[Show abstract][Hide abstract] ABSTRACT: 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 $N<N_c(d)$ from a second order region for $N>N_c(d)$. 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.
[Show abstract][Hide abstract] ABSTRACT: The crumpled-to-flat phase transition that occurs in D-dimensional polymerized phantom membranes embedded in a d-dimensional space is investigated nonperturbatively using a field expansion up to order 8 in powers of the order parameter. We get the critical dimension dcr(D) that separates a second-order region from a first-order one everywhere between D=4 and 2. Our approach strongly suggests that the phase transitions that take place in physical membranes are of first order in agreement with most recent numerical simulations.
Full-text · Article · Apr 2014 · Physical Review E
[Show abstract][Hide abstract] ABSTRACT: The behaviour of a d-dimensional vectorial N = 3 model at a m-axial Lifshitz critical point is investigated by means of a nonperturbative renormalization group approach that is free of the huge technical difficulties that plague the perturbative approaches and limit their computations to the lowest orders. In particular being systematically improvable, our approach allows us to control the convergence of successive approximations and thus to get reliable physical quantities in d=3. Copyright (C) EPLA, 2012
Full-text · Article · Feb 2012 · EPL (Europhysics Letters)
[Show abstract][Hide abstract] ABSTRACT: 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.
Preview · Article · Jan 2012 · International Journal of Modern Physics A
[Show abstract][Hide abstract] ABSTRACT: The non-perturbative renormalization group (NPRG), in its modern form, constitutes an efficient framework to investigate the physics of systems whose long-distance behavior is dominated by strong fluctuations that are out of reach of perturbative approaches. We present here the basic principles underlying the NPRG and illustrate its power in the context of two longstanding problems of condensed matter and soft matter physics: the nature of the phase transition occuring in frustrated magnets in three dimensions and the phase diagram of polymerized phantom membranes.
No preview · Article · Nov 2011 · Modern Physics Letters B
[Show abstract][Hide abstract] ABSTRACT: Anisotropic D-dimensional polymerized phantom membranes are investigated within a nonperturbative renormalization group framework. One focuses on the transition between a high-temperature, crumpled phase and a low-temperature, tubular phase where the membrane is flat along one direction and crumpled along the other ones. While the upper critical dimension--D(uc)=5/2--is close to D=2, the weak-coupling perturbative approach is qualitatively and quantitatively wrong. We show that our approach is free of the problems encountered within the perturbative framework and provides physically meaningful critical quantities.
Full-text · Article · Mar 2011 · Physical Review Letters
[Show abstract][Hide abstract] ABSTRACT: 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<Nc, the results obtained at the stable
FP are similar to those obtained at the spurious FPs of the O(N) and cubic
models. We conclude from this analysis that the stable FP found for N 3, we conclude that the transitions for
XY and Heisenberg frustrated magnets are of first order.
Full-text · Article · Dec 2010 · Condensed Matter Physics
[Show abstract][Hide abstract] ABSTRACT: We show that the critical behaviour of two- and three-dimensional frustrated magnets cannot reliably be described from the known five- and six-loops perturbative renormalization group results. Our conclusions are based on a careful re-analysis of the resummed perturbative series obtained within the zero momentum massive scheme. In three dimensions, the critical exponents for XY and Heisenberg spins display strong dependences on the parameters of the resummation procedure and on the loop order. This behaviour strongly suggests that the fixed points found are in fact spurious. In two dimensions, we find, as in the O(N) case, that there is apparent convergence of the critical exponents but towards erroneous values. As a consequence, the interesting question of the description of the crossover/transition induced by Z2 topological defects in two-dimensional frustrated Heisenberg spins remains open. Comment: 18 pages, 30 figures
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: We analyze the validity of perturbative renormalization group estimates obtained within
the fixed dimension approach of frustrated magnets. We reconsider the resummed five-loop
β-functions obtained within the minimal subtraction scheme without
ε-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 occurring at three
dimensions and provides elements towards a unified picture of all existing theoretical
approaches to frustrated magnets.
Full-text · Article · Mar 2008 · Journal of Statistical Mechanics Theory and Experiment
[Show abstract][Hide abstract] ABSTRACT: The Comment of A. Pelissetto and E. Vicari (cond-mat/0610113) on our article (cond-mat/0609285) is based on misunderstandings of this article as well as on unfounded implicit assumptions. We clarify here the controversial points and show that, contrary to what is asserted by these authors, our paper is free of any contradiction and agrees with all well-established theoretical and experimental results. Also, we maintain that our work reveals pathologies in the (treatment of) perturbative approaches performed at fixed dimensions. In particular, we emphasize that the perturbative approaches to frustrated magnets performed either within the minimal substraction scheme without epsilon-expansion or in the massive scheme at zero momentum exhibit spurious fixed points and, thus, do not describe correctly the behaviour of these systems in three dimensions.
[Show abstract][Hide abstract] ABSTRACT: We analyze the validity of perturbative estimations obtained at fixed dimensions in the study of frustrated magnets. To this end we consider the five-loop beta-functions obtained within the minimal subtraction scheme and exploited without epsilon-expansion both for frustrated magnets and for the well-controlled ferromagnetic systems with a cubic anisotropy. Comparing the two cases it appears 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 the non-Gaussian character of this fixed point at the upper critical dimension d=4. Our work confirms the weak first order nature of the phase transition and constitutes a step towards a unified picture of existing theoretical approaches to frustrated magnets.
[Show abstract][Hide abstract] ABSTRACT: Critical scaling and universality in the short-time dynamics for antiferromagnetic models on a three-dimensional stacked triangular lattice are investigated using Monte Carlo simulation. We have determined the critical point by searching for the best power law for the order parameter as a function of time and measured the critical exponents. Our results indicate that it is possible to distinguish weak first-order from second-order phase transitions and confirm that XY antiferromagnetic systems undergo a (weak) first-order phase transition accompanied by pseudocritical scaling.
Full-text · Article · Aug 2006 · Physical Review E
[Show abstract][Hide abstract] ABSTRACT: 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, France
Preview · Article · Jan 2004 · Journal of Physics Condensed Matter
[Show abstract][Hide abstract] ABSTRACT: 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.
Full-text · Article · Jan 2004 · Physical Review B
[Show abstract][Hide abstract] ABSTRACT: 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. 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. Comment: 58 pages, 15 PS figures
[Show abstract][Hide abstract] ABSTRACT: 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 nu=0.632 and to an anomalous dimension eta=0.033 which is significantly improved compared with lower orders calculations.
[Show abstract][Hide abstract] ABSTRACT: 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 $\partial^2$ of the derivative expansion. This approach allows us to select optimized cut-off functions and to improve the accuracy of the critical exponents $\nu$ and $\eta$. 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. Comment: 13 pages, 9 PS figures, published version