Theoretical Studies On Quantum Breathers In Low-dimensional Nonlinear Lattices

Intrinsic localized modes or discrete breathers are spatially-localized and time-periodic nonlinear excitations which occur in Hamiltonian nonlinear lattice systems with translational symmetry. In principle, both the discreteness and the nonlinearity play a key role in the existence of discrete breathers. In condensed matter physics, there are many systems that contain the discreteness of crystal structure and the nonlinear interaction,such as nonlinear oscillator lattices and spin lattices. So far, a huge number of works on discrete breathers in these two types of lattices have been reported. However, these works have been focused on studying discrete breathers in the framework of classical mechanics. As is well known, quantum effect or quantum dynamics is very important in many cases. Thus, two urgent problems arise, i.e., how to study discrete breathers in the framework of quantum mechanics and how to use quantum picture to describe them. All works in the present thesis focus on these two issues. We shall investigate quantum counterpart of discrete breathers(i.e., quantum breathers) by making use of semiclassical and full quantum approaches, respectively.For the semiclassical case, we adopt the coherent state method. First, the existence and property of discrete breathers in a one-dimensional quantum Ising–Heisenberg ferromagnetic lattice with single-ion uniaxial anisotropy are studied. By means of the semidiscrete multiple-scale method, the equation of motion is reduced to the standard one-dimensional nonlinear Schr?dinger equation. It is shown that, the Brillouin zone center, a bright intrinsic localized spin-wave mode can occur below the bottom of the linear spin-wave spectrum. And, we find that, at the Brillouin zone boundary, a dark intrinsic localized spin-wave mode exists above the top of the linear spin-wave spectrum,which is different from the dark intrinsic localized spin-wave resonant mode. Second, we present a study on discrete breathers in two-dimensional ferromagnet with easy-plane anisotropy. With the help of the semidiscrete multiple-scale method that we developed, the equation of motion can be reduced to the two-dimensional nonlinear Schr?dinger equation. Our results show that the system can support the two-dimensional spin discrete breathers with the rotation symmetry whose eigenfrequency is above the top of the linear spin-wave spectrum. In fact, we give a semiclassical scheme to seek discrete breather solutions in two-dimensional quantum nonlinear lattices by this work.For the full quantum case, we develop a full quantum approach based on the time-dependent Hartree approximation and the semidiscrete multiple-scale method, and present a new picture to depict quantum breathers. Initially, we try to study analytically quantum breathers in the β-Fermi-Pasta-Ulam model by adopting this approach. In this work, we have successfully constructed quantum breather states, and can obtain the energy level formula of quantum breathers, which clearly means the energy of such quantum breathers is quantized. Afterward, we also study quantum breathers in one-dimensional ferromagnets via the full quantum approach, which consists of the two works. In one work, we study quantum breathers in the ferromagnetic chain with Dzyaloshinsky-Moriya interaction. It is shown that the introduction of the Dzyaloshinsky-Moriya interaction changes extreme points of the linear spin-wave frequency and causes the shifting of the wave number corresponding to the appearance of the quantum breathers. Actually, this result reveals a new physical mechanism in ferromagnetic systems. Besides, we find that the magnitude of the Dzyaloshinsky-Moriya interaction also influences the degree of the localization of quantum breathers. In another work, we cosider a ferromagnetic XXZ spin chain in an oblique magnetic field. It is found that, for quantum breathers in this system, both energy and magnetic moment of the system are quantized. This is a new finding. In addition, our results display that the appearance and degree of the localization of quantum breathers can be controlled via varying the value of the oblique angle, which have important potential applications in quantum information storage.