N -band Hopf insulator

Abstract

We study the generalization of the three-dimensional two-band Hopf insulator to the case of many bands, where all the bands are separated from each other by band gaps. The obtained Z classification of such a N -band Hopf insulator and the formulation of its bulk-boundary correspondence are our main results. We find that the quantized boundary effect can only be measured in a non-equilibrium state.

Topologically localized insulators

“…This follows directly from the arguments presented in Ref. 28: when the slab is fully filled with electrons, the electrons are "inert" and do not respond to an external magnetic field. Since the boundary has non-zero quantized Hall conductance, it follows from the Streda formula that the filling of the boundary changes by an integer amount (= n TLI ) when the flux Φ 0 threads the slab.…”

“…Topology of N -band Hopf insulator crucially depends on the presence of translational symmetry. For example, inclusion of perturbations that reduce the translational symmetry down to one of its subgroups can trivialize N -band Hopf insulators [28]. Nevertheless, we show that every N -band Hopf insulator can be deformed to a TLI phase with the same bulk topological invariant.…”

“…, N . Such deformation of the model does not change the unitary (S3); The resulting, translationally invariant model, corresponds to topologically non-trivial N -band Hopf insulator [28]. Nevertheless, TLIs and N -band Hopf insulators belong to different phases of matter: the constraint of fully-localized bulk implies that TLIs are insulating at an arbitrary filling, whereas N -band Hopf insulators are insulating only at integer fillings.…”

“…This distinction is important, since it implies that the boundary Fermi energy E ∂ F of N -band Hopf cannot be varied independently of that of the bulk. As a consequence, its quantized boundary response can be measured only in a transient state [28].…”

A topological principle for photovoltaics: Shift current in intrinsically polar insulators

“…Currently all known examples of delicate topological insulators are intrinsically noncentric, 8 in the sense that the topological distinction between trivial and nontrivial insulators (in the standard sense) is only meaningful for space groups without centrosymmetry. [32,33,37,55,56] This offers a rich playing field to search for nontrivial optical invariants. To give a flavor of the possibilities, the reverting Thouless pump (RTP) has been theoretically explored in a wide variety of -symmetric Hamiltonians, [32,33,37] where an -fold rotational symmetry plays a role analogous to the reflection sym- 7 If the assumption of a monatomic basis is relaxed, the generalized -band intercellular shift ̆ (0)∈ℤ has the meaning of the net change in the primitive unit cell label when all inter-band optical excitations are accounted for.…”