Field Theory Approach to Equilibrium Critical Phenomena

Transcription

1 Field Theory Approach to Equilibrium Critical Phenomena Uwe C. Täuber Department of Physics (MC 0435), Virginia Tech Blacksburg, Virginia 24061, USA Renormalization Methods in Statistical Physics and Lattice Field Theories Montpellier, August 2015

2 Lecture 1: Critical Scaling: Mean-Field Theory, Real-Space RG Ising model: mean-field theory Real-space renormalization group Landau theory for continuous phase transitions Scaling theory Lecture 2: Momentum Shell Renormalization Group Landau Ginzburg Wilson Hamiltonian Gaussian approximation Wilson s momentum shell renormalization group Dimensional expansion and critical exponents Lecture 3: Field Theory Approach to Critical Phenomena Perturbation expansion and Feynman diagrams Ultraviolet and infrared divergences, renormalization Renormalization group equation and critical exponents Recent developments

3 Lecture 1: Critical Scaling: Mean-Field Theory, Real-Space RG

4 Ferromagnetic Ising model Principal task of statistical mechanics: understand macroscopic properties of matter (interacting many-particle systems): thermodynamic phases and phase transitions Phase transitions at temperature T > 0 driven by competition between energy E minimization and entropy S maximization: minimize free energy F = E T S Example: Ising model for N spin variables σ i = ±1 with ferromagnetic exchange couplings J ij > 0 in external field h: H({σ i }) = 1 2 N J ij σ i σ j h i,j=1 Goal: partition function Z(T, h, N) = {σ i =±1} e H({σ i })/k B T, free energy F (T, h, N) = k B T ln Z(T, h, N), thermal averages: A({σi }) 1 = A({σ i }) e H({σ i })/k B T Z(T, h, N) {σ i =±1} N i=1 σ i

5 Curie Weiss mean-field theory Mean-field approximation: replace effective local field with average: h eff,i = H = h+ J ij σ j h+ σ Jm, J = J(x i ), m = σ i i j i More precisely: σ i = m + (σ i σ i ) σ i σ j = m 2 + m (σ i σ i + σ j σ j ) + (σ i σ i )(σ j σ j ) Neglect fluctuations / spatial correlations H Nm2 J ( h+ 2 Jm ) N ( σ i, Z e Nm2 J/2k B T 2 cosh h + Jm ) N k B T i=1 yields Curie Weiss equation of state m(t, h) = 1 ( ) Fmf N h T,N = tanh h + J m(t, h) k B T Solution for large T : disordered, paramagnetic phase m = 0 T < T c = J/k B : ordered, ferromagnetic phase m 0 Spontaneous symmetry breaking at critical point T c, h = 0

6 Mean-field critical power laws Expand equation of state near T c : τ = T T c T c 1 and h J m 1: h τm + m3 k B T c 3 critical isotherm: T = T c : h k BT c 3 m 3 coexistence curve: h = 0, T < T c : m ±( 3τ) 1/2 isothermal susceptibility: ( ) m χ T = N N 1 h k B T c τ + m 2 T N k B T c { 1/τ 1 τ > 0 1/2 τ 1 τ < 0 Power law singularities in the vicinity of the critical point Deficiencies of mean-field approximation: predicts transition in any spatial dimension d, but Ising model does not display long-range order at d = 1 for T > 0 experimental critical exponents differ from mean-field values origin: diverging susceptibility indicates strong fluctuations

7 Real-space renormalization group: Ising chain Partition sum for h = 0, K = J/kB T : N Z (K, N) = e K i=1 σi σi+1 a K a K... K σi-1 σi K K K σi+1... {σi =±1} decimation of σi, σi+2,... { } 2 cosh 2K σ σ i 1 i+1 = +1 K σi (σi 1 +σi+1 ) e = = e 2g +K σi 1 σi+1 2 σi 1 σi+1 = 1 σi =±1 ( ) 1 N Z (K, N) = Z K = ln cosh 2K, 2 2 ℓ decimations: N (ℓ) = N/2ℓ, a(ℓ) = 2ℓ a, RG recursion: K (ℓ) = 12 ln cosh 2K (ℓ 1) Fixed points phases, phase transition: I K = 0 stable T =, disordered K = unstable T = 0, ordered ( ) 1 2 T 0: expand K 1 ( K )ln dk dℓ (ℓ) ln22 K 1 (ℓ)2 2K Correlation length: K ℓ = ln(2ξ/a) 0 ξ(t ) 2a e 2J/kB T ln 2 I

