Notes on Renormalization Group: ϕ 4 theory and ferromagnetism Yi Zhou (Dated: November 4, 015) In this lecture, we shall study the ϕ 4 theory up to one-loop RG and discuss the application to ferromagnetic transition. I. QUARTIC PERTURBATION TO GAUSSIAN MODEL: THE ϕ 4 THEORY Now we consider the quartic perturbation to the Gaussian model δs = 1 4! 1 4! 4 k <Λ i=1 k <Λ d D k i (π) D u( k 4 k3 k k1 )ϕ ( k 4 )ϕ ( k 3 )ϕ( k )ϕ( k 1 )δ D ( k 4 + k 3 k k 1 ) u(431)ϕ (4)ϕ (3)ϕ()ϕ(1), where the coupling function obeys the symmetry condition The RG transformation can be done through Linked expansion will give rise to u(431) = u(341) = u(431). e S ϕ <] = e S0ϕ<] e δsϕ<,ϕ] 0 e S0 δs. δs = δs 1 ( δs δs ) +. Since δs u, the above is a perturbation expansion and can be evaluated through Feynman diagrams. Linked expansion where c denotes connected average. ln e U = U + 1 U c + + ( 1)n n! U n c, The leading term has the form δs = 1 u(431) (ϕ < + ϕ ) 4 4! (ϕ < + ϕ ) 3 (ϕ < + ϕ ) (ϕ < + ϕ ) 1. k <Λ 0 There are possible monomials, 8 terms with odd number of fast ϕ modes, 1 term with all fast ϕ modes, 1 term with all slow ϕ < modes (called tree-level), 6 terms with two fast ϕ and two slow ϕ < modes. The first set will vanish by symmetry and the second set will make a constant contribution, independent of ϕ <, to the effective action. Consider the third set will all slow modes, say, the tree-level terms in field theory, and rewrite it in terms of new momenta and fields, we find that δs 4,tree = 1 4 d D k i 4! k < Λ i=1 (π) D u( k 4 k3 k k1 )ϕ ( k 4 )ϕ ( k 3 )ϕ( k )ϕ( k 1 )δ D ( k 4 + k 3 k k 1 ) = 1 4 4! ζ4 b 3D d D k i k <Λ i=1 (π) D u( k 4/b, k 3/b, k /b, k 1/b)ϕ ( k 4)ϕ ( k 3)ϕ ( k )ϕ ( k 1)δ D ( k 4 + k 3 k k 1) = 1 4 d D k 4! b4 D i k <Λ (π) D u( k 4/b, k 3/b, k /b, k 1/b)ϕ ( k 4)ϕ ( k 3)ϕ ( k )ϕ ( k 1)δ D ( k 4 + k 3 k k 1). i=1
So that Carrying out the Taylor expansion, we see that b 4 D u( k 4/b, k 3/b, k /b, k 1/b) = u ( k 4, k 3, k, k 1). u = u 0 + u 1 k + u k +, u 0 = b 4 D u 0, u 1 = b 3 D u 1, u = b D u, and so on and so forth. It is clear that when D = 4, only u 0 is marginal and other terms are irrelevant. This is why the ϕ 4 theory in D = 4 is described by a coupling constant and not a coupling function. For D 4, all the ϕ 4 terms are irrelevant. Homework: Analyze the relevance of operators ϕ n, n 4 in different dimensions D. II. ONE-LOOP RG FOR THE ϕ 4 THEORY According to previous tree-level analysis, higher gradient terms are irrelevant, we consider the following simplified model, in D-dimension, 1 Sϕ] = d D r ( ϕ) + 1 rϕ + ] 4! ϕ4 hϕ, where h is the external magnetic field. By seperating ϕ = ϕ < + ϕ, up to the one-loop, we have Sϕ] = S < ϕ < ] + S ϕ ] + S I ϕ <, ϕ ], with 1 S < ϕ < ] = d D r ( ϕ <) + r ϕ < + ] 4! ϕ4 < hϕ <, 1 S ϕ ] = d D r ( ϕ ) + r ] ϕ, S I ϕ <, ϕ ] = d D r ϕ 4 <ϕ + ], where we only keep the ϕ <ϕ term at one-loop level. (It is also easy to verify that other terms will lead to two-loop contribution at leading order.) Thus, the effective action is given by the following connected diagrams contribution S ϕ < ] = S < ϕ < ] + S I ϕ <, ϕ ] 1 where the sbuscript c denotes a connected average. The first order average reads S I ϕ <, ϕ ] = 4 The integral can be computed by expanding r to the first order, SI ϕ <, ϕ ] c, d D p 1 d D p (π) D r + p < (π) D ϕ <( p)ϕ < ( p). d D p (π) D 1 r + p = I 1 ri,
