Study On The Spin-wave Excitations And Topological Properties In Low-dimensional Flat-band Strongly Correlated Systems

Band structure with non-trivial topology has been one of the most active fields in condensed matter physics since the discovery of topological insulators.The topological band structure is not unique to the electronic fermi system.In recent years,the bosonic elementary excitations are also found to have the same topology band structure,such as topological photons,topological phonons and topological magnon.In this doctoral dissertation,after a systematic introduction to the basic theory of topological magnon and flat-band,we focus mainly on what I have finished,which include the following two topics on the ferromagnetic spin wave excitations of two strongly correlated models:(1)Topological magnons in a one-dimensional itinerant flat-band ferromagnetRecently,the search of topological magnons has attracted much attention,not only because of fundamental interest but also due to its possible applications in spintronics.The previous proposals share a common feature:they are based on local spin models.Then,a natural question follows:is there any other scenario that topological magnons can arise?Specifically,can itinerant spin systems host topological magnons?We consider a one-dimensional Tasaki model with a flat band,which can be viewed as a quarter filled periodic Anderson model with impurities locating in between and hybridizing with the nearest neighbor conducting electrons,together with a Hub-bard repulsion for these electrons.we determine the spin-wave excitations of the model by the projected exact diaganolization method.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 7r,leads to the occurrence of a topolog-ical transition.After this transition,there always exist in-gap edge magnonic modes which is consistent with the bulk-edge correspondence.This is the first proposal of itinerant topological magnons.The Hubbard interaction driven transition reveals a new mechanism to realize non-trivial acoustic magnon bands.(2)Ferromagnetism and spin wave excitations in nearly flat-band topological Hub-bard modelsEarly,Mielke and Tasaki proved the itinerant ferromagnetism of a class of flat-band Hubbard model.They prove that the ground states of the models exhibit ferro-magnetism when the low energy flat band is half-filling or nearly half-filling.The flat bands constructed by Mielke and Tasaki are topologically trivial.Since the discovery of topological insulators,one would expect to find the fractional quantum Hall effect(FQHE)in topological insulators.A topologically non-trivial flat band is considered to be the key to the realization of the FQHE effect.Therefore,the topological flat band has attracted extensive research interest.Previous work on topological flat band focus mainly on the fractionalized charge excitation,but rarely on the magnetic excitation.We consider a 1/4-filled topological flat-band Hubbard model whose hopping ter-m is a nearly flat-band 7r-flux Chern or Z2 topological insulator at half-filling.Then the sufficiently large repulsion Hubbard interactions make the ground state a ferromag-netic insulator.When the Hubbard interactions U is smaller than the band gap,we project the system onto a low-energy nearly flat band,where the Hilbert space dimen-sion increases linearly with the number of lattice points,and then determine the spin-1 excitation spectra of the models by the exact diaganolization method.Since spin-1 ex-citation energy must be larger than or equal to zero,we obtain a critical Uc.When U>Uc,the ground state is the completely spin-polarized ferromagnetic state.The low energy excitations consist of two types of collective modes identified as the spin-wave excitations while the higher energy excitations denote the Stoner continuum.We find that there is a minimum value in the spin-wave excitation spectrum,which is similar to the roton mode in antiferromagnetism.For the Chern insulator,the excitation spectrum is gapless due to spontaneous broken symmetry while for the Z2topological insulator,the excitation spectrum is gapped.When U is much smaller than the gap,our results can be regarded as a rigorous solution of the system.Our spin-wave excitation spectra are obviously different from the results of the bosonization method.This is because the bosonization method ignores the contributions of spin-wave interaction and the nearly flat-band to the spin-wave excitation spectrum,and our results precisely include their contributions.