Renormalization group approach to QCD phase transitions

Effective scalar theories for QCD are proposed to investigate the deconfining and chiral phase transitions. The orders of the phase transitions are determined by infrared stabilities of the fixed points. It is found that the transitions inSU (3) gauge theories are of 1st order for any number of massless flavors. The cases ofSU (2) andSU (4) gauge theories are also discussed.     

Renormalization group treatment of phase transitions in ferromagnetic superconductors

We investigate the nature of the phase transition from the superconducting to the uniformly ordered magnetic phase in ferromagnetic superconductors by renormalization group methods. For this purpose Halperin, Lubensky, Ma's application of the-expansion to pure superconductors is extended to include the influence of magnetic order as well. Our treatment shows that in the vicinity of the ferromagnetic to superconducting phase-boundary critical fluctuations of the magnetization are absorbed into fluctuations of the magnetic field. As a consequence the transition always is first order. An additional indication to the dominance of electromagnetic effects is the irrelevance of the coupling corresponding to spin flip scattering.     

Renormalization group theory for temperature-driven first-order phase transitions in scalar models

We study the scaling and universal behavior of temperature-driven first-order phase transitions in scalar models. These transitions are found to exhibit rich phenomena, though they are controlled by a single complex-conjugate pair of imaginary fixed points of ? ³ theory. Scaling theories and renormalization group theories are developed to account for the phenomena, and three universality classes with their own hysteresis exponents are found: a field-like thermal class, a partly thermal class, and a purely thermal class, designated, respectively, as Thermal Classes I, II, and III. The first two classes arise from the opposite limits of the scaling forms proposed and may cross over to each other depending on the temperature sweep rate. They are both described by a massless model and a purely massive model, both of which are equivalent and are derived from ? ³ theory via symmetry. Thermal Class III characterizes the cooling transitions in the absence of applied external fields and is described by purely thermal models, which include cases in which the order parameters possess different symmetries and thus exhibit different universality classes. For the purely thermal models whose free energies contain odd-symmetry terms, Thermal Class III emerges only at the mean-field level and is identical to Thermal Class II. Fluctuations change the model into the other two models. Using the extant three- and two-loop results for the static and dynamic exponents for the Yang–Lee edge singularity, respectively, which falls into the same universality class as ? ³ theory, we estimate the thermal hysteresis exponents of the various classes to the same precision. Comparisons with numerical results and experiments are briefly discussed.     

Renormalization group approach to the phase diagram of two-dimensional Heisenberg spin systems

A renormalization group method is used to analyse the phase diagram of a quantum-mechanical Hamiltonian version of the O(2) and O(3) Heisenberg spin systems in two dimensions. It shows a phase transition at non-zero coupling for the O(2) model and no evidence of it for the O(3) model.     

Renormalization group and high-temperature series for the two-dimensional ising model

We employ high-temperature series to investigate a two-parameter class of renormalization group transformations for the two-dimensional Ising model on the triangular lattice. For the static case we identify an optimal organization of the high-temperature expansion and an optimal transformation matrix and thus find, in second order, =0.96 and the magnetic eigenvaluey=2-/2=1.76.From recursion relations for flip rates we find the dynamic exponent to be the same for all transformations in our two-parameter class,z=2.32.Our fixed-point flip rates do not describe a Markov process even though the corresponding master equation for the single-event probability displays no explicit memory effects. The non-Markovian nature shows up only in a violation of the Markovian detailed balance conditions.     

Renormalization group functions for the radiative symmetry breaking scheme

We obtain the renormalization group (RG) functions for the massless scalar field theory where symmetry breaking occurs radiatively. After obtaining the effective potential for the radiative symmetry breaking scheme by finite transformations for the classical field and coupling constant, we obtain the corresponding RG functions from that of the minimal subtraction (MS) scheme.     

Renormalization group equations for a two-dimensional nonlinear sigma-model with torsion

The renormalization group equations in the one-loop approximation are formulated for a two-dimensional nonlinear -model with torsion on a semisimple group manifold. Solutions of these equations are investigated for several particular cases.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 37–40, September, 1988.     

Renormalization group approach towards the QCD phase diagram

The idea of the functional renormalization group and one-loop improved renormalization group flows are reviewed. The associated flow equations and nonperturbative approximations schemes for its solutions are discussed. These techniques are then applied to the strong interaction in the framework of an effective quark meson model, which is introduced in great detail. The renormalization group analysis of the two flavor quark meson model is extended to finite temperature and quark chemical potential which allows for an analysis of the chiral phase diagram beyond the mean field approximation. The text was submitted by the authors in English.     

Isoperimetric phase transitions of two-dimensional droplets

We consider two-dimensional assemblies of particles governed by hamiltonians depending on the area and the perimeter of their convex hull. Provided the hamiltonian is quadratically homogeneous in the coordinates, we find an exact formula for the free energy. Phase transitions resulting from the competition between area and perimeter can easily be produced and explicitly dealt with. We illustrate those features by a simple example undergoing a second-order transition.     

