Time Reversal Polarization and a Z 2 Adiabatic Spin Pump

By Liang Fu, C. L. Kane. 2006

688 citations recorded by the source corpus.

We introduce and analyze a class of one-dimensional insulating Hamiltonians that, 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 that we define, and transmits a finite, though nonquantized, 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 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 ${Z}_{2}$ classification of the quantum spin Hall effect is valid in the presence of electron electron interactions.

Subfields: Atomic and Molecular Physics, and Optics

View Paper: https://www.semanticscholar.org/paper/dac3e5ead98d1251eba936b3d3d5080bd555e0d0

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In Atomic and Molecular Physics, and Optics

Citations in the corpus

Failure of Nielsen-Ninomiya Theorem and Fragile Topology in Two-Dimensional Systems with Space-Time Inversion Symmetry: Application to Twisted Bilayer Graphene at Magic Angle (2018, methodology): “where we used the definition of the time-reversal polarization in the last step [72].”
A Short Course on Topological Insulators: Band-structure topology and edge states in one and two dimensions (2015, neutral): “e Hamiltonian Hˆ(k) should have a corresponding topological invariant (generalized winding number). The bulk Z 2 invariant is notoriously difﬁcult to calculate. The original deﬁnition of the invariant[19, 11] uses a smooth gauge in the whole Brillouin zone, that is hard to construct [31], which make”
Topological photonics (2018, background): “This Z2 topological invariant keeps the same value even when spin-mixing terms are added, provided time-reversal symmetry is maintained (Kane and Mele, 2005b; Fu and Kane, 2006; Sheng et al., 2006; Roy, 2009).”
Z2Pack: Numerical Implementation of Hybrid Wannier Centers for Identifying Topological Materials (2016, methodology): “In the quantum spin Hall phase, they are constrained to be odd, while they are bound to be even in the Z2-even phase [95], as illustrated in Fig.”