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## A comparison of the renormalization group and the variational approach on an approximation of a non trivial second order dierential equation

We consider the following example of a non linear oscillator: y00(t) + y(t) + y(t)3 = 0: (2.2.57) This equation can be set into a set of two rst order equations that are linear in . Hence a direct renormalization approach would lead us nowhere. Instead we reparametrize the problem in the following way: the 0th term in gives a simple linear oscillator equation whose solution is y0(t) = Acos(t) , hence instead of performing an expansion on y (t) and y0 (t) we may trade variables to A; . Notice here that the exact underlying equations for A and in terms of Eq.(2.2.57) are not known here and are not even clear how to dene at this stage. We thus proceed in the usual manner by expanding y = y0 + y1 + ::: where the rst order in gives: y00 1 (t) + y1(t) + y0(t)3 = 0.

### Exact RG equations

As we have explained in the previous chapter, renormalisable interacting eld theories naturally lead to large logarithms log p2=p2 ref which then limit the range of validity of approximations. This is xed by using the so-called renormalization procedure which is simply the act of imposing FSS on approximate solutions. The innitesimal form of this self-similarity is given by the beta-functions dgi dt = i(fgjg) of the dierent couplings that appear in the initial action as in Sec.2.2.6. Failure to impose this leads to non nonsensical divergences for quantities that can be measured and are thus nite. However, at this stage we do not have the exact equation for the renormalization ow of a given theory and thus it is dicult to construct approximation schemes that go beyond perturbation theory. Hence, in this chapter we will show how the exact equation may be found by looking for a transitivity property of the path integral itself. Doing this we adapt the Wilsonian renormalization picture where instead of locating the divergencies and removing them by demanding that the couplings vary with scale, we seek an FSS relation directly on the partition function, or equivalently, the free energy, as it contains all of the information of the theory. Once an FSS relation is found we may seek an innitesimal form of the group action which will be as usual an autonomous dierential equation, but it will now act on functionals such as the free energy. By imposing FSS and also choosing from the beginning a regime of validity given by UV and IR cutos we may evade all previous discussions on divergences or large logarithms.

Hence we now begin our derivation of the RG ows. We rst need to consider an object that contains all of the information of the theory both perturbative and non perturbative. This can be the eective action but it is much easier to rst consider the partition function: Z[J] = Z DPu[J; ] = Z De(S()J).

#### Diusive nature of the RG ow

In Sec.(2.3.1) we saw that the RG equation can be set into a heat equation thereby showing its diusive nature. A concrete example of this may be given by the Gaussian xed-point through the central limit theorem which we now show using Kadano blocking.

Consider N identical and independent stochastic variables xk, with probability distribution p, where each variable xk is associated to a stochastic object at position k on a one dimensional lattice. The Kadano RG blocking procedure applied to this problem consists in dening a hierarchy of probability distributions pl as we move from the microscopic variables xk to macroscopic variables xk;l. This can be done by dening p0 = p; xk;0 = xk then successively taking averages of nearest neighbors. More precisely at level l = 1 we take bxk;1 = xk+xk+1 2 where the index k in xk;1 only takes macroscopic steps k = 1; 3; 5; : : : as opposed to the microscopic steps of xk. We then rescale k such that xk;1 = bx2k1;1. The probability distribution of xk;1 is then p1 = L:p = def p p where represents a convolution product. At level l = 2 we iterate this procedure on the blocks xk;1 dening bxk;2 = xk;1+xk+2;1 2 = xk+xk+1+xk+2+xk+3 4 where we may once more rescale k such that we may dene xk;2 with probability distribution p2 = L:p1 = p1. Zooming out by taking larger and larger levels l, the probability distribution of pl for l ! 1 tends towards a xed-point of the L operator which is the Gaussian function. Hence, regardless of the microscopic probability distribution p, as one zooms out taking averages of the stochastic variables, the probability distribution of the average becomes Gaussian. This is regardless of the microscopic probability distribution p as long as it veries the rather general set of hypotheses of the central limit theorem. More concretely for condensed matter systems, if we consider a system of nearest neighbor interactions, then at suciently high temperatures the system is weakly correlated. Hence, we expect the system to be composed of small blocks of the size of the correlation length where the components of the system remain correlated. Each block may then be regarded as a random variable and if the system is homogeneous these random variables t the criteria of the central limit theorem which then allows us to predict a Gaussian probability distribution in the infrared limit without any information on the type of interactions at hand.

