This article is devoted to the study of the critical properties of classical XY and Heisenberg frustrated magnets in three dimensions. We first analyze the experimental and numerical situations. We show that the unusual behaviors encountered in these systems, typically nonuniversal scaling, are hardly compatible with the hypothesis of a second order phase transition. Moreover, the fact that the scaling laws are significantly violated and that the anomalous dimension is negative in many cases provides strong indications that the transitions in frustrated magnets are most probably of very weak first order. We then review the various perturbative and early nonperturbative approaches used to investigate these systems. We argue that none of them provides a completely satisfactory description of the three-dimensional critical behavior. We then recall the principles of the nonperturbative approach -the effective average action method -that we have used to investigate the physics of frustrated magnets. First, we recall the treatment of the unfrustrated -O(N ) -case with this method. This allows to introduce its technical aspects. Then, we show how this method unables to clarify most of the problems encountered in the previous theoretical descriptions of frustrated magnets. Firstly, we get an explanation of the long-standing mismatch between different perturbative approaches which consists in a nonperturbative mechanism of annihilation of fixed points between two and three dimensions. Secondly, we get a coherent picture of the physics of frustrated magnets in qualitative and (semi-) quantitative agreement with the numerical and experimental results. The central feature that emerges from our approach is the existence of scaling behaviors without fixed or pseudo-fixed point and that relies on a slowing-down of the renormalization group flow in a whole region in the coupling constants space. This phenomenon allows to explain the occurence of generic weak first order behaviors and to understand the absence of universality in the critical behavior of frustrated magnets.

We study the optimization of nonperturbative renormalization group equations truncated both in fields and derivatives. On the example of the Ising model in three dimensions, we show that the Principle of Minimal Sensitivity can be unambiguously implemented at order ∂ 2 of the derivative expansion. This approach allows us to select optimized cutoff functions and to improve the accuracy of the critical exponents ν and η. The convergence of the field expansion is also analyzed. We show in particular that its optimization does not coincide with optimization of the accuracy of the critical exponents.

On the example of the three-dimensional Ising model, we show that nonperturbative renormalization group equations allow one to obtain very accurate critical exponents. Implementing the order ∂ 4 of the derivative expansion leads to ν = 0.632 and to an anomalous dimension η = 0.033 which is significantly improved compared with lower orders calculations.Many problems in high-energy as well as in statistical physics call for nonperturbative methods. On the one hand, several physical systems are described by field theories in their strong coupling regime so that the usual perturbative techniques become troublesome. They fail either because only the first orders of perturbation are computed and do not suffice, or because, even when high orders are known, standard resummation techniques do not provide converged results. On the other hand, some phenomena such as confinement in QCD or phase transitions induced by topological defects are genuinely nonperturbative.Apart from some methods restricted to specific dimensions or situations, very few nonperturbative techniques are available. During the last years, the Wilson approach 1 to the renormalization group (RG) has been turned into an efficient tool 2,3,4 . This nonperturbative RG can be implemented in very general situations and, in particular, in any dimension, so that it has allowed one to study several issues difficult to tackle within a perturbative framework among which the three-dimensional Gross-Neveu model 5 , frustrated magnets 6 , the randomly dilute Ising model 7 , and the Abelian Higgs model 8 .This method relies on a nonperturbative renormalization of the effective action Γ, i.e. the Gibbs free energy. It consists in building an effective action Γ k at the running scale k by integrating out only fluctuations greater than k. At the scale k = Λ, Λ −1 denoting the spacing of the underlying lattice, Γ k coincides with the Hamiltonian H since no fluctuation has yet been taken into account while, at k = 0, it coincides with the standard effective action Γ since all fluctuations have been integrated out. Thus, Γ k continuously interpolates between the microscopic Hamiltonian H and the free energy Γ. The running effective action Γ k follows an exact equation which controls its evolution with the running scale k 2 :where t = ln(k/Λ) and Γ(2) k [φ] is the second functional derivative of Γ k with respect to the field φ(q). In Eq. (1), R k (q) is an infrared cutoff function which suppresses the propagation of the low-energy modes without affecting the high-energy ones.Although exact, Eq. (1) is a functional partial integrodifferential equation which cannot be solved exactly. To handle it, one has to truncate Γ k . A natural and widely used truncation is the derivative expansion, which consists in expanding Γ k in powers of ∂φ, keeping only the lowest order terms. Physically, this truncation rests on the assumption that the long-distance physics of a given model is well described by the lowest derivative terms, the higher ones corresponding to less relevant operators. U...

