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Journal of Physics A Mathematical and General

Phase transition of quantum Ising spin models on g-letter generalized Thue-Morse aperiodic chains

Zhifang Lin, Ruibao Tao

Published May 7, 1992• 9 citations
Publisher

Abstract

The quantum Ising spin model in a transverse field with couplings ordered according to the g-letter generalized Thue-Morse sequence is considered. Exact analytical results for the critical line, energy gap and dispersion relation of the low-energy excitations are obtained. It is shown that the quantum Ising spin models can undergo a magnetic phase transition with the critical behaviour the same as in the periodic case.

DOI
10.1088/0305-4470/25/9/021

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The quantum Ising spin model in a transverse field with couplings ordered according to the g-letter generalized Thue-Morse sequence is considered. Exact analytical results for the critical line, energy gap and dispersion relation of the low-energy excitations are obtained. It is shown that the quantum Ising spin models can undergo a magnetic phase transition with the critical behaviour the same as in the periodic case.
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.
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.
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 present a detailed study of the finite one-dimensional quantum Ising chain in a transverse field in the presence of boundary magnetic fields coupled with the order-parameter spin operator. We consider two magnetic fields located at the boundaries of the chain that have the same strength and that are aligned in the same or in the opposite direction. We derive analytic expressions for the gap in all phases for large values of the chain length L , as a function of the boundary field strength. We also investigate the behaviour of the chain in the quantum ferromagnetic phase for oppositely aligned fields, focusing on the magnet-to-kink transition that occurs at a finite value of the magnetic field strength. At this transition we compute analytically the finite-size crossover functions for the gap, the magnetisation profile, the two-point correlation function, and the density of fermionic modes. As the magnet-to-kink transition is equivalent to the wetting transition in two-dimensional classical Ising models, our results provide new analytic predictions for the finite-size behaviour of Ising systems in a strip geometry at this transition.
This chapter is devoted to a discussion of quantum phase transitions in regularly alternating spin-1/2 Ising chain in a transverse field. After recalling some generally-known topics of the classical (temperature-driven) phase transition theory and some basic concepts of the quantum phase transition theory I pass to the statistical mechanics calculations for a one-dimensional spin-1/2 Ising model in a transverse field, which is the simplest possible system exhibiting the continuous quantum phase transition. The essential tool for these calculations is the Jordan-Wigner fermionization. The latter technique being completed by the continued fraction approach permits to obtain analytically the thermodynamic quantities for a `slightly complicated' model in which the intersite exchange interactions and on-site fields vary regularly along a chain. Rigorous analytical results for the ground-state and thermodynamic quantities, as well as exact numerical data for the spin correlations computed for long chains (up to a few thousand sites) demonstrate how the regularly alternating bonds/fields effect the quantum phase transition. I discuss in detail the case of period 2, swiftly sketch the case of period 3 and finally summarize emphasizing the effects of periodically modulated Hamiltonian parameters on quantum phase transitions in the transverse Ising chain and in some related models.