Fractional Chern Insulators In Singular Lattices

The discovery of quantum Hall states is a miracle in condensed matter physics which introduces the notion of "topology" into physics.The first quantum anomalous Hall or Chern insulator model has been proposed based on the honeycomb lattice by Haldane in 1988 which can realize the quantum Hall states without magnetic field.The Haldane model has two non-trivial topological bands labeled by Chern numbers.Chern insulator states have been proposed and investigated on two-dimensional flat surfaces or lattices with different geometries like the plane,cylinder,torus and so on.At the same time,we can study quantum anomalous Hall or Chern insulator states on two-dimensional singular surfaces(such as conical and helicoid-like surfaces).Such singular geometries can be constructed based on the disk geometry and a defined unit sector with n-fold rotational symmetry through through 'cutting and gluing".Here,we study the Chern insulator states on the Kagome lattice and its Hamiltonian has 6-fold rotational symmetry.Other singular geometries with arbitrary n-fold rotational symmetry can be constructed through "cutting and gluing" the unit sectors.There are some defects emerge near the center of the lattices because of the operator of "cutting and gluing".The singular geometry induces novel and intriguing features of Chern insulator states,such as in-gap and in-band core states,charge fractionalization,and multiple branches of edge excitations.There are some core states observed at the center of the singular lattices because of the defects.Considering many free fermions filling in the bands,there are multiple branches of edge excitations in these systems because of more than one occupation choices for fermions meeting core states.Each branch of edge excitations has the same quasi-degeneracy sequence.Charge fractionalization is one of the key features of the fractional quantum Hall states.However,we have observed the fractional charge in Chem insulator states in singular lattices.Because of the defects in singular lattices,there are some core states localized in the singular parts and these change the charge distribution.The fractional charge is related to the n-fold rotational symmetry.Similarly,we can propose the Chern insulator states in 2-D quasicrystals.We construct the 2-D quasicrystal lattice based on some pentagons and diamonds with 5-fold rotational symmetry in a disk geometry.Considering adding the staggered-flux phases,we obtain the Chem insulator in the quasicrystals.Topological properties of Chem insulator states in the quasicrystals can be described based on the real-space Chern number and the transport properties.The fractional platform of Hall Hall conductivity in the fractional quantum Hall states has been observed and it stems from the strong correlative interactions between electrons.Laughlin proposed a trial wave function to describe the fractional quantum Hall states which is called as Laughlin wave function.Laughlin states with a sim-ple math expression but profound implications can be used to describe the ground state(GS)of fractional quantum Hall effect.Recently,the the fractional quantum anomalous Hall or fractional Chern insulator states have been proposed in the toplogical flat bands with the interacting electrons.Meanwhile,fractional Chern insulator states studied with various boundary conditions,like the torus,cylinder and disk geometries.One of the most palpable features for the fractional Chern insulator states is the edge excitation-s which are characterized by the chiral Luttinger liquid theory.In fact,the fractional Chem insulator states obey the generalized Pauli principle.In terms of the general-ized Pauli principle and single particle states of Chern insulators,the optimal trial wave functions for the Fractional Chern states can be constructed.And These edge excitation spectra have been predicted and approximately obtained based on the generalized Pauli principle.The quantum Hall states can be studied in the curved surfaces,like the cone.The wave function of quantum Hall states on the cone is related to the geometry of the cone.We found that,the quantum Hall states on the cone also obey the generalized Pauli principle.Here,we propose the geometry-dependent fractional Chem insulator s-tates that interacting particles are bounded on 2D singular lattices with arbitrary n-fold rotational symmetry.Based on the generalized Pauli principle,we construct trial wave functions for the singular-lattice fractional Chern insulator states states with the aid of an effective projection approach,and compare them with the exact diagonalization results.High wave-function overlaps show that the singular-lattice fractional Chern in-sulator states states are certainly related to the geometric factor.More interestingly,we observe some exotic degeneracy sequences of edge excitations in these singular-lattice fractional Chern insulator states states,and provide an explanation that two branches of edge excitations mix together.In this thesis,we introduce our work with five parts.These five parts are organized as follows.The first part is introduction.In this part,we will review some concepts about the quantum Hall states and the Chern insulator states.We also will introduce abreast of advances,such as the fractional quantum Hall states,Chern insulators and fractional Chern insulator states.In the second part,we will recall how to construct the trial wave functions for the fractional Chern insulator states based on the generalized Pauli principle.In the third part,we first tell how to propose the Chern insulator states in the singular lattices.And we will obtain some intriguing features,such as in-gap and in-band core states,charge fractionalization,and multiple branches of edge excitations.Meanwhile,we construct the Chern insulator states in the quasicrystals.This topologi-cal character can be described with the real space Chern number and transport property.In the third part,based on the generalized Pauli principle,we construct the trial wave function for fractional Chern insulator states in the singular lattices filling with hard-core bosons.Comparing with the results through exact diagonalization technology,we found that these fractional Chern insulator states are related to the geometric factors.We have also observed many branches of edge excitations and some exotic fraction-al Chern insulator states.We summarize our results in the last part,and the research prospect is presented in the following days.