Notes on Renormalization Group: Introduction

Transcription

1 Notes on Renormalization Group: Introduction Yi Zou (Dated: November 19, 2015) We introduce brief istory for te development of renormalization group (RG) in pysics. Kadanoff s scaling ypoteis is present to understand te critical penomena. We also present te general framework for tree-stage RG transformations to obtain RG flow equations (Gell-Mann- Low equations). Te idea of fixed point is presented as well as relevant, irrelevant and marginal coupling. I. BRIEF HISTORY Wy renormalization group (RG)? Renormalization group plays an essential role in modern pysics, including quantum field teory, statistical mecanics, condensed matter pysics, and even dissipative quantum tunnelling. Quantum electrodynamics (QED): A quantum field teory Te problem of infinities: Ultraviolet (UV) divergence (Weisskopf) Renormalization: Feynman, Scwinger, Tomonaga, Dyson, et al. Quantum field teory and renormalization group: Peterman and Strückberg, Gell-Mann and Low, Bogoliubov and Sirkov Tey consider QED in te limit of potons and massless electrons. In a massive teory, te renormalized carge can be defined troug te electric interaction of particles at rest, say, Coulomb interaction. Tis definition is no longer applicable to massless particles, wic always travel at te speed of ligt. Ten it is necessary to introduce an arbitary mass or energy scale µ to define te renormalized carge e. Tis effective carge ten depends on µ. Te set of transformations of pysical constants accociated wit te cange in scale µ is called renormalization group. Standard model Critical penomena: Oter infinities (Infrared (IR) divengence) Pases and pase transition Mean field teory Fluctuations Te non-decoupling of scales Te cutoff as scale paramter Kadanoff and Wilson s renormalization group Scaling ypotesis Continuum limit Gaussian fixed point and mean field teory Go beyond Guassian fixed point Te concept of effective field teory More is different Emergnet penomena: Different pysical laws will appear at different scale Renormalizability: A criterion to construct effective field teories

2 2 A. Te cutoff as scale parameter: Effective field teory at criticality To ensure tat we remain in te low-energy domain, we would like to take te cutoff to be infinite, but tis cannot be done by declaration. In te absence of external fields, ta action of te system does not contain an intrinsic energy scale apart from te cutoff. Tus, te cutoff disappears from te action wen we reduce all quantities to dimensionless form. Te only way to tell weter it is finite or infinite is to calculate some pysical quantity wit dimension, suc as te correlation lengt, from te teory. Te cutoff is infinite wen te correlation lengt diverges, in wic case te system is said to be at a critical point. To approac te limit of infinite cutoff, terefore, we must adjust te parameters so as to make te system go critical. ϕ(x)ϕ(y) Ce x y / ξ x y Here te correlation lengt ξ is measured in te same unit as x. Using Λ 1 as unit for distance, we ave were ξ is dimensionless: x y ξ = x y Λ ξ ξ = Λ ξ. Ignoring te patological case ξ = 0, we see tat an infinite cutoff corresponds to te limit ξ. II. CRITICAL PHENOMENA A. Critical exponents For continuous pase transitions, te critical exponents will not vary along te pase boundary, wic caracterize te universality class for pase transitions. Te critical exponents for ferromagnetic systems are list in te following table. TABLE I: Critical exponents for ferromagnetic systems, were t = (T T c )/T 0, = (H H c )/H 0, T 0 and H 0 are nonuniversal constants. critical exponent associated quantity singular part α eat capacity C H t α β magnetization m t β γ susceptibility χ t γ δ magnetization m 1/δ η correlation function G( r) e r/ξ r d 2+η ν correlation lengt ξ t ν B. Scaling ypotesis Widom made te following omogeneity assumption for te singular form of free energy in te vicinity of a pase transition, ( ) f(t, ) = t 2 α g f t,

3 were g f is a scaling funciton for free energy (wic is different for t > 0 and t < 0), and te exponents α and depend on te critical point under consideration. Using tis form of free energy, one can determine a series of critical exponents using te two independent exponents α and. Heat capacity: E f ( ) ( ) ( ) t = (2 α)t1 α g f t t 1 α g f t t 1 α g E t, 3 and ( ) C H t α g C t, ( were g ) ( C t = (1 ) ( α)ge t g ) E t. Magnetization: In te limit x 0, g m (x) is a constant, so tat m(t, ) = f ( ) ( ) t2 α g f t t 2 α g m t m(t, = 0) t 2 α g m (0), i.e., β = 2 α. On te oter and, if x, g m (x) x p, we must ave p = 2 α. Hence i.e., δ = /(2 α ) = /β. Susceptibility: So tat i.e., γ = 2 + α 2. A number of exponent identities follows, for example, m(t = 0, ) p = 2 α, ( ) χ(t, ) t 2 α 2 g χ t χ(t, = 0) t 2 α 2 g χ (0), and α + 2β + γ = 2, δ 1 = γ/β. Correlation lengt: Close to criticality, te correlation lengt ξ is te most imporant lengt scale, and is solely responsible for singular contribution to termodynamic quantities. Since ln Z is dimensionless and extensive, ln Z = Ld ξ d g s + Ld a d g a. From te assumption of te sigular form for f(t, ), we ave g s ξ d = t2 α g f, and ξ(t, ) t α 2 d gξ ( t ),