3 with where I α = d D p (π) D 1 p α = Ω D Λ D α 1 b 1 dpp D 1 α = Ω DΛ D α D α (1 bα D ), Ω D = (π) D Γ( D )(π)d is the volume of the D-dimensional unit sphere. So that by the momenta and field scaling, S () ϕ ] = b r + Ω D (D ) (1 b D ) rω ] D (D 4) (1 b4 D ) d D rϕ. The second order will lead to the contribution proportional to ϕ 4 <, 1 SI ϕ <, ϕ ] c d D rϕ 4 d D q 1 < (π) D (r + q ) = I d D rϕ 4 < + O( r). Evaluating the integral and rescaling, we have the S (4) ϕ ] as follows, Finally, the linear term after rescaling reads S (4) ϕ ] = b 4 D ( 4! Ω D S (1) ϕ ] = hb 1+ D 1 b 4 D D 4 d D rϕ. Thus, to the one-loop order, the coupling constants scale as follows, r b r + Ω D (D ) (1 b D ) rω D b 4 D ( 3 Ω D h hb 1+ D. 1 b 4 D D 4 ), ) d D rϕ 4. ] (D 4) (1 b4 D ), ϵ-expansion: It is interesting that ϵ = 4 D can serves an expansion paramter. We set D = 4 ϵ and evaluate the above to leading order of ϵ. With Ω 4 ϵ Ω 4 = 1 8π. Therefore, r b r + 3π (1 b ) r ] π ln b, ) (1 + ϵ ln b) ( 3 π ln b, h hb 3 ϵ/. Setting b = e l, we have the Gell-Mann-Low equations dr d dh = r + π r 8π, = ϵ 3 π, = 6 ϵ h. We find that one-loop correction will not affect the scaling behavior of h. We shall consider the simple case with h = 0 and analyze the RG flow for r and at first. The β-function β() = ϵ 3 π exhibits different behaviors for ϵ 0 and ϵ < 0. When ϵ < 0, there is only one trivial fixed point. While for ϵ 0, there exists a non-trivial fixed point.
Fixed point: The fixed points are given by r + π r 8π = 0, ϵ 3 π = 0. Gaussian fixed point: a trival fixed point is Gaussian fixed point given by r = = 0. The W -matrix is ( ) W =. ϵ Non-Gaussian fixed point: a nontrivial fixed point is given by r = ϵ 6, 4 Around this fixed point, we have = π 3 ϵ. The critical exponent can be computed as follows, y t = ϵ 3, y h = 6 ϵ. α = ϵ 3, β = 1 ϵ 6, γ = 1 + ϵ 6, δ = 3 + ϵ, η = 0, ν = 1 + ϵ. Phase diagram ε <0 ε 0 Paramagnetic Paramagnetic Ferromagnetic r Ferromagnetic r FIG. 1: Phase digrams of the ϕ 4 model from ϵ-expansion. 1] John Cardy, Scaling and renormalization group in statistical physics, Cambridge Univsersity Press (1996).
] M. E. Peskin and D. V. Schroeder, An introduction to quantum field theory, Westview Press (1995). 3] Alexander Altland and Ben Simons, Condensed Matter Field theory, Cambridge Univsersity Press (010). 4] R. Shankar, Renormalization-group approach to interacting fermions, Review of Modern Physics, 66, 19 (1994). 5] John B. Kogut, Rev. Mod. Phys. 51, 659 (1979). 5