Critical exponents of the Ising model with quenched structural disorder and long-range interactions at spatial dimension <i>d</i> = 3

We analyze the behavior of several renormalization group functions at infrared fixed points for $SU(N)$ gauge theories with fermions in the fundamental and two-indexed representations. This includes the beta function of the gauge coupling, the anomalous dimension of the gauge parameter and the anomalous dimension of the mass. The scheme in which the analysis is performed is the minimal momentum subtraction scheme through third-loop order. Because of the fact that scheme dependence is inevitable once the perturbation theory is truncated we compare to previous identical studies done in the minimal subtraction scheme and the modified regularization invariant scheme. We find only mild to moderate scheme dependence.

Effective and asymptotic criticality of structurally disordered magnets

The critical behavior of a complex N-component order parameter Ginzburg-Landau model with isotropic and cubic interactions describing antiferromagnetic and structural phase transitions in certain crystals with complicated ordering is studied in the framework of the four-loop renormalization group (RG) approach in $(4-\ve)$ dimensions. By using dimensional regularization and the minimal subtraction scheme, the perturbative expansions for RG functions are deduced and resummed by the Borel-Leroy transformation combined with a conformal mapping. Investigation of the global structure of RG flows for the physically significant cases N=2 and N=3 shows that the model has an anisotropic stable fixed point governing the continuous phase transitions with new critical exponents. This is supported by the estimate of the critical dimensionality $N_c=1.445(20)$ obtained from six loops via the exact relation $N_c={1/2} n_c$ established for the complex and real hypercubic models.

We study spatial anisotropy effects on the bulk and finite-size critical behavior of the O(n) symmetric anisotropic phi;{4} lattice model with periodic boundary conditions in a d -dimensional hypercubic geometry above, at, and below Tc. The absence of two-scale factor universality is discussed for the bulk order-parameter correlation function, the bulk scattering intensity, and for several universal bulk amplitude relations. The anisotropy parameters are observable by scattering experiments at Tc. For the confined system, renormalization-group theory within the minimal subtraction scheme at fixed dimension d for 2<d<4 is employed. In contrast to the epsilon=4-d expansion, the fixed- d finite-size approach keeps the exponential form of the order-parameter distribution function unexpanded. For the case of cubic symmetry and for n=1 , our perturbation approach yields excellent agreement with the Monte Carlo (MC) data for the finite-size amplitude of the free energy of the three-dimensional Ising model at Tc by Mon [Phys. Rev. Lett. 54, 2671 (1985)]. The epsilon expansion result is in less good agreement. Below Tc, a minimum of the scaling function of the excess free energy is found. We predict a measurable dependence of this minimum on the anisotropy parameters. The relative anisotropy effect on the free energy is predicted to be significantly larger than that on the Binder cumulant. Our theory agrees quantitatively with the nonmonotonic dependence of the Binder cumulant on the ferromagnetic next-nearest-neighbor (NNN) coupling of the two-dimensional Ising model found by MC simulations of Selke and Shchur [J. Phys. A 38, L739 (2005)]. Our theory also predicts a nonmonotonic dependence for small values of the antiferromagnetic NNN coupling and the existence of a Lifshitz point at a larger value of this coupling. The nonuniversal anisotropy effects in the finite-size scaling regime are predicted to satisfy a kind of restricted universality. The tails of the large- L behavior at T++Tc violate both finite-size scaling and universality even for isotropic systems as they depend on the bare four-point coupling of the phi4 theory, on the cutoff procedure, and on subleading long-range interactions.

Recently, it has been found that an effective long-range interaction is realized among local bistable variables (spins) in systems where the elastic interaction causes ordering of the spins. In such systems, generally we expect both long-range and short-range interactions to exist. In the short-range Ising model, the correlation length diverges at the critical point. In contrast, in the long-range interacting model the spin configuration is always uniform and the correlation length is zero. As long as a system has non-zero long-range interactions, it shows criticality in the mean-field universality class, and the spin configuration is uniform beyond a certain scale. Here we study the crossover from the pure short-range interacting model to the long-range interacting model. We investigate the infinite-range model (Husimi-Temperley model) as a prototype of this competition, and we study how the critical temperature changes as a function of the strength of the long-range interaction. This model can also be interpreted as an approximation for the Ising model on a small-world network. We derive a formula for the critical temperature as a function of the strength of the long-range interaction. We also propose a scaling form for the spin correlation length at the critical point, which is finite as long as the long-range interaction is included, though it diverges in the limit of the pure short-range model. These properties are confirmed by extensive Monte Carlo simulations.

Renormalization group approaches to polymers in disordered media

We have studied a Fermi system with attractive U(r)-symmetric interaction at the finite temperatures by the quantum field renormalization group (RG) method. The RG functions have been calculated in the framework of dimensional regularization and minimal subtraction scheme up to five loops. It has been found that for r≥4 the RG flux leaves the system's stability region – the system undergoes a first order phase transition. To estimate the temperature of the transition to superconducting or superfluid phase the RG analysis for composite operators has been performed using three-loops approximation. The result of this analysis shows that for 3D systems estimated phase transition temperature is higher then well known theoretical estimations based on continuous phase transition formalism.

