arXiv subjects

L. Turban

Publications and source records attributed to L. Turban.

48 records · Page 3Linked to original sources

arXiv1997

Surface Magnetization of Aperiodic Ising Quantum Chains

We study the surface magnetization of aperiodic Ising quantum chains. Using fermion techniques, exact results are obtained in the critical region for quasiperiodic sequences generated through an irrational number as well as for the automatic binary Thue-Morse sequence and its generalizations modulo p. The surface magnetization exponent keeps its Ising value, beta_s=1/2, for all the sequences studied. The critical amplitude of the surface magnetization depends on the strength of the modulation and also on the starting point of the chain along the aperiodic sequence.

cond-mat.stat-mech↗

arXiv1997

Marginal Extended Perturbations in Two Dimensions and Gap-Exponent Relations

The most general form of a marginal extended perturbation in a two-dimensional system is deduced from scaling considerations. It includes as particular cases extended perturbations decaying either from a surface, a line or a point for which exact results have been previously obtained. The first-order corrections to the local exponents, which are functions of the amplitude of the defect, are deduced from a perturbation expansion of the two-point correlation functions. Assuming covariance under conformal transformation, the perturbed system is mapped onto a cylinder. Working in the Hamiltonian limit, the first-order corrections to the lowest gaps are calculated for the Ising model. The results confirm the validity of the gap-exponent relations for the perturbed system.

hep-th↗

arXiv1997

Inhomogeneous Systems with Unusual Critical Behaviour

The phase transitions and critical properties of two types of inhomogeneous systems are reviewed. In one case, the local critical behaviour results from the particular shape of the system. Here scale-invariant forms like wedges or cones are considered as well as general parabolic shapes. In the other case the system contains defects, either narrow ones in the form of lines or stars, or extended ones where the couplings deviate from their bulk values according to power laws. In each case the perturbation may be irrelevant, marginal or relevant. In the marginal case one finds local exponents which depend on a parameter. In the relevant case unusual stretched exponential behaviour and/or local first order transitions appear. The discussion combines mean field theory, scaling considerations, conformal transformations and perturbation theory. A number of examples are Ising models for which exact results can be obtained. Some walks and polymer problems are considered, too.

cond-mat.stat-mech↗

arXiv1997

Fractal Dimensions of Confined Clusters in Two-Dimensional Directed Percolation

The fractal structure of directed percolation clusters, grown at the percolation threshold inside parabolic-like systems, is studied in two dimensions via Monte Carlo simulations. With a free surface at y=\pm Cx^k and a dynamical exponent z, the surface shape is a relevant perturbation when k<1/z and the fractal dimensions of the anisotropic clusters vary continuously with k. Analytic expressions for these variations are obtained using a blob picture approach.

cond-mat.stat-mech↗

arXiv1997

Surface Magnetization of Aperiodic Ising Systems: a Comparative Study of the Bond and Site Problems

We investigate the influence of aperiodic perturbations on the critical behaviour at a second order phase transition. The bond and site problems are compared for layered systems and aperiodic sequences generated through substitution. In the bond problem, the interactions between the layers are distributed according to an aperiodic sequence whereas in the site problem, the layers themselves follow the sequence. A relevance-irrelevance criterion introduced by Luck for the bond problem is extended to discuss the site problem. It involves a wandering exponent for pairs, which can be larger than the one considered before in the bond problem. The surface magnetization of the layered two-dimensional Ising model is obtained, in the extreme anisotropic limit, for the period-doubling and Thue-Morse sequences.

cond-mat.stat-mech↗

arXiv1997

Surface Shape and Local Critical Behaviour in Two-Dimensional Directed Percolation

Two-dimensional directed site percolation is studied in systems directed along the x-axis and limited by a free surface at y=\pm Cx^k. Scaling considerations show that the surface is a relevant perturbation to the local critical behaviour when k<1/z where z=ν_\parallel/νis the dynamical exponent. The tip-to-bulk order parameter correlation function is calculated in the mean-field approximation. The tip percolation probability and the fractal dimensions of critical clusters are obtained through Monte-Carlo simulations. The tip order parameter has a nonuniversal, C-dependent, scaling dimension in the marginal case, k=1/z, and displays a stretched exponential behaviour when the perturbation is relevant. The k-dependence of the fractal dimensions in the relevant case is in agreement with the results of a blob picture approach.

