We analyse the universal features of the critical behaviour of frustrated spin systems with noncollinear order. By means of the field theoretical renormalization group approach, we study the 3D model of a frustrated magnet and obtain pseudo-ε expansions for its universal order parameter marginal dimensions. These dimensions govern accessibility of the renormalization group transformation fixed points, and, hence, define the scenario of the phase transition.
We analyse the universal features of the critical behaviour of frustrated spin systems with noncollinear order. By means of the field theoretical renormalization group approach, we study the 3D model of a frustrated magnet and obtain pseudo-ε expansions for its universal order parameter marginal dimensions. These dimensions govern accessibility of the renormalization group transformation fixed points, and, hence, define the scenario of the phase transition.
We review results concerning the critical behavior of spin systems at equilibrium. We consider the Ising and the general O(N)-symmetric universality class. For each of them, we review the estimates of the critical exponents, of the equation of state, of several amplitude ratios, and of the two-point function of the order parameter. We report results in three and two dimensions. We discuss the crossover phenomena that are observed in this class of systems. In particular, we review the field-theoretical and numerical studies of systems with medium-range interactions. Moreover, we consider several examples of magnetic and structural phase transitions, which are described by more complex Landau-Ginzburg-Wilson Hamiltonians, such as N-component systems with cubic anisotropy, O(N)-symmetric systems in the presence of quenched disorder, frustrated spin systems with noncollinear or canted order, and finally, a class of systems described by the tetragonal Landau-Ginzburg-Wilson Hamiltonian with three quartic couplings. The results for the tetragonal Hamiltonian are original, in particular we present the six-loop perturbative series for the β-functions and the critical exponents.
We analyze the critical behavior of two-dimensional N-vector spin systems with noncollinear order within the five-loop renormalization-group (RG) approximation. The structure of the RG flow is studied for different N leading to the conclusion that the chiral fixed point governing the critical behavior of physical systems with N=2 and N=3 does not coincide with that given by the 1/N expansion. We show that the stable chiral fixed point for N<N∗, including N=2 and N=3, turns out to be a focus. We give a complete characterization of the critical behavior controlled by this fixed point, also evaluating the subleading crossover exponents. The spiral-like approach of the chiral fixed point is argued to give rise to unusual crossover and near-critical regimes that may imitate varying critical exponents seen in numerous physical and computer experiments.
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 and 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 and show how it enables to clarify most of the problems encountered in the previous theoretical descriptions of frustrated magnets. First, 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 agreement with the numerical and experimental results. The central feature that emerges from our approach is the existence of scaling behaviors without fixed or pseudofixed point and that relies on a slowing down of the renormalization group flow in a whole region in the coupling constants space. This phenomenon allow us to explain the occurrence of generic weak first order behaviors and to understand the absence of universality in the critical behavior of frustrated magnets.
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 < Nc in frustrated models is spurious.Since Nc > 3, we conclude that the transitions for XY and Heisenberg frustrated magnets are of first order.
Some physical systems, although dierent by their microscopic nature, in the vicinity of critical points can have similar thermodynamic and structural properties governed by universal power laws.Such systems are grouped into the so-called universality classes.The latter are determined by such global characteristics as the type of interaction, space dimensionality, symmetry, and the number of components of the order parameter.Therefore, it is interesting to investigate how the mutual inuence of these characteristics can aect the critical features of the system.The goal of our research is to analyze the changes in the critical behavior of a many-particle magnetic system under the inuence of two competing factors: long-range interaction and weak structural disorder.Using the example of an n-vector model in a d-dimensional space, we will investigate ferromagnetic ordering in a structurally disordered magnet with the long-range interaction decaying with the distance x as J(x) ∼ x -d-σ , with the control parameter σ.The analysis is carried out using the eld-theoretical renormalization group methods, which makes it possible to identify the universality classes of the system under consideration, their change with global parameters, and to determine the universal characteristics of critical behavior (critical exponents and marginal dimensions).It will be shown that there exists such a region of parameters (d, n, σ), where the long-range interaction and structural disorder lead to a synergistic eect and the appearance of a new, random long-range universality class.We obtain the renormalization group functions in the three-loop approximation, and calculate the correlation length critical exponent ν(ϵ ′ , n) as a perturbation theory series in ϵ ′ = 2σ -d.Quantitative estimates are obtained using asymptotic series resummation methods.
Abstract We revisit the critical behavior of classical frustrated systems using the nonperturbative renormalization group (NPRG) equation. Our study is performed within the local potential approximation of this equation to which is added the flow of the field renormalization. Our flow equations are functional to avoid possible artifacts coming from the field expansion of the fixed point potential which consists in keeping only a limited number of coupling constants. We explain in detail our numerical implementation, its advantages and the difficulties encountered in the vicinity of d = 2. For N-component spins, the function N c ( d ) separating the regions of first and second order transitions in the (d, N) plane is computed for d between 4 and 2.3. Our results confirm what was previously found with cruder approximations of the NPRG equation and contradict both the fixed dimension perturbative approach and some of the results obtained within the conformal bootstrap approach.