This is a somewhat trivial renormalization ow which leads to the high temperature xed-point where the system is completely decorrelated. A more interesting renormalization ow is the ow between dierent scale invariant theories. In this case the system loses information as well but the probability distribution is not necessarily Gaussian as scale invariant theories have innite variance and thus the central limit theorem does not apply. Hence, the renormalization group can be seen as a generalization of the central limit theorem.

This is most easily seen in two dimensions via the c-theorem. The c-theorem states that an object called the central charge decreases along the renormalization ow. This object in turn measures, in some sense, the number of degrees of freedom in the system. Hence, as we expect the information content of a theory to diminish with the number of degrees of freedom this is another example of the diusive nature of the RG ow 32.

For example, a system of N free Majorona fermions has central charge N/2 and hence via bosonization, where two free Majorona fermions can be seen as a system of one boson,33 we may deduce that a system of N free bosons has central charge N. This then leads to the fact that a theory of one massless free boson can, through the c-theorem, lead to a theory of one free massless majorona fermion via the renormalization group as one would have cUV cIR = 1 1=2 = 1=2 > 0. In fact, this situation does arise with uni axial ferromagnets, that is, the Ising model. Both the massless boson and massless fermion theory are scale invariant and correspond to adjusting the temperature of the system to the critical temperature T = Tc but the boson xed-point is unstable whereas the free fermion one is stable.

As the RG ows from smaller to larger scales, this dissipative eect is a fortunate feature when working on statistical mechanics as one does not need to know great detail of the UV physics. However, this poses a real problem when the objective is to instead decipher the UV from the IR. Indeed, moving in the opposite direction one never knows when the ow of the couplings will lead to new physics.

However, everything is not lost along this river ow as, for example, the symmetries of the initial UV

problem are also conserved as long as they act linearly on the elds34. Moreover, not all quantities are universal as, for example the critical temperature at which a phase transition happens depends on the physics at the UV scale. Examples of universal quantities instead are critical exponents or ratios of correlation amplitudes or masses. Fortunately however, non universal quantities can and have [54] been obtained in the framework of the non perturbative renormalization group (NPRG).

**Phase transition and stability analysis**

Now that we have obtained an exact equation the question is : how can we obtain quantitative results ? Perhaps the simplest quantitative results one can obtain are critical exponents. There are quite a few critical exponents but fortunately they are not all independent and in the simplest type of second order transition there are just two independent exponents commonly obtained from experiments 35. The two which are commonly discussed are the exponents and . Both exponents are related to the connected correlation function: < (r) (0) >c=< (r) (0) > < (r) >< (0) > .

**FSS as a convergence accelerator**

In [73] it was found that imposing FSS on sequences leads to faster convergence. This in turn can be understood by minimizing the Cauchy dierence jfn+p fnj for the sequence at hand. We will give a perturbative example here but the method can be extended to achieve non perturbative results, in particular using variational techniques and minimal sensitivity as is done in NPRG. Hence following [73] we consider the function: 1 1 g.

The rst terms in the Taylor expansion are f0 (g) = 1 + g and f1 (g) = 1 + g + g2. We can express f1 as a function of f0 as g = (f0 1) thus we have: f1 = f0 + (1 f0)2.

**Applying FSS in absence of a small coupling**

In the present section we shall study how FSS can be used in absence of an explicit small parameter 53.

This in turn is more closely related to the NPRG framework as the latter also lacks such a parameter. More precisely we will study the logistic map which is known to be closely related to the renormalization group. We will not dwell into the many fascinating aspects of this map, instead we will focus on an FSS approximation scheme to derive universal quantities. This section follows closely [75] where we have added more details.

