Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

Kohmoto (1985) pointed out in Topological Invariant and the Quantization of the Hall Conductance how TKNN's calcuation of Hall conducance is related to topology, in which topologically nontriviality is said to be equivalent to impossiblility choosing a global phase of Bloch function $u_k (r)$ in Brillouin zone. As shown in the Figure, we can choose two distinct gauges in sector I and II, and the curvature is the loop integral of phase mismatch on boundary $\partial H$.

enter image description here

What is the simplest possible Bloch function that is

  • topologically nontrivial, and
  • an eigenstate of Bloch Hamiltonian?

Bloch Hamiltonian: $H(k_x,k_y) = \frac{1}{2m}(-i\partial + {\bf k}+e{\bf A}(x,y))^2 + U(x,y)$ where $U$ is lattice periodic.

share

1 Answer 1

up vote 2 down vote accepted
+50

Surprisingly, according to Immanuel Bloch's group (no relation to F. Bloch!), the simplest topological Bloch function is the 1D staggered lattice. The topological invariant is the Zak phase, the Barry phase accrued by walking across the edge of the Brillouin zone. The article will explain it better than I can: Direct Measurement of the Zak phase in Topological Bloch Bands

share
    
Dear emarti, In the future please link to abstract page rather than pdf file, e.g., arxiv.org/abs/1212.0572 –  Qmechanic Jan 3 '13 at 14:44

This site is currently not accepting new answers.

Not the answer you're looking for? Browse other questions tagged .