Kane, Charles L
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Publication Time Reversal Polarization and a Z2 Adiabatic Spin Pump(2006-11-08) Fu, Liang; Kane, Charles LWe introduce and analyze a class of one-dimensional insulating Hamiltonians that, when adiabatically varied in an appropriate closed cycle, define a “Z2 pump.” For an isolated system, a single closed cycle of the pump changes the expectation value of the spin at each end even when spin-orbit interactions violate the conservation of spin. A second cycle, however, returns the system to its original state. When coupled to leads, we show that the Z2 pump functions as a spin pump in a sense that we define, and transmits a finite, though nonquantized, spin in each cycle. We show that the Z2 pump is characterized by a Z2 topological invariant that is analogous to the Chern invariant that characterizes a topological charge pump. The Z2 pump is closely related to the quantum spin Hall effect, which is characterized by a related Z2 invariant. This work presents an alternative formulation that clarifies both the physical and mathematical meaning of that invariant. A crucial role is played by time reversal symmetry, and we introduce the concept of the time reversal polarization, which characterizes time reversal invariant Hamiltonians and signals the presence or absence of Kramers degenerate end states. For noninteracting electrons, we derive a formula for the time reversal polarization that is analogous to Berry’s phase formulation of the charge polarization. For interacting electrons, we show that Abelian bosonization provides a simple formulation of the time reversal polarization. We discuss implications for the quantum spin Hall effect, and argue in particular that the Z2 classification of the quantum spin Hall effect is valid in the presence of electron electron interactions.Publication Majorana Fermoins and Non-Abelian Statistics in Three Dimensions(2010-01-25) Teo, Jeffrey C.Y.; Kane, Charles LWe show that three dimensional superconductors, described within a Bogoliubov–de Gennes framework, can have zero energy bound states associated with pointlike topological defects. The Majorana fermions associated with these modes have non-Abelian exchange statistics, despite the fact that the braid group is trivial in three dimensions. This can occur because the defects are associated with an orientation that can undergo topologically nontrivial rotations. A feature of three dimensional systems is that there are ‘‘braidless’’ operations in which it is possible to manipulate the ground state associated with a set of defects without moving or measuring them. To illustrate these effects, we analyze specific architectures involving topological insulators and superconductors.