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Time-Reversal Invariant MomentaΒΆ

Time-Reversal Invariant Momenta are points at the Brillouin zone center and corners that obey: \(-\mathbf{k} = \mathbf{k} \hspace{0.75cm} \text{mod} \hspace{0.25cm} \mathbf{G}\), where \(G\) is a reciprocal lattice vector. There's \(2^d\)of them in d -dimensions. In 3D, they can be written as

\(\Lambda_{n_1n_2n_3} = \frac{n_1}{2} \mathbf{b_1} + \frac{n_2}{2} \mathbf{b_2} + \frac{n_3}{2} \mathbf{b_3}\)

Where \(n_1n_2n_3 = 0,1\) and \(b_i\) are the primitive reciprocal lattice vectors. Below are 2 figures showing these points in 2 and 3 dimensions for a square lattice. These lattices can be smoothly deformed into any other shape, so the definition of the TRIM does not change.

TRIM 2D TRIM 3D
TRIM_2D_Square_Lattice.png TRIM_3D_Square_Lattice.png

Useful Links :

  1. Time-Reversal Invariant Momenta - Quantum Tinkering