Symmetry Protected Topological Hopf Insulator and its Generalizations

Abstract

We study a class of $3d$ and $4d$ topological insulators whose topological nature is characterized by the Hopf map and its generalizations. We identify the symmetry ${\mathcal{C}}^{\prime }$, a generalized particle-hole symmetry that gives the Hopf insulator a ${\mathbb{Z}}_2$ classification. The $4d$ analogue of the Hopf insulator with symmetry ${\mathcal{C}}^{\prime }$ has the same ${\mathbb{Z}}_2$ classification. The minimal models for the $3d$ and $4d$ Hopf insulator can be heuristically viewed as "Chern-insulator$⋊S^1$" and "Chern-insulator$⋊T^2$" respectively. We also discuss the relation between the Hopf insulator and the Weyl semimetals, which points the direction for its possible experimental realization.

4 pages, 3 figures

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• Topological Defects and Gapless Modes in Insulators and Superconductors
• Topological surface states in three-dimensional magnetic insulators
• Symmetry Protected Topological States of Interacting Fermions and Bosons
• Homotopy Theory of Strong and Weak Topological Insulators

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