We consider a natural generalization of the lattice model for a periodic array of two layers, A and B, of spinless electrons proposed by Fu [Phys. Rev. Lett. 106, 106802 (2011)] as a prototype for a crystalline insulator. This model has time-reversal
symmetry and broken inversion symmetry. We show that when the intralayer next-nearest-neighbor hoppings ta2, a = A, B vanish, this model supports a Weyl semimetal phase for a wide range of the remaining model parameters. When the effect of ta2 is considered, topological crystalline insulating phases take place within the Weyl semimetal one. By mapping to an effective Weyl Hamiltonian we derive some analytical results for the phase diagram as well as for the structure of the nodes in the spectrum of the Weyl semimetal.
Following Diracs ideas concerning the quantization of constrained systems, we suggest to replace the free centre of mass Hamiltonian H_{CM} by another operator which commutes with all the elements of the algebra generated via the commutation relation
s by H_{CM} and the constraints which fix the centre of mass position. We show that the new Hamiltonian is a multiple of the identity operator and, as a result, its unique effect is to raise the internal energy levels by a constant amount.
We have reanalyzed the long-term optical light curve (LC) of the symbiotic star Z Andromedae, covering 112--yr of mostly visual observations. Two strictly periodic and one quasi-periodic cycles can be identified in this LC. A P1=7550 d quasi periodic
ity characterizes the repetition time of the outburst episodes of this symbiotic star. Six such events have been recorded so far. During quiescence states of the system, i.e. in time intervals between outbursts, the LC is clearly modulated by a stable coherent period of P2=759.1 d. This is the well known orbital period of the Z And binary system that have been measured also spectroscopically. A third coherent period of P3=658.4 d is modulating the intense fluctuations in the optical brightness of the system during outbursts. We attribute the trigger of the outbursts phenomenon and the clock that drives it, to a solar type magnetic dynamo cycle that operates in the convection and the outer layers of the giant star of the system. We suggest that the intense surface activity of the giant star during maximum phases of its magnetic cycle is especially enhanced in one or two antipode regions, fixed in the atmosphere of the star and rotating with it. Such spots could be active regions around the North and the South poles of a general magnetic dipole field of the star. The P3 periodicity is half the beat of the binary orbital period of the system and the spin period of the giant. The latter is then either 482 or 1790 d. If only one pole is active on the surface of the giant, P3 is the beat period itself, and the spin period is 352 d. It could also be 5000 d if the giant is rotating in retrograde direction. We briefly compare these findings in the LC of Z And to similar modulations that were identified in the LC of two other prototype symbiotics, BF Cyg and YY Her.
In a trapped Bose-Einstein condensate, subject to the action of an alternating external field, coherent topological modes can be resonantly excited. Depending on the amplitude of the external field and detuning parameter, there are two principally di
fferent regimes of motion, with mode locking and without it. The change of the dynamic regime corresponds to a dynamic phase transition. This transition can be characterized by an effective order parameter defined as the difference between fractional mode populations averaged over the temporal period of oscillations. The behavior of this order parameter, as a function of detuning, pumping amplitude, and atomic interactions is carefully analyzed. A special attention is payed to numerical calculations for the realistic case of a quadrupole exciting field and the system parameters accessible in current experiments.