4 4 i.e., 2 α = dν. Correlation functions: At te critical point, On te oter and So tat χ G( r) 1 r d 2+η. ξ d d rg( r) d d 1 r r d 2+η ξ2 η t ν(2 η). γ = (2 η)ν. Te critical system as an additional dilation symmetry, G critical (λ r) = λ p G critical ( r), wic implies scale invariance or self-similarity. It is not in general possible to see directly ow suc a symmetry requrirment constrains te effective Hamiltonian. We sall instead precrible a less direct route by following te effects of te dilation operation on te effective energy, namely, a process known as renormalization group. Homework: Solve one-dimensional Ising model using transfer matrix metod. Find out te transition temperature T c, and compute te critical exponents α, β, and γ. III. RENORMALIZATION GROUP: GENERAL THEORY Suppose we ave an effective field teory defined torug te action, S[ϕ] = a g a O a [ϕ], were ϕ is some field, g a are coupling constants and O a [ϕ] a certain set of operators. Renormalization is a sceme to derive a set of Gell-Mann-Low equations describing te cange of te coupling constants {g a } as fast fluctuation modes of te teory are sucessively integrated out. Tere are a number of metodologically different procedures by wic te set of flow equations can be obtained from te microscopic teory. In general, RG transformations contains te following tree stages. Subdivision of te field manifold, te practice may be different in different situations, for instance We may proceed a generalized block spin sceme, and integrate over all degrees of freedom witin a certain structural unit in real space. We may separate fast and slow field modes (ϕ > and ϕ < ) in momentum space, and integrate over a sell of Λ/b p < Λ, were Λ is te momentum cutoff and b > 1. We may decide to integrate over all te ig-lying degrees of freedom λ 1 p. In tis case, te encountered divergent integrals may be andled wit dimensional regularization, for example. We may use te sort-distance real space cutoff underlying te socalled operator product expansion (OPE). RG transformation, te central part of RG, is to actually integrate over sort-range fluctuations. For RG in momentum space, a frequently used approximation sceme is te loop expansion.

5 5 Following te procedure, an integration over te fast degrees of freedom gives rise to an action S [ϕ < ] = a g ao a[ϕ < ], were coupling constants of te remaining slow fields are altered. Notice tat te integration over fast field fluctuations may lead to te generation of new operators. Rescaling, to recover te form of S[ϕ] wit te same set of {O a [ϕ]}. One ten rescales momentum/frequency so tat te rescaled field ϕ fluctuates on te same scales as te original field ϕ, i.e., one sets q bq, ω b z ω, were te dynamical exponent z depends on te effective dispersion. We will also designate a dimension L d ϕ to for te field ϕ, so as to compensate for te factor b x arising after te renormalization of te operator. Te rescaling ϕ b d ϕ ϕ is known as field renormalization. It renders te leading operator in te action scale invarinat. As a result of all tese manipulations, we obtain a renormalized action S[ϕ] = a g ao a [ϕ], wic is entirely described by te set of canged coupling contants, i.e. te effect of te RG transformation is fully encapsulated in te mapping g = R(g). Letting l ln b, we ave te Gell-Mann-Low equation dg dl = R(g), were R(g) = lim l 0 ( R(g) g)/l is te generalized β-function. A. Analysis of te Gell-Mann-Low equation Te Gell-Mann-Low equation represents te principle result of an RG analysis. Fixed points, were te RG flow becomes stationary, R(g ) = 0. At a fixed point, te system becomes self-similar, namely, does not cange under rescaling. One can analyze te RG flow close to a fixed point g by expanding were te matrix W is given by Ten te matrix W can be diagonalized as follows, R(g) W (g g ), W ab = R a g b g=g. ϕ T αw = λ α ϕ T α.

6 6 Scaling field. Defining we ave dv α dl v α = ϕ T α (g g ), = ϕ T α dg dl = ϕt α R(g) = ϕ T αw (g g ) = λ α ϕ T α (g g ) = λ α v α. Under renormalization, te coefficients v α cange by a mere scaling factor λ α, tereby tey are called scaling fields. Relevant scaling field. For λ α > 0, te flow is directly away from te fixed point. Te associated scaling field v α is said to be relevant. Irrelevant scaling field. For λ α < 0, te flow is attracted by te fixed point. Te associated scaling field is said to be irrelevant. Marginal scaling field. For λ α = 0, te scaling field is uncanged under te flow, are termed marginal. Tere exist different types of fixed points. Stable fixed point. All te scaling fields are irrelevant or, at worst, marginal. Tese points define wat we migt call stable pase of matter. Unstable fixed point. Complementary to te stable fixed points, tere are unstable fixed points, wose scalling fields are all relevant. Tere is te gneric class of fixed points wit bot relevant and irrelevant scaling fields. Tese points are of particular interest because tey can be associated wit pase transition. [1] Jon Cardy, Scaling and renormalization group in statistical pysics, Cambridge Univsersity Press (1996). [2] M. E. Peskin and D. V. Scroeder, An introduction to quantum field teory, Westview Press (1995). [3] Alexander Altland and Ben Simons, Condensed Matter Field teory, Cambridge Univsersity Press (2010). [4] Jon B. Kogut, Rev. Mod. Pys. 51, 659 (1979). [5] Jean Zinn-Justin, Pase Transition and Renormalization Group, Oxford Univsersity Press (2002).