8 Real-space RG for the Ising square lattice βh({σ i }) = K n.n. (i,j) σ i σ j βh ({σ i }) = A + K +L n.n. (i,j) σ i σ j + M σ i σ j σ k σ l n.n.n. (i,j) (i,j,k,l) a a σ i σ j K K L K 2 cosh K(σ 1 + σ 2 + σ 3 + σ 4 ) = = e A K (σ 1 σ 2 +σ 2 σ 3 +σ 3 σ 4 +σ 4 σ 1 )+L (σ 1 σ 3 +σ 2 σ 4 )+M σ 1 σ 2 σ 3 σ 4 List possible configurations for four nearest neighbors of given spin: σ 1 σ 2 σ 3 σ cosh 4K = e A +2K +2L +M 2 cosh 2K = e A M 2 = e A 2L +M 2 = e A 2K +2L +M

9 RG recursion relations 1 ln cosh 4K 2K 2 + O(K 4 ) 4 K 1 L = = ln cosh 4K K A = L + ln 4 cosh 2K ln 2 + 2K 2 2 M = A ln 2 cosh 2K 0 drop [ ]2 [ ]2 a(ℓ) = 2ℓ/2 a, K (ℓ) 2 K (ℓ 1) + L(ℓ 1), L(ℓ) K (ℓ 1) K = K = 0 = L stable T = : disordered paramagnet I K = = L stable T = 0: ordered ferromagnet I Kc = 1/3, L c = 1/9 unstable: critical fixed point ( (ℓ) ) ( ) ( (ℓ 1) ) δk = K (ℓ) Kc 4/3 1 δk Linearize RG flow: = 2/3 0 δl(ℓ) = L(ℓ) L c δl(ℓ 1) ( ) with eigenvalues λ1/2 = 31 2 ± 10 and associated eigenvectors: ( (ℓ) ) ( ) ( ) ( ) 3 3 K 1/3 ℓ ℓ + c1 λ1 + c2 λ2 1/ L(ℓ) I

10 Critical point scaling Utilize linearized RG flow to analyze critical behavior: λ 1 > 1 relevant direction; λ 2 < 1 irrelevant direction Critical line: c 1 = 0, set L c = 0 (n.n. Ising model), l = 0 ( ) ( ) ( ) Kc 1/3 3 + c 0 1/9 2 1 c = 9( 10+2) K c ; mean-field: K c = 0.25; exact: K c = Relevant eigenvalue determines critical exponent: l 1: λ l 2 0, δk (l) e l ln λ 1 (K K c ) correlations: ξ (l) = 2 l/2 ξ ξ = ξ (l) δk (l) K K c ln 2/2 ln λ 1 ξ (l) a ξ(t ) T T c ν, ν = ln 2 2 ln compare mean-field theory: ν = 1 2 ; exact (L. Onsager): ν = 1 Real-space renormalization group approach: difficult to improve systematically, no small parameter successful applications to critical disordered systems

11 General mean-field theory: Landau expansion Expand free energy (density) in terms of order parameter (scalar field) ϕ near a continuous (second-order) phase transition at T c : f (ϕ) = r 2 ϕ2 + u 4! ϕ h ϕ r = a(t T c ), u > 0; conjugate field h breaks Z(2) symmetry ϕ ϕ f r > 0 r = 0 r < 0 f (ϕ) = 0 equation of state: φ - 0 φ+ φ h(t, ϕ) = r(t ) ϕ + u 6 ϕ3 φ Stability: f (ϕ) = r + u 2 ϕ2 > 0 h > 0 Critical isotherm at T = T c : h(t c, ϕ) = u 6 ϕ3 0 T c h < 0 T Spontaneous order parameter for r < 0: ϕ ± = ±(6 r /u) 1/2

12 Thermodynamic singularities at critical point Isothermal order parameter susceptibility: ( ) h V χ 1 T = = r + u ϕ 2 ϕ2 χ { T 1/r V = 1 r > 0 1/2 r 1 r < 0 T divergence at T c, amplitude ratio 2 χ T C h=0 0 T c T 0 T c Free energy and specific heat vanish for T T c ; for T < T c : f (ϕ ± ) = r 4 ϕ2 ± = 3r 2 ( 2 ) 2u, C f h=0 = VT T 2 = VT 3a2 u h=0 discontinuity at T c T

