General Bounded Corner States In The Two-dimensional Off-diagonal Aubry-andré-harper Model With Flat Bands

In recent years,the higher-order topological states,which go beyond the conventional bulk-boundary correspondence,have attracted significantly growing interest by the condensed matter physics community.As the simplest example of higher-order topological states,the corner states in two-dimensional systems have been experimentally observed in many experimental platforms and attracted more attention.In the process of investigating corner states in two-dimensional systems,a new class of corner states called general bounded corner states,which are not located at the corner points but around the corners,has been observed.On the other hand,the translationally invariant lattice systems with flat bands in the momentum-space energy spectrums have drawn a great deal of attention in recent decades because they can act as a good platform to study strong correlated phenomena.The compact localized state corresponding to the flat band has almost become an integral part of the investigation of flat bands for physically interpreting the appearance of the flat bands in a real-space point of view,but the real-space energy spectrum and field profiles of eigenstates for finite lattice system with flat bands have been rarely investigated.The nearest-neighbor hopping amplitudes of the two-dimensional(2D)off-diagonal Aubry-Andre-Harper(AAH)lattice system are modulated by the cosine function,thus the energy spectrum of the 2D off-diagonal AAH model exhibits more abundant phenomena.In this thesis,we investigate the real-space energy spectrum and the realspace eigenstates of a 2D off-diagonal AAH model with flat bands and the effect of the intracellular next-nearest-neighbor hopping on the system.The specific research contents are as follows:Based on a 2D off-diagonal AAH model with only nearest-neighbor hoppings,we investigated the effect of phase factor of cosine function on the energy spectrum of the system.For certain values of phase factor,triply degenerate zero-energy flat bands emerge in the momentum-space energy spectrum of this lattice system.By studying the energy spectrum of finite 2D off-diagonal AAH model with flat bands,it is found that the system has some general bound corner states.This is because,for these values of the phase factor,some next-nearest-neighbor hopping amplitudes become zero,the system splits into isolated fragments.However,the absence of the energy gap results in that these general bounded corner states are not resistant to disorder.By partially adding intracellular next-nearest-neighbor hoppings,two flat bands with opposite energies split off from the original triply zero-energy flat bands.The Schrodinger equation is used to evaluate the the compact localized states corresponding to the flat bands,and the analytical solutions of two nonzero-energy flat bands are obtained by solving the momentum-space Hamiltonian.The results show that the energies of these two flat bands can be controlled by changing the value of intracellular next-nearest-neighbor hoppings.In addition,some robust general bounded corner states appear in real-space energy spectrum.By increasing the disorder strength,we numerically investigate the boundaries of the robustness for these general bounded corner states.Finally,based on superconducting circuit lattice,we propose a feasible scheme to realize the 2D off-diagonal AAH model.