The logistic map is dened by the following equation: xn+1 = f (xn) = rxn (1 xn) (2.4.11) For an initial condition 0 < x0 < 1 and r < 3 = r1 the sequence converges to the xed-point x = 11=r. However, for slightly larger values of r this xed-point becomes unstable and the system oscillates between two values x 1 and x 2 as the sequence x 1 ! f (x 1 ) = x 2 ! f (x 2 ) = x 1 ! f (x 1 ) = x 2 ! etc. Thus, x 1 = f (x 2 ) = f (f (x 1 )) = def f2 (x 1 ) such that (x 1 ; x 2 ) are xed-points of f2 (x) = f (f (x)). This is called period-doubling and for slightly larger values of r, above a value r2, this period doubling happens once more creating a sequence of period 4 : x 1 ! f (x 1 ) = x 2 ! f (x 2 ) =0.

**Table of contents :**

Contents

**1 Introduction **

**2 Introduction to the functional renormalization framework **

2.1 The role of correlations

2.1.1 Mean eld theory applied to gas-liquid and uni-axial ferromagnetic systems

2.1.2 Landau theory

2.1.3 Range of application of Landau theory

2.1.4 Going beyond mean eld theory

2.2 Perturbative Renormalization without eld theory: a rst conceptual step towards functional renormalization

2.2.1 A one loop calculation

2.2.2 Eective scale-dependent parameters

2.2.3 A divergent product

2.2.4 An exact solution from a rst order correction using the renormalization group

2.2.5 An improved approximation using the renormalization group :

2.2.6 Charge beta function

2.2.7 A comparison of the renormalization group and the variational approach on an approximation of a non trivial second order dierential equation

2.3 Non perturbative Renormalization

2.3.1 Exact RG equations

2.3.2 The ow as an interpolation function

2.3.3 The ow as an RG improved one loop calculation

2.3.4 Diusive nature of the RG ow

2.3.5 Phase transition and stability analysis

2.4 Approximation schemes

2.4.1 FSS as a convergence accelerator

2.4.2 Applying FSS in absence of a small coupling

2.4.3 NPRG approximation schemes

**3 Application of the functional renormalisation group to models **

3.1 O(N) models and the Bardeen-Moshe-Bander phenomenon

3.1.1 O(N) models

3.1.2 Multi-critical points of the O(N) model

3.1.2.1 Multi-critical points within the framework of Landau theory

3.1.2.2 Multicritical xed-points in the O (N) model

3.1.3 Bardeen-Moshe-Bander phenomenon using standard eld theory techniques

3.1.3.1 Large N analysis: leading order

3.1.3.2 Large N analysis: order 1=N

3.1.4 BMB phenomenon at the level of the LPA

3.1.5 Improving the LPA result

3.1.6 Generalization to all upper multicritical dimensions

3.1.7 Exact order 1/N equations

3.1.8 BMB phenomenon at order 2 of the derivative expansion

3.1.9 Physical interpretation of cusped xed-points

3.1.10 Extension of the BMB phenomenon to moderate N and non trivial homotopies in (N; d) space

**4 Conclusion **

**A Van der Waals Phase diagram **

**B Discussion on Euler product **

**C Counter terms **

**D Formal derivation of the ow **

**E FSS RG via rescalings **

**F Derivative expansion without an underlying eective action **

**G Fluctuation dissipation relations **

**H Multicritical phase diagram **

**I 1/N expansion for the tricritical function **

**J LPA Polchinski and ow equation **

**K Derivation of () at LPA **

**L LPA singular solutions as weak solutions **

**M Singular perturbation theory for the LPA **

**N Boundary layer analysis of xed-point SG **

**O LPA equivalence of Polchinski and ow for Litim regulator **

**P SG eigenvalues **

**Q Coupling to (d;N) space mapping for all multicritical dimensions **

**R Large N ow equations at order 2 of the derivative expansion **

**Bibliography**