The Functional Renormalization Group Method An Introduction

Transcription

1 The Functional Renormalization Group Method An Introduction A. Wipf Theoretisch-Physikalisches Institut, FSU Jena La Plata 21. Juli 2014 Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction 21. Juli / 60

2 1 Introduction 2 Scale-dependent Functionals 3 Derivation of Flow Equations 4 Functional Renormalization in QM 5 Scalar Field Theories 6 O(N) Models Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction 21. Juli / 60

3 9 Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction 21. Juli / 60

4 Introduction particular implementation of the renormalization group for continuum field theory, in momentum-space functional methods + renormalization group idea scale-dependent Schwinger functional or effective action conceptionally simple, technically demanding flow equations scale parameter k = adjustable screw of microscope large values of a momentum scale k: high resolution lowering k: decreasing resolution of the microscope known microscopic laws complex macroscopic phenomena non-perturbative Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction 21. Juli / 60

5 flow of Schwinger functional W k [j]: Polchinski equation flow of effective action Γ k [ϕ]: Wetterich equation flow from classical action S[ϕ] to effective action Γ[ϕ] applied to variety of physical systems strong interaction electroweak phase transition asymptotic safety scenario condensed matter systen e.g. Hubbard model, liquid He 4, frustrated magnets, superconductivity... effective models in nuclear physics ultra-cold atoms Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction 21. Juli / 60

6 Γ k=λ = S Γ k=0 = Γ Theory space Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction 21. Juli / 60

7 1 K. Aoki, Introduction to the nonperturbative renormalization group and its recent applications, Int. J. Mod. Phys. B14 (2000) C. Bagnus and C. Bervillier, Exact renormalization group equations: An introductionary review, Phys. Rept. 348 (2001) J. Berges, N. Tetradis and C. Wetterich, Nonperturbative renormalization flow in quantum field theory and statistical physics, Phys. Rept. 363 (2002) J. Polonyi, Lectures on the functional renormalization group methods, Central Eur. J. Phys. 1 (2003) 1. 5 J. Pawlowski, Aspects of the functional renormalisation group, Annals Phys. 322 (2007) Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction 21. Juli / 60

8 1 H. Gies, Introduction to the functional RG and applications to gauge theories, Lect. Notes Phys. 162, Renormalization group and effective field theory approaches to many-body systems, ed. by A. Schwenk, J. Polonyi (2012). 2 P. Kopietz, L. Bartosch and F. Schütz, Introduction to the functional renormalization group, Lecture Notes in Physics, Vol. 798, Springer, Berlin (2010). 3 A. Wipf, Statistical Approach to Quantum Field Theory, Lecture Notes in Physics, Vol. 864, Springer, Berlin (2013). Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction 21. Juli / 60

9 Scale-dependent functionals generating functional of (Euclidean) correlation functions Z [j] = Dφ e S[φ]+(j,φ), (j, φ) = d d x j(x)φ(x) Schwinger functional W [j] = log Z [j] connected correlation functions effective action = Legendre transform of W [j] Γ[ϕ] = (j, ϕ) W [j] with ϕ(x) = δw [j] δj(x) (1) one-particle irreducible correlation functions last equation in (1) j[ϕ], insert into first equation in (1) Γ: all properties of QFT in a most economic way Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction 21. Juli / 60

10 add scale-dependent IR-cutoff term S k to classical action in functional integral scale-dependent generating functional Z k [j] = Dφ e S[φ]+(j,φ) S k [φ] Scale-dependent Schwinger functional W k [j] = log Z k [j] (2) regulator: quadratic functional with a momentum-dependent mass, S k [φ] = 1 2 d d p (2π) d φ (p)r k (p)φ(p) 1 φ (p)r k (p)φ(p), 2 p one-loop structure of flow equation Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

11 conditions on cutoff function R k (p) should recover effective action for k 0: R k (p) k 0 0 for fixed p should recover classical action at UV-scale Λ: regularization in the IR: R k k Λ R k (p) > 0 for p 0 Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

