Renormalization group

The renormalization group (RG) describes the dependence of certain physical quantities on the length scale. Originally a concept of quantum field theory , its scope now extends to solid-state physics , continuum mechanics , cosmology and nanotechnology . The beta function and the Callan-Symanzik equations are related to the RG .

definition

Renormalization is the term used to describe various computational techniques that allow quantities measured on one length scale to be mapped onto quantities measured on another length scale. These computing techniques typically describe scale-invariant self-similar systems, such as B. Diffusion paths or percolation scluster. The renormalization group is the essential step in the renormalization. It describes scale invariance , deviations from it and transitions between different variants of the scale invariance.

The considered scale-invariant systems are all stochastic in nature. In quantum field theory this is based on quantum fluctuations, in classical physics mostly on thermal fluctuations. A measured variable in the diffusion example would be e.g. B. the number of diffusion steps until a path of length is covered on average . It is typical for this measured variable that it is a correlation function . ${\displaystyle L}$

The importance of computational techniques lies in the fact that they can often be applied according to a scheme and deliver results where other methods fail. For example, naive (regularized) perturbation calculations in quantum field theory and in critical phenomena provide a divergent perturbation series, while the renormalization group implicitly adds up perturbation calculation contributions and correctly expresses the scale invariance . The renormalization group describes in detail the change in the coupling constants when the length scale changes.

Easiest access: Kadanoff's block spin model

Kadanoff's block spin

The block spin model by Leo Kadanoff (1966) provides the easiest didactic access to the RG. For this purpose, one considers a two-dimensional lattice of spin degrees of freedom (but this can also be a model for lattices of atoms with completely different degrees of freedom than angular momentum) of the Ising model type , i.e. only directly adjacent spins interact with one another with a coupling strength . The system is described by a Hamilton function and has the mean temperature . ${\displaystyle \,J}$ ${\displaystyle \,H(T,J)}$ ${\displaystyle \,T}$

Now the spin grid is divided into blocks of - squares and new block variables are introduced by averaging over the state values ​​in the block. Often the new Hamiltonian has the same structure as the old one, only with new values and :    , ${\displaystyle 2\times 2}$ ${\displaystyle \,T}$ ${\displaystyle \,J}$ ${\displaystyle \quad H(T,J)\to H(T',J')}$

This process is now repeated, that is, the new spin block variables are summarized again by averaging (that would then be 4 spins or 16 spins from the original model) etc. The system is thus viewed on a constantly coarser scale. If the parameters under RG transformations no longer change significantly, one speaks of a fixed point of the RG. ${\displaystyle 2\times 2}$

In the specific case of the Ising model , originally introduced as a model for magnetic systems (with an interaction that makes a negative contribution to energy with parallel spins, and a positive contribution with anti-parallel spins ), the heat movement characterized by temperature affects the efforts to order the interaction ( characterized by) against. Here (and often also in similar models) there are three types of fixed points of the RG: ${\displaystyle -\,J}$ ${\displaystyle H}$ ${\displaystyle \,J}$ ${\displaystyle \,T}$ ${\displaystyle \,J}$

(a) and . On large scales, the order, ferromagnetic phase, predominates . ${\displaystyle \,T=0}$ ${\displaystyle \,J\to \infty }$

(b) and . Disorder on large scales. ${\displaystyle \,T\to \infty }$ ${\displaystyle \,J\to 0}$

(c) A point in between with and , at which a change of the scale does not change the physics of the system (scale invariance as in fractal structures), the point is a fixed point of the RG. At this so-called critical point , a phase transition takes place between the two phases (a), (b). In the case of ferromagnetism, it is called the Curie point . ${\displaystyle \,T=T_{c}}$ ${\displaystyle \,J=J_{c}}$

Elements of the RG theory

In general, let the system be described by a function of the state variables with the coupling constants describing the interaction . Depending on the application, this can be a distribution function (statistical mechanics), an effect , a Hamilton function, etc. a. but should fully describe the physics of the system. ${\displaystyle \,Z}$ ${\displaystyle \,\{s_{i}\}}$ ${\displaystyle \,\{J_{k}\}}$

