* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download 1.1 Construction of two band models
Atomic theory wikipedia , lookup
Perturbation theory wikipedia , lookup
Particle in a box wikipedia , lookup
Coupled cluster wikipedia , lookup
Hartree–Fock method wikipedia , lookup
Renormalization group wikipedia , lookup
Topological quantum field theory wikipedia , lookup
Scalar field theory wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Hydrogen atom wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Ising model wikipedia , lookup
Perturbation theory (quantum mechanics) wikipedia , lookup
Electron configuration wikipedia , lookup
Canonical quantization wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Dirac bracket wikipedia , lookup
Atomic orbital wikipedia , lookup
Molecular orbital wikipedia , lookup
1.1 Construction of two band models Tuesday, July 28, 2015 9:42 AM We first illustrate the form of our two band models and how to construct it from a simple square lattice with , and orbitals. The two band model takes the following form where . is the 2 by 2 Pauli matrix and the basis for this matrix depends on the details of the problem and will be illustrated below for our simple case. For the continuous limit ( ), the above Hamiltonian can be simplified as This is nothing but Dirac Hamiltonian in the 2+1D, but with a momentum dependent mass Now let's consider a square lattice with , the figure below. and . orbitals. The lattice, as well as orbitals on each site, is shown in Figure 1.1.1: A tight-binding model for a square lattice with both s and p orbitals. The tight-binding Hamiltonian of this system can be written as where is for sites, parameters for is for the distance of hopping, : With the Fourier transform are for orbitals ( and where For our case, we have Or in a matrix form, the Hamiltonian is written as Quantum anomalous Hall effe Page 1 ). Here we consider the following , we have Or in a matrix form, the Hamiltonian is written as For this Hamiltonian, we can see that linear coupling appears naturally between and orbitals. However, and orbitals are degenerate with each other at and there are three bands appearing in low energy physics for this system. To remove this degeneracy, we consider a coupling to the angular momentum of orbitals. We may consider the superposition of these two orbitals, which possess the angular momentum along the z direction. Here we have set to be . Let's denote the angular momentum operator along the z direction to be and we introduce a Zeeman type of Hamiltonian for the coupling on the basis . We can also transform the Hamiltonian and the corresponding Hamiltonian is written as at The above Hamiltonian is degenerate when two orbitals. Let's consider the limit the low energy physics is dominated by Hamiltonian is written as from the basis for and orbitals. Thus, we can neglect to the basis orbitals, but will split these , in which orbital and the total This recover the Hamiltonian for the two band model except that we need to redefine the Pauli matrices and neglect the identity matrix term. One can further show that in the above limit, the contribution from the orbital can be included in the second order perturbation theory, which is not essential. The above model can be realized in the optical lattice of cold atom systems and the Hamiltonian can be generated by rotating each lattice site by a laser. See the following references for more details. Congjun Wu, Phys. Rev. Lett. 101, 186807 (2008). W Zheng, H. Zhai, Phys. Rev. A 89, 061603(R) (2015) . Before we go to the detailed discussion of chiral edge mode and bulk topology of the above model, we first take a look at the bulk energy dispersion. The two band model can be written as where energy is given by and , or explicitly, . The eigen . Again let's consider the limit , in which the energy dispersion is . For this energy dispersion, energy gap closes when , corresponding to a topological phase transition between a normal insulator with and a so-called quantum anomalous Hall insulator ( or Chern insulator) with . Here we always assume . Intuitively, this transition is also called "band inversion", as illustrated in the figure below. The concept of band inversion plays a central role in searching for topological materials. We will show explicitly how band inversion determines topological structures of electronic states later. Quantum anomalous Hall effe Page 2 Figure 1.1.2: Band inversion. Problems: 1. Complete the derivation of the tight-binding model of sp orbitals in the momentum space. 2. Complete the derivation of effective two band model from our three orbital model up to the second order terms with the perturbation theory. Quantum anomalous Hall effe Page 3