Marginal Anisotropy in Layered Aperiodic Ising Systems

Abstract. We consider the behaviour of a critical system in the presence of a gradient perturbation of the couplings. In the direction of the gradient an interface region separates the ordered phase from the disordered one. We develop a scaling theory for the density profiles induced by the gradient perturbation which involves a characteristic length given by the width of the interface region. The scaling predictions are tested in the framework of the mean-field Ginzburg-Landau theory. Then we consider the Ising quantum chain in a linearly varying transverse field which corresponds to the extreme anisotropic limit of a classical two-dimensional Ising model. The quantum Hamiltonian can be diagonalized exactly in the scaling limit where the eigenvalue problem is the same as for the quantum harmonic oscillator. The energy density, the magnetization profile and the two-point correlation function are studied either analytically or by exact numerical calculations. Their scaling behaviour are in agreement with the predictions of the scaling theory.

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“…Rewriting σ x l in terms of diagonal fermions and using Wick's theorem, the local magnetization can be expressed as a determinant [55] …”

Section: −4mentioning

confidence: 99%

Gradient critical phenomena in the Ising quantum chain

Platini¹,

Karevski²,

Turban³

2007

J. Phys. A: Math. Theor.

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“…Details on their properties can be found in ref. [29,33]. We first performed preliminary runs (to be presented in the next section) for different values of the temperature, and then intensive Monte Carlo simulations at criticality on two-dimensional square lattices of sizes L × 4L (PF, TM, PS) or L × 3L (TF), using the Swendsen-Wang cluster algorithm [41] .…”

Section: Sequence Substitutions Wandering Exponentmentioning

confidence: 99%

“…The question one poses in this section is whether the small observed variations of the critical exponents in the secondorder phase transition regime results from a crossover effect or from a marginal behaviour. It is interesting here to [33,35], but the coupling distribution used here being different from the one used in previous works, we cannot directly compare the values of the exponents. We performed a few simulations for q = 2, and the results are given in Table 6.…”

Section: Crossover Effect or Marginal Variation Of The Exponents For mentioning

confidence: 99%

“…The critical behaviour is in agreement with Luck's criterion, leading to essential singularities or first-order surface transition when the perturbation is relevant and power laws with continuously varying exponents in the marginal situation with logarithmically diverging fluctuations. A strongly anisotropic behaviour has been recognized in this latter situation [33,34,35].…”

Section: Introductionmentioning

confidence: 99%

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Second-order phase transition induced by deterministic fluctuations in aperiodic eight-state Potts models

Chatelain¹,

Berche

Berche³

1999

Eur. Phys. J. B

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We investigate the influence of aperiodic modulations of the exchange interactions between nearest-neighbour rows on the phase transition of the two-dimensional eight-state Potts model. The systems are studied numerically through intensive Monte Carlo simulations using the Swendsen-Wang cluster algorithm for different aperiodic sequences. The transition point is located through duality relations, and the critical behaviour is investigated using FSS techniques at criticality. While the pure system exhibits a first-order transition, we show that the deterministic fluctuations resulting from the aperiodic coupling distribution are liable to modify drastically the physical properties in the neighbourhood of the transition point. For strong enough fluctuations of the sequence under consideration, a second-order phase transition is induced. The exponents β/ν, γ/ν and (1 − α)/ν are obtained at the new fixed point and crossover effects are discussed. Surface properties are also studied.

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“…Rewriting σ x l in terms of diagonal fermions and using Wick's theorem, the local magnetization can be expressed as a determinant [34] …”

Section: Magnetization Profilesmentioning

confidence: 99%

Conformal profiles in the Hilhorst-van Leeuwen model

Karevski¹,

Turban²,

Iglói³

2000

J. Phys. A: Math. Gen.

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Abstract. We study the critical energy and magnetization profiles for the Ising quantum chain with a marginal extended surface perturbation of the form A/y, y being the distance from the surface (Hilhorst-van Leeuwen model). For weak local couplings, A < Ac, the model displays a continuous surface transition with Adependent exponents, whereas, for A > Ac, there is surface order at the bulk critical point. If conformal invariance is assumed to hold with such marginal perturbations, it predicts conformal profiles with the same scaling form as for the unperturbed quantum chain, with marginal surface exponents replacing the unperturbed ones. The results of direct analytical and numerical calculations of the profiles confirm the validity of the conformal expressions in the regimes of second-and first-order surface transitions.

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Marginal Anisotropy in Layered Aperiodic Ising Systems

Cited by 20 publications

References 45 publications

Gradient critical phenomena in the Ising quantum chain

Gradient critical phenomena in the Ising quantum chain

Second-order phase transition induced by deterministic fluctuations in aperiodic eight-state Potts models

Conformal profiles in the Hilhorst-van Leeuwen model

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