Theoretical Investigations On The Topological Phases Of Matter In Strongly Correlated Electron Systems

Topological phases of matter exhibit exotic physical properties.Such phases have become one of the main streams in condensed matter physics since the observation of integer quantum Hall states and fractional quantum Hall states,and eventually have turned out to be one of the central fields in condensed matter physics after the discovery of topological insulators.Strong correlations have nontrivial effects on topological phases.The studies on correlated topological phases deepen our understandings of matter,and at the same time,face challenging problems remaining unanswered.Recently,motivated by the progress of ultracold atom experiments on the simulation of Chern insulators,much attention has been paid to correlated Chern insulators.Meanwhile,the concept of band topology in electron systems has been generalized to quantum magnets.Topological magnons have attracted enormous researches both theoretically and experimentally not only because of their fundamental interest but also due to their possible applications in spintronics.However,all previous researches focused on local spin magnetism and it has remained blank on the field of itinerant topological magnons up to now.Itinerant magnetism usually arises from strong interactions between electrons.Therefore,the search of itinerant topological magnons constitute a subfield of strong correlated topological phases of matter and provides a newborn supplement to it.In this dissertation,we study correlated Chern insulators and itinerant topological magnons,respectively:1.Correlated Chern insulators.We detect the topological properties of Chern insulators with strong Coulomb interactions by use of cluster perturbation theory and variational cluster approach.The common scheme in previous studies only involves the calculation of the interacting bulk Chern number within the natural unit cell by means of the so-called topological Hamiltonian.With close investigations on a prototype model,the half-filled Haldane Hubbard model,which is subject to both periodic and open boundary conditions,we uncover the unexpected failure of this scheme due to the explicit breaking of the translation symmetry.Instead,we assert that the faithful interacting bulk Chern number in the framework of quantum cluster approaches can be computed in the enlarged unit cell,which is free of the fault caused by the explicit translation symmetry breaking and consistent with the interacting bulk-edge correspondence.2.Itinerant topological magnons.This part contains two halves:In the first half,different from previous scenarios that topological magnons emerge in local spin models,we propose an alternative that itinerant electron magnets can host topological magnons.A one-dimensional Tasaki model with a flatband is considered as the prototype.This model can be viewed as a quarter-filled periodic Anderson model with impurities located in between and hybridizing with the nearest-neighbor conducting electrons,together with a Hubbard repulsion for these electrons.Focusing on the physics related to the flat band,the Hamiltonian of the system is projected onto this subspace.Based on this method,we perform large-scale numerical simulations on the spin-1 spectra of the system.By increasing the Hubbard interaction,the gap between the acoustic and optical magnons closes and reopens while the Berry phase of the acoustic band changes from 0 to ?,leading to the occurrence of a topological transition.After this transition,there always exist in-gap edge magnonic modes,which is consistent with the bulk-edge correspondence.The Hubbard interaction-driven transition reveals a new mechanism to realize nontrivial magnon bands.In the second half,we report the first theoretical proposal of two dimensional itinerant topological magnons by numerical and analytical investigations on the quarterfilled Haldane Hubbard model with a nearly flat electron band.We find Dirac magnons in the flatband limit with symmetric sublattice Hubbard interactions.Although the Hubbard imbalance opens trivial gaps for the Dirac magnons,the magnon gap induced by the nonflatness of the lower electron band is topological.Consistent with the bulk-edge correspondence,there always exist in-gap magnon states on magnetic domain walls.We propose the “flatband sublattice particle-hole bases”,based on which a general framework to effectively describe the itinerant spin wave excitations can be derived.We thus attribute the nontrivial magnon band topology to the “mass inversion mechanism”.Our work provides a powerful tool to understand the itinerant spin wave excitations and their topological properties.