the renormalization group (RG) idea

Transcription

1 the renormalization group (RG) idea Block Spin Partition function Z =Tr s e H. block spin transformation (majority rule) T (s, if s i ; s,...,s 9 )= s i > 0; 0, otherwise. b

2 Block Spin (block-)transformed Hamiltonian e H (s ) Tr s T (s ; s i )e H(s) block Partition function of transformed system Z = e H (s ) =Tr s Tr s T (s ; s i )e H(s) = Z block Free energy per site f ln Z = N = b d ln Z N = bd f Coupling constants set of coupling constants of the reduced Hamiltonian {K} = {K,K 2,...} Renormalization group transformation {K } = R b {K} Fixed point of the transformation {K } = R b {K } R b R b2 = R b b 2 (group property)

3 General Theory Linearization around the fixed point. K i K i j Eigenvalue and eigenstates of the matrix T scaling variable u a = i = λ a j T ij (K j K j ), i ea i T ij = λ a e a j e a i (K i K i )= i T ij = K i K j u a i ea i (K i K i ) e a i e a j (K j K j )=λ a u a K=K T ij (K j Kj ) j RG eigenvalues Group property R b R b2 = R b b 2 λ a b λ a b 2 = λ a b b 2. Thus, λ a = b y a y a : renormalization group eigenvalues ya > 0 : scaling variable u a is relevant. y a < 0 : scaling variable u a is irrelevant. y a =0: scaling variable u a is marginal.

4 RG flow (toy model) K 2 critical surface High T K Low T (K c,k 2c ) K Free energy For example, the Ising model. Two relevant scaling variables u t (T T c )/T c and u h h. Two relevant eigenvalues Block spin transformation revisited Z = e N f (K ) =Tr s e H N f (K )=Ng(K)+N f(k ) non-singular y t and y h. =Tr s e H = e Nf(K) f s (K) =b d f s (K ) singular part

5 Scaling behavior Near the fixed point (critical point) f s (u t,u h )=b d f s (b y t u t,b y h u h )=b nd f s (b ny t u t,b ny h u h ) Set b ny t u t = u t0 ( u t0 : small but arbitrary) f s (t, h) = t d/y t Φ ht y h/y t Φ(x) : universal scaling function Critical Exponents specific heat 2 f t 2 spontaneous Magnetization susceptibility f h ( t) (d y h)/y t h=0 2 f h 2 t d/yt 2 α =2 d/y t h=0 t (d 2y h)/y t h=0 β = d y h y t γ = 2y h d y t

6 Critical Exponents Set b ny h u h = u h0, f s (t, h) = h d/y h Φ th y t/y h critical isotherm M = f s h h d/yh δ = t=0 y h d y h Correlation Function spin-spin correlation function G(r r 2, H) = block spin correlation function G((r r 2 )/b, H )= renormalization 2 ln Z h(r ) h(r 2 ) 2 ln Z h (r ) h (r2 ) = h=0 h=0 2 ln Z h (r ) h (r2 ) G((r r 2 )/b, H )=b 2(d y h) G(r r 2, H) h=0 scaling function G(r, t) =b 2(d y h) G(r/b, tb y t ) G(r, t) =r 2(d y h) Ψ(r/ t y t )

7 Critical Exponents Correlation Length G(r, t) =r 2(d y h) Ψ(r/ t y t ) ξ t y t ν = y t correlation function at the critical point G(r, t = 0) r 2(d y h) η = d +2 2y h Wilson s momentum shell renormalization group

8 Landau-Ginzburg-Wilson Hamiltonian revisited Partition function Z[J] = Dϕ exp V I = λ 4! q 2 (q2 + a)ϕ( q)ϕ(q) J( q)ϕ(q) V I q,q 2,q 3 ϕ(q )ϕ(q 2 )ϕ(q 3 )ϕ( q q 2 q 3 ) UV cutoff : inverse of lattice spacing Λ c 0 c 0 momentum shell RG carry out the partial integral over all Fourier components with Λ/b < q < Λ. scale transformation x x = x/b, q q = bq Λ/b Λ field rescale m(x) m (x )=b ζ m(x), ϕ(q) ϕ (q )=b ζ d ϕ(q)