Exact five-loop renormalization group functions of θ4-theory with O( N)-symmetric and cubic interactions. Critical exponents up to ϵ5

The complete analysis of a model with three quartic coupling constants associated with $\mathrm{O}(2N)$-symmetric, cubic, and tetragonal interactions is carried out within the three-loop approximation of the renormalization-group (RG) approach in $D=4\ensuremath{-}2\ensuremath{\epsilon}$ dimensions. Perturbation expansions for RG functions are calculated using dimensional regularization and the minimal subtraction (MS) scheme. It is shown that for $N&gt;~2$ the model does possess a stable fixed point in three-dimensional space of coupling constants, in accordance with predictions made earlier on the base of the lower-order approximations. A numerical estimate for the critical (marginal) value of the order parameter dimensionality ${N}_{c}$ is given using the Pad\'e-Borel summation of the corresponding \ensuremath{\epsilon}-expansion series obtained. It is observed that a twofold degeneracy of the eigenvalue exponents in the one-loop approximation for the unique stable fixed point leads to the substantial decrease of the accuracy expected within three loops and may cause powers of $\sqrt{\ensuremath{\epsilon}}$ to appear in the expansions. The critical exponents \ensuremath{\gamma} and \ensuremath{\eta} are calculated for all fixed points up to ${\ensuremath{\epsilon}}^{3}$ and ${\ensuremath{\epsilon}}^{4},$ respectively, and processed by the Borel summation method modified with a conformal mapping. For the unique stable fixed point the magnetic susceptibility exponent \ensuremath{\gamma} for $N=2$ is found to differ in third order in \ensuremath{\epsilon} from that of an $\mathrm{O}(4)$-symmetric point. Qualitative comparison of the results given by \ensuremath{\epsilon} expansion, three-dimensional RG analysis, nonperturbative RG arguments, and experimental data is performed.

This is a short chapter summarizing the main results concerning the renormalization group in models of pure quantum gravity, without matter fields. The chapter starts with a critical analysis of non-perturbative renormalization group approaches, such as the asymptotic safety hypothesis. After that, it presents solid one-loop results based on the minimal subtraction scheme in the one-loop approximation. The polynomial models that are briefly reviewed include the on-shell renormalization group in quantum general relativity, and renormalization group equations in fourth-derivative quantum gravity and superrenormalizable models. Special attention is paid to the gauge-fixing dependence of the renormalization group trajectories.

Phase transitions and autocorrelation times in two-dimensional Ising model with dipole interactions

We consider the Ising model in a transverse field with long-range antiferromagnetic interactions that decay as a power law with their distance. We study both the phase diagram and the entanglement properties as a function of the exponent of the interaction. The phase diagram can be used as a guide for future experiments with trapped ions. We find two gapped phases, one dominated by the transverse field, exhibiting quasi-long-range order, and one dominated by the long-range interaction, with long-range Néel ordered ground states. We determine the location of the quantum critical points separating those two phases. We determine their critical exponents and central charges. In the phase with quasi-long-range order the ground states exhibit exotic corrections to the area law for the entanglement entropy coexisting with gapped entanglement spectra.

We present a field-theoretical treatment of the critical behavior of three-dimensional weakly diluted quenched Ising model. To this end we analyse in a replica limit n=0 5-loop renormalization group functions of the $\phi^4$-theory with O(n)-symmetric and cubic interactions (H.Kleinert and V.Schulte-Frohlinde, Phys.Lett. B342, 284 (1995)). The minimal subtraction scheme allows to develop either the $\epsilon^{1/2}$-expansion series or to proceed in the 3d approach, performing expansions in terms of renormalized couplings. Doing so, we compare both perturbation approaches and discuss their convergence and possible Borel summability. To study the crossover effect we calculate the effective critical exponents providing a local measure for the degree of singularity of different physical quantities in the critical region. We report resummed numerical values for the effective and asymptotic critical exponents. Obtained within the 3d approach results agree pretty well with recent Monte Carlo simulations. $\epsilon^{1/2}$-expansion does not allow reliable estimates for d=3.

On the critical properties of the three-dimensional random Ising model

We consider systems confined to a d-dimensional slab of macroscopic lateral extension and finite thickness L that undergo a continuous bulk phase transition in the limit L --> infinity and are describable by an O(n) symmetrical Hamiltonian. Periodic boundary conditions are applied across the slab. We study the effects of long-range pair interactions whose potential decays as bx-(d+sigma) as x --> infinity, with 2<sigma<4 and 2<d+sigma< or =6, on the Casimir effect at and near the bulk critical temperature Tc,infinity, for 2<d<4. These interactions decay sufficiently fast to leave bulk critical exponents and other universal bulk quantities unchanged--i.e., they are irrelevant in the renormalization group (RG) sense. Yet they entail important modifications of the standard scaling behavior of the excess free energy and the Casimir force Fc. We generalize the phenomenological scaling Ansätze for these quantities by incorporating these long-range interactions. For the scaled reduced Casimir force per unit cross-sectional area, we obtain the form LdFc/kBt approximately Xi0(L/xi infinity) + g omegaL -omega Xi omega (L/Xi infinity) + g sigma L -omega sigma Xi sigma (L/Xi infinity). Here Xi0, Xi omega, and Xi sigma are universal scaling functions; g omega and g sigma are scaling fields associated with the leading corrections to scaling and those of the long-range interaction, respectively; omega and omega sigma = sigma + eta - 2 are the associated correction-to-scaling exponents, where eta denotes the standard bulk correlation exponent of the system without long-range interactions; xi infinity is the (second-moment) bulk correlation length (which itself involves corrections to scaling). The contribution proportional variant g sigma decays for T not = Tc,infinity algebraically in L rather than exponentially, and hence becomes dominant in an appropriate regime of temperatures and L. We derive exact results for spherical and Gaussian models which confirm these findings. In the case d + sigma = 6, which includes that of nonretarded van der Waals interactions in d = 3 dimensions, the power laws of the corrections to scaling proportional to b of the spherical model are found to get modified by logarithms. Using general RG ideas, we show that these logarithmic singularities originate from the degeneracy omega = omega sigma = 4 - d that occurs for the spherical model when d + sigma = 6, in conjunction with the b dependence of g omega.