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arXiv:cond-mat/9411030 · Journal of Physics A Mathematical and General

Local critical behaviour at aperiodic surface extended perturbation in the Ising quantum chain

Dragi Karevski, Gábor Palágyi, L. Urban

Published Jan 7, 1995• 8 citations
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Abstract

The surface critical behaviour of the semi-infinite one-dimensional quantum Ising model in a transverse field is studied in the presence of an aperiodic surface-extended modulation. The perturbed couplings are distributed according to a generalized Fredholm sequence, leading to a marginal perturbation and varying surface exponents. The surface magnetic exponents are calculated exactly, whereas the expression of the surface energy density exponent is conjectured from a finite-size scaling study. The system displays surface order at the bulk critical point, above a critical value of the modulation amplitude. It may be considered as a discrete realization of the Hilhorst-van Leeuwen model.

DOI
10.1088/0305-4470/28/1/011

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The surface critical behaviour of the semi-infinite one-dimensional quantum Ising model in a transverse field is studied in the presence of an aperiodic surface-extended modulation. The perturbed couplings are distributed according to a generalized Fredholm sequence, leading to a marginal perturbation and varying surface exponents. The surface magnetic exponents are calculated exactly, whereas the expression of the surface energy density exponent is conjectured from a finite-size scaling study. The system displays surface order at the bulk critical point, above a critical value of the modulation amplitude. It may be considered as a discrete realization of the Hilhorst-van Leeuwen model.
We study the influence of an aperiodic extended surface perturbation on the surface critical behaviour of the two-dimensional Ising model in the extreme anisotropic limit. The perturbation decays as a power κ of the distance l from the free surface with an oscillating amplitude Al=(−1)flAA_l = ( - 1)^{f_l } AAl​=(−1)fl​A where f l = 0, 1 follows some aperiodic sequence with an asymptotic density equal to 1/2 so that the mean amplitude vanishes. The relevance of the perturbation is discussed by combining scaling arguments of Cordery and Burkhardt for the Hilhorst-van Leeuwen model and Luck for aperiodic perturbations. The relevance-irrelevance criterion involves the decay exponent κ, the wandering exponent ω which governs the fluctuation of the sequence and the bulk correlation length exponent ν. Analytical results are obtained for the surface magnetization which displays a rich variety of critical behaviours in the (κ, ω)-plane. The results are checked through a numerical finite-size-scaling study. They show that second-order effects must be taken into account in the discussion of the relevance-irrelevance criterion. The scaling behaviours of the first gap and the surface energy are also discussed.
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 < A c , the model displays a continuous surface transition with A -dependent exponents, whereas, for A > A c , 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.
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.
We consider random extended surface perturbations in the transverse field Ising model decaying as a power of the distance from the surface towards a pure bulk system. The decay may be linked either to the evolution of the couplings or to their probabilities. Using scaling arguments, we develop a relevance-irrelevance criterion for such perturbations. We study the probability distribution of the surface magnetization, its average and typical critical behaviour for marginal and relevant perturbations. According to analytical results, the surface magnetization follows a log-normal distribution and both the average and typical critical behaviours are characterized by power-law singularities with continuously varying exponents in the marginal case and essential singularities in the relevant case. For enhanced average local couplings, the transition becomes first order with a nonvanishing critical surface magnetization. This occurs above a positive threshold value of the perturbation amplitude in the marginal case.
The quantum version of the Hilhorst-van Leeuwen model i.e. an inhomogeneous Ising model in a transverse field with a marginal perturbation of the bulk interaction varying like l -1 with the distance l from the surface is studied numerically on a finite chain with symmetric interactions using fermion techniques and finite-size scaling at the bulk critical point. The surface exponents for the energy and the magnetisation are respectively x e =2 alpha +2, x m = alpha +1/2 when alpha >or= alpha c and x e =- alpha +1/2, x m =0 and x' m =- alpha -1/2 when alpha <or= alpha c where alpha c =-1/2 is the critical value of the perturbation amplitude below which the surface becomes ordered at the bulk critical point.