Kramers Theorem

Kramer's Theorem¶

Source

Proof¶

For spin-1/2 systems, we have \(\Theta^2 = -1\) where \(\Theta\) is the time-reversal operator. In such a time-reversal invariant system, \(|\psi\rangle\) and \(\Theta |\psi\rangle\) have the same energy (since \([H,\Theta] =0\)). Thus, if we can prove that these two states are different, then we will have proved Kramer's theorem since there will be two different states with the same energy!

Let us assume the two states are the same, i.e,

\(\Theta |\psi \rangle = e^{i\phi}|\psi\rangle, \phi \in \mathcal{R}.\)

Then \(\Theta^2 = \Theta e^{i\phi}|\psi\rangle = e^{-i\phi}\Theta|\psi\rangle = e^{-i\phi}e^{i\phi}|\psi\rangle \Rightarrow \Theta^2 = 1 \Rightarrow\Leftarrow\)

(Proof by contradiction)

Solid-State Physics¶

Consider a system with Bloch states \(|n,\textbf{k},\sigma \rangle\) where n is the band index, \(\textbf{k}\) is the crystal momentum and \(\sigma\) is the spin. For any such state, there is another state \(\Theta |n,\textbf{k},\sigma \rangle\) with the same energy.

\(\Theta |n,\textbf{k},\sigma \rangle = |n,-\textbf{k},-\sigma \rangle\)

(Note that we use a notation where there are two bands labeled by the same n, one with spin-up and one with spin-down). In general, such a Kramer's doublet are located at different momenta \(\textbf{k}\) and \(-\textbf{k}\). For example, consider the system shown in Fig. 1 below.

Fig 1: The band diagram in the first Brillouin zone of a spin-1/2 particle with time-reversal symmetry in 1-dimension. The yellow line shows a constant energy cut, with four states having the same energy. There are two Kramer's pairs here, denoted by yellow dots. There is one pair at \(\alpha_{\pm}\) and another at \(\beta_{\pm}\). These points are known as time-reversal conjugate momenta. Moreover, note the special green points at the center and the boundary of the zone (denoted by green dots), which are their own time-reversal conjugates. These points are known as the time-reversal invariant moment (TRIM).