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Physical review. B, Condensed matter

Attractors in quantum Ising models

Miroslav Kolář, Mazhar Ali, Indubala I. Satija

Published Dec 1, 1989• 10 citations
Publisher

Abstract

We study the quantum Ising model in a transverse field where the nearest-neighbor exchange interaction can assume two possible values which follow generalized Fibonacci sequences. While the generalized Fibonacci sequences can be quasiperiodic or aperiodic, in this work we consider a class of aperiodic sequences whose trace maps are dissipative. The spectral properties of such systems can be studied by studying the attractors and their fractal basins underlying the dissipative map. We study the thermodynamical as well as spectral properties of the model both at and away from the critical point. The detailed analysis of the energy spectra provide a way to classify these sequences in terms of their degree of randomness. The results are compared with previously studied quasiperiodic (nondissipative) and Thue-Morse models.

DOI
10.1103/physrevb.40.11083

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We study the quantum Ising model in a transverse field where the nearest-neighbor exchange interaction can assume two possible values which follow generalized Fibonacci sequences. While the generalized Fibonacci sequences can be quasiperiodic or aperiodic, in this work we consider a class of aperiodic sequences whose trace maps are dissipative. The spectral properties of such systems can be studied by studying the attractors and their fractal basins underlying the dissipative map. We study the thermodynamical as well as spectral properties of the model both at and away from the critical point. The detailed analysis of the energy spectra provide a way to classify these sequences in terms of their degree of randomness. The results are compared with previously studied quasiperiodic (nondissipative) and Thue-Morse models.
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 report a comprehensive investigation of the low-energy properties of antiferromagnetic quantum XXZ spin chains with aperiodic couplings. We use an adaptation of the Ma-Dasgupta-Hu renormalization-group method to obtain analytical and numerical results for the low-temperature thermodynamics and the ground-state correlations of chains with couplings following several two-letter aperiodic sequences, including the quasiperiodic Fibonacci and other precious-mean sequences, as well as sequences inducing strong geometrical fluctuations. For a given aperiodic sequence, we argue that in the easy-plane anisotropy regime, intermediate between the XX and Heisenberg limits, the general scaling form of the thermodynamic properties is essentially given by the exactly known XX behavior, providing a classification of the effects of aperiodicity on XXZ chains. We also discuss the nature of the ground-state structures and their comparison with the random-singlet phase characteristic of random-bond chains.
The heat conductivity of nonperiodic quantum Ising chains whose ends are connected with heat baths at different temperatures are studied numerically by solving the Lindblad master equation. The chains are subjected to a uniform transverse field h, while the exchange coupling J{m} between the nearest-neighbor spins takes the two values J{A} and J{B} arranged in Fibonacci, generalized Fibonacci, Thue-Morse, and period-doubling sequences. We calculate the energy-density profile and energy current of the resulting nonequilibrium steady states to study the heat-conducting behavior of finite but large systems. Although these nonperiodic quantum Ising chains are integrable, it is clearly found that energy gradients exist in all chains and the energy currents appear to scale as the system size ~N{α}. By increasing the ratio of couplings, the exponent α can be modulated from α > -1 to α < -1 corresponding to the nontrivial transition from the abnormal heat transport to the heat insulator. The influences of the temperature gradient and the magnetic field to heat conduction have also been discussed.