Instability of renormalization-group pathologies under decimation

We investigate the stability and instability of pathologies of renormalization group transformations for lattice spin systems under decimation. In particular we show that, even if the original renormalization group transformation gives rise to a non-Gibbsian measure, Gibbsianness may be restored by applying an extra decimation transformation. This fact is illustrated in detail for the block spin transformation applied to the Ising model. We also discuss the case of another non-Gibbsian measure with nicely decaying correlations functions which remains non-Gibbsian after arbitrary decimation.     

Some numerical results on the block spin transformation for the 2D Ising model at the critical point

We study the block spin transformation for the 2D Ising model at the critical temperatureT _c . We consider the model with the constraint that the total spin in each block is zero. An old argument by Cassandro and Gallavotti strongly supports the Gibbsianness of the transformed measure, provided that such model has a critical temperatureT _c lower thanT _c . After describing a possible rigorous approach to the problem, we present numerical evidence that indeedT _c <T _c and study the Dobrushin-Shlosman uniqueness condition.     

Finite-size scaling corrections for the eight-vertex model

Finite-size scaling corrections are calculated analytically for two of the maximal eigenvalues of the transfer matrix in the isotropic eight-vertex model. The valuec=1 for the conformal anomaly of the Virasoro algebra is confirmed.     

Some remarks on the stochastic model of crystal growth

Conclusions In this paper further possibilities of extending of the stochastic theory of crystal growth as well as possibilities of solving the basic equations are presented. The building of kinetic processes on the solidification front in the stochastic model gives us the possibility of studying the influence of kinetic processes in connection with temperature distribution during crystal growth.The approximations mentioned in chapter 2 considerably accelerate the numerical solution of our problem. Tests in concrete cases show that mentioned approximations do not significantly influence the results if the conditions referred to are fulfilled.     

Some remarks on BRS transformations,anomalies and the cohomology of the Lie algebra of the group of gauge transformations

We show that ghosts in gauge theories can be interpreted as Maurer-Cartan forms in the infinite dimensional group ? of gauge transformations. We examine the cohomology of the Lie algebra of ? and identify the coboundary operator with the BRS operator. We describe the anomalous terms encountered in the renormalization of gauge theories (triangle anomalies) as elements of these cohomology groups.     

Some remarks on a model for internal structure of black holes

Some clarifications about a model for the internal structure of black holes developed in an earlier paper are presented to remove confusion in the interpretation of certain results.     

On the character of peculiarities in the position-space renormalization-group transformations

The image magnetic field is shown to be a discontinuous function of the object magnetic field near the zero-field point for T < T_c in the thermodynamic limit in the position-space renormalization-group procedure as applied to an infinite-range interaction model.     

A renormalization-group study of an Ising spin-glass with annealed vacancies

Systems driven and characterized by fluctuations in density and magnetization can be realistically modeled using the Blume–Emery–Griffiths model; a spin-1 Ising model with bilinear, biquadratic, and crystal-field interactions. In this study, renormalization-group techniques are used on an exactly solvable system in which frustration is present due to competing ferromagnetic and antiferromagnetic interactions. Thus, this calculation models a spin-glass system with annealed vacancies. To determine the effects of these competing bilinear interactions, an exactly solvable frustrated hierarchical model has been constructed, similar to those introduced to study spin glasses [S.R. McKay, A.N. Berker, S. Kirkpatrick, Phys. Rev. Lett. 48 (1982) 767]. Phase diagrams have been calculated for a series of planes of constant biquadratic coupling while varying the temperature and concentration of annealed vacancies in the system. In addition, a phase diagram was produced for constant concentration of annealed vacancies as the biquadratic coupling (i.e. clustering bias) was varied. Each phase diagram reveals three qualitatively unique basins of attraction, each corresponding to a phase distinguished by a unique renormalization-group trajectory. The sink of each trajectory is interpreted to determine the nature of each phase: dense paramagnetic, dilute paramagnetic and spin-glass.     