13 Scaling hypothesis for free energy Postulate: (sing.) free energy generalized homogeneous function: ( ) h f sing (τ, h) = τ 2 α ˆf ± τ, τ = T T c T c two-parameter scaling, with scaling functions ˆf ±, ˆf ± (0) = const. Landau theory: critical exponents α = 0, = 3 2 Specific heat: C h=0 = VT T 2 c Equation of state: ϕ(τ, h) = ( fsing h ( 2 ) f sing τ 2 = C ± τ α h=0 ) Coexistence line h = 0, τ < 0: τ ( ) = τ 2 α h ˆf ± τ ϕ(τ, 0) = τ 2 α ˆf (0) τ β, β = 2 α

14 Scaling relations Critical isotherm: τ dependence in ˆf ± must cancel prefactor, as x : ˆf ±(x) x (2 α )/ ϕ(0, h) h (2 α )/ = h 1/δ, δ = β Isothermal susceptibility: ( ) χ τ ϕ V = = χ ± τ γ, γ = α + 2( 1) h τ, h=0 Eliminate scaling relations: = β δ, α + β(1 + δ) = 2 = α + 2β + γ, γ = β(δ 1) only two independent (static) critical exponents Mean-field: α = 0, β = 1 2, γ = 1, δ = 3, = 3 2 (dim. analysis) Experimental exponent values different, but still universal: depend only on symmetry, dimension..., not microscopic details

15 Thermodynamic self-similarity in the vicinity of T c Temperature dependence of the specific heat near the normal- to superfluid transition of He 4, shown in successively reduced scales From: M.J. Buckingham and W.M. Fairbank, in: Progress in low temperature physics, Vol. III, ed. C.J. Gorter, , North-Holland (Amsterdam, 1961).

16 Selected literature: J.J. Binney, N.J. Dowrick, A.J. Fisher, and M.E.J. Newman, The theory of critical phenomena, Oxford University Press (Oxford, 1993). N. Goldenfeld, Lectures on phase transitions and the renormalization group, Addison Wesley (Reading, 1992). S.-k. Ma, Modern theory of critical phenomena, Benjamin Cummings (Reading, 1976). G.F. Mazenko, Fluctuations, order, and defects, Wiley Interscience (Hoboken, 2003). R.K. Pathria, Statistical mechanics, Butterworth Heinemann (Oxford, 2nd ed. 1996). A.Z. Patashinskii and V.L. Pokrovskii, Fluctuation theory of phase transitions, Pergamon Press (New York, 1979). L.E. Reichl, A modern course in statistical physics, Wiley VCH (Weinheim, 3rd ed. 2009). F. Schwabl, Statistical mechanics, Springer (Berlin, 2nd ed. 2006). U.C. Täuber, Critical dynamics A field theory approach to equilibrium and non-equilibrium scaling behavior, Cambridge University Press (Cambridge, 2014), Chap. 1.

17 Lecture 2: Momentum Shell Renormalization Group

18 Landau Ginzburg Wilson Hamiltonian Coarse-grained Hamiltonian, order parameter field S(x): [ r H[S] = d d x 2 S(x) [ S(x)]2 + u ] 4! S(x)4 h(x) S(x) r = a(t T 0 c ), u > 0, h(x) local external field; gradient term [ S(x)] 2 suppresses spatial inhomogeneities Probability density for configuration S(x): Boltzmann factor P s [S] = exp( H[S]/k B T )/Z[h] canonical partition function and moments functional integrals: Z[h] = D[S] e H[S]/kBT, ϕ = S(x) = D[S] S(x) P s [S] Integral measure: discretize x x i, D[S] = i ds(x i) or employ Fourier transform: S(x) = d d q S(q) e iq x (2π) d D[S] = q ds(q) V = q,q 1 >0 d Re S(q) d Im S(q) V

19 Landau Ginzburg approximation Most likely configuration Ginzburg Landau equation: 0 = δh[s] δs(x) = [ r 2 + u 6 S(x)2] S(x) h(x) Linearize S(x) = ϕ + δs(x) δh(x) ( r 2 + u 2 ϕ2) δs(x) Fourier transform Ornstein Zernicke susceptibility: χ 0 (q) = 1 r + u 2 ϕ2 + q 2 = 1 ξ 2 + q 2, ξ = { 1/r 1/2 r > 0 1/ 2r 1/2 r < 0 Zero-field two-point correlation function (cumulant): C(x x ) = S(x) S(x ) S(x) 2 = (k B T ) 2 δ2 ln Z[h] δh(x) δh(x ) Fourier transform C(x) = d d q (2π) d C(q) e iq x fluctuation response theorem: C(q) = k B T χ(q) h=0

