Time Reversal Polarization and a Z_2 Adiabatic Spin Pump

TL;DR

This paper introduces a $Z_2$ topological invariant for a class of one-dimensional insulators that, when adiabatically cycled, can pump spin in a manner related to the quantum spin Hall effect, with implications for interacting systems.

Contribution

It defines a $Z_2$ topological invariant for spin pumps and clarifies its physical and mathematical meaning, especially in relation to time reversal symmetry and the quantum spin Hall effect.

Findings

The $Z_2$ pump changes spin expectation values at ends during a cycle.

Coupled to leads, it transmits a finite, non-quantized spin per cycle.

The $Z_2$ invariant characterizes the pump and relates to the quantum spin Hall effect.

Abstract

We introduce and analyze a class of one dimensional insulating Hamiltonians which, when adiabatically varied in an appropriate closed cycle, define a "$Z_2$ 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 $Z_2$ pump functions as a spin pump in a sense which we define, and transmits a finite, though non quantized spin in each cycle. We show that the $Z_2$ pump is characterized by a $Z_2$ topological invariant that is analogous to the Chern invariant that characterizes a topological charge pump. The $Z_2$ pump is closely related to the quantum spin Hall effect, which is characterized by a related $Z_2$ invariant. This work presents an alternative formulation which 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 non interacting electrons we derive a formula for the time reversal polarization which is analogous to the 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 $Z_2$ classification of the quantum spin Hall effect is valid in the presence of electron electron interactions.