Loïc Turban

Anomalous diffusion in aperiodic environments

We study the Brownian motion of a classical particle in one-dimensional inhomogeneous environments where the transition probabilities follow quasiperiodic or aperiodic distributions. Exploiting an exact correspondence with the transverse-field Ising model with inhomogeneous couplings, we obtain many analytical results for the random walk problem. In the absence of global bias the qualitative behavior of the diffusive motion of the particle and the corresponding persistence probability strongly depend on the fluctuation properties of the environment. In environments with bounded fluctuations the particle shows normal diffusive motion and the diffusion constant is simply related to the persistence probability. On the other hand, in a medium with unbounded fluctuations the diffusion is ultraslow and the displacement of the particle grows on logarithmic time scales. For the borderline situation with marginal fluctuations both the diffusion exponent and the persistence exponent are continuously varying functions of the aperiodicity. Extensions of the results to disordered media and to higher dimensions are also discussed. @S1063-651X~99!04402-5#

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 critical exponent

Effective-medium approximation for quenched bond-disorder in the Ising model

{\mathrm{\ensuremath{\beta}}}_{\mathit{s}}

Log-periodic corrections to scaling: exact results for aperiodic Ising quantum chains

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

Exactly solvable spin-12 quantum chains with multispin interactions

{\mathrm{\ensuremath{\beta}}}_{\mathit{s}}

Evolution of the magnetization after a local quench in the critical transverse-field Ising chain

is a continuous function of 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.

Disordered Potts model on the diamond hierarchical lattice: Numerically exact treatment in the large- q limit

Abstract An effective-medium approach to quenched disorder in Ising systems is developed. In the percolation limit the method leads to accurate values for the critical concentration and gives a correct critical behaviour for T c ( p ). The results of Harris are recovered in the dilute limit. The method provides an accurate interpolation between both limits.

Surface-induced disorder and aperiodic perturbations at first-order transitions

Log-periodic amplitudes of the surface magnetization are calculated analytically for two Ising quantum chains with aperiodic modulations of the couplings. The oscillating behaviour is linked to the discrete scale invariance of the perturbations. For the Fredholm sequence, the aperiodic modulation is marginal and the amplitudes are obtained as functions of the deviation from the critical point. For the other sequence, the perturbation is relevant and the critical surface magnetization is studied.

Common Trends in the Critical Behavior of the Ising and Directed Walk Models

Abstract A generalized spin- 1 2 quantum chain with m (n) spin interactions for the z (x) component of the spins is shown to be self-dual. Through duality, the m arbitrary, n = 2 version of the model is transformed into m independent Ising chains in a transverse field, allowing us to get the exact values of the critical exponents for any m (v = s = z = 1; β = m/8; nx = m/4) and the spectrum. The anisotropic XY chain is a particular case with = 2.

Marginal extended perturbations in two-dimensions and gap exponent relations

We study the time evolution of the local magnetization in the critical Ising chain in a transverse field after a sudden change of the parameters at a defect. The relaxation of the defect magnetization is algebraic and the corresponding exponent, which is a continuous function of the defect parameters, is calculated exactly. In finite chains the relaxation is oscillating in time and its form is conjectured on the basis of precise numerical calculations.