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The field-theoretical model describing multicritical phenomena with two coupled order parameters with n_{||} and n_{\perp} components and of O(n_{||}) \oplus O(n_{\perp}) symmetry is considered. Conditions for realization of different types of multicritical behaviour are studied within the field-theoretical renormalization group approach. Surfaces separating stability regions for certain types of multicritical behaviour in parametric space of order parameter dimensions and space dimension d are calculated using the two-loop renormalization group functions. Series for the order parameter marginal dimensions that control the crossover between different universality classes are extracted up to the fourth order in \varepsilon=4-d and to the fifth order in a pseudo-\varepsilon parameter using the known high-order perturbative expansions for isotropic and cubic models. Special attention is paid to a particular case of O(1) \oplus O(2) symmetric model relevant for description of anisotropic antiferromagnets in an external magnetic field., Comment: 10 pages, 3 figures, Proceedings of 4-th Conference on Statistical Physics: Modern Trends and Applications, July 3-6, 2012 Lviv, Ukraine

This article concludes a series of papers (R. Folk, Yu. Holovatch, and G. Moser, Phys. Rev. E 78, 041124 (2008); 78, 041125 (2008); 79, 031109 (2009)) where the tools of the field theoretical renormalization group were employed to explain and quantitatively describe different types of static and dynamic behavior in the vicinity of multicritical points. Here, we give the complete two loop calculation and analysis of the dynamic renormalization-group flow equations at the multicritical point in anisotropic antiferromagnets in an external magnetic field. We find that the time scales of the order parameters characterizing the parallel and perpendicular ordering with respect to the external field scale in the same way. This holds independent whether the Heisenberg fixed point or the biconical fixed point in statics is the stable one. The non-asymptotic analysis of the dynamic flow equations shows that due to cancelation effects the critical behavior is described - in distances from the critical point accessible to experiments - by the critical behavior qualitatively found in one loop order. Although one may conclude from the effective dynamic exponents (taking almost their one loop values) that weak scaling for the order parameter components is valid, the flow of the time scale ratios is quite different and they do not reach their asymptotic values., Comment: 22 pages, 7 figures. Revised version, references updated, Fig. 5 modified, typos corrected

A complete two loop renormalization group calculation of the multicritical dynamics at a tetracritical or bicritical point in anisotropic antiferromagnets in an external magnetic field is performed. Although strong scaling for the two order parameters (OPs) perpendicular and parallel to the field is restored as found earlier, in the experimentally accessible region the effective dynamical exponents for the relaxation of the OPs remain different since their equal asymptotic values are not reached., Comment: 6 pages, 2 figures; some additions, corrected typos

We discuss the static and dynamic multicritical behavior of three-dimensional systems of $O(n_\|)\oplus O(n_\perp)$ symmetry as it is explained by the field theoretical renormalization group method. Whereas the static renormalization group functions are currently know within high order expansions, we show that an account of two loop contributions refined by an appropriate resummation technique gives an accurate quantitative description of the multicritical behavior. One of the essential features of the static multicritical behavior obtained already in two loop order for the interesting case of an antiferromagnet in a magnetic field ($n_\|=1$, $n_\perp=2$) are the stability of the biconical fixed point and the neighborhood of the stability border lines to the other fixed points leading to very small transient exponents. We further pursue an analysis of dynamical multicritical behavior choosing different forms of critical dynamics and calculating asymptotic and effective dynamical exponents within the minimal subtraction scheme., Comment: 14 pages, Submitted to the Proceedings of the Conference "Statphys'09" dedicated to the 100-th anniversary of N.N.Bogolyubov (23.06-25.06.2009, Lviv, Ukraine)

We calculate the relaxational dynamical critical behavior of systems of $O(n_\|)\oplus O(n_\perp)$ symmetry including conservation of magnetization by renormalization group (RG) theory within the minimal subtraction scheme in two loop order. Within the stability region of the Heisenberg fixed point and the biconical fixed point strong dynamical scaling holds with the asymptotic dynamical critical exponent $z=2\phi/\nu-1$ where $\phi$ is the crossover exponent and $\nu$ the exponent of the correlation length. The critical dynamics at $n_\|=1$ and $n_\perp=2$ is governed by a small dynamical transient exponent leading to nonuniversal nonasymptotic dynamical behavior. This may be seen e.g. in the temperature dependence of the magnetic transport coefficients., Comment: 6 figures