Some remarks on a model of supersymmetric quantum mechanics

We develop a recently proposed model within supersymmetric quantum mechanics that puts a group structure on the creation and annihilation operators. We apply the scheme to a variety of quantum mechanical problems and work out a two-term energy recursion equation when the overall group structure isU(1, 1).     

Multifractal properties of aperiodic Ising model on hierarchical lattices: role of the geometric fluctuations

The role of the geometric fluctuations on the multifractal properties of the local magnetization of aperiodic ferromagnetic Ising models on hierarchical lattices is investigated. The geometric fluctuations are introduced by generalized Fibonacci sequences. The local magnetization is evaluated via an exact recurrent procedure encompassing real space renormalization group decimation. The symmetries of the local magnetization patterns induced by the aperiodic couplings is found to be strongly (weakly) different, with respect to the ones of the corresponding homogeneous systems, when the geometric fluctuations are relevant (irrelevant) to change the critical properties of the system. At the criticality, the measure defined by the local magnetization is found to exhibit a non-trivial F(α) spectra being shifted to higher values of α when relevant geometric fluctuations are considered. The critical exponents are found to be related with some special points of the F(α) function and agree with previous results obtained by the quite distinct transfer matrix approach. Received 2 April 2001 and Received in final form 14 August 2001     

On the finite-size scalling equation for the spherical model

The mean spherical model with an arbitrary interaction potential, the Fourier transform of which has a long-wavelength exponent , 0<2, is considered under periodic boundary conditions and fully finite geometry ind dimensions, when <d<2. A new form of the finite-size scaling equation for the spherical field in the critical region is derived, which relates the temperature shift to Madelung-type lattice constants. The method of derivation makes use of the Poisson summation formula and a Laplace transformation of the momentumspace correlation function.On leave of absence from Institute of Mechanics and Biomechanics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria.     

Fractal Failure of Quasilocality for a Majority Rule Transformation on a Tree

We provide a transformation of the Ising model on a Cayley tree leading to non-Gibbsianness at any temperature, i.e. even within the uniqueness regime. We also introduce a new type of pathologies of renormalized Gibbs measures, called the fractal failure of quasilocality, and exhibit a concrete example.     

Finite-size effects on the phase transition in a four- and six-fermion interaction model

We consider four- and six-fermion interacting models at finite temperature and density. We construct the corresponding free energies and investigate the appearance of first- and second-order phase transitions. Finite-size effects on the phase structure are investigated using methods of quantum field theory on toroidal topologies.     

Fermi-liquid quasi-particles and -superconductivity in the two-dimensional Hubbard model

A frequency- and momentum-renormalization-group acceleration together with an analytical approach is used to obtain the retarded Green's function in the self-consistent and conserving fluctuation-exchange (FLEX) approximation for the two-dimensional Hubbard model in the normal state and in the superconducting state. The analytical expressions for this approach are given. For band-fillings near half filling the self-energy in the normal state exhibits Fermi-liquid behaviour for, low temperatures and frequencies near the chemical potential, if the momentum is chosen near the Fermi-surface. However due to the presence of large many body effects the observed Fermi-liquid region near the chemical potential and near the Fermi-surface is very small. Results for the single particle density of states, the occupation number and the spectral function are presented. The superconducting state with symmetry is obtained for U = 2 to U = 6 and a (U, n)-phase diagram for the transition temperature T_c is presented. A maximum T_c/t of 0.0275 is obtained for U = 6 near half filling.     

Bounds for the anisotropic Ising model

Rigorous upper bounds are found for the magnetisation, susceptibility, critical temperature and crossover exponent in an anisotropic Ising system.     

Some remarks on the Gribov ambiguity

The set of all connections of a principal bundle over the 4-sphere with compact nonabelian Lie group under the action of the group of gauge transformations is studied. It is shown that no continuous choice of exactly one connection on each orbit can be made. Thus the Gribov ambiguity for the Coloumb gauge will occur in all other gauges. No gauge fixing is possible.