Topological insulator

Definition

Let $d \ge 1$ and let $T^d = \R^d / \Lambda^*$ be the Brillouin zone torus of a crystal. A band insulator is a continuous family $k \mapsto H(k)$ of Hermitian $N \times N$ matrices on $T^d$, the Bloch Hamiltonians (see band structure). It comes with a Fermi energy $E_F$ that is not an eigenvalue of any $H(k)$. Fix a symmetry class, that is, a set of symmetry conditions on $H$ such as time-reversal symmetry $\Theta H(k) \Theta^{-1} = H(-k)$ with an antiunitary $\Theta$. The insulator is topologically trivial if it can be deformed to a $k$-independent Hamiltonian, after possibly adding extra $k$-independent bands. The deformation must pass through band insulators of the same symmetry class, so $E_F$ never becomes an eigenvalue. A topological insulator is a band insulator that is not topologically trivial.

In the narrow sense, a topological insulator is one of class AII in dimension $d = 2$ or $3$ whose strong $\Z_2$ invariant is nonzero. Class AII means spin-$1/2$ electrons with time-reversal symmetry, $\Theta^2 = -1$.