Topological Invariants And Entanglement In Two-dimensional Topological Systems

In the past thirty years, the condensed matter physics community has been fas-cinated by topological phases of matter, for instance, the integer quantum Hall effect, the fractional quantum Hall effect,the quantum anomalous Hall effect, the quantum spin Hall effect, and the three-dimensional topological insulators. These topological states of quantum matter are usually distinguished by use of some global topological quantum numbers rather than certain local order parameters. The topological aspect of the integer quantum Hall effect with periodic potentials was first discussed by Thou-less, Kohmoto, Nightingale, and Nijs (TKNN). In their famous work, a topological expression for the Hall conductivity was given by the Chern number over the magnetic Brillouin zone. Their result was then generalized to the fractional quantum Hall effect. For the quantum spin Hall systems which are characterized by the Z2invariant, with the extension of the idea, the well-defined spin Chern number can be used to char-acterize trivial and non-trivial bulk band topology. However,calculation of the Chern number in the presence of disorder is usually based upon the integral of partial deriva-tives of electron wave functions over the boundary phases. Numerical implementation involves hundreds of times of exact diagonalization for a given disorder configuration, which is very time-consuming even for noninteracting electron systems. Therefore, de-velopment of efficient numerical approaches to direct calculation of the Chern number is highly desirable. On the other hand, in recent years, quantum entanglement, which reveals the phase information of the quantum-mechanical ground-state wavefunction, has been used as a tool to characterize the topological phases. It has been pointed out that, the existence of topological entanglement entropy in a fully gapped system, such as fractional quantum Hall and the gapped Z2spin liquid, indicates existence of long-range quantum entanglement (topological order). Another important progress is the demonstration that the entanglement spectrum reveals the gapless edge spectrum for fractional quantum Hall systems, Chern insulators, topological insulators and even to spin systems. Although such surprising connections have been established between the quantum entanglement and topological quantum phases in condensed matters recently, there exists a deficiency in the description of the Z2topological insulators by using this approach, and the entanglement entropy is still difficult to measure experimentally because it is not a observable. Moreover, application of Z2index to disordered sys-tems is impossible so far, and we need to define new topological index to demonstrate topological phase transitions in disordered systems.In this dissertation, we study the topological invariants and entanglement proper-ties in two-dimensional non-interacting topological systems. The dissertation consists of four chapters:In chapter one, we give an introduction for the related theoretical and experimental background, theoretical methods and a brief outline of some fundamental conceptions.In chapter two, A fast and efficient coupling-matrix method is designed to cal-culate the Chern number in two-dimensional finite crystalline and disordered systems. only one time exact diagonalization for the actual system is needed without loss of ac-curacy. A transparent coupling-matrix formulation will be given, from which the Chern number can be very efficiently computed, compared with the existing approaches. To show its effectiveness, we apply the approach to the Haldane model and the lattice Hof-stadter model, the quantized Chern numbers being correctly obtained. The calculated Chern number is found well quantized provided the Fermi level lies within the energy gap. The calculated results, in particular the disorder-induced phase transition, are in good agreement with the known results. Especially, the disorder-induced topological phase transition is well reproduced, when the disorder strength is increased beyond the critical value. and the obtained critical disorder strength is in good agreement with the result previously obtained from the Hall conductivity calculation. We expect the method to be widely applicable to the study of topological quantum numbers.In chapter three,We study the relationship between bipartite entanglement, sub-system particle number and topology in a half-filled free fermion system. It is pro-posed that the spin-projected particle numbers can distinguish the quantum spin Hall state from other states even though sz is not conserved, and can be used to establish a new topological index for the system. Furthermore, we apply the new topological in-variant to disordered system and show that a topological phase transition occurs when the disorder strength is increased beyond a critical values. Moreover, we also reveal a relationship between the entanglement entropy and the subsystem particle number fluc-tuation. They share several common properties.They both satisfy the same area law, and are dominated by the boundary excitations with each zero mode having a maxi-mal contribution. The connection between the two quantities is universal, regardless of whether the system has a nontrivial band topology.As a result, the subsystem particle number fluctuation, as an observable quantity,provides a lower-bound estimation of the entanglement entropy, and has the advantage that it can be measured experimentally..The last chapter presents a summary of this dissertation, and then gives some outlook for the investigation.