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Journal of Mathematical Physics

Phase transition in aperiodic quantum Ising models

J. Q. You, J. R. Yan, Jialin Zhong

Published Nov 1, 1992• 4 citations
Publisher

Abstract

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.

DOI
10.1063/1.529837

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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 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.
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 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.
A quantum Ising spin chain with two-valued nearest-neighbour couplings arranged in a quasi-periodic sequence is considered, in the presence of a uniform transverse field. The scaling properties of the energy spectrum of the model are analytically determined. The system can undergo a phase transition both when the two couplings have equal sign (ferro- or antiferromagnetic couplings) and when they have opposite sign (frustrated system). The critical point is explicitly obtained; the critical behaviour is shown to be the same as in the periodic case.
A quantum Ising model in a transverse field is of the simplest quantum many-body systems used for studying universal properties of quantum phase transitions. Interestingly, it is well known that such phase transitions can be mapped to topological phase transitions in toric code models. Therefore, one can expect that well-known properties of the transverse Ising model are used for characterizing topological phase transitions in toric code models. In this paper, we consider the magnetization of the Ising model and show, while it is a local order parameter, it is mapped to a nonlocal order parameter on the topological side. Consequently, we introduce a string order parameter for topological phase transitions in toric code models defined on different lattices with different dimensions in the presence of a uniform magnetic field. Since such a string order parameter is dual of the Ising order parameter with well-known properties, it leads to a topological (nonlocal) characterization of phase transition in the toric code model in uniform field. Our results show that although topological phase transitions do not follow a symmetry-breaking mechanism, it might still be possible to use concepts of symmetry-breaking phase transitions for topological ones by finding suitable mappings.