2 research outputs found
Topological Description of (Spin) Hall Conductances on Brillouin Zone Lattices : Quantum Phase Transitions and Topological Changes
It is widely accepted that topological quantities are useful to describe
quantum liquids in low dimensions. The (spin) Hall conductances are typical
examples. They are expressed by the Chern numbers, which are topological
invariants given by the Berry connections of the ground states. We present a
topological description for the (spin) Hall conductances on a discretized
Brillouin Zone. At the same time, it is quite efficient in practical numerical
calculations for concrete models. We demonstrate its validity in a model with
quantum phase transitions. Topological changes supplemented with the transition
is also described in the present lattice formulation.Comment: proceeding of EP2DS-1
An edge index for the Quantum Spin-Hall effect
Quantum Spin-Hall systems are topological insulators displaying
dissipationless spin currents flowing at the edges of the samples. In
contradistinction to the Quantum Hall systems where the charge conductance of
the edge modes is quantized, the spin conductance is not and it remained an
open problem to find the observable whose edge current is quantized. In this
paper, we define a particular observable and the edge current corresponding to
this observable. We show that this current is quantized and that the
quantization is given by the index of a certain Fredholm operator. This
provides a new topological invariant that is shown to take same values as the
Spin-Chern number previously introduced in the literature. The result gives an
effective tool for the investigation of the edge channels' structure in Quantum
Spin-Hall systems. Based on a reasonable assumption, we also show that the edge
conducting channels are not destroyed by a random edge.Comment: 4 pages, 3 figure