Research On The Topological Properties Of Unconventional Lattice Models

In this dissertation I studied the topological property in two kinds of un-conventional lattice models.One is the transition from a nodal-loop phase to a nodal-chain phase in a periodically modulated optical lattice.The other is the geometrical meaning of winding number and its characterization of topological phases in one-dimensional chiral non-Hermitian systems.We propose to study the transition from a nodal loop to nodal chain phase in a tunable two-dimensional ?-flux lattice with periodical modulation potential.The Hamiltonian describes a periodically modulated optical lattice system under artificial magnetic fluxes and the tunable modulation phase factor provides ad-ditionally an artificial dimension of external parameter space.We demonstrate that this lattice system is able to describe a semimetal with either nodal loop or nodal chain Fermi surface in the extended three-dimensional Brillouin zone.By changing the strength of modulation potential V,we realize the transformation between the nodal-loop and nodal-chain semimetal.We unveil the geometrical meaning of winding number and utilize it to char-acterize the topological phases in one-dimensional chiral non-Hermitian systems.While chiral symmetry ensures the winding number of Hermitian systems being in-tegers,it can take half integers for non-Hermitian systems.We give a geometrical interpretation of the half integers by demonstrating that the winding number v of a non-Hermitian system is equal to half of the summation of two winding numbers v1 and v2 associated with two exceptional points respectively.The winding num?bers v1 and v2 represent the times of real part of the Hamiltonian in momentum space encircling the exceptional points and can only take integers.We further find that the difference of v1 and v2 is related to the second winding number or energy vorticity.By applying our scheme to a non-Hermitian Su-Schrieffer-Heeger model and an extended version of it,we show that the topologically different phases can be well characterized by winding numbers.Furthermore,we demonstrate that the existence of left and right zero-mode edge states is closely related to the winding number v1 and v2.