Time-reversal symmetry

Question

For a quantum system with time-reversal symmetry, other than the absence of a magnetic field, can we infer anything else about the system?

Answer 1

We can infer some general topological properties of the system for special cases. This has been used in the recent "trending topic" of condensed matter physics, topological insulators. For simplicity, I will restrict myself in the following to $2\text{D}$ systems although one can generalize everything to $3\text{D}$.

The time-reversal operator is an antiunitary operator that admits the following representation:

$\hat{\Theta} = \exp \left(i\pi\hat{S}_y/\hbar \right)K$

where $K$ means complex conjugation and $\hat{S}_y$ is the spin operator along the $\hat{y}$ axis. Consider a fermionic Hamiltonian for spin $s=1/2$ electrons. Then

$\hat{\Theta} =-\hat{1}$ [*]

In this case, Kramers theorem applies:

Let $\hat{\mathcal{H}}$ be a $T$-invariant (fermionic) Hamiltonian. Then, all the eigenstates of the Hamiltonian are twofold degenerate.

The proof of this statement is simple once you have understood [*]. As a consequence, $T$-invariant fermionic systems must have topologically protected twofold degenerate states. The $T$-invariant Hamiltonian satisfies

$ \hat{\Theta}\hat{\mathcal{H}} (\mathbf{k}) =\hat{\mathcal{H}} (-\mathbf{k}) \hat{\Theta}$

and can be classified by a new topological index, called the $\mathbb{Z}_2$ index. The $\mathbb{Z}_2$ index, $\nu$, is an integer given by the number of edge states modulo $2$ and distinguishes the $\nu = 0$ or insulating phase from the $\nu = 1$, the topological insulator. Thus, the equivalence classes of $T$-invariant Hamiltonians for insulators can be classified by its $n = 0$ Thouless-Kohmoto-Nightingaleden Nijs invariant [i.e. its $C=0$, first Chern index] and the additional index $\nu$. This gives a $\mathbb{Z} \times \mathbb{Z}_2$ symmetry for the $2\text{D}$ band structures.

After all, what can we infer from the Hamiltonian? For example, that we have time reversal invariant electronic states with a bulk electronic band gap that supports the transport of charge and spin in gapless edge states. This is exactly what we get in the so-called quantum spin Hall phase.