The quantum Ising model in a transverse field with couplings following the Fibonacci sequence is considered. Exact analytical results for the critical line, energy gap, and ground-state singularity are presented. The main features of the multifractal structure of the Cantor-set energy spectrum are discussed.
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 1D quasiperiodic quantum Ising model in a transverse field, which exhibits long-range order above a critical coupling and describes the phase diagram of the 2D classical Ising-model quasiperiodic in one direction. We numerically determine the correlation length, magnetization, and energy spectrum of this system. The site dependency of the magnetization is described by a new order parameter, the width of the magnetization. The scalings \ensuremath{\delta} and f(\ensuremath{\alpha}) associated with the Cantor spectrum are found to exhibit special features at the onset of long-range order.
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 chain with both the exchange couplings and the transverse fields arranged in a hierarchical way is considered. Exact analytical results for the critical line and dispersion relation of the low-energy excitations are obtained. It is shown that only when R1=R2>Rc, where R1 and R2 are the hierarchical parameters for the exchange couplings and the transverse fields, respectively, does the system preserve a logarithmic singularity in the specific heat.