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arXiv:cond-mat/9706106 · Journal of Physics A Mathematical and General

Aperiodic Ising quantum chains

Joachim Hermisson, Uwe Grimm, Michael Baake

Published Nov 7, 1997• 51 citations
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Abstract

Some years ago, Luck proposed a relevance criterion for the effect of aperiodic disorder on the critical behaviour of ferromagnetic Ising systems. In this article, we show how Luck's criterion can be derived within an exact renormalization scheme for Ising quantum chains with coupling constants modulated according to substitution rules. Luck's conjectures for this case are confirmed and refined. Among other outcomes, we give an exact formula for the correlation length critical exponent for arbitrary two-letter substitution sequences with marginal fluctuations of the coupling constants.

DOI
10.1088/0305-4470/30/21/009

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Some years ago, Luck proposed a relevance criterion for the effect of aperiodic disorder on the critical behaviour of ferromagnetic Ising systems. In this article, we show how Luck's criterion can be derived within an exact renormalization scheme for Ising quantum chains with coupling constants modulated according to substitution rules. Luck's conjectures for this case are confirmed and refined. Among other outcomes, we give an exact formula for the correlation length critical exponent for arbitrary two-letter substitution sequences with marginal fluctuations of the coupling constants.
We study thermodynamic properties, specific heat and susceptibility, of XY quantum chains with coupling constants following arbitrary substitution rules. Generalizing an exact renormalization group transformation, originally formulated for Ising quantum chains, we obtain exact relevance criteria of Harris-Luck type for this class of models. For two-letter substitution rules, a detailed classification is given of sequences leading to irrelevant, marginal or relevant aperiodic modulations. We find that the relevance of the same aperiodic sequence of couplings in general will be different for XY and Ising quantum chains. By our method, continuously varying critical exponents may be calculated exactly for arbitrary (two-letter) substitution rules with marginal aperiodicity. A number of examples are given, including the period-doubling, three-folding and precious mean chains. We also discuss extensions of the renormalization approach to a special class of long-range correlated random chains, generated by random substitutions.
We study thermodynamic properties, surface magnetization, specific heat and susceptibility of XY quantum chains with coupling constants following arbitrary substitution sequences. Generalizing an exact renormalization group (RG) transformation, originally formulated for Ising quantum chains, we obtain exact relevance criteria of Harris-Luck-type for this class of models. For two-letter substitution rules, a detailed classification is given of sequences leading to irrelevant, marginal or relevant aperiodic modulations. We find that the relevance of the same aperiodic sequence of couplings in general will be different for XY and Ising quantum chains. By our method, continuously varying critical exponents may be calculated exactly for arbitrary (two-letter) substitution rules with marginal aperiodicity. A number of examples are given, including the period-doubling, three-folding and precious mean chains. We also discuss extensions of the renormalization approach to a special class of long-range correlated random chains, generated by random substitutions.
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 employ an adaptation of a strong-disorder renormalization group technique in order to analyze the ferro–paramagnetic quantum phase transition of Ising chains with aperiodic but deterministic couplings under the action of a transverse field. In the presence of marginal or relevant geometric fluctuations induced by aperiodicity, for which the critical behavior is expected to depart from the Onsager universality class, we derive analytical and asymptotically exact expressions for various critical exponents (including the correlation length and the magnetization exponents, which are not easily obtainable by other methods), and shed light onto the nature of the ground-state structures in the neighborhood of the critical point. The main results obtained by this approach are confirmed by finite-size scaling analyses of numerical calculations based on the free-fermion method.
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.