12 possible cut-offs are the exponential regulator: R k (p) = e p2 /k 2 1, ( the optimized regulator: R k (p) = (k 2 p 2 ) θ k 2 p 2), the quartic regulator: R k (p) = k 4 /p 2, the sharp regulator: R k (p) = the Callan-Symanzik regulator: R k (p) = k 2 p 2 p 2 θ ( k 2 p 2) p2, Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

13 exponential cutoff function and its derivative 2k 2 k k R k k 2 R k p 2 k 2 2k 2 Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

14 Polchinski equation partial derivative of W k in (2) is given by k W k [j] = 1 2 d d xd d y φ(x) k R k (x, y)φ(y) k relates to connected two-point function G (2) k (x, y) δ2 W k [j] δj(x)δj(y) = φ(x)φ(y) k ϕ(x)ϕ(y) Polchinski equation k W k [j] = 1 2 = 1 ( 2 tr k R k G (2) d d xd d y k R k (x, y)g (2) k (y, x) k S k [ϕ] k ) k S k [ϕ] Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

15 Scale dependent effective action average field of the cutoff theory with j ϕ(x) = δw k[j] δj(x) (3) fixed source average field depends on cutoff fixed average field source depends on cutoff modified Legendre transformation: Γ k [ϕ] = (j, ϕ) W k [j] S k [ϕ] (4) solve (3) for j = j[ϕ] use solution in (4) Γ k not Legendre transform of W k [j] for k > 0! Γ k need not to be convex, but Γ k 0 is convex Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

16 Derivation of Wetterich equation vary effective average action δγ k δϕ(x) = δj(y) ϕ(y) + j(x) δϕ(x) δwk [j] δj(y) terms chancel effective equation of motion δj(y) δϕ(x) δ S k[ϕ] δϕ(x) δγ k δϕ(x) = j(x) δ δϕ(x) S k[ϕ] = j(x) (R k ϕ)(x) flow equation: ϕ fixed, j depends on scale, differentiate Γ k k Γ k = d d Wk [j] x k j(x)ϕ(x) k W k [j] j(x) kj(x) k S k [ϕ] Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

17 two red contributions cancel k W k [j]: only scale dependence of the parameters k Γ k = k W k [j] k S k [ϕ] = k W k [j] 1 d d xd d y ϕ(x) k R k (x, y)ϕ(y) 2 use Polchinski equation k Γ k = 1 d d xd d y k R k (x, y) G (2) 2 k (y, x) (5) second derivative of W k vs. second derivative of Γ k : ϕ(x) = δw k[j] and j(x) = δγ k δj(x) δϕ(x) + d d y R k (x, y)ϕ(y) Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

18 chain rule δ(x y) = d d z δϕ(x) δj(z) δj(z) δϕ(y) = d d z G (2) k (x, z) { Γ (2) k + R k } (z, y) Hence G (2) k = 1 Γ (2), Γ (2) k k + R (x, y) = δ 2 Γ k k δϕ(x)δϕ(y) insert into (5) Wetterich equation ( ) k Γ k [ϕ] = 1 2 tr k R k Γ (2) k [ϕ] + R k (6) Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

19 non-linear functional integro-differential equation full propagator enters flow equation Polchinski and Wetterich equations = exact FRG equations Polchinski: simple polynomial structure favored in structural investigations Wetterich: second derivative in the denominator stabilizes flow in (numerical) solution mainly used in explicit calculations. in practice: truncation = projection onto finite-dim. space difficult: error estimate for flow improve truncation, optimize regulator, check stability Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

