Table 1 The inversion symmetry breaking perturbation term for each of the IR modes added to the effective Hamiltonian. \(M={(m-D{k}^{2})}^{2}\), \(\tilde{{k}_{i}}={v}_{i}{k}_{i}\) and \(| {\tilde{k}}_{ij}| =\sqrt{{\tilde{k}}_{i}^{2}+{\tilde{k}}_{j}^{2}}\).

B_1u

M_c

cτ^x

\({M}^{2}+{\tilde{k}}_{z}^{2}+{(| {\tilde{k}}_{xy}| \pm c)}^{2}\)

(k_xσ^y + k_yσ^x)τ^z

k_x-k_y plane

B_2u

G_b

cτ^yσ^x

\({M}^{2}+{\tilde{k}}_{y}^{2}+{(| {\tilde{k}}_{xz}| \pm c)}^{2}\)

(k_zσ^x + k_xσ^z)τ^z

k_x-k_z plane

B_3u

M_a

cτ^yσ^y

\({M}^{2}+{\tilde{k}}_{x}^{2}+{(| {\tilde{k}}_{yz}| \pm c)}^{2}\)

(k_yσ^z + k_zσ^y)τ^z

k_y-k_z plane