Now we consider block transformations of the state variables , where the number of is smaller than that of . One tries now to write solely as a function of the new status variables . If this is possible simply by changing the parameters of the theory , it is called a renormable theory. ${\displaystyle \{s_{i}\}\to \{{\tilde {s}}_{i}\}}$ ${\displaystyle {\tilde {s}}_{i}}$ ${\displaystyle \,s_{i}}$ ${\displaystyle \,Z}$ ${\displaystyle {\tilde {s}}_{i}}$ ${\displaystyle \{J_{k}\}\to \{{\tilde {J}}_{k}\}}$

Most of the fundamental theories of elementary particle physics, such as quantum electrodynamics , quantum chromodynamics , the electroweak interaction , can be renormalized (but not gravity). In solid state physics and continuum physics, too, many theories can (approximately) be renormalized (e.g. superconductivity, theory of the turbulence of liquids).

The parameters are changed by a so-called beta function , which generates a flow of the RG (RG flow) in the room. The change from under this flow is described by the term running coupling constant . One is particularly interested in the fixed points of the RG flow, which describe the phase transitions between the macroscopic phases of the system. ${\displaystyle {\tilde {J}}_{k}=\beta (J_{i})}$ ${\displaystyle J}$ ${\displaystyle J}$

Since information is constantly being lost in the RG transformations, they generally have no inverse and thus actually do not form groups in the mathematical sense (only semigroups ). The name renormalization group has nevertheless become common.

Relevant and irrelevant operators, universality classes

Consider the behavior of the observables ( given by operators in quantum mechanics ) under an RG transformation: ${\displaystyle A}$

• if it always increases with the transition to larger scales, one speaks of relevant observables ${\displaystyle A}$
• if it always decreases with the transition to larger scales, one speaks of irrelevant observables and ${\displaystyle A}$
• if neither is true, one speaks of marginal observables.

Only relevant operators are important for the macroscopic behavior, and in practice it turns out that in typical systems, after a sufficient number of renormalization steps, only very few operators are “left over” because only they are relevant (although one often assigns an infinite number of operators do, on a microscopic basis the number of observables is typically of the order of the number of molecules in a mole ).

This also explains the astonishing similarity of the critical exponents to one another in the most diverse systems with second order phase transitions, whether it is magnetic systems, superfluids or alloys : the systems become relevant observables due to the same number and the same types (with regard to the scaling behavior) described, they belong to the same universality class .

This quantitative and qualitative justification for the division of the phase transition behavior into universality classes was one of the main successes of the RG.

Momentum space RG

In practical use, there are two types of RG: the RG in real space , as discussed above in Kadanoff's block spin picture, and the momentum space -RG, in which the system is viewed in different wavelengths or frequency scales . A type of integration is usually carried out using the modes of high frequency or short wavelength. In this form, the RG was originally used in particle physics. Since one usually starts from a perturbation theory around the system of free particles, this usually no longer works for strongly correlated systems.

An example for the application of the momentum space RG is the classical renormalization of the mass and charge of the free particles in the QED. In this theory, a naked positive charge is surrounded by a cloud of electron-positron pairs that are constantly generated from the vacuum and immediately destroyed. Since the positrons are repelled by the charge and the electrons are attracted, the charge is ultimately shielded, and the size of the observed charge depends on how close you get to it (sliding coupling constant) or, in the Fourier-transformed image, on which momentum scale you are moves.