Frustrated magnets are a notorious example where the usual perturbative methods are in conflict. Using a nonperturbative Wilson-like approach, we get a coherent picture of the physics of Heisenberg frustrated magnets everywhere between d = 2 and d = 4. We recover all known perturbative results in a single framework and find the transition to be weakly first order in d = 3. We compute effective exponents in good agreement with numerical and experimental data.PACS No: 75.10.Hk, 11.10.Hi, 11.15.Tk Understanding the effect of competing interactions in three dimensional classical spin systems is one of the great challenges of condensed matter physics. However, after twenty five years of investigations, the nature of the universality class for the phase transition of the simplest frustrated model, the antiferromagnetic Heisenberg model on a triangular lattice (AFHT model), is still a strongly debated question [1] . Due to frustration, the ground state of the AFHT model is given by a canted configuration -the famous 120 • structure -that implies a matrix-like order parameter [2] and thus, the possibility of a new universality class. Experiments performed on materials supposed to belong to the AFHT universality class display indeed exponents different from those of the standard O(N ) universality class: for VCl These results however call for several comments. First, the exponents violate the scaling relations, at least by two standard deviations. Second, they differ significantly from those obtained by Monte Carlo (MC) simulations performed either directly on the AFHT model (ν ≃ 0.59(1), γ ≃ 1.17(2), β ≃ 0.29(1), α ≃ 0.24(2)), and on models supposed to belong to the same universality class: AFHT with rigid constraints (ν = 0.504(10), γ = 1.074(29), β = 0.221(9), α = 0.488(30)), dihedral (i.e. V 3,2 Stiefel) models (ν ≃ 0.51(1), γ ≃ 1.13(2), β ≃ 0.193(4), α ≃ 0.47(3)). See Ref.[9] for a review, and references therein. Finally, the anomalous dimensions η obtained by means of scaling relations is found to be negative in experiments as well as in MC simulations, a result forbidden by first principles for second order phase transitions [10] . All these results are hardly compatible with the assumption of universality. It has been proposed that the exponents are, in fact, effective exponents characterizing a very weakly first order transition, the so-called "almost second order phase transition [11][12][13] ".From the theoretical point of view the situation is also very unsatisfactory since one does not have a coherent picture of the expected critical behaviour of the AFHT model between two and four dimensions. On the one hand, the weak coupling expansion performed on the suitable Landau-Ginzburg-Wilson (LGW) model in the vicinity of d = 4 leads to a first order phase transition due to the lack of a stable fixed point [14][15][16] . On the other hand, the low temperature expansion performed around two dimensions on the Non-Linear Sigma (NLσ) model predicts a second order phase transition of the standard O(4)/O(3) universal...

Polymerized phantom membranes are revisited using a nonperturbative renormalization group approach. This allows one to investigate both the crumpling transition and the low-temperature, flat, phase in any internal dimension D and embedding dimension d, and to determine the lower critical dimension. The crumpling phase transition for physical membranes is found to be of second order within our approximation. A weak first-order behavior, as observed in recent Monte Carlo simulations, is however not excluded. 11.10.Hi, 11.15.Tk Membranes form a particularly rich and exciting domain of statistical physics in which the interplay between two-dimensional geometry and thermal fluctuations has led to a lot of unexpected behaviors going from flat to tubular and glassy phases (see [1,2,3, 4] for reviews). Roughly speaking, membranes fall into two groups [4]: fluid membranes, in which the building monomers are free to diffuse. The connectivity is thus not fixed and the membrane displays a vanishing shear modulus. In contrast, in polymerized membranes the monomers are tied together through a potential which leads to a fixed connectivity and to elastic forces. While fluid membranes are always crumpled, polymerized membranes, due to their nontrivial elastic properties, exhibit a phase transition between a crumpled phase at high temperature and a flat phase at low temperature with orientational order between the normals of the membrane [4,5,6,7]. Amazingly, due to the existence of long-range forces mediated by phonons, the correlation functions in the flat phase display a nontrivial infrared scaling behavior [8,9,10]. Accordingly, the lower critical dimension above which an order can develop appears to be smaller than 2 [10], in apparent violation of the Mermin-Wagner theorem.Let us consider the general case of D-dimensional non self-avoiding (phantom) membranes embedded in a d-dimensional space. Early ǫ-expansion [7] performed at one-loop order on the Landau-Ginzburg-Wilson-type model relevant to study the crumpling transition of polymerized membranes has led to predict that just below the upper critical dimension D = 4, the crumpling transition is of second order for d > d cr = 219 while it is of first order for d < d cr . This leaves however open the question of the nature of the transition in the physical (D = 2, d = 3) situation, the case ǫ = 2 being clearly out of reach of such a one-loop order computation. On the numerical side former Monte Carlo (MC) studies (see [11, 12] for reviews) predict a second-order behavior while more recent simulations [13,14] rather favor first-order behaviors. There is however no definite conclusion and no explanation for these versatile results.In parallel to the investigation of the crumpling transition, an effective elastic field theory has been used to probe the flat, low-temperature, phase of membranes [4,5,8,10]. An ǫ-expansion has been performed A flaw of the previous approaches to polymerized membranes is that, due to their perturbative character, they are unable to treat all asp...

