Time Reversal Symmetry¶

The action of time reversal on a wave function then contains two parts: first take complex conjugation in a certain basis, then apply a linear transformation in this basis. That is, we can write.

$$ T = U K $$ where, K denotes complex conjugation and U denotes some unitary transformation. Then time reversal acts on the operators as,

$$ TOT^{-1} = UKOKU^{\dagger} = UO^*U{\dagger} $$ That is, the action of time reversal on operators contains two parts: first take complex conjugation of the operator written in certain basis, then conjugate the operator by some unitary transformation in this basis.

Time reversal breaking model¶

Let \(\hat{H}(\mathbf{k})\) be the Hamiltonian considered which breaks time-reversal symmetry but preserves inversion symmetry, then we have the following condition to be satisfied,