Field theoretical renormalization group, technical aspects

The most widespread variant of the renormalization group has its origin in quantum field theory and is a cornerstone of theoretical physics, with many applications in other areas as well. The starting point is a Lagrange function for a field theory and the corresponding path integral . A number of technical aspects in combination result in a great variety. examples are

• Regularization . A regularization is necessary because otherwise the disturbance series terms diverge. The idea today is that in quantum field theory there is in fact something like a cutoff, e.g. B. in the Planck length . In practice i. d. Usually dimensional regularization is the method of choice.
• Different derivations. Multiplicative or additive renormalization.
• Renormalization conditions or minimum subtraction.
• Consideration of the critical point only or consideration of relevant and irrelevant terms (mass terms, external fields, approximation to the critical point).
• Difference between quantum field theory (Limes of small wavelengths) and solid state physics (Limes of large wavelengths)
• Scale invariance at the critical dimension or below the critical dimension. Development according to or numerical calculation directly at . ${\displaystyle d_{c}}$ ${\displaystyle \varepsilon =d-d_{c}}$ ${\displaystyle d<d_{c}}$

Despite the diversity, the computing technology is always the same in essence. The simplest example can be used to understand the essential technical points.

The essence based on an example

The starting point is the Lagrangian function of the model at the critical temperature (without mass term and without magnetic field term ) ${\displaystyle \varphi ^{4}}$ ${\displaystyle \propto \varphi ^{2}}$ ${\displaystyle \propto \varphi }$

${\displaystyle L=\int d^{d}x\left\{{\frac {1}{2}}\left(\nabla \varphi \right)^{2}+{\frac {u}{4!}}\varphi ^{4}\right\}.}$

As a sum of monomials, the Lagrangian can be invariant by rescaling the fields, coordinates, and coupling constants with any scale factor . Here it is ${\displaystyle b}$

${\displaystyle {\begin{aligned}x&\rightarrow x/b,\\\varphi &\rightarrow \varphi b^{\left[\varphi \right]},\\u&\rightarrow ub^{\left[u\right]}.\end{aligned}}}$

By convention, the rescaling exponent is always used for the coordinates . The two terms of the Lagrangian thus provide two equations from which the scaling exponents and result. Here is with the (upper) critical dimension . It should be noted that the coupling constant for the critical dimension is dimensionless. ${\displaystyle [x]=-1}$ ${\displaystyle [\varphi ]=1-\varepsilon /2}$ ${\displaystyle [u]=\varepsilon }$ ${\displaystyle \varepsilon =d-d_{c}}$ ${\displaystyle d_{c}=4}$ ${\displaystyle u}$

The scale invariance of the Lagrange function in the critical dimension does not directly imply a scale invariance of the physical quantities, because these are determined from the path integral with the Lagrange function in the exponent. In order for the path integral to make sense, a regularization is required, which implicitly brings another length scale into play. The regularized path integral supplies the physical quantities. The naive scale invariance of the Lagrangian is i. A. at least modified by fluctuations. A generic starting point of the renormalization group is the assumption that the scale invariance remains asymptotically in a modified form, i.e. This means that the 2- and 4-point vertex functions of the effective Lagrangian are also scale-invariant, albeit with modified scale exponents. By convention, the scale exponent of is written in the form , which is also referred to as the critical exponent . ${\displaystyle d_{c}}$ ${\displaystyle \varphi }$ ${\displaystyle [\varphi ]=1-\varepsilon /2+\eta /2}$ ${\displaystyle \eta }$

By "removing" the nontrivial parts of the scale exponents from the vertex functions and using a field renormalization factor, the "renormalized" vertex functions are obtained, ${\displaystyle \Gamma _{2}}$ ${\displaystyle \Gamma _{4}}$ ${\displaystyle Z=(k/\mu _{0})^{\eta }}$

${\displaystyle {\begin{aligned}\Gamma _{2}^{\left(R\right)}\left(k,u\right)&=Z\,\Gamma _{2}\left(k,u\right),\\\Gamma _{4}^{\left(R\right)}\left(k,u\right)&=Z^{2}\Gamma _{4}\left(k,u\right).\end{aligned}}}$

