossover (electron gas with attractive interactions) mean field theory (BCS-theory)
y g0 y y H = ²~ c c~ c c c ~ c~ k ~k¾ k¾ ¡ V ~k+~p ¡~k ¡k0 k0+~p ~ ~ ~ Xk¾ k;Xk0;~p
flow equations of order parameter FRG needs additionally Ward identities (relations between vertex functions
28 Applications
interacting fermions y 1 y y H = ²~ c c~ + V~ ~ c c c ~ c~ k ~k¾ k¾ 2V k;k0 ;~p ~k+~p ¡~k ¡k0 k0+~p ~k¾ ~k;~k ;~p X X0 1 ¡S[Ã;ù]¡S1[Ã;ù] ¹ ¹ ¹ = [Ã; ù]e S1[Ã; Ã] = V ÃÃÃà Z D 2 k;k ;q Z Z 0 Hubbard-Stratonovich transformation ¹ ¹ e¡S1[Ã;Ã] = [Á; Á¤]e¡S0[Á;Á¤]¡S0[Ã;Ã;Á;Á¤] D Z compare: ¤ 1 ¡1 ¤ 4 2 S0[Á; Á ] = V ÁkÁk x 1 x 2 2 ¡ 2! ¡ 2 ¡ix y Zk e = ¡1 dx e 0 ¹ ¤ S [Ã; Ã; Á; Á ] = i Ãk+qÃkÁq R k q Z Z 29 Hubbard Stratonovich transformation
interacting fermions bosons and fermions with Yukawa type interaction 1 S [Ã; ù] = V ùÃÃù S0[Ã; Ã;¹ Á; Á¤] = i à à Á 1 2 k+q k q Zk;k0;q Zk Zq
1 S [Á; Á¤] = V ¡1Á¤ Á 0 2 k k Zk
FRG of coupled bosons and fermions 30 Summary
phase transitions critical exponents @f ¯ m = lim (Tc T ) ¡ h!0 @h / ¡ universality classes (same symmetry same properties at critical point) mean field theory Wilsonian Renormalisation Group
functional Renormalization group 31 Exercise 2: Real-space RG of the 1D Ising model
model N H = J s s ¡ i i+1 i=1 X transfer matrix method to calculate partition function eg e¡g Z = Tr[T N ] T = g = ¯J e¡g eg µ ¶ calculate trace in diagonal basis
y 1 1 1 T = U TU~ U = p2 1 1 µ ¡ ¶ 32 Exercise 2: Real-space RG
keep only every bs spin and derive effective model with new coupling
N Z = Tr[T N ] = Tr[T b] b eg0 e¡g0 T b = T 0 = e¡g0 eg0 derive recursionµ relation¶ (RG transformation) 0 b g (g) = Artanh(tanh (g)) 33 Exercise 2: Real-space RG
variable transformation b b 0 (1 + y) (1 y) ¡2g 0 ¡2g y (y) = ¡ ¡ y = e y = e (1 + y)b + (1 y)b ¡ infinitesimal transformation (differential equation) dy 1 y2 1 + y b = e±l 1 + ±l = ¡ ln( ) ¼ dl 2 1 y ¡ fixed points and flow dy = 0 dl 34 Literature
S.K. Ma: Modern Theory of Critical Phenomena (Benjamin/Cummings, Reading, 1976)
N. Goldenfeld: Lectures on Phase Transitions and the Renormalization Group (Addison-Weseley, Reading, 1992)
J. Cardy: Scaling and Renormalization in Statistical Physics (Cambridge University Press, Cambridge, 1996)
J. Zinn-Justin: Quantum Field Theory and Critical Phenomena (Oxford Science Publication, Oxford, 2002)
P. Kopietz, L. Bartosch, F. Schütz: Introduction to the Functional Renormalization Group (Springer, Heidelberg, 2010) 35