Universal characterizing topological insulator and topological semi-metal with Wannier functions Ye Xiong
arXiv:1504.06007v1 [cond-mat.quant-gas] 22 Apr 2015
Department of Physics and Institute of Theoretical Physics , Nanjing Normal University, Nanjing 210023, P. R. China∗
Peiqing Tong Department of Physics and Institute of Theoretical Physics , Nanjing Normal University, Nanjing 210023, P. R. China∗ Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing Normal University, Nanjing 210023, P. R. China and Kavli Institute for Theoretical Physics China, CAS, Beijing 100190, China† The nontrivial evolution of Wannier functions (WF) for the occupied bands is a good starting point to understand topological insulator. By modifying the definition of WFs from the eigenstates of the projected position operator to those of the projected modular position operator, we are able to extend the usage of WFs to Weyl metal where the WFs in the old definition fails because of the lack of band gap at the Fermi energy. This extension helps us to universally understand topological insulator and topological semi-metal in a same framework. Another advantage of using the modular position operators in the definition is that the higher dimensional WFs for the occupied bands can be easily obtained. We show one of their applications by schematically explaining why the winding numbers ν3D = ν2D for the 3D topological insulators of DIII class presented in Phys. Rev. Lett. 114, 016801(2015). PACS numbers: 73.43.-f, 03.65.Vf, 71.20.Nr, 71.10.Ca
I.
INTRODUCTION
The discovery of quantum Hall effect [1], for the first time, sheds light on the topological properties of electronic bands hidden behind the 2-dimensional (2D) Fermi gas in magnetic field. Thouless et al. (referred as TKNN) calculated the Hall conductance by Kubo formula [2] and expressed it as an integral in the whole first Brillouin zone (FBZ). Then Simon et al. recognized the integral as the first Chern class of the U(1) principal fiber bundle on the torus of the FBZ [3–5]. Besides this route, Laughlin raised another way to understand quantum Hall effect by a gauge invariant argument (referred as the Laughlin argument on a cylinder) [6]. These two ways are equivalent but the latter one is more profound in explaining the robustness of the Hall conductance against local disorders because it is based on the global gauge arguments of the 2D electron system. For topological insulators (TI), e. g., 2D spin Hall system [7] and 3D strong or weak TI [8–10], there are also two routes, parallel to those for quantum Hall effect, to understand their topological properties. One is the topological invariant, likes the Hall conductance in the TKNN theorem, that can be expressed as an integral in a portion of the FBZ [9–14]. The other route, similar to the Laughlin argument, is based on the evolution of the Wannier centers (WC) of the localized Wannier
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functions (WF) (along one direction) as a function of the momentum (in the perpendicular directions). These two routes are also equivalent. For instance, in Ref. [7], the Z2 topological invariant in the 2D spin Hall system is defined by the switch partners of the WFs, which can be schematically illustrated by the evolution of WCs in the x direction as a function of momentum in the y direction, ky . In Ref. [15–17], the topological invariant for 2D and 3D TIs are also associated with the evolution of the WCs as a function of ky (and kz ) as well. Beside the schematic illustration of the nontrivial topological bands, WFs are also a good one-particle basis to study the many-body wave functions of the fractional topological insulator by mimicking the combination of these WFs as that of the Landau orbitals playing roles in the Laughlin state in the fractional quantum Hall effect [18, 19], or introducing pump process to axion coupling directly. [20] Besides TI, another group of topological materials, called topological semi-metal (TM), such as Weyl metal, has attracted great interest for their exotic surface states and bulk behaviors in recent years[21–23]. Weyl metal does not have topological invariant defined in the 3D FBZ because bands touch at the Weyl points so that there is no band gap separating the occupied and the empty bands. One usually needs to split the 3D FBZ into a series of 2D slides and the topological invariant, like Chern numbers, can only be defined on these 2D slides that do not cross the Weyl points. It is also impossible to directly adopt the evolution of WCs to understand TM because the discussion of WCs has a prerequisite condition: the WCs of WFs must be meaningful. But as discussed by Kivelso et al., the WFs became extended as long as the
2 band gap at the Fermi energy became zero [24–27]. So the traditional WCs of WFs become meaningless in Weyl metals. In this paper, we adopt the modular operators, e.g. ˆx L, where x x ˆ = x ˆmod + N ˆ is the position operator in the x direction and xˆmod is the modular position operator, which can be expressed as xmod = x mod L in the representation of x ˆ [28]. Here L is the length of a unit cell. We change the definition of WFs, |WFi, from the eigenstates of Pˆ x ˆPˆ to those of Pˆ xˆmod Pˆ , where Pˆ is the project operator on the occupied states. The WCs of these WFs measured in each unit cell are the eigenvalues of the projected modular position operator. We find that the standard deviation of projected modular position operator, hWF|Pˆ x ˆ2mod Pˆ |WFi is zero for every WF |WFi even when the gap between the occupied bands and the empty bands goes to zero. As a result, the WCs of these newly defined WFs in each unit cell are still meaningful in both TM and TI. By this way, we can universally illustrate the topological properties of TI and TM by evaluating the evolution of WCs. Although the physical model may be in 2D and 3D, the WFs defined above are 1-dimensional (1D) as they are the eigenstates of the operator xˆmod and the momenta ky and kz are only taken as parameters. We find that such WCs of the 1D WFs are equivalent to the phase accumulated by the Bloch states along the Wilson loop in the kx direction, which has been introduced by Yu et al. and Soluyanov et al. individually [15, 16]. One may regard the extension of the 1D WFs as a new representation of their conclusions, but we should emphasize that beside the application of the 1D WFs to TM, from the present starting point the 2D or higher dimensional WFs can be easily obtained. Previously, such higher dimensional WFs are seldom used in TI due to the reasons such as [Pˆ x ˆPˆ , Pˆ yˆPˆ ] 6= 0 that causes the absence of 2D WFs as common eigenstates of P x ˆPˆ and P yˆPˆ . As a contrast, we will show that such obstacle is removed in our consideration because [Pˆ x ˆmod Pˆ , Pˆ yˆmod Pˆ ] = 0. Here yˆ and yˆmod are the position and modular position operators in the y direction, respectively. This paper is organized as the following: In section II, we show that the states in each band are also the eigenstates of the modular position operators with a certain degenerated eigenvalue xmod . We then show that 3D TI, as well as 3D TM, can be universally illustrated in the same framework with (xmod ,ky ,kz ). In section III, we extend our WFs from 1D to higher dimensions. We show their applications by illustrating in a plausible way that why the winding numbers ν2D = ν3D in two models of the DIII class. In section IV we present the conclusions.
II.
1D MODULAR WANNIER FUNCTIONS IN MULTI-BAND SYSTEMS.
Traditionally, the 1D WFs for the occupied bands are defined as the eigenstates of the projected position oper-
ator, x ˜ = Pˆ x ˆPˆ ,
(1)
whereP xˆ is the position operator in the x direction and Pˆ = ǫi