The constant wave vector is introduced for dimensional reasons. The vertex function actually depends on 3 wave vectors, but for the purpose of renormalization it is sufficient to consider a symmetrical situation where the three wave vectors point from the corners of a tetrahedron to the center and have the same amount (other conventions are also possible). ${\displaystyle \mu _{0}}$ ${\displaystyle \Gamma _{4}}$

The perturbation calculation provides for the vertex functions and power series in the non-renormalized dimensionless coupling constant . These power series are at the critical point; H. at divergent and useless at first. The next step is to set up the normalization condition ${\displaystyle \Gamma _{2}}$ ${\displaystyle \Gamma _{4}}$ ${\displaystyle {\bar {u}}=uk^{-\varepsilon }}$ ${\displaystyle k\to 0}$

${\displaystyle {\frac {\partial }{\partial k^{2}}}\Gamma _{2}^{\left(R\right)}\left(k,u\right)=1.}$

In principle, the factor is determined as a power series in . The highlight of the whole action is the definition of a dimensionless renormalized coupling constant ${\displaystyle Z}$ ${\displaystyle {\bar {u}}}$

${\displaystyle u_{R}\left({\bar {u}}\right)=k^{-\epsilon }\Gamma _{4}^{\left(R\right)}\left(k,u\right).}$

This dimensionless renormalized coupling constant changes as a function of the wave vector i. d. Usually only slowly, is often small and strives to U. against a fixed point. The trick is therefore to transform the power series in to power series in . In other words, the inverse function is determined . The river then plays a decisive role ${\displaystyle {\bar {u}}}$ ${\displaystyle u_{R}}$ ${\displaystyle {\bar {u}}(u_{R})}$

${\displaystyle k\left({\frac {du_{R}}{dk}}\right)_{u}=\beta \left(u_{R}\right)}$

the renormalized coupling constant when changing the length scale with constant . If necessary, the condition supplies the fixed point of the renormalized coupling constant . With and , of course, you also know the physical quantities and . ${\displaystyle u}$ ${\displaystyle \beta (u_{R})=0}$ ${\displaystyle u_{R}}$ ${\displaystyle u_{R}}$ ${\displaystyle Z}$ ${\displaystyle \Gamma _{2}}$ ${\displaystyle \Gamma _{4}}$

Remarks

• It is by no means self-evident that the calculation method described works. A basic requirement is the scale invariance of the Lagrangian in the critical dimension.
• In solid-state physics the Limes is of interest , in quantum field theory the Limes . ${\displaystyle k\to 0}$ ${\displaystyle k\to \infty }$
• In quantum field theory, the case is of interest ; H. the Limes . In this case, the critical exponents disappear, but logarithmic scaling factors remain. ${\displaystyle d=d_{c}=4}$ ${\displaystyle \varepsilon \to 0}$
• The example contains some seemingly arbitrary conventions. The critical behavior is independent of this (universality).
• The field theoretical renormalization group enables systematic series developments according to the renormalized coupling constants. The power series are only asymptotically convergent, but this is often sufficient for small coupling constants.
• Physical quantities can be obtained as a power series in or (numerically) directly with a given dimension (e.g. for ). ${\displaystyle \varepsilon }$ ${\displaystyle d=3}$
• Other fields or z. B. a mass term in the Lagrangian require additional factors. ${\displaystyle Z}$
• The algebra is simplified if the Lagrangian is by having replaced . ${\displaystyle u}$ ${\displaystyle u/K_{d}}$ ${\displaystyle K_{d}=2^{-d+1}\pi ^{-d/2}/\Gamma (d/2)}$

Functional renormalization group

A functional renormalization group (FRG) is a method for calculating the effective potential of a field theory for a variable length scale. An FRG takes into account relevant, marginal and irrelevant couplings. An exact determination of the effective potential is thus i. d. Usually just as impossible as with other techniques. However, a FRG allows a wide variety of parameterizations and is independent of (only asymptotically convergent) perturbation series developments.

There are at least three FRG variants, one based on the Wilsonian elimination renormalization group (Wegner and Houghten), a variant with a variable UV cutoff (Polchinski) and a variant with an infrared regulator (Wetterich). The variant with an IR regulator is the easiest to use.

