Renormalization group

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In theoretical physics, renormalization group (RG) refers to a set of techniques and concepts related to the change of physics with the observation scale. It was initially devised within particle physics, but nowadays its applications are extended to solid state physics, fluid mechanics and even cosmology.

Kadanoff's Blocking Picture

This section introduces pedagogically the picture of RG which may be easiest to grasp: Kadanoff's blocks. It was devised by Leo P. Kadanoff in 1966, when RG already had a long history behind it.

Let us consider a 2D solid, a set of atoms in a perfect square array, as depicted in the figure. Let us assume that atoms interact among themselves only with their nearest neighbours, and that the system is at a given temperature Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} . The strength of their interaction is measured by a certain coupling constant Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} . The physics of the system will be described by certain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle formula} , say Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(T,J)} .

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Now we proceed to divide the solid into blocks of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\times 2} squares. Now we attempt to describe the system in terms of block variables, i.e.: some magnitudes which describe the average behaviour of the block. Also, let us assume that, due to a lucky coincidence, the physics of block variables is described by a formula of the same kind, but with different values for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(T',J')} .

Perhaps the initial problem was too hard to solve, since there were too many atoms. Now, in the renormalized problem we have only one fourth of them. But why should we stop now? Another iteration of the same kind leads to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(T'',J'')} , and only one sixteenth of the atoms. We are increasing the observation scale with each RG step.

Of course, the best idea is to iterate until there is only one very big block. Since the number of atoms in any real sample of material is very large, this is more or less equivalent to finding the long term behaviour of the RG transformation which took Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (T,J)\to (T',J')} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (T',J')\to (T'',J'')} . Usually, when iterated many times, this RG transformation leads to a certain number of fixed points.

Let us be more concrete and consider a magnetic system (e.g.: the Ising model), in which the J coupling constant denotes the trend of neigbour spins to be parallel. Physics is dominated by the tradeoff between the ordering J term and the disordering effect of temperature. For many models of this kind there are three fixed points:

(a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T=0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J\to\infty} . This means that, at the largest size, temperature becomes unimportant, i.e.: the disordering factor vanishes. Thus, in large scales, the system appears to be ordered. We are in a ferromagnetic phase.

(b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T\to\infty} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J\to 0} . Exactly the opposite, temperature has its victory, and the system is disordered at large scales.

(c) A nontrivial point between them, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T=T_c} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J=J_c} . In this point, changing the scale does not change the physics, because the system is in a fractal state. It corresponds to the Curie phase transition, and is also called a critical point.

So, if we are given a certain material with given values of T and J, all we have to do in order to find out the large scale behaviour of the system is to iterate the pair until we find the corresponding fixed point.

Elements of RG theory

In more technical terms, let us assume that we have a theory described by a certain function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} of the state variables Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{s_i\}} and a certain set of coupling constants Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{J_k\}} . This function may be a partition function, an action, a hamiltonian, etc. It must contain the whole description of the physics of the system.

Now we consider a certain blocking transformation of the state variables Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{s_i\}\to \{\tilde s_i\}} , the number of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde s_i} must be lower than the number of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_i} . Now let us try to rewrite the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Z} function only in terms of the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde s_i} , but keeping it invariant under the change. If this is achievable by a certain change in the parameters, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{J_k\}\to \{\tilde J_k\}} , the the theory is said to be renormalizable.

By some reason, all fundamental theories of physics but gravity (QED, QCD and electro-weak interaction) are exactly renormalizable. Also, most theories in condensed matter physics are approximately renormalizable, from superconductivity to fluid turbulence.

The change in the parameters is implemented by a certain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} -function: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{\tilde J_k\}=\beta(\{ J_k \})} , which is said to induce a renormalization flow (or RG flow) on the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} -space. The values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J} under the flow are called running coupling constants.

As it was stated in the previous paragraph, the most important information in the RG flow are its fixed points. The possible macroscopic states of the system, at a large scale, are given by this set of fixed points.

Since the RG transformations are lossy (i.e.: the number of variables decreases - see as an example in a different context, Lossy data compression), there need not be an inverse for a given RG transformation. Thus, the renormalization group is, in practice, a semigroup.

Relevant and irrelevant operators. Universality classes

Let us consider a certain observable Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} of a physical system undergoing a RG transformation. The magnitude of the observable may be (a) always increasing, (b) always decreasing or (c) have fluctuations, but no definite average trend. In the first case, the observable is said to be a relevant observable; in the second, irrelevant and in the third, marginal.

A relevant operator is needed to describe the macroscopic behaviour of the system, but not an irrelevant observable. Marginal observables always give trouble when deciding whether to take them into account or not. A remarkable fact is that most observables are irrelevant, i.e.: the macroscopic physics is dominated by only a few observables in most systems. In other terms: microscopic physics contains Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \approx 10^{23}} variables, and macroscopic physics only a few.

Before the RG, there was an astonishing empirical fact to explain: the coincidence of the critical exponents (i.e.: the behaviour near a second order phase transition) in very different phenomena, such as magnetic systems, superfluid transition, alloy physics... This was called universality and is successfully explained by RG, just showing that the differences between all those phenomena are related to irrelevant observables.