20 Quadratic action at the cutoff Γ Λ [ϕ] = 1 2 d d x ϕ( + m 2 Λ )ϕ, solution of the FRG-equation Γ k [ϕ] = Γ Λ [ϕ] + 1 ( + m 2 ) 2 log det Λ + R k + mλ 2 + R Λ (7) last term for optimized cutoff 3d: 4d: ( 1 6π 2 mλ 3 arctan m ) Λ(k Λ) mλ 2 + kλ + k 3 m2 Λ (Λ k) + 3 Λ3 3 1 (m 4Λ 64π 2 log m2 Λ + k 2 mλ m2 Λ2 Λ (Λ 2 k 2) + k 4 ) 2 Λ4 2, Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

21 Functional renormalization in QM anharmonic oscillator S[q] = dτ ( ) 1 2 q2 + V (q), here LPA (local potential approximation) ( ) 1 Γ k [q] = dτ 2 q2 + u k (q) (8) low-energy approximation leading order in gradient expansion scale-dependent effective potential u k neglected: higher derivative terms, mixed terms q n q m Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

22 flow equation contains Γ (2) k = τ 2 + u k (q) LPA: sufficient to consider a constant q momentum space dτ u k(q) k = 1 2 = 1 2 dτdτ R k k (τ τ ) dτ dp 2π 1 τ 2 + u k (q) + R k k R k (p) p 2 + u k (q) + R k(p) (τ τ) choose optimal regulator function ( R k (p) = (k 2 p 2 ) θ k 2 p 2) = k R k (p) = 2kθ(k 2 p 2 ) non-linear partial differential equation for u k : k u k (q) = 1 π k 2 k 2 + u k (q) Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

23 minimum of u k (q) not ground state energy differs by q-independent contribution free particle limit fixes subtraction in flow equation k u k (q) = 1 ( k 2 ) π k 2 + u k (q) 1 = 1 u k (q) π k 2 + u k (q) (9) assume u Λ (q) even u k (q) even polynomial ansatz u k (q) = n=0,1, (2n)! a 2n(k) q 2n, Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

24 scale-dependent couplings a 2n Insert into (9), compare coefficients of powers of q 2 infinite set of coupled ode s da 0 dk = 1 π a 1 2 0, 0 = k 2, + a 2 da 2 dk = k2 π a 4 2 0, da 4 dk = 2 k2 0 π da 6 dk = 2 k2 0 π.. ( a 6 6a ) ( ) a 8 30a 4 a a initial condition: a 2n at cutoff = parameters in classical potential projection onto space of polynomials up to given degree n,, Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

25 e.g. crude truncation a 6 = a 8 = = 0: finite set of ode s use standard notation a 0 = E, a 2 = ω 2 and a 4 = λ, truncated system of flow equations de k dk = ω2 k π 0, dωk 2 dk = λ k k2 π 2 0, dλ k dk = 6k 2 λ 2 k π 3 0 solve numerically (eg. with octave) initial conditions E Λ = 0, ω Λ = 1, varying λ at the cutoff scale scale-dependent couplings E k and ω 2 k hardly change for k ω variation near typical scale k ω Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

26 E k ω 2 k from above: from above: λ = 2.0 λ = 1.0 λ = 0.5 λ = λ = 2.0 λ = 1.0 λ = 0.5 λ = k The flow of the couplings E k and ωk 2 (E Λ = 0, ωλ 2 = 1). k Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

27 ω = ω k=0 > 0 effective potential minimal at origin ground state energy: E 0 = min(u k=0 ) energy of first excited state E 1 = E 0 + u k=0 (0) = E 0 + ω k=0 already good results with simple truncation Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

28 energies for different λ different truncations und regulators units of ω ground state energy energy of first excited state cutoff optimal optimal Callan exact optimal optimal Callan exact order 4 order order 4 result order 4 order order 4 result λ = λ = λ = λ = λ = λ = λ = λ = λ = λ = λ = λ = Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

29 Recall flow equation in LPA: k u k (q) = 1 π u k (q) k 2 + u k (q) negative ω 2 in V : local maximum at 0 and two minima denominator minimal where u k minimal (maximum of u k) denominator positive for large scales denominator remains positive during the flow flow equation u k (q) increases toward infrared if u k (q) is positive u k (q) decreases toward infrared if u k (q) is negative double-well potential flattens during flow, becomes convex convexity expected on general grounds Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