On the topological characterization of two-dimensional phase transitions

It is shown that for the two-dimensional planar-rotator model the variance of the number of vortices inside a large loop is proportional to the perimeter of the loop at all temperatures. A similar result holds for the variance of the charge inside a loop in the lattice Coulomb gas model. Hence a topological distinction between the high and low temperature phases of these models (area law versus perimeter law) is not possible.     

Renormalization group and the ultraviolet problem in the Luttinger model

The Luttinger model describes a non-local interacting relativistic theory for spinless and massless fermions. Albeit the exact solution is already known, the perturbative approach to the model via the renormalization group is useful on account of the connection to the study of more realistic models' behaviour near the Fermi surface. In this work we show that the effective potential describing the interaction on the physical scalep ₀ ^?1 is analytical in the coupling constants, and has an exponential decay on that scale. Besides the physical motivation of this approach, the problem is also technically interesting, since it is an example of a trivially superrenormalizable theory, as far as the ultraviolet region is concerned; nevertheless the proof is quite delicate, as the convergence of the perturbative series does not follow from the superficial bounds (which would give logarithmic and linear divergences), but is due to accidental compensations furnished by the particular symmetry properties of the model.     

Kinetics of phase transitions with singular multiplicative noise

The behavior of the most probable values of the order parameter x and the amplitude p of conjugate force fluctuations is studied for a stochastic system with a noise amplitude depending on x as |x|^a. It is shown that the phase half-plane x>0 for the canonical pair x, p is divided into isolated regions of large, intermediate, and small values of x. In the first region, the trajectories converge to values of x, p → ∞ as the time t → ∞, and the probability of their realization is negligibly small. In the intermediate region, the configuration point tends to the attraction center corresponding to a stationary ordered state. In the region , the trajectories converge to the point x=p=0 for 0<a<1/2 and to x=0, p → ∞ for 1/2<a≤1. In the former case, the probability of realization of trajectories is finite, while, in the latter case, it is negligibly small, and an absorbing state can be formed.     

Renormalization group phase boundary of the three-dimensional bond-dilute Ising ferromagnet

The phase boundary (as well as the thermal-type critical exponents) associated to the quenched bond-dilute spin-1/2 Ising ferromagnet in the simple cubic lattice is approximately calculated within a real space renormalization group framework in two different versions. Both lead to qualitatively satisfactory critical frontiers, although one of them provides an unphysical fixed point (which seems to be related to the three-dimensionality of the system) besides the expected pure ones; its effects tend to disappear for increasingly large clusters. Through an extrapolation procedure the (unknown) critical phase boundary is approximately located.     

Dynamical phase transitions in the two-dimensional ANNNI model

We study the phase diagram of the two-dimensional anisotropic next-nearest neighbor Ising (ANNNI) model by comparing the time evolution of two distinct spin configurations submitted to the same thermal noise. We clearly see several dynamical transitions between ferromagnetic, paramagnetic, antiphase, and floating phases. These dynamical transitions seem to occur rather close to the transition lines determined previously in the literature.     

Renormalization group flow of two-dimensional hierarchical Heisenberg model of Dyson-Wilson type

Two-dimensionalO(N)-invariant hierarchical Heisenberg models (of Dyson-Wilson type) are investigated by the real space renormalization group method. It is established that ifN2, then the effective actions at long distance scales are driven into the high temperature region by the iterative use of the block spin transformations thus concluding the non-existence of phase transitions in the system. The correlation functions are also obtained and they decay at the speed of massive gaussian field model. The driving force is geometrical and an a priori one and stronger than the boundary effect which is bounded byO(1) in the present system. Thus the hierarchical formulas fail to exhibit the Kosterlitz-Thouless transitions (at least forN=2).     

To the theory of phase transitions in relaxors

It is shown that the process of charge carrier localization on impurity centers is of importance. Localized carriers produce local electric fields, thereby stimulating the appearance of an induced polarization near the phase transition point. The direction of this polarization is dictated by the spatial distribution of the centers occupied by charge carriers. The temperature dependence of the dielectric constant is determined by the dynamics of the local-center occupation with decreasing temperature. The dispersion of the dielectric constant is determined by the vibrational characteristics of the local states forming near a localized charge. The phase transitions in relaxors are considered in the framework of the thermodynamic approach.     

Renormalization group equations for the Kobayashi-Maskawa matrix

The renormalization group equations for the parametrization-convention independent quadratic parameters |V _ij |² of the KM matrix are derived. Numerical analysis of these equations shows that the heavy quark family (t, b) tends to mix with the lighter families (c, s) and (u, d) with increasing energy, although the variation is very much slow. The CP-nonconservation effects are shown to get larger with energy.