One-dimensional Non-periodic System Of Quantum Phase Transitions

The half-filled Hubbard chains with Fibonacci. Harper and Frenkel-Kontorova modulating site potentials are studied by using two self-consistent mean-field approximations. A new order parameter is introduced to describe charge density order. We also calculated von Neumann entropy and fidelity of the ground states. We find that the behaviors of charge density wave parameter and spin density wave parameter of nonperiodic Hubbard model are similar with behaviors in periodic Hubard model, and the distribution of charge are nonperiodic. But nonperiodic Hubbard model are different from an uniform model, there is no charge density wave state in uniform model. And the results show that von Neumann entropy and fidelity can identify a charge density wave(CDW)/spin density wave(SDW) transition for nonperiodic models.Then by using the concept of concurrence, block entropy, Renyi entropy, nonadditive entropy and fidelity, nonuniform (Fibonacci, General Fibonacci and random) quantum Ising chains in a transverse field are studied. It is found that the behaviors of concurrence for different nonuniform chains at the critical point are different. As similar as uniform quantum Ising chain, concurrence of fibonacci quantum Ising chain exhibit a logarithmic divergence at critical point. But in random and some general fibonacci quantum Ising chain, concurrence does not diverge logarithmically with system size. Further more, we found that there are a large peak and a small bump on each side of the critical point in general fibonacci chain because of subsystem construction. And block entropy, Renyi entropy, nonadditive entropy also grow with system size, derivative of logarithm of fidelity diverge at critical point.Next, we use Berry phase to study quantum phase transition of a general one dimensional XY model. Berry phase in XY model is associated with the expected value of the corresponding general momentum. This idea is applied to aperiodic XY models and quantum phase transitions in the models can be retrieved from the studies of Berry phase and its derivatives. Different from a second order phase transition in uniformed XY chain, a KT like phase transition is obtained in random XY model and general fibonacci model. The employed Berry curvatures help us conclude that quantum phase transition in XY chain is ascribed to the appearance of 'magnetic monopole' in parameter space.Finally, we study the mutual entropy of Ising chains in a transverse field. The results in two periodic and random condition are given. We find that as same as in uniform quantum Ising chain, the mutual entropy is symmetrical with increasing number of spins of subchain. Mutual entropy decays exponential with increasing temperature. And it decays quiker at the vicinity of criticality.