Quantum Spin Hall Effect in Three Dimensional Materials: Lattice Computation of Z₂ Topological Invariants and Its Application to Bi and Sb

Abstract

We derive an efficient formula for Z 2 topological invariants characterizing the quantum spin Hall effect. It is defined in a lattice Brillouin zone, which enables us to implement numerical calculations for realistic models even in three dimensions. Based on this, we study the quantum spin Hall effect in Bi and Sb in quasi-two and three dimensions using a tight-binding model. Quantum spin Hall (QSH) effect [1][2][3][4][5] has been attracting much current interest as a new device of spintronics. [6][7][8][9] It is a topological insulator [10][11][12] analogous to the quantum Hall (QH) effect, but it is realized in time-reversal (T ) invariant systems. While QH states are specified by Chern numbers, 13,14) QSH states are characterized by Z 2 topological numbers, 2) which suggests that Z 2 invariants would have deep relationship with the Z 2 anomaly of the Majorana fermions. 15,16) Graphene has been expected to be in the QSH phase. 1,2)However, recent calculations have suggested that the spinorbit coupling in graphene is too small to reveal the QSH effect experimentally. 17,18) Recently, it has been pointed out that Bi thin film is another plausible material for QSH effect.19) Also by the idea of adiabatic deformation of the diamond lattice, it has been conjectured that Bi in three dimensions (3D) is in a topological phase. 20)While systems in two dimensions (2D) are characterized by a single Z 2 topological invariant, four independent Z 2 invariants are needed in 3D. [20][21][22][23] This makes it difficult to investigate realistic models, in which complicated many-band structure is involved. Therefore, for the direct study of Bi in 3D as well as for the search for other materials, to establish a simple and efficient computational method of Z 2 invariants in 3D is an urgent issue to be resolved.In this paper, we present a method of computing Z 2 invariants based on the formula derived by Fu and Kane 24) together with the recent development of computing Chern numbers in a lattice Brillouin zone. [25][26][27] This method is based on recent developments in lattice gauge theories [30][31][32][33][34][35] but simple enough to compute Z 2 invariants even for realistic 3D systems. Based on this, we study a tight-binding model for Bi and Sb.First, we derive a lattice version of the Fu-Kane formula.24) To this end, we restrict our discussions, for simplicity, to systems in 2D, where a single Z 2 invariant is relevant. Let T be the time-reversal transformation T ¼ i 2 K, and assume that the Hamiltonian in the momentum space HðkÞ transforms under T as T HðkÞT À1 ¼ HðÀkÞ. Let ðkÞ ¼ ðj1ðkÞi; . . . ; j2MðkÞiÞ denote the 2M dimensional ground state multiplet of the Hamiltonian: HðkÞjnðkÞi ¼ E n ðkÞjnðkÞi. 11,12) Assuming that the many-body energy gap is finite, we focus on topological invariants under the Uð2MÞ transformation ðkÞ ! ðkÞUðkÞ; UðkÞ 2 Uð2MÞ:As discussed, 2,27) the pfaffian defined by pðkÞ ¼ pf É y ðT ÉÞ characterizes the topological phases of T invariant systems. To be precise, the systems belong to topological insu...

Topological insulators from the perspective of first‐principles calculations

“…For the compounds without inversion symmetry, several methods are proposed to calculate 2 Z invariants [40][41][42][43]. Considering the simplicity for first-principles calculations, here we briefly introduce the proposal of Fukui et al [40].…”

“…Considering the simplicity for first-principles calculations, here we briefly introduce the proposal of Fukui et al [40]. Firstly, 2 Z formula of QSH state can be expressed with the Berry connection and the Berry curvature, shown by Fu and Kane,…”

Chern number of thin films of the topological insulatorBi2Se3

“…As long as these two gaps do not close, the spin Chern number remains unchangeable under smooth deformations of the system Hamiltonian, including those breaking time-reversal symmetry. While some numerical computations 8,22,23 have been done before, an explicit analytical calculation of the spin Chern number will be very useful for understanding the physical implications of this topological invariant, which, however, has not been performed so far.…”

Equivalent expression ofZ2topological invariant for band insulators using the non-Abelian Berry connection

“…Interestingly the Z 2 topological invariant can also be generalized to the 3D band insulators with time reversal symmetry 5,6,15 . In this case, there are four independent Z 2 topological numbers: one strong topological index and three weak topological indices [15][16][17][18][19] . The 3D time reversal invariant band insulators can be classified as normal insulators, weak topological insulators (WTI) and strong topological insulators (STI) according to the values of these four Z 2 topological indices.…”

“…i) Compute the Z 2 numbers using the integration of both Berry's connection and curvature over half of the Brillouin Zone (BZ). In order to do so, one has to set up a mesh in the k-space and calculate the corresponding quantities on the lattice version of the problem 18,41,42 . Since the calculation involves the Berry's connection, one has to numerically fix the gauge on the half BZ, which is not easy for the realistic wave functions obtained by first principle calculation.…”