We calculate the relaxational dynamical critical behavior of systems of $O(n_\|)\oplus O(n_\perp)$ symmetry by renormalization group method within the minimal subtraction scheme in two loop order. The three different bicritical static universality classes previously found for such systems correspond to three different dynamical universality classes within the static borderlines. The Heisenberg and the biconical fixed point lead to strong dynamic scaling whereas in the region of stability of the decoupled fixed point weak dynamic scaling holds. Due to the neighborhood of the stability border between the strong and the weak scaling dynamic fixed point corresponding to the static biconical and the decoupled fixed point a very small dynamic transient exponent, of $\omega_v^{{\cal B}}=0.0044$, is present in the dynamics for the physically important case $n_\|=1$ and $n_\perp=2$ in $d=3$., Comment: 8 figures

We calculate the static critical behavior of systems of $O(n_\|)\oplus O(n_\perp)$ symmetry by renormalization group method within the minimal subtraction scheme in two loop order. Summation methods lead to fixed points describing multicritical behavior. Their stability boarder lines in the space of order parameter components $n_\|$ and $n_\perp$ and spatial dimension $d$ are calculated. The essential features obtained already in two loop order for the interesting case of an antiferromagnet in a magnetic field ($n_\|=1$, $n_\perp=2$) are the stability of the biconical fixed point and the neighborhood of the stability border lines to the other fixed points leading to very small transient exponents. We are also able to calculate the flow of static couplings, which allows to consider the attraction region. Depending on the nonuniversal background parameters the existence of different multicritical behavior (bicritical or tetracritical) is possible including a triple point., Comment: 6 figures

We study the relaxational critical dynamics of the three-dimensional random anisotropy magnets with the non-conserved n-component order parameter coupled to a conserved scalar density. In the random anisotropy magnets the structural disorder is present in a form of local quenched anisotropy axes of random orientation. When the anisotropy axes are randomly distributed along the edges of the n-dimensional hypercube, asymptotical dynamical critical properties coincide with those of the random-site Ising model. However structural disorder gives rise to considerable effects for non-asymptotic critical dynamics. We investigate this phenomenon by a field-theoretical renormalization group analysis in the two-loop order. We study critical slowing down and obtain quantitative estimates for the effective and asymptotic critical exponents of the order parameter and scalar density. The results predict complex scenarios for the effective critical exponent approaching an asymptotic regime., Comment: 8 figures, style files included

The critical dynamics of superconductors in the charged regime is reconsidered within field-theory. For the dynamics the Ginzburg-Landau model with complex order parameter coupled to the gauge field suggested earlier [Lannert et al. Phys. Rev. Lett. 92, 097004 (2004)] is used. Assuming relaxational dynamics for both quantities the renormalization group functions within one loop approximation are recalculated for different choices of the gauge. A gauge independent result for the divergence of the measurable electric conductivity is obtained only at the weak scaling fixed point unstable in one loop order where the time scales of the order parameter and the gauge field are different., Comment: 9 pages, style file included

The critical dynamics of model C in the presence of disorder is considered. It is known that in the asymptotics a conserved secondary density decouples from the nonconserved order parameter for disordered systems. However couplings between order parameter and secondary density cause considerable effects on non-asymptotic critical properties. Here, a general procedure for a renormalization group treatment is proposed. Already the one-loop approximation gives a qualitatively correct picture of the diluted model C dynamical criticality. A more quantitative description is achieved using two-loop approximation. In order to get reliable results resummation technique has to be applied., Comment: 21 page, 4 figures, submitted to the proceedings of the conference "Renormalization Group 2005", Helsinki, Finland, 30 August - 3 September 2005

We study the purely relaxational critical dynamics with non-conserved order parameter (model A critical dynamics) for three-dimensional magnets with disorder in a form of the random anisotropy axis. For the random axis anisotropic distribution, the static asymptotic critical behaviour coincides with that of random site Ising systems. Therefore the asymptotic critical dynamics is governed by the dynamical exponent of the random Ising model. However, the disorder influences considerably the dynamical behaviour in the non-asymptotic regime. We perform a field-theoretical renormalization group analysis within the minimal subtraction scheme in two-loop approximation to investigate asymptotic and effective critical dynamics of random anisotropy systems. The results demonstrate the non-monotonic behaviour of the dynamical effective critical exponent $z_{\rm eff}$., Comment: 11 pages, 4 figures, style file included

We consider the influence of quenched disorder on the relaxational critical dynamics of a system characterized by a non-conserved order parameter coupled to the diffusive dynamics of a conserved scalar density (model C). Disorder leads to model A critical dynamics in the asymptotics, however it is the effective critical behavior which is often observed in experiments and in computer simulations and this is described by the full set of dynamical equations of diluted model C. Indeed different scenarios of effective critical behavior are predicted., Comment: 4 pages, 5 figures