20 Scaling hypothesis for correlation function Scaling ansatz, defines Fisher exponent η and correlation length ξ: C(τ, q) = q 2+η Ĉ ± (qξ), ξ = ξ ± τ ν Thermodynamic susceptibility: χ(τ, q = 0) ξ 2 η τ ν(2 η) = τ γ, γ = ν(2 η) Spatial correlations for x : C(τ, x) = x (d 2+η) C± (x/ξ) ξ (d 2+η) τ ν(d 2+η) S(x)S(0) S 2 = ϕ 2 ( τ) 2β hyperscaling relations: β = ν (d 2 + η), 2 α = dν 2 Mean-field values: ν = 1 2, η = 0 (Ornstein Zernicke) Diverging spatial correlations induce thermodynamic singularities!

21 Gaussian approximation High-temperature phase, T > T c : neglect nonlinear contributions: d d [ q 1 ( H 0 [S] = r + q 2 ) ] (2π) d S(q) 2 h(q)s( q) 2 Linear transformation Gaussian integral: S(q) = S(q) h(q), r+q 2 q... = d d q (2π) d and Z 0 [h] = D[S] exp( H 0 [S]/k B T ) = ( 1 h(q) 2 ) ( = exp 2k B T q r + q 2 D[ S] r + q 2 ) exp q 2k B T S(q) 2 S(q)S(q ) 0 = (k BT ) 2 (2π) 2d δ 2 Z 0 [h] Z 0 [h] δh( q) δh( q ) h=0 = C 0 (q) (2π) d δ(q + q ), C 0 (q) = k BT r + q 2

22 Gaussian model: free energy and specific heat F 0 [h] = k B T ln Z 0 [h] = 1 2 Leading singularity in specific heat: ( 2 ) F 0 C h=0 = T T 2 Vk B (atc 0 ) 2 2 h=0 q ( h(q) 2 r + q 2 + k BT V ln 2π k BT r + q 2 q 1 (r + q 2 ) 2. d > 4: integral UV-divergent; regularized by cutoff Λ (Brillouin zone boundary) α = 0 as in mean-field theory d = d c = 4: integral diverges logarithmically: Λξ 0 k 3 (1 + k 2 dk ln(λξ) ) 2 d < 4: with k = q/ r = qξ, surface area K d = 2πd/2 C sing Vk B(aT 0 c ) 2 ξ 4 d 2 d π d/2 Γ(d/2) 0 Γ(d/2) : ) k d 1 (1 + k 2 ) 2 dk T T c 0 4 d 2 diverges; stronger singularity than in mean-field theory.

23 Renormalization group program in statistical physics Goal: critical (IR) singularities; perturbatively inaccessible. Exploit fundamental new symmetry: divergent correlation length induces scale invariance. Analyze theory in ultraviolet regime: integrate out short-wavelength modes / renormalize UV divergences. Rescale onto original Hamiltonian, obtain recursion relations for effective, now scale-dependent running couplings. Under such RG transformations: Relevant parameters grow: set to 0: critical surface. Certain couplings approach IR-stable fixed point: scale-invariant behavior. Irrelevant couplings vanish: origin of universality. Scale invariance at critical fixed point infer correct IR scaling behavior from (approximative) analysis of UV regime derivation of scaling laws. Dimensional expansion: ϵ = d c d small parameter, permits perturbational treatment computation of critical exponents.

24 Wilson s momentum shell renormalization group RG transformation steps: (1) Carry out the partition integral over all Fourier components S(q) with wave vectors Λ/b q Λ, where b > 1: eliminates short-wavelength modes (2) Scale transformation with the same scale parameter b > 1: x x = x/b, q q = b q Λ/b Λ Accordingly, we also need to rescale the fields: S(x) S (x ) = b ζ S(x), S(q) S (q ) = b ζ d S(q) Proper choice of ζ rescaled Hamiltonian assumes original form scale-dependent effective couplings, analyze dependence on b Notice semi-group character: RG transformation has no inverse

