| Landau’s symmetry breaking theory was once believed to describe all the possiblephases of matter, until the discovery of low dimensional topologically ordered phases,where different phases can have the same symmetry. Topologically ordered phases opena whole new world to be explored and provide many of the most exciting issues in con-densed matter physics. Correlated many body systems play an important role in low di-mensionaltopologicallyorderedphases,ononehand,insystemsrepresentedbyfractionalquantum hall effect, strongly correlated interactions are necessary to form topological or-ders; on the other hand, in systems represented by2d topological insulator, topologicalorders can be defined in free systems, but there is a constraint that it should be stableagainst weak correlated interactions, and the maximal strength of interactions allowed bystability constraint often lies in the strong interaction regions. There are no general meth-ods to deal with these systems, both mean filed theory and perturbation theory are inap-propriate, so numerical methods are very important. In this thesis, we will use numericalmethods to simulate two typical types of topologically ordered states in low dimensionalstrongly correlated systems.In the first part, we first give a brief review of the most typical1d topological states,the haldane gapped states. Then we construct a spin model in S=2spin chain whichcan describe two topologically distinct valence bond solid (VBS) states in a unified form.When tuning parameters from one VBS state to the other, there will undergo a quantumphase transition. However, generally the ground state can no longer be solved exactly.So we introduce an imaginary time evolution algorithm based on matrix product states,which is equivalent to density matrix renormalization method. By using this numericalalgorithm, we calculate the ground state energy density and its second derivatives, andfind the phase transitions between two VBS states are second order type. Furthermore,entanglement spectrum are employed to study how topological order changes in the phasetransition and confirm that topological order remains unchanged in each VBS phase. Wealso calculate the central charges of underlying conformal field theory on critical line andfind it equals to2. Then we conjecture the possible effective field theory descriptionis SU(2)4Wess-Zumino-Witten model. In this work, we find a continuous topologicalquantum phase transition (TQPT) between two states attached with same spin value for the first time, and confirm some basic concepts in Landau’s second order phase transitiontheory can be used in this TQPT. Moreover, this work may provide some help for thefurther investigation of topological orders around TQPT critical point.Inthesecondpart,duringananalysisofthepresentresearchstatementontopologicalinsulator in2d, we point the main drawback or limitations about existing few researchworks focusing on interaction effects on topological insulators. Consequently, in order togiveareliableandcompleteresults,weintroduceafermionicdeterminantquantummontecarlo (DQMC) algorithm, which needs no prior assumptions and has no bias. At halffilling,we demonstrate there is no fermion minus sign problem in Kane-Mele-Hubbard(KMH) model and DQMC are applicable to this model. By using DQMC algorithm, weinvestigatethezerotemperaturephasediagramofKMHmodelandfixthephaseboundarybetween topological band insulator phase and antiferromagnetic Mott insulator phase.Also, we discuss the instability of helical edges which can be induced by infinitesimaltwo-particle backscattering term on edges, and give the instability parameter region. Atlast, we check a proposal about the nature of spin liquid phase, which is neighboring thetopological insulator phase. In this work, for the first time, we give a reliable result abouthow strongly correlated interactions will affect the stability of2d topological insulators,as well as the safe upper limit of interaction strength when topological order remainsunbroken by using DQMC numerical simulation... |
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