Wave Functions For Fractional Chern Insulators

Recently, fractional Chern insulators (FCIs), also called fractional quantum anomalous Hall (FQAH) states, have been theoretically established in lattice systems with topological flat bands. These systems exhibit similar fractional-ization phenomena(such as the fractional charge, fractional hall conductivity fractional statistics and so on) to the conventional fractional quantum Hall (FQH) systems. Using the mapping relationship between theFQHstates and the FCI/FQAHstates, we construct the many-body wave functions for the Feminonic and Bosonic FCI/FQAH states on disk geometry with the aid of the generalized Pauli principle (GPP) and Jack polynomials. Compared with the ground state by the exact diagonalization method, the wave-function overlap is higher than 0.97, even when the Hilbert space dimension is as large as 3 x 106. We also use the GPP and the Jack polynomials to construct edge excitations for the fermion-ic FCI/FQAHstates. The quasi-degeneracy sequences of Fermionic FCI/FQAH systems reproduce the prediction of the chiral Luttinger liquid theory, comple-menting the exact diagonalization results with larger lattice sizes and more par-ticles, For example, we can obtain the wave functions for the FCIs lattice with more than 1000 sites filling more than 15 particles that the exact diagonalization can not deal with.There are six parts of the paper are organized as follows. In Sec.1, we review the origin and evolution of Chern insulators and the wave functions for the FCIs. IN Sec.2, we will introduce the Laughlin wave function and then analyze the form of the Laughlin wave function with the aid of the math. From the Laughlin wave function, we can obtain the GPP and tell you the math expression of the GPP. In this part, we may explain some newfangled physics and these theories are very very important. In Sec.3, we obtain the single particle states and the energies of the CIs on the disk geometry numerically. We choose the topological flat bands(TFBs) parameters and diagonalize the Hamiltonian matrix. At the sane time, we introduce some concepts about TFBs. The Sec.4 is the core part in this article and in this part we share some significant results. We construct the wave function for Fermionic FCI on disk geometry with the aid of the GPP and the Jacks. And we can test the rationality the wave function with the ED result. After these, we can estimate the energies for the FCIs and obtain the order of edge excitation. We find the order of edge excitation for FCIs are the same as the Wen’s prediction. In Sec.5, we can also construct the exact wave functions for the FCIs with hard-core Bosons in terms of the GPP and the Jacks, but there are also some details different from the Fermionic situation because of the hard-core peculiarity. Here, we propose an effective projection approach to construct the FCI states filling with the hard-core bosons and we obtain the high wave function overlap value. In the last part, we summarize our thesis and discuss some new problems which can be solved in the future. Some results in this thesis can be find in the Journal of Physics named’ New Journal of Physics’ which is under the flag of Institute of Physics(IOP).