An exact renormalization group approach to frustrated magnets

Frustrated magnets are a notorious example where usual perturbative methods fail. Having recourse to an exact renormalization group approach, one gets a coherent picture of the physics of Heisenberg frustrated magnets everywhere between d=2 and d=4: all known perturbative results are recovered in a single framework, their apparent conflict is explained while the description of the phase transition in d=3 is found to be in good agreement with the experimental context.Comment: 4 pages, Latex, invited talk at the Second Conference on the Exact Renormalization Group, Rome, September 2000, for technical details see http://www.lpthe.jussieu.fr/~tissie

Functional renormalization group approach to non-collinear magnets

A functional renormalization group approach to $d$-dimensional, $N$-component, non-collinear magnets is performed using various truncations of the effective action relevant to study their long distance behavior. With help of these truncations we study the existence of a stable fixed point for dimensions between $d= 2.8$ and $d=4$ for various values of $N$ focusing on the critical value $N_c(d)$ that, for a given dimension $d$, separates a first order region for $NN_c(d)$. Our approach concludes to the absence of stable fixed point in the physical - $N=2,3$ and $d=3$ - cases, in agreement with $\epsilon=4-d$-expansion and in contradiction with previous perturbative approaches performed at fixed dimension and with recent approaches based on conformal bootstrap program.Comment: 16 pages, 8 figure

Critical properties of a continuous family of XY noncollinear magnets

Monte Carlo methods are used to study a family of three dimensional XY frustrated models interpolating continuously between the stacked triangular antiferromagnets and a variant of this model for which a local rigidity constraint is imposed. Our study leads us to conclude that generically weak first order behavior occurs in this family of models in agreement with a recent nonperturbative renormalization group description of frustrated magnets.Comment: 5 pages, 3 figures, minor changes, published versio

Monte Carlo Calculation of the Spin-Stiffness of the Two-Dimensional Heisenberg Model

Using a collective-mode Monte Carlo method (the Wolff-Swendsen-Wang algorithm), we compute the spin-stiffness of the two-dimensional classical Heisenberg model. We show that it is the relevant physical quantity to investigate the behaviour of the model in the very low temperature range inaccessible to previous studies based on correlation length and susceptibility calculations.Comment: 6 pages, latex, 3 postscript figures appended, DIM preprint 93-3

Frustrated magnets in three dimensions: a nonperturbative approach

Frustrated magnets exhibit unusual critical behaviors: they display scaling laws accompanied by nonuniversal critical exponents. This suggests that these systems generically undergo very weak first order phase transitions. Moreover, the different perturbative approaches used to investigate them are in conflict and fail to correctly reproduce their behavior. Using a nonperturbative approach we explain the mismatch between the different perturbative approaches and account for the nonuniversal scaling observed.Comment: 7 pages, 1 figure. IOP style files included. To appear in Journal of Physics: Condensed Matter. Proceedings of the conference HFM 2003, Grenoble, Franc

Auxiliary fields approach to shift-symmetric theories: the φ4\varphi^4 derivative theory and the crumpled-to-flat transition of membranes at two-loop order

We introduce a technique relying on the use of auxiliary fields in order to eliminate explicit field-derivatives that plague the high orders renormalization group treatment of shift-symmetric, derivative, theories. This technique simplifies drastically the computation of fluctuations in such theories. This is illustrated by deriving the two-loop renormalization group equations and the three-loop anomalous dimension of the $\varphi^4$ derivative theory in $D=4-\epsilon$, which is also relevant to describe the crumpled-to-flat transition of polymerized membranes. Some features of this transition are provided.Comment: 7 page

Spontaneous versus explicit replica symmetry breaking in the theory of disordered systems

We investigate the relation between spontaneous and explicit replica symmetry breaking in the theory of disordered systems. On general ground, we prove the equivalence between the replicon operator associated with the stability of the replica symmetric solution in the standard replica scheme and the operator signaling a breakdown of the solution with analytic field dependence in a scheme in which replica symmetry is explicitly broken by applied sources. This opens the possibility to study, via the recently developed functional renormalization group, unresolved questions related to spontaneous replica symmetry breaking and spin-glass behavior in finite-dimensional disordered systems.Comment: 16 page

Fixed points in frustrated magnets revisited

We analyze the validity of perturbative renormalization group estimates obtained within the fixed dimension approach of frustrated magnets. We reconsider the resummed five-loop beta-functions obtained within the minimal subtraction scheme without epsilon-expansion for both frustrated magnets and the well-controlled ferromagnetic systems with a cubic anisotropy. Analyzing the convergence properties of the critical exponents in these two cases we find that the fixed point supposed to control the second order phase transition of frustrated magnets is very likely an unphysical one. This is supported by its non-Gaussian character at the upper critical dimension d=4. Our work confirms the weak first order nature of the phase transition occuring at three dimensions and provides elements towards a unified picture of all existing theoretical approaches to frustrated magnets.Comment: 18 pages, 8 figures. This article is an extended version of arXiv:cond-mat/060928

Analysis of the 3d massive renormalization group perturbative expansions: a delicate case

The effectiveness of the perturbative renormalization group approach at fixed space dimension d in the theory of critical phenomena is analyzed. Three models are considered: the O(N) model, the cubic model and the antiferromagnetic model defined on the stacked triangular lattice. We consider all models at fixed d = 3 and analyze the resummation procedures currently used to compute the critical exponents. We first show that, for the O(N) model, the resummation does not eliminate all non-physical (spurious) fixed points (FPs). Then the dependence of spurious as well as of the Wilson-Fisher FPs on the resummation parameters is carefully studied. The critical exponents at the Wilson-Fisher FP show a weak dependence on the resummation parameters. On the contrary, the exponents at the spurious FP as well as its very existence are strongly dependent on these parameters. For the cubic model, a new stable FP is found and its properties depend also strongly on the resummation parameters. It appears to be spurious, as expected. As for the frustrated models, there are two cases depending on the value of the number of spin components. When N is greater than a critical value Nc, the stable FP shows common characteristic with the Wilson-Fisher FP. On the contrary, for N 3, we conclude that the transitions for XY and Heisenberg frustrated magnets are of first order