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...

We report an initial experimental survey of spin-Hall torques generated by the rare-earth metals Gd, Dy, Ho, and Lu, along with comparisons to first-principles calculations of their spin Hall conductivities. Using spin torque ferromagnetic resonance (ST-FMR) measurements and DC-biased ST-FMR, we estimate lower bounds for the spin-Hall torque ratio, ξSH, of ≈ 0.04 for Gd, ≈ 0.05 for Dy, ≈ 0.14 for Ho, and ≈ 0.014 for Lu. The variations among these elements are qualitatively consistent with results from first principles (density functional theory, DFT, in the local density approximation with a Hubbard-U correction). The DFT calculations indicate that the spin Hall conductivity is enhanced by the presence of the partially-filled f orbitals in Dy and Ho, which suggests a strategy to further strengthen the contribution of the f orbitals to the spin Hall effect by shifting the electron chemical potential.

We report spin-torque ferromagnetic resonance studies of the efficiency of the damping-like (ξ DL ) spin−orbit torque exerted on an adjacent ferromagnet film by current flowing in epitaxial ( 001) and ( 110) IrO 2 thin films. IrO 2 possesses Dirac nodal lines (DNLs) in the band structure that are gapped by spin− orbit coupling, which could enable a very high spin Hall conductivity, σ SH . We find that the (001) films do exhibit exceptionally high ξ DL ranging from 0.45 at 293 K to 0.65 at 30 K, which sets the lower bounds of σ SH to be 1.9 × 10 5 and 3.75 × 10 5 Ω −1 m −1 , respectively, 10 times higher and of opposite sign than the theoretical prediction. Furthermore, ξ DL and σ SH are substantially reduced in anisotropically strained (110) films. We suggest that this high sensitivity to anisotropic strain is because of changes in contributions to σ SH near the DNLs.

Spin Hall effect (SHE), a mechanism by which materials convert achargecurrent into aspincurrent, invokes interesting physics and promises to empower transformative, energy-efficient memory technology. However, fundamental questions remain about the essential factors that determine SHE. Here, we solve this open problem, presenting a comprehensive theory of five rational design principles for achievinggiantintrinsic SHE in transition metal oxides. Arising from our key insight regarding the inherently geometric nature of SHE, we demonstrate that two of these design principles are weak crystal fields and the presence of structural distortions. Moreover, we discover that antiperovskites are a highly promising class of materials for achieving giant SHE, reaching SHE values anorder of magnitudelarger than that reported for any oxide. Additionally, we derive three other design principles for enhancing SHE. Our findings bring deeper insight into the physics driving SHE and could help enhance and externally control SHE values.