For the FRG with IR regulator, a compact formula can be derived in a few formal steps within the framework of quantum field theory , which is the starting point for specific applications (Wetterich). In order to simplify the notation, the de Witt notation is recommended, where the field is a vector whose index specifies a point in space and, if necessary, a field index. The first step is to put a regulator term into effect ${\displaystyle \varphi }$ ${\displaystyle S}$

${\displaystyle S_{R}\left(\mu \right)={\frac {1}{2}}\varphi \cdot R\left(\mu \right)\cdot \varphi }$

add where the matrix depends on a wave vector scale (examples below). The generating function of the related correlation functions is then ${\displaystyle R}$ ${\displaystyle \mu }$

${\displaystyle W\left(J,\mu \right)=\ln \int {\mathcal {D}}\varphi \exp \left(-S-S_{R}\left(\mu \right)+J\cdot \varphi \right),}$

where denotes an external field. The expectation of is and the 2-point correlation function is given by ${\displaystyle J}$ ${\displaystyle \varphi }$ ${\displaystyle {\overline {\varphi }}_{a}=\partial W/\partial J_{a}}$

${\displaystyle {\widetilde {G}}_{a,b}\left(\mu \right)={\frac {\partial ^{2}W\left(J,\mu \right)}{\partial J_{a}\partial J_{b}}}=\left\langle \varphi _{a}\varphi _{b}\right\rangle -{\overline {\varphi }}_{a}{\overline {\varphi }}_{b}.}$

The generating function of the 1-particle irreducible vertex functions is the Legendre transform according to the usual scheme ${\displaystyle {\widetilde {\Gamma }}\left(\mu ,{\overline {\varphi }}\right)}$

${\displaystyle {\widetilde {\Gamma }}\left({\overline {\varphi }},\mu \right)=J\cdot {\overline {\varphi }}-W\left(J,\mu \right).}$

Differentiate on the Wave Vector Scale and Use the Definition of List ${\displaystyle \mu }$ ${\displaystyle {\widetilde {G}}_{a,b}}$

${\displaystyle {\frac {\partial }{\partial \mu }}{\widetilde {\Gamma }}=-{\frac {\partial }{\partial \mu }}W={\frac {\partial }{\partial \mu }}\left\langle S_{R}\left(\mu \right)\right\rangle ={\frac {1}{2}}{\frac {\partial }{\partial \mu }}\left\langle \varphi _{a}R_{a,b}\varphi _{b}\right\rangle ={\frac {1}{2}}\left({\widetilde {G}}_{a,b}+{\overline {\varphi }}_{a}{\overline {\varphi }}_{b}\right){\frac {\partial }{\partial \mu }}R_{a,b}.}$

The renormalization group differential equation follows from this as

${\displaystyle \mu {\frac {\partial }{\partial \mu }}\Gamma \left({\overline {\varphi }},\mu \right)={\frac {1}{2}}Tr\left(\left(\Gamma _{a,b}^{\left(2\right)}\left({\overline {\varphi }},\mu \right)+R_{a,b}\right)^{-1}\mu {\frac {\partial }{\partial \mu }}R_{a,b}\right),}$

where the effective potential is denoted without the artificial and the propagator is also written in a form that makes the artificial contribution explicit. stands for the trace of a matrix. ${\displaystyle \Gamma ={\widetilde {\Gamma }}-{\frac {1}{2}}{\overline {\varphi }}\cdot R\cdot {\overline {\varphi }}}$ ${\displaystyle S_{R}}$ ${\displaystyle {\widetilde {G}}=1/{\widetilde {\Gamma }}_{2}=1/\left(\Gamma _{2}+R\right)}$ ${\displaystyle S_{R}}$ ${\displaystyle Tr\left(\dots \right)}$