cond-mat.stat-mech↗

arXiv1997

Surface Magnetization and Critical Behavior of Aperiodic Ising Quantum Chains

We consider semi-infinite two-dimensional layered Ising models in the extreme anisotropic limit with an aperiodic modulation of the couplings. Using substitution rules to generate the aperiodic sequences, we derive functional equations for the surface magnetization. These equations are solved by iteration and the surface magnetic exponent can be determined exactly. The method is applied to three specific aperiodic sequences, which represent different types of perturbation, according to a relevance-irrelevance criterion. On the Thue-Morse lattice, for which the modulation is an irrelevant perturbation, the surface magnetization vanishes with a square root singularity, like in the homogeneous lattice. For the period-doubling sequence, the perturbation is marginal and the surface magnetic exponent varies continuously with the modulation amplitude. Finally, the Rudin-Shapiro sequence, which corresponds to the relevant case, displays an anomalous surface critical behavior which is analyzed via scaling considerations: Depending on the value of the modulation, the surface magnetization either vanishes with an essential singularity or remains finite at the bulk critical point, i.e., the surface phase transition is of first order.

cond-mat.stat-mech↗

arXiv1997

Marginal Anisotropy in Layered Aperiodic Ising Systems

Two-dimensional layered aperiodic Ising systems are studied in the extreme anisotropic limit where they correspond to quantum Ising chains in a transverse field. The modulation of the couplings follows an aperiodic sequence generated through substitution. According to Luck's criterion, such a perturbation becomes marginal when the wandering exponent of the sequence vanishes. Three marginal sequences are considered: the period-doubling, paper-folding and three-folding sequences. They correspond to bulk perturbations for which the critical temperature is shifted. The surface magnetization is obtained exactly for the three sequences. The scaling dimensions of the local magnetization on both surfaces vary continuously with the modulation factor. The low-energy excitations of the quantum chains are found to scale as L^z with the size L of the system. This is the behaviour expected for a strongly anisotropic system, where z is the ratio of the exponents of the correlation lengths in the two directions. The anisotropy exponent z is equal to the sum of the scaling dimensions of the local magnetization on the two surfaces. The anisotropic scaling behaviour is verified numerically for other surface and bulk critical properties as well.

cond-mat.stat-mech↗

arXiv1997

Anisotropic Scaling in Layered Aperiodic Ising Systems

The influence of a layered aperiodic modulation of the couplings on the critical behaviour of the two-dimensional Ising model is studied in the case of marginal perturbations. The aperiodicity is found to induce anisotropic scaling. The anisotropy exponent z, given by the sum of the surface magnetization scaling dimensions, depends continuously on the modulation amplitude. Thus these systems are scale invariant but not conformally invariant at the critical point.

cond-mat.stat-mech↗

arXiv1997

Exact Renormalization-Group Study of Aperiodic Ising Quantum Chains and Directed Walks

We consider the Ising model and the directed walk on two-dimensional layered lattices and show that the two problems are inherently related: The zero-field thermodynamical properties of the Ising model are contained in the spectrum of the transfer matrix of the directed walk. The critical properties of the two models are connected to the scaling behavior of the eigenvalue spectrum of the transfer matrix which is studied exactly through renormalization for different self-similar distributions of the couplings. The models show very rich bulk and surface critical behaviors with nonuniversal critical exponents, coupling-dependent anisotropic scaling, first-order surface transition, and stretched exponential critical correlations. It is shown that all the nonuniversal critical exponents obtained for the aperiodic Ising models satisfy scaling relations and can be expressed as functions of varying surface magnetic exponents.

cond-mat.stat-mech↗

arXiv1997

Corner Exponents in the Two-Dimensional Potts Model

The critical behavior at a corner in two-dimensional Ising and three-state Potts models is studied numerically on the square lattice using transfer operator techniques. The local critical exponents for the magnetization and the energy density for various opening angles are deduced from finite-size scaling results at the critical point for isotropic or anisotropic couplings. The scaling dimensions compare quite well with the values expected from conformal invariance, provided the opening angle is replaced by an effective one in anisotropic systems.

cond-mat.stat-mech↗

arXiv1996

Common trends in the critical behavior of the Ising and directed walk models

We consider layered two-dimensional Ising and directed walk models and show that the two problems are inherently related. The information about the zero-field thermodynamical properties of the Ising model is contained into the transfer matrix of the directed walk. For several hierarchical and aperiodic distributions of the couplings, critical exponents for the two problems are obtained exactly through renormalization.

cond-mat↗