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Register a new userUniversal critical behavior of noisy coupled oscillators: A renormalization group study
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We show that the synchronization transition of a large number of noisy coupled oscillators is an example for a dynamic critical point far from thermodynamic equilibrium. The universal behaviors of such critical oscillators, arranged on a lattice in a $d$-dimensional space and coupled by nearest neighbors interactions, can be studied using field theoretical methods. The field theory associated with the critical point of a homogeneous oscillatory instability (or Hopf bifurcation of coupled oscillators) is the complex Ginzburg-Landau equation with additive noise. We perform a perturbative renormalization group (RG) study in a $4-epsilon$ dimensional space. We develop an RG scheme that eliminates the phase and frequency of the oscillations using a scale-dependent oscillating reference frame. Within a Callan-Symanzik RG scheme to two-loop order in perturbation theory, we find that the RG fixed point is formally related to the one of the model $A$ dynamics of the real Ginzburg-Landau theory with an O(2) symmetry of the order parameter. Therefore, the dominant critical exponents for coupled oscillators are the same as for this equilibrium field theory. This formal connection with an equilibrium critical point imposes a relation between the correlation and response functions of coupled oscillators in the critical regime. Since the system operates far from thermodynamic equilibrium, a strong violation of the fluctuation-dissipation relation occurs and is characterized by a universal divergence of an effective temperature. The formal relation between critical oscillators and equilibrium critical points suggests that long-range phase order exists in critical oscillators above two dimensions.
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We study the universal thermodynamic properties of systems consisting of many coupled oscillators operating in the vicinity of a homogeneous oscillating instability. In the thermodynamic limit, the Hopf bifurcation is a dynamic critical point far from equilibrium described by a statistical field theory. We perform a perturbative renormalization group study, and show that at the critical point a generic relation between correlation and response functions appears. At the same time the fluctuation-dissipation relation is strongly violated.
We show that the recent renormalization-group analysis of Lifshitz critical behavior presented by Leite [Phys. Rev. B {bf 67}, 104415 (2003)] suffers from a number of severe deficiencies. In particular, we show that his approach does not give an ultraviolet finite renormalized theory, is plagued by inconsistencies, misses the existence of a nontrivial anisotropy exponent $theta e 1/2$, and therefore yields incorrect hyperscaling relations. His $epsilon$-expansion results to order $epsilon^2$ for the critical exponents of $m$-axial Lifshitz points are incorrect both in the anisotropic ($0<m<d$) and the isotropic cases ($m=d$). The inherent inconsistencies and the lack of a sound basis of the approach makes its results unacceptable even if they are interpreted in the sense of approximations.
We investigate the critical behavior that d-dimensional systems with short-range forces and a n-component order parameter exhibit at Lifshitz points whose wave-vector instability occurs in a m-dimensional isotropic subspace of ${mathbb R}^d$. Utilizing dimensional regularization and minimal subtraction of poles in $d=4+{mover 2}-epsilon$ dimensions, we carry out a two-loop renormalization-group (RG) analysis of the field-theory models representing the corresponding universality classes. This gives the beta function $beta_u(u)$ to third order, and the required renormalization factors as well as the associated RG exponent functions to second order, in u. The coefficients of these series are reduced to m-dependent expressions involving single integrals, which for general (not necessarily integer) values of $min (0,8)$ can be computed numerically, and for special values of m analytically. The $epsilon$ expansions of the critical exponents $eta_{l2}$, $eta_{l4}$, $ u_{l2}$, $ u_{l4}$, the wave-vector exponent $beta_q$, and the correction-to-scaling exponent are obtained to order $epsilon^2$. These are used to estimate their values for d=3. The obtained series expansions are shown to encompass both isotropic limits m=0 and m=d.
Using the density-matrix renormalization-group method we study the surface critical behaviour of the magnetization in Ising strips in the subcritical region. Our results support the prediction that the surface magnetization in the two phases along the pseudo-coexistence curve also behaves as for the ordinary transition below the wetting temperature for the finite value of the surface field.
The two-dimensional ferromagnetic anisotropic Ashkin-Teller model is investigated through a real-space renormalization-group approach. The critical frontier, separating five distinct phases, recover all the known exacts results for the square lattice. The correlation length $( u_T)$ and crossover $(phi)$ critical exponents are also calculated. With the only exception of the four-state Potts critical point, the entire phase diagram belongs to the Ising universality class.
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