25 Momentum shell RG: Gaussian model ( < H 0 [S < ] + H 0 [S > ] = + where < q... = q <Λ/b q > q ) [ ] r + q 2 S(q) 2 h(q) S( q) 2 d d q..., > (2π) d q Choose ζ = d 2 2 r r = b 2 r,... = Λ/b q Λ d d q (2π) d... h(q) h (q ) = b ζ h(q), h(x) h (x ) = b d ζ h(x) r, h both relevant critical surface: r = 0 = h Correlation length: ξ ξ = ξ/b ξ r 1/2 : ν = 1 2 Correlation function: C (x ) = b 2ζ C(x) η = 0 Add other couplings: c d d x ( 2 S) 2 : c c = b d 4 2ζ c = b 2 c, irrelevant u d d x S(x) 4 : u u = b d 4ζ u = b 4 d u; relevant for d < 4, (dangerously) irrelevant for d > 4, marginal at d = d c = 4 v d d x S(x) 6 : v v = b 6 2d v, marginal for d = 3; irrelevant near d c = 4: v = b 2 v

26 Momentum shell RG: general structure General choice: ζ = d 2+η 2 τ = b 1/ν τ, h = b (d+2 η)/2 h Only two relevant parameters τ and h Few marginal couplings u i u i = ui + b x i u i, x i > 0 Other couplings irrelevant: v i v i = b y i v i, y i > 0 After single RG transformation: { f sing (τ, h, {u i }, {v i }) = b d f sing (b 1/ν τ, b d ζ h, ui + u } { i vi }) b x, i b y i After sufficiently many l 1 RG transformations: ) f sing (τ, h, {u i }, {v i }) =b ld f sing (b l/ν τ, b l(d+2 η)/2 h, {ui }, {0} Choose matching condition b l τ ν = 1 scaling form: ( f sing (τ, h) = τ dν ˆf± h/ τ ν(d+2 η)/2) Correlation function scaling law: use b l = ξ/ξ ± ( C(τ, x, {u i }, {v i }) = b 2lζ C b l/ν τ, x ) b l, {u i }, {0} C ± (x/ξ) x d 2+η

27 Perturbation expansion Nonlinear interaction term: H int [S] = u S(q 1 )S(q 2 )S(q 3 )S( q 1 q 2 q 3 ) 4! q i <Λ Rewrite partition function and N-point correlation functions: Z[h] = Z 0 [h] e H int[s], 0 i S(q i ) = i S(q i) e H int[s] 0 e H int [S] 0 contraction: S(q)S(q ) = S(q)S(q ) 0 = C 0 (q) (2π) d δ(q + q ) Wick s theorem: S(q 1 )S(q 2 )... S(q N 1 )S(q N ) 0 = = S(q i1 (1))S(q i2 (2))... S(q in 1 (N 1))S(q in (N)) permutations i 1 (1)...i N (N) compute all expectation values in the Gaussian ensemble

28 First-order correction to two-point function Consider S(q)S(q ) = C(q) (2π) d δ(q + q ) for h = 0; to O(u): [ S(q)S(q ) 1 u ] S(q 1 )S(q 2 )S(q 3 )S( q 1 q 2 q 3 ) 4! q i <Λ Contractions of external legs S(q)S(q ): terms cancel with denominator, leaving S(q)S(q ) 0 The remaining twelve contributions are of the form q i <Λ S(q)S(q 1) S(q 2 )S(q 3 ) S( q 1 q 2 q 3 )S(q ) = = C 0 (q) 2 (2π) d δ(q + q ) p <Λ C 0(p) [ C(q) = C 0 (q) 1 u ] 2 C 0(q) C 0 (p) + O(u 2 ) p <Λ re-interpret as first-order self-energy in Dyson s equation: C(q) 1 = r + q 2 + u 1 2 r + p 2 + O(u2 ) p <Λ Notice: to first order in u, there is only mass renormalization, no change in momentum dependence of C(q) 0

29 Wilson RG procedure: first-order recursion relations Split field variables in outer (S > ) / inner (S < ) momentum shell: simply re-exponentiate terms u S 4 <e H 0[S] contributions such as u S 3 < S > e H 0[S] vanish terms u S 4 >e H 0[S] const., contribute to free energy contributions u S 2 < S 2 >e H 0 : Gaussian integral over S > With S d = K d /(2π) d = 1/2 d 1 π d/2 Γ(d/2) and η = 0 to O(u): r = b 2[ r + u ] 2 A(r) = b [r 2 + u Λ 2 S p d 1 ] d Λ/b r + p 2 dp u = b 4 d u [1 3u ] [ 2 B(r) = b 4 d u 1 3u Λ 2 S p d 1 ] dp d (r + p 2 ) 2 r 1: fluctuation contributions disappear, Gaussian theory r 1: expand d 2 1 b2 d d 4 1 b4 d A(r) = S d Λ r S d Λ + O(r 2 ) d 2 d 4 d 4 1 b4 d B(r) = S d Λ d 4 + O(r) Λ/b

