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**.

I think the answer should be 'no'.

Because when we introduce the antiunitary time-reversal(TR) opeartor $T$ for spin-system, it should satisfy $T\mathbf{S}_iT^{-1}=-\mathbf{S}_i$ since angular-momentum should be sign reversed under TR(due to the *classical correspondence*). Thus, spin-spin interactions like $\mathbf{S}_i\cdot\mathbf{S}_j$ are *invariant* under TR.

The TR operator $T$ for the $N$-spin-$1/2$ system has a form $T=(-i)^N\sigma_1^y\sigma_2^y...\sigma_N^yK$, where $K$ is the conjugation operator. You can easily check that $T$ is antiunitary and satisfy $T\mathbf{S}_iT^{-1}=-\mathbf{S}_i$. Furthermore, $T^2=(-1)^N$, so for odd-number spin system(including single spin case), if the Hamiltonian has TR symmetry, we will arrive at the well-known **Kramers theorem**.

## Best Answer

Kramers theorem is a very general result that in fermionic systems with time-reversal symmetry, energy levels are at least double degenerate.

Circularly polarized light is an eigenstate of angular momentum. Angular momentum is odd under time reversal ($T$), since $\mathbf x\times\mathbf p$ is an angular momentum and $\mathbf p \mapsto -\mathbf p$ under $T$.

Therefore $T$ makes left-circular polarized light into right-circular polarized light and vice versa. Thus a Hamiltonian containing an external circular-polarized field is not $T$-invariant, and Kramers theorem does not apply.