30 solution of partial differential equation, ω 2 = 1, λ = 1 u k u k=0.5 V v q Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

31 energies of ground state and first excited state: less good, less stable fourth-order polynomials inaccurate results for weak couplings numerical solution of the flow equation does better decreasing λ (increasing barrier) increasingly difficult to detect splitting induced by instanton effects: must go beyond leading order LPA Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

32 energies for ω 2 = 1 and varying λ optimized regulator, units of ω ground state energy energy of first excited state optimal optimal pde exact optimal optimal pde exact order 4 order 12 order 4 order 12 λ = λ = λ = λ = λ = λ = λ = λ = λ = λ = λ = Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

33 Scalar Field Theory QM = 1-dimensional field theory Now: Euclidean scalar field theory in d dimensions L = 1 2 ( µφ) 2 + V (φ) first local potential approximation ( ) 1 Γ k [ϕ] = d d x 2 ( µϕ) 2 + u k (ϕ) second functional derivative: Γ (2) k = + u k (ϕ) flow of effective potential: may assume constant average field k u k (q) = 1 2 d d p k R k (p) (2π) d p 2 + u k (q) + R k(p) (10) Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

34 optimized regulator: volume of the d-dimensional ball divided by (2π) d, µ d = 1 (4π) d/2 Γ(d/2 + 1) p-integration can be done flow equation k d+1 k u k (ϕ) = µ d k 2 + u, (11) (ϕ) k dimensions enters via k d+1 and µ d nonlinear partial differential equation polynomial ansatz for even potential Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

35 flow equations for infinite set of couplings k da 0 dk = +µ dk d+2 0, 0 = 1 k 2 + a 2, k da 2 dk = µ dk d a 4, k da ( ) 4 dk = µ dk d a 6 6a4 2 0, k da 6 dk = µ dk d ( a 8 30a 4 a a ), Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

36 Fixed points K 2 T < T c (K 1,K 2 ):fixedpoint (K 1c,K 2c ): criticalpoint T > T c line of critical points K 1 Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

37 critical hyper-surface on which ξ = RG trajectory moves away from critical surface If flow begins on critical surface stays on surface most critical points are not fixed point d 3 : expect a finite set of isolated fixed points fixed point K = (K 1, K 2,... ) RG flow in the vicinity of fixed point K = K + δk linearize flow around fixed point K i = Ki + δk i ( ) = R i K j + δk j = K i + R i K δk δk j + O(δK 2 ) j Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

38 linearized RG transformation, δk i = j M j i δk j, M j i = R i K j K eigenvalues and left-eigenvectors Φ α of matrix M Φ j αmj i = λ α Φ i α = b yα Φ i α j subset of {Φ α } span space tangential to critical surface at K every λ α defines a critical exponent y α consider the new variables g α = i Φ i αδk i. Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

39 We have g α = i Φ i αδk i = ij Φ i αm j i δk j = j b yα Φ j αδk j = b yα g α. (12) y α > 0: deviation g α increases, flow moves point K + g α away from the fixed point K relevant perturbation y α < 0: deviation g α decreases, flow carries point K + g α towards the fixed point K irrelevant perturbation y α = 0: marginal coupling relevant couplings determine important scaling laws all TD critical exponent functions of relevant exponents relevant couplings and exponents determine IR-physics Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

40 Fixed point analysis for scalar models introduce the dimensionless field and potential, ϕ = k (d 2)/2 µ d χ and u k (ϕ) = k d µ d v k (χ) flow equation in terms of dimensionless quantities k k v k + dv k d 2 2 χv k = v k, v k = v k χ... at a fixed point: k v k = 0 fixed point equation for effective potential (ode) dv d 2 2 χv = v Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