30 Differential RG flow, fixed points, dimensional expansion Differential RG flow: set b = e δl with δl 0: d r(l) = 2 r(l) + ũ(l) dl 2 S dλ d 2 r(l)ũ(l) S d Λ d 4 + O(ũ r 2, ũ 2 ) 2 dũ(l) = (4 d)ũ(l) 3 dl 2 ũ(l)2 S d Λ d 4 + O(ũ r, ũ 2 ) Renormalization group fixed points: d r(l)/dl = 0 = dũ(l)/dl Gauss: u 0 = 0 Ising: u I S d = 2 3 (4 d)λ4 d, d < 4 Linearize δũ(l) = ũ(l) ui : d dl δũ(l) (d 4)δũ(l) u0 stable for d > 4, u I stable for d < 4 Small expansion parameter: ϵ = 4 d = d c d ui emerges continuously from u 0 = 0 Insert: ri = 1 4 u I S dλ d 2 = 1 6 ϵλ2 : non-universal, describes fluctuation-induced downward T c -shift RG procedure generates new terms S 6, 2 S 4, etc; to O(ϵ 3 ), feedback into recursion relations can be neglected

31 Critical exponents Deviation from true T c : τ = r ri T T c Recursion relation for this (relevant) running coupling: [ d τ(l) = τ(l) 2 ũ(l) ] dl 2 S dλ d 4 Solve near Ising fixed point: τ(l) = τ(0) exp [( 2 ϵ ) ] 3 l Compare with ξ(l) = ξ(0) e l ν 1 = 2 ϵ 3 Consistently to order ϵ = 4 d: ν = ϵ 12 + O(ϵ2 ), η = 0 + O(ϵ 2 ) Note at d = d c = 4: ũ(l) = ũ(0)/[1 + 3 ũ(0) l/16π 2 ] logarithmic corrections to mean-field exponents Renormalization group procedure: Derive scaling laws. Two relevant couplings independent critical exponents. Compute scaling exponents via power series in ϵ = d c d.

32 Selected literature: J.J. Binney, N.J. Dowrick, A.J. Fisher, and M.E.J. Newman, The theory of critical phenomena, Oxford University Press (Oxford, 1993). J. Cardy, Scaling and renormalization in statistical physics, Cambridge University Press (Cambridge, 1996). M.E. Fisher, The renormalization group in the theory of critical behavior, Rev. Mod. Phys. 46, (1974). N. Goldenfeld, Lectures on phase transitions and the renormalization group, Addison Wesley (Reading, 1992). S.-k. Ma, Modern theory of critical phenomena, Benjamin Cummings (Reading, 1976). G.F. Mazenko, Fluctuations, order, and defects, Wiley Interscience (Hoboken, 2003). A.Z. Patashinskii and V.L. Pokrovskii, Fluctuation theory of phase transitions, Pergamon Press (New York, 1979). U.C. Täuber, Critical dynamics A field theory approach to equilibrium and non-equilibrium scaling behavior, Cambridge University Press (Cambridge, 2014), Chap. 1. K.G. Wilson and J. Kogut, The renormalization group and the ϵ expansion, Phys. Rep. 12 C, (1974).

33 Lecture 3: Field Theory Approach to Critical Phenomena

34 Perturbation expansion O(n)-symmetric Hamiltonian (henceforth set k B T = 1): n [ r H[S] = d d x 2 S α (x) [ S α (x)] 2 + u n ] S α (x) 2 S β (x) 2 4! α=1 β=1 Construct perturbation expansion for ij S α i S α j : ij Sα i S α j e H int[s] 0 ij = Sα i S α j l=0 e H int [S] 0 l=0 l! q Diagrammatic representation: α Propagator C 0 (q) = 1 r+q 2 Vertex u 6 ( H int [S]) l l! ( H int [S]) l 0 α α β β β 0 = C 0 (q) = u 6 Generating functional for correlation functions (cumulants): Z[h] = exp d d x h α S α, S α i = δ(ln)z[h] (c) δh α i α i i h=0 δ αβ

35 Vertex functions Connected Feynman diagrams: u u Dyson equation: + + propagator self-energy: u u u + = + Σ + Σ Σ +... = + Σ C(q) 1 = C 0 (q) 1 Σ(q) Generating functional for vertex functions, Φ α = δ ln Z[h]/δh α : Γ[Φ] = ln Z[h] + d d x h α Φ α, Γ (N) {α i } = N δγ[φ] δφ α i h=0 α i 4 4 Γ (2) (q) = C(q) 1, S(q i ) c = C(q i ) Γ (4) ({q i }) i=1 one-particle irreducible Feynman graphs Perturbation series in nonlinear coupling u loop expansion i=1 u u