We study analytically one-dimensional interacting spinless fermions in a Fibonacci potential. We show that the effects of the quasiperiodic modulation are intermediate between those of a commensurate potential and a disordered one. The system exhibits a metal-insulator transition whose position depends both on the strength of the correlations and on the position of the Fermi level. Consequently, the conductivity displays a power-law-like size and frequency behavior characterized by a nontrivial exponent. PACS numbers: 71.30. + h, 61.44.Br, Since the discovery of quasicrystals in 1984 by Shechtman et al. [1], the electronic properties of quasiperiodic systems have been intensively studied. These metallic alloys are notably characterized by a low electrical conductivity s which increases when either temperature or disorder increases [2]. The very low temperature behavior of s is still an open question and depends on the materials. For example, in AlCuFe [3] and AlCuRu [4], a finite conductivity at zero temperature is expected, whereas recent results [5] seem to confirm a Mott's variable range hopping mechanism for i-AlPdRe down to 20 mK.Many theoretical works have attempted to understand how the quasiperiodic order could induce such exotic behaviors. In particular, the case of independent electrons in one-dimensional (1D) systems has been deeply investigated for different structures (Harper model, Fibonacci chain, . . . [6,7]), giving rise to singular continuous spectra with an infinite number of gaps. Moreover, the corresponding eigenstates are neither extended nor localized but critical, and are known to be responsible of anomalous diffusion [8,9]. For higher dimensional systems, similar studies had also displayed complex and intricated spectra, with analogous characteristics of the electronic states [10-13]. However, given the complexity of these problems due to the geometry alone, the interactions between electrons have often been neglected. Even in 1D incommensurate structures, few results have been obtained [14][15][16][17][18].In this Letter, we investigate the effect of the interactions considering a Hubbard-like model for spinless fermions embedded in a Fibonacci potential. We take the correlations into account using a bosonization technique whereas the quasiperiodicity is treated perturbatively. Using a renormalization group approach, we show that the quasicrystalline system displays a metal-insulator transition (MIT) induced by the interactions. The corresponding critical MIT point is found to be strongly dependent on the Fermi level. In marked contrast with the simple cases of disordered or commensurate potentials, the quasiperi-odicity leads to a power-law-like dependence of the conductivity either in size or frequency with an exponent depending both on the interactions and on the position of the Fermi level. Though our analysis is performed for a Fibonacci potential, we stress that these results can be extended to any potential having a nonflat dense Fourier spectrum.Let us consider a model of interac...

Though much of the recent work on 2D quantum canted antiferromagnets has been devoted to the study of disordered states, there is evidence that some of these systems have Neel ground states, even for spin y. Following Chakravarty, Halperin, and Nelson, we study the quantum nonlinear sigma model suited to these systems and obtain for the correlation length ^'=C^lA/(kBT)^^^e " * , where A and B depend on the spin stiffnesses and spin-wave velocities renormalized by quantum fluctuations. We discuss experiments on ^He adsorbed on Grafoil.PACS numbers: 75.10.Jm, 74.65.+n, 75.50.Ee Since at the classical level the effect of frustration is to reduce the order, frustrated quantum spin systems are expected to exhibit nonconventional disordered ground states [l]. Such an effect would be of importance for some theories of high-r^ superconductivity [2,3]. For this reason, quantum frustrated spin models have recently been the object of intensive numerical as well as analytical work [4][5][6][7][8]. Though most of the theoretical interest has been devoted to the study of states not having longrange order, it is likely that some frustrated models may exhibit long-range order in their ground states [6-8], even for spin y. To our knowledge, there have been no studies of the low-temperature properties of frustrated quantum antiferromagnets assuming a noncollinear ordered ground state (which will be referred to as a Neel state in the following). In particular, the expression for the correlation length for these systems is still lacking. Such an expression would be of interest in discussing experiments on doped La2Cu04 [9,10] and on ^He [11] adsorbed on a graphite substrate. In this work, we extend the analysis of Chakravarty, Halperin, and Nelson (CHN) [12,13] to quantum canted Heisenberg models with zero net magnetization. We study the quantum nonlinear sigma (QNLa) model suited to these systems and give the general expression of their correlation length.Years ago, Halperin and Saslow [14] showed that the long-distance physics of frustrated quantum Heisenberg models may be described by a hydrodynamical theory, provided the total magnetization is zero. They predicted the existence of three spin-wave excitations resulting from the complete breaking of the 0(3) spin-rotation group. The linear spectrum and the interactions of these spin-wave excitations are described by a QNLa model whose action is entirely determined by symmetries. This action was first derived from the lattice by Dombre and Read [15] in the special case of the D = 2 triangular Heisenberg antiferromagnet (AFT). They wrote it in terms of a rotation matrix of SO(3) which is the relevant order parameter for such a model:where g(x,r) € SO(3), 9^, = (9o,8/) =(9/9r,d/ax/), / = 1,2, p is the inverse temperature, and /^^ =diag(/7ip, P2ft.P}fi\ iU=0,±, are diagonal matrices which are a priori independent. It is important to notice that action (l) is not particular to the AFT model. In fact, if not for the anisotropy between space and time directions, action (1) would be th...

XY frustrated magnets exhibit an unusual critical behavior: they display scaling laws accompanied by nonuniversal critical exponents and a negative anomalous dimension. This suggests that they undergo weak first-order phase transitions. We show that all perturbative approaches that have been used to investigate XY frustrated magnets fail to reproduce these features. Using a nonperturbative approach based on the concept of effective average action, we are able to account for this nonuniversal scaling and to describe qualitatively and, to some extent, quantitatively the physics of these systems.