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Journal of Physics A Mathematical and General

Aperiodic Ising quantum chain and its relation with a classical Ising chain in an inhomogeneous field

Dragi Karevski

Published Jul 7, 1997• 0 citations
Publisher

Abstract

A formal correspondence beetwen the surface magnetization of an Ising quantum chain, perturbed by the paper-folding aperiodic sequence, and the partition function of a classical Ising chain in an inhomogeneous external field is derived. The perturbation is marginal and the critical exponent associated with the surface magnetization is a continuous function of the perturbation amplitude. We obtain this exponent by analysing the classical chain.

DOI
10.1088/0305-4470/30/13/002

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A formal correspondence beetwen the surface magnetization of an Ising quantum chain, perturbed by the paper-folding aperiodic sequence, and the partition function of a classical Ising chain in an inhomogeneous external field is derived. The perturbation is marginal and the critical exponent associated with the surface magnetization is a continuous function of the perturbation amplitude. We obtain this exponent by analysing the classical chain.
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.
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.
Abstract. — 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, paperfolding 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, xms and xms, 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 here simply equal to xms+xms. The anisotropic scaling behaviour is verified numerically for other surface and bulk critical properties as well. Résumé. — On étudie des systèmes d’Ising apériodiques en couches à deux dimensions dans la limite anisotrope extrême où ils correspondent à des chaînes quantiques d’Ising en champ
Surface and bulk critical properties of an aperiodic spin chain are investigated in the framework of the phenomenological Ginzburg - Landau theory. According to Luck's criterion, the mean field correlation length exponent leads to a marginal behaviour when the wandering exponent of the sequence is . This is the case of the Fibonacci sequence that we consider here. We calculate the bulk and surface critical exponents for the magnetizations, critical isotherms, susceptibilities and specific heats. These exponents continuously vary with the amplitude of the perturbation. Hyperscaling relations are used in order to obtain an estimate of the upper critical dimension for this system.
We study the surface magnetization and critical behavior of a hierarchical quantum Ising chain. The surface magnetization exponent ${\mathrm{\ensuremath{\beta}}}_{\mathit{s}}$ is calculated exactly by using scaling arguments. It is found that, although the size of the fluctuations in the exchange couplings may be bounded or unbounded depending on the value of the hierarchical parameter r, the surface critical behavior stays nonuniversal with ${\mathrm{\ensuremath{\beta}}}_{\mathit{s}}$ varying continuously with the value of r. This result is rather different from the cases where the aperiodicity of the couplings is generated by substitution rules and different types of critical behavior are observed for bounded and unbounded fluctuations in the couplings.