36 Explicit results Two-point vertex function to two-loop order: Γ (2) (q) = r + q 2 + n ( ) n u 6 n u2 u k + u k 1 r + k 2 u u 1 k r + k 2 1 r + k 2 1 k r + k k (r + k 2 ) 2 u 1 r + (q k k ) 2 four-point vertex function to one-loop order: Γ (4) ({q i = 0}) = u n + 8 u (r + k 2 ) 2 u u k u

37 Ultraviolet and infrared divergences Fluctuation correction to four-point vertex function: d < 4 : u d d k 1 (2π) d (r + k 2 ) 2 = u r 2+d/2 2 d 1 π d/2 Γ(d/2) 0 x d 1 (1 + x 2 ) 2 dx effective coupling u r (d 4)/2 as r 0: infrared divergence fluctuation corrections singular, modify critical power laws Λ 0 k d 1 (r + k 2 ) 2 dk { ln(λ 2 /r) d = 4 Λ d 4 d > 4 } as Λ ultraviolet divergences for d > d c = 4: upper critical dimension Power counting in terms of arbitrary momentum scale µ: [x] = µ 1, [q] = µ, [S α (x)] = µ 1+d/2 ; [r] = µ 2 relevant, [u] = µ 4 d marginal at d c = 4 only divergent vertex functions: Γ (2) (q), Γ (4) ({q i = 0}) field dimensionless at lower critical dimension d lc = 2

38 Dimension regimes and dimensional regularization dimension perturbation O(n)-symmetric critical interval series Φ 4 field theory behavior d d lc = 2 IR-singular ill-defined no long-range UV-convergent u relevant order (n 2) 2 < d < 4 IR-singular super-renormalizable non-classical UV-convergent u relevant exponents d = d c = 4 logarithmic IR-/ renormalizable logarithmic UV-divergence u marginal corrections d > 4 IR-regular non-renormalizable mean-field UV-divergent u irrelevant exponents Integrals in dimensional regularization: even for non-integer d, σ: d d k (2π) d k 2σ Γ(σ + d/2) Γ(s σ d/2) (τ + k 2 ) s = 2 d π d/2 Γ(d/2) Γ(s) in effect: discard divergent surface integrals τ σ s+d/2 UV singularities dimensional poles in Euler Γ functions

39 Renormalization Susceptibility χ 1 = C(q = 0) 1 = Γ (2) (q = 0) = τ = r r c r c = n u 6 r c + k 2 + O(u2 ) = n + 2 u K d Λ d 2 6 (2π) d d 2 k (non-universal) T c -shift: additive [ renormalization χ(q) 1 = q 2 + τ 1 n + 2 ] 1 u 6 k k 2 (τ + k 2 ) Multiplicative renormalization: absorb UV poles at ϵ = 0 into renormalized fields and parameters: SR α = Z 1/2 S S α Γ (N) R = Z N/2 S Γ (N) τ R = Z τ τ µ 2, u R = Z u u A d µ d 4, A d = Γ(3 d/2) 2 d 1 π d/2 Normalization point outside IR regime, τ R = 1 or q = µ: O(u R ) : Z τ = 1 n + 2 u R 6 ϵ, Z u = 1 n + 8 u R 6 ϵ O(uR 2 ) : Z S = 1 + n + 2 ur ϵ

40 Renormalization group equation Unrenormalized quantities cannot depend on arbitrary scale µ: 0 = µ d dµ Γ(N) (τ, u) = µ d [ ] Z N/2 dµ S Γ (N) R (µ, τ R, u R ) renormalization group equation: [ µ µ + N 2 γ S + γ τ τ R + β u τ R u R with Wilson s flow and RG beta functions: γ S = µ ln Z S = n + 2 µ 0 72 u2 R + O(u3 R ) γ τ = µ ln τ R µ 0 τ = 2 + n + 2 u R + O(uR 2 6 ) β u = µ [ u R = u R d 4 + µ ] ln Z u µ 0 µ 0 [ = u R ϵ + n + 8 ] u R + O(uR 2 6 ) ] Γ (N) R (µ, τ R, u R ) = 0 ß u 0 u R