41 constant solution dv = 1 trivial Gaussian fixed point are there non-gaussian fixed points? answer depends on the dimension d of spacetime even classical potential v k even as well: flow equation for w k (ϱ) v k (χ) = w k (ϱ), with ϱ = χ2 2 k k w k (ϱ) + dw k (ϱ) (d 2) ϱw k (ϱ) = w k (ϱ) + 2ϱw k (ϱ) fixed point equation dw (ϱ) (d 2) ϱw (ϱ) = w (ϱ) + 2ϱw (ϱ) Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

42 2d theories: many fixed-point solutions [Morris 1994] also true for 2d Yukawa theories [Synatschke et al.] polynomial truncation to order m: w (m) = flow equation for couplings m c n ϱ n n=0 k k c 0 = dc 0 + 0, 0 = (1 + c 1 ) 1, k k c 1 = 2c 1 6c 2 2 0, k k c 2 = (d 4)c 2 15c c k k c 3 = (2d 6)c 3 28c c 2 c c , k k c 4 = (3d 8)c 4 45c (336c 2 c c3) c2c c Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

43 Scalar fields in three dimensions expect nontrivial fixed point in d = 3 first: polynomial truncation set c k = 0 for k > m insert into above system of equations with lhs = 0 m algebraic equations for the m + 1 fixed-point couplings 0 = f 0 (c 0, c 1 ) = f 1(c 1, c 2 ) = = f m 1(c 1,..., c m) polynomials in c 0, c 2,..., c m and 0 = 1/(1 + c 1 ) prescribe c 1 (= slope at origin) and thus 0 solve the system for c 0, c 2, c 3,..., c m in terms of c 1 algebraic computer program solution for m up to 42 Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

44 explicit expression for the lowest fixed-point couplings c0 = c1 c 2 = c 1 (1 + c 1 )2 3 c3 = c 1 (1 + c 1 )3 (1 + 13c1 ) 45 c4 = c 2 1 (1 + c 1 )4 (1 + 7c1 ), 21.. P k polynomial of order k c m = c 2 1 (1 + c 1 )m P m 3 (c 1 ), Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

45 trivial solution (Gaussian fixed point w = 1) search for other fixed points: set c m = 0 P m 3 (c 1 ) = 0 c 0 = 1 3, 0 = c 2 = c 3 = c 4 =... polynomials P k has many real roots c1 for each m choose c1 such that convergence for large m the approximating polynomials converge to a power series with maximal radius of convergence example: m = 20 c 1 = m = 42 c 1 = calculate other ck fixed point solution Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

46 With n! multiplied fixed-point coefficients c n c 0 c 1 c 2 c 3 c 4 c 5 c 6 m = m = c 7 c 8 c 9 c 10 c 11 c 12 c 13 m = m = c 14 c 15 c 16 c 17 c 18 c 19 c 20 m = m = c k stable when one increases polynomial order m (m 2k) Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

47 Polynomial approximations vs. numerical solution numerics: shooting method with seventh-order Runge-Kutta 6 w m = 30 m = 10 numerical solution m = 40 m = Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

48 fine-tune slope at origin w (0) Polynomial of degree 42 w (0) Critical exponents flow equation in the vicinity of fixed-point solution w set w k = w + δ k, linearize the flow in small δ k linear differential equation for the small fluctuations k k δ k = dδ k + (d 2) ϱδ k ( dw (d 2) ϱw ) 2 ( δ k + 2ϱδ ) insert the polynomial approximation for fixed-point solution polynomial ansatz for the perturbation k δ k (ϱ) = m 1 n=0 d n ϱ n ϱ = χ2 2 Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

49 linear system for the coefficients d m d 0 d 1 k k. = M (c 0 ) d m 1 d 0 d 1. d m 1 critical exponents = eigenvalues of m-dimensional matrix M up to order m = 46 with algebraic computer program Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