Thus, many macroscopic phenomena may be grouped into a small set of universality classes, described by the set of relevant observables.

Real and Momentum Space RG

RG, in practice, comes in two main flavours. The Kadanoff picture explained above refers mainly to the so-called real-space RG. The technique which has a longer history, although is harder to grasp at a first attempt, is momentum-space RG. Within this framework, the degrees of freedom under consideration are the Fourier modes of a given field, and a RG transformation proceeds by integrating out a certain set of high momentum. Since high momentum is related to short length scales, the main picture is the same as that of real space RG.

Momentum space RG usually starts out with a perturbative series, i.e.: the idea that the true physics of our system is close to that of a free system, so we may find out the differences between the values of observables by summing a series of terms with decreasing magnitude, classified according to powers of some of the coupling constant. This approach has proved very successful for some theories, including most of particle physics, but fails for systems whose physics is very far away from any free system, i.e.: systems with strong correlations.

As an example of the physical meaning of RG in particle physics we will give a short description of charge renormalization in quantum electrodynamics (QED). Let us suppose we have a point positive charge of a certain true (or bare) magnitude. The electromagnetic field around it has a certain energy, and thus may produce some pairs of (e.g.) electrons-positrons, which will be annihilated very quickly. But in their short life, the electron will be attracted by the charge, and the positron will be repelled. Since this happens continuously, these pairs are effectively screening the charge from abroad. Therefore, the measured strength of the charged will depend on how close to it our probes may enter. We have a dependence of a certain coupling constant (the electric charge) with distance.

Energy, momentum and length scales are related, according to Heisenberg's uncertainty principle. The higher the energy or momentum scale we may reach, the lower the length scale we may probe. Therefore, the momentum-space RG practicioners sometimes claim to integrate out high momenta or high energy from their theories.

History of the Renormalization Group

Of course, the idea of scale invariance is old and venerable in physics. Scaling arguments were commonplace for the pythagorean school, Euclid and up to Galileo. They became popular again at the end of the 19th century, perhaps the first example being the idea of enhanced viscosity of Osborne Reynolds, as a way to explain turbulence.

RG made its appearance in physics in very different guise. An article by E.C.G. Stueckelberg and A. Peterman in 1953 and another one by M. Gell-Mann and F.E. Low in 1954 opened the field, but as a mathematical trick to get rid of the infinities in quantum field theory. As a pure technique, it obtained maturity with the book by N.N. Bogoliubov and D.V. Shirkov in 1959. The RG term was inherited from this time and, although most people agree that it is incorrect, no other alternative has been proposed so far.

The technique was developed further by R.P. Feynman, J. Schwinger and S. Tomonaga, who obtained the Nobel prize for their contributions to quantum electrodynamics. They devised the theory of mass and charge renormalization.

But real understanding of the physical meaning of the technique came with Leo P. Kadanoff's paper in 1966. The new blocking idea reached maturity with Kenneth Wilson's solution of the Kondo problem in 1974. He was awarded the Nobel prize of this contribution in 1982. The old-style RG in particle physics was reformulated in 1970 in more physical terms by C.G. Callan and K. Symanzik. In this field, momentum space RG is a very mature tool, its only failure being the non-renormalizability of gravity. Momentum space RG also became a highly developed tool in solid state physics, but its success was hindered by the extensive use of perturbation theory, which, as we have stated before, prevented the theory from reaching success in strongly correlated systems.

In order to study these strongly correlated systems, variational approaches are a better alternative. During the 80's some real space RG techniques were developed in this sense, being the most successful the density matrix RG (DMRG), developed by S.R. White and R.M. Noack in 1992.

See also

momentum-space RG.

real-space RG technique up to date.

References

Historical papers

  • E.C.G. Stueckelberg, A. Peterman (1953): Helv. Phys. Acta, 26, 499. M. Gell-Mann, F.E. Low (1954): Phys. Rev. 95, 5, 1300. The origin of renormalization group
  • N.N. Bogoliubov, D.V. Shirkov (1959): The theory of quantized fields, Interscience.

The first text-book on RG.

  • L.P. Kadanoff (1966): Scaling laws for Ising models near Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T_c} , Physica

2, 263. The new blocking picture.

  • C.G. Callan (1970): Phys. Rev. D 2, 1541. K. Symanzik (1970): Comm. Math. Phys.

18, 227. The new view on momentum-space RG.

  • K.G. Wilson (1975): The renormalization group: critical phenomena and the Kondo

problem, Rev. Mod. Phys. 47, 4, 773. The main success of the new picture.

  • S.R. White (1992): Density matrix formulation for quantum renormalization groups,

Phys. Rev. Lett. 69, 2863. The most successful variational RG method.


Didactical reviews

  • N. Goldenfeld (1993): Lectures on phase transitions and the renormalization group.

Addison-Wesley.

applications. Ph.D. thesis, UCM. [1].

es:Grupo de Renormalización ja:繰り込み群 ru:Ренормализационная группа sl:Renormalizacijska grupa