41 Method of characteristics Susceptibility χ(q) = Γ (2) (q) 1 : ( χ R (µ, τ R, u R, q) 1 = µ 2 ˆχ R τ R, u R, q ) 1 µ solve RG equation: method of characteristics µ µ(l) = µ l [ l χ R (l) 1 = χ R (1) 1 l 2 exp 1 ] γ S (l ) dl l u(l) u(1) τ(1) with running couplings, initial values τ(1) = τ R, ũ(1) = u R : l d τ(l) = τ(l) γ τ (l), l dũ(l) = β u (l) dl dl Near infrared-stable RG fixed point: β u (u ) = 0, β u(u ) > 0 τ(l) τ R l γ τ, χ R (τ R, q) 1 µ 2 l 2+γ S ˆχR (τ R l γ τ, u, q µ l ) 1 matching l = q /µ scaling form with η = γ S, ν = 1/γ τ τ(l)

42 Critical exponents Systematic ϵ = 4 d expansion: [ β u = u R ϵ + n u 0 = 0, u H = IR stability: β u(u ) > 0 u R + O(u 2 R ) ] 6 ϵ n O(ϵ2 ) d > 4: Gaussian fixed point u0 η = 0, ν = 1 2 (mean-field) d < 4: Heisenberg fixed point uh stable η = n (n + 8) 2 ϵ2 + O(ϵ 3 ), ν 1 = 2 n + 2 n + 8 ϵ + O(ϵ2 ) d = d c = 4: logarithmic corrections: u R ũ(l) = 1 n+8 6 u R ln l, τ(l) τ R l 2 (ln l ) (n+2)/(n+8) ξ τ 1/2 R (ln τ R ) (n+2)/2(n+8) Accurate exponent values: Monte Carlo simulations; or: Borel resummation; non-perturbative exact (numerical) RG ß u 0 u R

43 Non-perturbative RG, critical dynamics Non-perturbative RG: numerically solve exact RG flow equation for effective potential Γ = Γ k 0 t Γ k = 1 [ 2 Tr Γ (2) 1 k k(q)] (q) + R t R k (q) q with appropriately chosen regulator R k, t = ln(k/λ) Critical dynamics: relaxation time t c (τ) ξ(τ) z τ zν with dynamic critical exponent z; time scale separation Langevin equations for order parameter and conserved fields: t S α (x, t) = F α [S](x, t) + ζ α (x, t), ζ α (x, t) = 0 ζ α (x, t)ζ β (x, t ) = 2L α δ(x x ) δ(t t ) δ αβ map onto Janssen De Dominicis response functional: A[S] ζ = D[S] A[S] P[S], P[S] D[i S] e A[ S,S] A[ S, tf S] = d d x dt [ ( S α t S α F [S]) α S ] α L α Sα 0 α

44 Non-equilibrium dynamic scaling Field theory representations for non-equilibrium dynamical systems: Coarse-grained effective Langevin description: Janssen De Dominicis functional Interacting / reacting particle systems: Doi Peliti field theory from stochastic master equation Non-equilibium quantum dynamics: Keldysh Baym Kadanoff Green function formalism All contain additional field encoding non-equilibrium dynamics anisotropic (d + 1)-dimensional field theory: dynamic exponent(s) RG fixed points dynamic scaling properties, characterize: non-equilibrium stationary states / phases universality classes for non-equilibrium phase transitions non-equilibrium relaxation and aging scaling features properties of systems displaying generic scale invariance

45 Selected literature: D.J. Amit, Field theory, the renormalization group, and critical phenomena, World Scientific (Singapore, 1984). M. Le Bellac, Quantum and statistical field theory, Oxford University Press (Oxford, 1991). C. Itzykson and J.M. Drouffe, Statistical field theory, Vol. I, Cambridge University Press (Cambridge, 1989). A. Kamenev, Field theory of non-equilibrium systems, Cambridge University Press (Cambridge, 2011). G. Parisi, Statistical field theory, Addison Wesley (Redwood City, 1988). P. Ramond, Field theory A modern primer, Benjamin Cummings (Reading, 1981). U.C. Täuber, Critical dynamics A field theory approach to equilibrium and non-equilibrium scaling behavior, Cambridge University Press (Cambridge, 2014). A.N. Vasil ev, The field theoretic renormalization group in critical behavior theory and stochastic dynamics, Chapman & Hall / CRC (Boca Raton, 2004). J. Zinn-Justin, Quantum field theory and critical phenomena, Clarendon Press (Oxford, 1993).