Wave Packet Diffusions And Quantum Phase Transitions In One-dimensional Complex Systems

In recent years, an impressive technological progress in experiments with cold atoms allows one to create optical lattices with controlled potential distribution. In particular, the experimental studies on Bose-Einstein condensates(BECs) expansions in the optical lattices have been one of the most active fields. The evolution of BECs can be described by the Gross-Pitaevskii(GP) equation, which includes a nonlinear term that represents the mean-field interaction. Therefore, the theoretical work on the wave packet dynamics in one-dimensional complex systems have also attracted a lot of interest. On the other hand, how the quasiperiodic structure affects the quantum phase transition is also a very interesting subject. In this paper, we study some dynamical problems in one-dimensional complex systems. Besides, we also study the quantum phase transitions in the quasiperiodic transverse field Ising chain.Firstly, we study the hyperdiffusion phenomenon in one-dimensional tight-binding lat-tices. Transient quantum hyperdiffusion, namely, faster-than-ballistic wave packet spreading for a certain time scale, is found to be a typical feature in tight-binding lattices if a sublat-tice with on-site potential is embedded in a uniform lattice without on-site potential. The strength of the sublattice on-site potential, which can be periodic, disordered, or quasiperi-odic, must be below certain threshold values for quantum hyperdiffusion to occur. This is explained by an energy band mismatch between the sublattice and the rest uniform lattice and by the structure of the underlying eigenstates. Cases with a quasiperiodic sublattice can yield remarkable hyperdiffusion exponents that are beyond three. A phenomenological explanation of hyperdiffusion exponents is also discussed.Then, we study the spreading of an initially localized wave packet in one-dimensional linear and nonlinear generalized Fibonacci (GF) lattices. For linear GF lattices of the first class, both the second moment and the participation number grow with time. For linear GF lattices of the second class, in the regime of a weak on-site potential, wave packet spreading is close to ballistic diffusion, whereas in the regime of a strong on-site potential, it displays stairlike growth in both the second moment and the participation number. Nonlinear GF lattices are then investigated in parallel. For the first class of nonlinear GF lattices, the second moment of the wave packet still grows with time, but the corresponding participation number does not grow simultaneously. For the second class of nonlinear GF lattices, an analogous phenomenon is observed for the weak on-site potential only. For a strong on-site potential, neither the second moment nor the participation number grows with time.Thirdly, we study numerically the spreading of an initially localized wave packet in a one-dimensional uncorrelated disordered chain with a nonadiabatic electron-phonon interac-tion. The nonadiabatic electronphonon coupling is taken into account in the time-dependent Schr " odinger equation by a delayed cubic nonlinearity. In the adiabatic regime, Anderson localization is destroyed and a subdiffusive spreading of wave packet takes place by mod-erate nonlinearity. In the nonadiabatic regime, the dynamical behavior becomes obviously different. We find that short delay time suppresses delocalization strongly. However, large delay time gives a stronger exponent of spreading. An explanation of this delay induced effect is also discussed.Finally, we study the quantum phase transitions in the quasiperiodic transverse field Ising chain. By using the average magnetic moments Mz and concurrence C,it is found that there is only one phase transition point for bounded and critical quasiperiodic Ising chains. Both the differential coefficients of Mz and C exhibit logarithmic divergence with the system size at the phase transition point, which are similar to those of the uniform Ising chain. For the unbounded quasiperiodic Ising chains, it is found that phase transition points occur in two regions. Moreover, both the differential coefficients of Mz and C do not diverge logarithmically with system size at the phase transition point, which are similar to those of the disordered Ising chain. Further study shows that the two phase transition regions are produced due to the two groups exist in the unbounded quasiperiodic Ising chains.