50 m ν = 1/ω 1 ω 2 ω 3 ω 4 ω convergence two negative exponents ω 0 = 3 and ω 1 = 1/ν ω 0 ground state energy, unrelated to critical behavior ω 2, ω 3, ω 4,... all positive (irrelevant) LPA-prediction: ν = (high-t expansion: ν = 0.630) Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

51 Wave function renormalization next-to-leading in derivative expansion wave function renormalization Z k (p, ϕ) difficult non-linear parabolic partial differential RG-equations first step: neglect field and momentum dependence ( ) 1 Γ k [ϕ] = d d x 2 Z k ( µ ϕ) 2 + u k (ϕ). second functional derivative Γ (2) k = Z k + u k (ϕ) flow equation (simplification for R k Z k R k ): d d x ( 1 2 ( kz k ) ( µ ϕ) 2 + k u k (ϕ) ) = 12 tr ( ) k (Z k R k ) Z k (p 2 + R k ) + u k (ϕ) Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

52 simple: flow of effective potential: k u k = Z k Z k k 2 + u k, Z k = µ ) d d + 2 k (k d+2 Z k. more difficult: flow of Z k project flow on operator ( φ) 2 must admit non-homogeneous fields [p 2, u k (ϕ)] 0 final answer ( ) 2 k k Z k = µ d k d+2 Z k a , 0 = Z k k 2 + a 2 see A. Wipf, Lecture Notes in Physics 864 anomalous dimension η = k k log Z t Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

53 Linear O(N) models scalar field φ R N O(N) invariant potential L = 1 2 ( µφ) 2 + V (φ) fixed-point analysis: dimensionless quantities χ and ν k invariant dimensionless composite field N ϱ = 1 2 i=1 χ 2 i set ν k (χ) = w k (ϱ) Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

54 flow equation in LPA (optimized regulator) k k w k + dw k (d 2) ϱw k = N w k contribution of the N 1 Goldstone modes contribution of massive radial mode w k + 2ϱw k large N: Goldstone modes give dominant contribution linearize about fixed-point solution: w k = w + δ k fluctuation δ k obeys the linear differential equation k k δ k = dδ k + (d 2) ϱδ k (N 1) δ k (1 + w ) 2 δ k + 2ϱδ k (1 + w + 2ϱw ) 2 Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

55 proceed as before: polynomial truncation to high order (40) slope at origin of fixed-point solution find always Wilson-Fisher fixed point eigenvalue ω 0 = 3 of the scaling operator 1 not listed N w (0) ν = 1/ω ω ω extract asymptotic formulas w (0) , ν N N Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

56 Large N Limit rather simple flow equation (t = log(k/λ) t = k k ) t w k = (d 2) ϱw k dw k + N 1 + w k t w = (d 2) ϱw 2w N (1 + w ) 2 w can be solved exactly with methods of characteristics analytic relation between fixed point solution and perturbation in s(t, ρ) w (ρ) + e ωt δ(ρ) result: w(t, ϱ) w (ϱ) + const e ωt w (ϱ) (ω+d)/2. Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

57 if perturbation regular all critical exponents ω {2n d n = 0, 1, 2,... } one-parameter family of fixed point solutions w N = π/4 4 /N 1 Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

58 flow of dimensionless eff. potential for critical κ λ Λ = 1, κ Λ = κ crit (κ Λ : dimensionless minimum of V ) w k w k (0) 1 k/λ k = Λ /N Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

59 flow of dimensionful eff. potential above critical κ λ Λ = 1, κ Λ = 1.3κ crit broken phase u k /(µ 3 Λ 3 N) k 0 /(µ 3 ΛN) k Λ Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60

60 flow of dimensionful eff. potential below critical κ λ Λ = 1, κ Λ = 0.5κ crit symmetric phase 1 u k /(µ 3 Λ 3 N) 0.5 k 0 k Λ /(µ 3 ΛN) Andreas Wipf (FSU Jena) The Functional Renormalization Group Method An Introduction21. Juli / 60