The meaning and the interpretation of the FRG differential equation result from the choice of regulator , i. H. of the propagator. Typical IR cutoff functions (expressed in space) are or . These features quickly disappear for and reach for value . This means that degrees of freedom with short wavelengths experience no change, while degrees of freedom with long wavelengths are given a finite mass and are suppressed. The FRG differential equation describes what happens when you add more and more degrees of freedom with long wavelengths. In this way, for example, you can reach a critical point at which wavelengths of any length have to be taken into account. ${\displaystyle R}$ ${\displaystyle k}$ ${\displaystyle R\left(\mu ,k\right)=k^{2}/\left(e^{k^{2}/\mu ^{2}}-1\right)}$ ${\displaystyle R\left(\mu ,k\right)=\left(\mu ^{2}-k^{2}\right)\theta \left(\mu ^{2}-k^{2}\right)}$ ${\displaystyle k\gg \mu }$ ${\displaystyle k\ll \mu }$ ${\displaystyle \mu ^{2}}$ ${\displaystyle \mu \rightarrow 0}$

History of the RG

There have been scaling considerations in physics since antiquity and at a prominent place e.g. B. with Galileo . The RG first appeared in 1953 in the treatment of renormalization in quantum electrodynamics by ECG Stueckelberg and André Petermann and in 1954 by Murray Gell-Mann and Francis Low . The theory was expanded by the Russian physicists NN Bogoljubow and DV Shirkov , who wrote a textbook about it in 1959.

A real physical understanding, however, was only achieved through the work of Leo Kadanoff in 1966 (block spin transformation), which was then successfully implemented by Nobel Prize winner (1982) Kenneth Wilson in 1971 for the treatment of so-called critical phenomena in the vicinity of continuous phase transitions and then in 1974 successively - constructive solution to the Kondo problem . Among other things, he received the Nobel Prize in 1982 for the first achievement. The old RG of particle physics was also reformulated around 1970 by Curtis Callan and Kurt Symanzik . In particle physics, the momentum space RG was mainly used and expanded. It was also widely used in solid state physics, but was not applicable to strongly correlated systems. Here they were in the 1980s with the local space-RG method successful as that of Steven R. White (1992) introduced the density matrix -RG (density matrix RG, DMRG).

literature

Original work

• ECG Stueckelberg and A. Petermann: La renormalisation des constants dans la theory de quanta . In: Helvetica physica acta . Volume 26, 1953 p. 499.
• M. Gell-Mann and FE Low: Quantum Electrodynamics at small distances . In: Physical Review . Volume 95, 1954, p. 1300. (Introduction of the concept by Stueckelberg / Peterman and Gell-Mann / Low)
• NN Bogoliubov and DV Shirkov : The theory of quantized fields . Interscience, 1959. (first textbook treatment)
• LP Kadanoff : Scaling laws for Ising models near ${\displaystyle T_{c}}$ . In: Physics (Long Island City, NY) Volume 2, 1966, p. 263 (the picture of the block-spin transformations)
• CG Callan : Broken scale invariance in scalar field theory . In: Physical Review D . Volume 2, 1970, p. 1541. (Online)
• K. Symanzik : Small distance behavior in field theory and power counting . In: Communications in Mathematical Physics . Volume 18, 1970, p. 227. (Online) (here and in Callan's work mentioned above, the RG is introduced in momentum space)
• KG Wilson : The renormalization group. Critical phenomena and the Kondo problem . In: Reviews of modern physics . Volume 47, No. 4, 1975, p. 773. (Online) (successful application of the RG to the Kondo effect )
• SR White : Density matrix formulation for quantum renormalization groups . In: Physical Review Letters . Volume 69, 1992, p. 2863. (Often used RG variation method)
• Franz Wegner , Anthony Houghton: Renormalization Group Equations for Critical Phenomena . In: Physical Review A , Volume 8, 1973, p. 401 (Functional Renormalization Group)
• Joseph Polchinski : Renormalization and Effective Lagrangians . In: Nuclear Phys. B , Volume 231, 1984 pp. 269-295
• Christof Wetterich : Exact evolution equation for the effective potential . In: Phys. Lett. B , Volume 301, 1993 p. 90. Arxiv

