Homotopy of periodic two by two matrices

Abstract

We describe the homotopy classes of 2 by 2 periodic simple (=non-degenerate) matrices with various symmetries. This turns out to be an elementary exercise in the homotopy of closed curves in three dimensions. The matrices represent gapped Bloch Hamiltonians in 1D with a two-dimensional Hilbert space per unit cell.

Figures (11)

• Figure 1: A periodic array with two atoms per unit cell. The red ellipse shows a unit cell.
• Figure 2: The two red ellipses correspond to two different choices of unit cell. Changing the unit cell is a gauge transformation (=diagonal unitary) acting on $H(k)$.
• Figure 3: The one dimensional Brillouin zone is a circle parameterized by the angular variable $k$. The time-reversal, Particle-Hole and reflection symmetry, relate $k$ with $-k$ and leave invariant the two points $k=0$ and $k=\pi$.
• Figure 4: The curve $\gamma_-$ associated with time-reversal symmetric Hamiltonians is anchored on the punctured $x-z$ plane. The rest of the loop is fixed by symmetry of reflection in the plane.
• Figure 5: The curve $\gamma_-$ associated with Particle-Hole symmetric Hamiltonians with $C^2=\mathds{1}$ is anchored on the punctured $x$-axis. $\gamma_+$ is fixed by the symmetry as the $\pi$ rotation of $\gamma_-$ around the $x$-axis.

Theorems & Definitions (1)

• Example 3.1