9 Perturbation expansion Z[J] = Dϕ < Dϕ > e S 0 V I + J ϕ = Dϕ < e S 0(ϕ < )+ J < ϕ < Dϕ > e S 0(ϕ > )+ J > ϕ > V I (ϕ < +ϕ > ) = Z 0 [ J > ] Dϕ < e S 0(ϕ < )+ J < ϕ < cumulant expansion e V I 0 = Z 0 [ J > ] Dϕ < exp S 0 (ϕ < )+ J < ϕ < +ln e V I 0 For J=0, 0 Z0 [J > = 0] ϕ > (q)ϕ > (q ) 0 = q 2 + a δ(q + q ) Dϕ > e S 0(ϕ > ) Cumulant expansion ln e V I 0 = V I First term ( ϕ = ϕ < + ϕ > ) V I 0 = λ 4!,2,3 V 2 I 0 V I ϕ(q )ϕ(q 2 )ϕ(q 3 )ϕ( q q 2 q 3 ) 0 one and/or three ϕ < = 0. four ϕ < : V I (ϕ < ), no ϕ < : non-singular Terms with two large momentum fields λ 2 < q ϕ < (q)ϕ < ( q) > p p 2 + a

10 2nd cumulant 2 V 2 I 0 = λ2 2(4!) 2 ϕ ϕ 2 ϕ 3 ϕ 4 ϕ 5 ϕ 6 ϕ 7 ϕ 8 δ( )δ( ) λ 2 6 p (p 2 + a) 2 ϕ 4 < higher order contribution to mass Partition Function Z[J] Dm < exp dx 2 m <( 2 λ + a)m < + 4! m4 < a = a + λ 2 S da(a) =a + λ 2 S d λ = λ 3 2 λ2 S d B(a) =λ Λ 3 2 λs d Λ/b p d p 2 + a dp Λ Λ/b p d dp (p 2 + a) 2 S d = K d (2π) d = 2 d π d/2 Γ(d/2)

11 Integral For small a d 2 b2 d A(a) =Λ d 2 d 4 b4 d B(a) =Λ d 4 d 4 b4 d rλ d 4 + O(r 2 ), For large a Contribution from λ is negligible. free theory < =b d = Rescale dx 2 m <( 2 + a)m < + λ dx 4 m4 < 2 b 2ζ m( b a)m + b 4ζ λ 4! m4 dx 2 m( 2 + a )m + λ 4! m4 ζ = d 2 2, a = b 2 a + λ 2 S d 2 b2 d dλ d 2 λ = b 4 d λ S d Λ d 4 3λ 2 aλ 2 S d 4 b4 d dλ d 4 b 4 d d 4,

12 Differential RG flow Set b = e δ with δ. Differential RG flow equation dr d =2r + u 2 ru 2, du d =(4 d)u 3 2 u2, a = Λ 2 r, fixed points ( =4 d) (r,u )= u = S d Λ d 4 λ (0, 0), Gaussian fixed point, ( /6, 2/3) + O( 2 ), Wilson-Fisher fixed point Stability of fixed points Gaussian fixed point du d = u d>4 : stable, d<4 : unstable Wilson-Fisher fixed point d d (u u )= (u u ) d>4 : unstable, d<4 : stable

13 RG eigenvalue At the Wilson-Fisher fixed point (d<4) d d (r r )=(2 /3)(r r ) y t =2 /3 ν =/y t = O(2 ), η = O( 2 ) At the Gaussian fixed point (d>4) dr d =2r y t =2 ν = 2, η = 0 : Mean Field Theory III. Linear Response Theory

14 Fluctuation and response spin fluctuation = susceptibility χ 2 ln Z h 2 2 = Z Tr s i e βh Z Tr i m 2 m 2. fluctuation response specific heat = energy fluctuation i s i e βh 2 c 2 ln Z β 2 = Z TrH2 e βh H 2 H 2. 2 Z TrHe βh Relaxation Dynamics Slightly out of equilibrium case m(r) t = Γ δs δm(r) S = dx +ξ(r, t) 2 ( m)2 + am 2 + bm 4 hm ξ =0, ξ(r, t)ξ(r,t ) = Dδ(r r )δ(t t ) P ξ (ξ(r, t)) exp 2D thermal fluctuations (noise) noise strength Hohenberg and Halperin, model A ξ(r, t) 2 drdt

15 Steady State Fokker-Planck equation P m (m(r),t)=δ(m(r) m(r, t, {ξ}) ξ t P m(m(r),t)= dr δ δm(r ) Γ δs δm(r ) P m + D 2 δp m δm(r ) steady state = equilibrium state P eq m exp 2ΓS D =exp( βs) D =2Γk B T fluctuation-dissipation relation