Topological classification of crystalline insulators with space group symmetry

Abstract

We show that in crystalline insulators point group symmetry alone gives rise to a topological classification based on the quantization of electric polarization. Using C3 rotational symmetry as an example, we first prove that the polarization is quantized and can only take three inequivalent values. Therefore, a Z3 topological classification exists. A concrete tight-binding model is derived to demonstrate the Z3 topological phase transition. Using first-principles calculations, we identify graphene on BN substrate as a possible candidate to realize the Z3 topological states. To complete our analysis we extend the classification of band structures to all 17 two-dimensional space groups. This work will contribute to a complete theory of symmetry conserved topological phases and also elucidate topological properties of graphene like systems. PACS numbers: 71.15.Mb,73.22.Pr,77.22.Ej Since the celebrated discovery of the quantum Hall effect [1], topological classification of electronic states has emerged as a powerful concept in condensed matter physics. The quantum Hall insulators are distinguished from ordinary insulators by a topological index, the TKNN number, which gives the quantized Hall conductance [2,3]. For a long time, the TKNN number was thought to be the only topological index describing non-degenerate electronic ground states. Recently, it was realized that in crystalline insulators new topological indices can be defined in the presence of discrete symmetries, which has led to the identification of a slew of new topological states. For example, the quantum spin Hall insulators are characterized by a nontrivial Z 2 index [4], which is protected by time-reversal symmetry [5,6]. Similarly, magnetic translation symmetry can also give rise to a Z 2 classification in antiferromagnetic insulators [7]. Another interesting proposal is the so-called topological crystalline insulators, in which a Z 2 index can be defined and is protected by both time-reversal and certain point group symmetries [8,9].In this Letter we show that in crystalline insulators point group symmetry alone can give rise to a new topological classification based on the quantization of electric polarization. Our idea is inspired by a beautiful result due to Zak [10], i.e., in one-dimensional (1D) systems with inversion symmetry, the Berry phase of the Bloch bands can be either 0 or π [11]. This quantization of the Berry phase naturally leads to a Z 2 classification in 1D. In higher dimensions, we find that the role of Zak's phase is replaced by a closely related quantity, the electric polarization [12,13], which is quantized in the presence of point group symmetry. The generalization to higher dimensions is expected to display a richer spectrum of possibilities because of the enlarged symmetry class compared to 1D.For the sake of definiteness, 2D crystals with C 3 rota-tional symmetry are used as an example in the following discussion. We first prove, from general symmetry argument, that the polarization is quantized and can only take three i...

Engineering polar discontinuities in honeycomb lattices

“…In view of the wellestablished experimental technology in growing single-layer graphene and BN, and the recent achievements [18][19][20][21][22] in obtaining sharp graphene/BN lateral heterostructures, it is of great interest to exploit these materials to engineer a polar discontinuity. Although pristine graphene is not an insulator and does not support a bulk polarization, its functionalized forms (graphane 41 and fluorographene 42,43 ) are insulators and their formal polarization is constrained by symmetry to be zero 12,13 . Moreover, we have seen above that functionalized BN acquires a non-trivial bulk polarization (m ¼ 1).…”

“…1a) where the discontinuity has to be considered with vacuum. According to the interface theorem 15 , the polarization charge density is related to the bulk formal polarization 9 , which, for non-centrosymmetric honeycomb crystals, is constrained by symmetry to have quantized values and to point along one of the equivalent armchair directions [12][13][14] :…”

“…In this respect, non-centrosymmetric honeycomb lattices offer a very promising playground, owing to the quantized and topological nature of their bulk polarization 12,13 . A suggestion in this direction has been put forward in ref.…”

Symmetry protected topological superfluid3He-B

“…5.3. Recently, numerous works have extended the topological classification of bulk wave functions to include the point group symmetry and magnetic symmetry in topological insulators [213,214,215,216,217,218,219] and in topological superconductors [51,56,220,221,222,223,224,225,226,227]. The topological classification enlarged from the Altland-Zirnbauer tenfold ways has been proposed in Refs.…”

Electromagnetic Response of Three-Dimensional Topological Crystalline Insulators

“…An odd number of Dirac cones, and the corresponding Hall effect, can never occur in a purely 2D system with the same symmetries without interactions. Indeed, the surface quantum Hall effect is actually a signature of a bulk EM response: the topological magneto-electric effect [3,5] with a response coefficient determined by a Z 2 topological invariant [3].After the periodic table was complete, and after many exciting materials predictions and discoveries [6][7][8][9][10][11][12][13], the classification of topological crystalline phases (TCIs) with point/space-group symmetries, such as reflection and discrete rotation, was initiated and continues to be an active area of research [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30]. One highlight of this line of research was the prediction and experimental confirmation of a 3D TCI phase in PbSnTe [31][32][33][34].…”