Theoretical Study Of Novel Quantum States And Fractionalized Excitations In Strongly Correlated Systems

The strongly correlated system is one of the central topics of recent condensed matter physics studies.Novel quantum states like Luttinger liquids,quantum spin liq-uids etc.emerge in strongly correlated systems due to the correlation effect between electrons,some of them have unique physical properties like fractionalized excitations and topological orders.In this paper,we study two instances of low dimensional d<3strongly correlated systems.One is a 1D fermionic system with high spin,the other is a novel quantum state in 2D Mott insulators——Kitaev quantum spin liquid.Now we introduce the respectively.1.Recently,the 1D fermionic systems with large spin and SU(N)(N from 1 to 10)symmetry have been realized in cold atom systems with alkaline earth atoms,however,the corresponding theoretical researches are far from enough.On the one hand,the existing theoretical studies mainly focus on N=2 systems,and there are few pieces of research on the systems with N>2.On the other hand,the existing methods have their own limitations.For this reason,a complete and consistent picture of the basic properties of the 1D fermionic SU(N)systems with arbitrary interaction and at differ-ent temperature regimes is still lacking.For this purpose,we studied the metallic phase of 1D fermionic SU(N)Hubbard model mainly in the framework of Luttinger liquid theory combined with Bethe Ansatz solution and discuss supplemental approaches in the regime when these two methods are not available.In the low-temperature regime,we gave a complete description of the system with arbitrary interactions for the first time and the physical properties like Green's functions(GF),momentum distributions and tunneling density of states(DOS)are also studied.In the strong interaction case,there exists a spin-incoherent regime E_spin<T?E_cdue to the intense spin-charge separation.The Luttinger liquid description of the spin degrees of freedom fails,we therefore calculated the GF and DOS by generalizing the path integral approach in the study of SU(2)case to SU(N)case here.We found them demonstrates significant dif-ferences between the spin-incoherent regime and Luttinger liquid regime.For example,the GF decays exponentially with distance in the former case while they decay as pow law in the latter case;the DOS in the former case diverges in zero energy limit while approaches zero in the latter case.We also compared our theoretical results with the re-sults of the cold atom experiments on alkaline earth atoms and found they qualitatively consistent with each other.2.The quantum spin liquids attract great attention because of their fascinating properties like fractionalized excitations,topological orders,etc.,and their close re-lation to quantum computations in recent years.The Kitaev model is the minimal toy model that holds spin liquid ground states.From the theoretical point of view,as a spin model,its properties in magnetic fields is of fundamental interest.Moreover,the recent experimental studies of Kitaev materials in which the dominant interaction is Kitaev interaction like?-Ru Cl₃suggest that they have magnetically ordered ground states and an external magnetic field could drive the system to a possible spin liquid state.Even though there are non-Kitaev interactions in real materials,the study of the pure Kitaev model in a magnetic field could enlighten the experiments.We mainly use the mean field theory(MFT)to study the responses and topologi-cal phase transitions of Kitaev model in different directions of the magnetic field.We found that a local(001)-direction magnetic field can generate a pair of dynamical flux excitations on the ground state of Kitaev model.Ans we discussed the correspondence between KSL with a pair of dynamical flux excitations and a Kondo problem in detail.In addition,we pointed out the magnetic fields can break the equivalents of ferromag-netic(FM)KSL and anti-ferromagnetic(AFM)KSL in the ground state,they behave differently in the magnetic fields.There is no phase transition before FM KSL is po-larized by high magnetic fields while there exists a topological nontrivial intermediate phase of AFM KSL before it is polarized.In the uniform(001)-direction magnetic field case,we discussed the phase transitions of AFM KSL with different coupling constants with the increase of the magnetic field.We found that there exists a robust gapless regime in the intermediate magnetic field and two gapless phases are connected by a topological phase transition.In the uniform(111)-direction magnetic field case,we studied isotropic AFM KSL and found there is a novel topological state with Chern number C=4 and Abelian anyon excitations in the field regime h_c1<h<h_c2before the system is polarized by a higher magnetic field.This indicates the magnetic field can be used to tune the topological order and the anyon excitations in this case.In par-ticular,we pointed out the SU(2)MFT that was widely used before was confined to local moment solution for any finite magnetic field,it can not be used to describe a spin liquid state.Therefore,we proposed a Z₂MFT that can be used in the finite magnetic field case and discussed its validity in detail.We think the Z₂MFT is a better way to construct an unconfined Z₂spin liquid.