Review article

• KG Wilson : The renormalization group . In: Spectrum of Science . October 1979. (Online)
• ME Fisher : The renormalization group in the theory of critical phenomena , In: Reviews of Modern Physics , 46, 597 (1974); Renormalization group theory: its basis and formulation in statistical physics . In: Reviews of Modern Physics 70, 653 (1998)
• DV Shirkov: Evolution of the Bogoliubov Renormalization Group. A mathematical introduction and historical overview with a stress on group theory and the application in high-energy physics. 1999. arXiv.org:hep-th/9909024 (Online)
• B. Delamotte: A hint of renormalization. A pedestrian introduction to renormalization and the renormalization group . In: American Journal of Physics . Volume 72, 2004, p. 170
• HJ Maris, LP Kadanoff : Teaching the renormalization group. A pedestrian introduction to the renormalization group as applied in condensed matter physics. In: American Journal of Physics . Volume 46, June 1978, pp. 652-657.
• LP Kadanoff : Application of renormalization group techniques to quarks and strings . In: Reviews of Modern Physics 49, 267 (1977)
• Karen Hallberg, Density matrix renormalization: A review of the method and its applications , published in David Senechal, Andre-Marie Tremblay and Claude Bourbonnais (eds.), Theoretical Methods for Strongly Correlated Electrons, CRM Series in Mathematical Physics, Springer, New York , 2003 (and Hallberg New Trends in Density Matrix Renormalization , Advances in Physics 2006 ); Hallberg, Ingo Peschel , Xiaoqun Wang, Matthias Kaulke (editors) Density Matrix Renormalization , Lecturenotes in Physics 1999; Ullrich Schollwöck: The density-matrix renormalization group . In: Reviews of Modern Physics , 77, 259 (2005), online here: (Online)
• Eleftherios Kirkinis: The Renormalization Group: A Perturbation Method for the Graduate Curriculum . In: SIAM Review . 54, No. 2, 2012, pp. 374-388. doi : 10.1137 / 080731967 .
• Ramamurti Shankar: Renormalization-group approach to interacting fermions . In: Reviews of Modern Physics . 66, 1994, p. 129. arxiv : cond-mat / 9307009 . doi : 10.1103 / RevModPhys.66.129 . bibcode : 1994RvMP ... 66..129S .

Books

• Daniel J. Amit: Field theory, the renormalization group, and critical phenomena. World Scientific 1984.
• Shang-keng Ma: Modern theory of critical phenomena. Addison-Wesley, Frontiers in Physics 1982.
• N. Goldenfeld: Lectures on phase transitions and the renormalization group. Addison-Wesley, 1993.
• L. Ts. Adzhemyan, NV Antonov and AN Vasiliev: The Field Theoretic Renormalization Group in Fully Developed Turbulence. Gordon and Breach, 1999, ISBN 90-5699-145-0 .
• Gérard Toulouse , Pierre Pfeuty: Introduction to the Renormalization Group and to Critical Phenomena. Wiley 1977.
• J. Zinn-Justin : Quantum Field Theory and Critical phenomena. Oxford 1990.
• J. Zinn Justin: Renormalization and renormalization group. From the discovery of UV divergences to the concept of effective field theories. In: C. de Witt-Morette, J.-B. Zuber (Ed.): Proceedings of the NATO ASI on Quantum Field Theory: Perspective and Prospective. 15-26. June 1998, Les Houches, France, Kluwer Academic Publishers, NATO ASI Series C 530, pp. 375-388 (1999). Online here: PostScript .
• Giovanni Gallavotti , G. Benfatto: Renormalization Group. Princeton University Press, 1995.

Web links

• Renormalization group on arxiv.org
• Shirkov Fifty years of the renormalization group. In: Cern Courier, 2001.