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Physical review. B, Condensed matter

Phase transition in aperiodic quantum Ising chains constructed with arbitrary substitution rules

Zhifang Lin, Ruibao Tao

Published Nov 1, 1992• 5 citations
Publisher

Abstract

We consider the aperiodic quantum Ising chains constructed with arbitrary substitution rules and initial or zero-order strings. It is shown that whether the spin systems can undergo a magnetic phase transition in the strict sense that the correlation length diverges at criticality depends not only on the substitution rules, but also on the initial strings. More precisely, when the characteristic polynomial associated with the substitution rule has a root of absolute value smaller than 1, the corresponding spin system can exhibit a phase transition analogous to the periodic case. On the other hand, when all roots of the above-mentioned characteristic polynomial are greater than or equal to 1 in absolute value, whether the spin system can exhibit a phase transition or behaves as a random Ising case may depend on the zero-order string.

DOI
10.1103/physrevb.46.10808

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We consider the aperiodic quantum Ising chains constructed with arbitrary substitution rules and initial or zero-order strings. It is shown that whether the spin systems can undergo a magnetic phase transition in the strict sense that the correlation length diverges at criticality depends not only on the substitution rules, but also on the initial strings. More precisely, when the characteristic polynomial associated with the substitution rule has a root of absolute value smaller than 1, the corresponding spin system can exhibit a phase transition analogous to the periodic case. On the other hand, when all roots of the above-mentioned characteristic polynomial are greater than or equal to 1 in absolute value, whether the spin system can exhibit a phase transition or behaves as a random Ising case may depend on the zero-order string.
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.
Phase transition in the aperiodic quantum Ising models were studied, in which the transverse magnetic fields are uniform and the couplings are arranged according to the aperiodic sequences that are generated by two types of two-tile inflation rules with initial condition {ApBq}: A → A′=Am11Bm12, B →B′=Am21Bm22, m12, m21≠0, and A → A′=Am11Bm12, B → B′=Bm21Am22, m12, m22≠0. When the inflation rules have the Pisot–Vijayaraghavan (PV) property in number theory, the generated aperiodic systems undergo phase transition at the critical points. When the inflation rules do not have the PV property, it is found that only by changing the initial conditions, the inflation rules can produce aperiodic systems exhibiting phase transition as well as aperiodic systems which do not exhibit any phase transition.
We study the one-dimensional quantum Ising model in a transverse magnetic field where the exchange couplings are ordered according to the Thue-Morse (TM) sequence. At zero temperature, this model is equivalent to a two-dimensional classical Ising model in a magnetic field with TM aperiodicity along one direction. We compute the order parameter (magnetization) of the chain and the scaling behavior of the energy spectrum when the system undergoes a phase transition. Analogous to the quasiperiodic (QP) quantum Ising chain, the onset of long-range order is signaled by a nonanaliticity in the exponent \ensuremath{\delta} which describes the scaling of the total bandwidth with the size of the chain. The critical spin-coupling can be computed analytically and it is found to be lower than the QP case. Furthermore, the energy bands are found to be narrower than the corresponding QP chain. The former and latter results are consistent with the fact that the present structure has a degree of ordering intermediate between QP and random.
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.