Symmetry condition for group velocity¶
Ref. [BIR74]
Group action on modified eigenvectors
\[\begin{split} g f_{\mu}(\kappa; \mathbf{q}\omega_{\mathbf{q}}\nu)
&= \sum_{\mu'\nu'} f_{\mu'}(\kappa'; \mathbf{q}\omega_{\mathbf{q}}\nu') \Gamma^{ (\mathbf{q} \omega_{\mathbf{q}}) }(g)_{\kappa'\mu'\nu', \kappa\mu\nu}
\quad (g \in \mathcal{G}^{\mathbf{q}}) \\
\omega_{\mathbf{q}}^{2}
&= \sum_{ \kappa \mu }\sum_{ \kappa' \mu' } f_{\mu}(\kappa; \mathbf{q}\omega_{\mathbf{q}}\nu)^{\ast} \Phi_{\mu\mu'}(\kappa\kappa'; \mathbf{q}) f_{\mu'}(\kappa'; \mathbf{q}\omega_{\mathbf{q}}\nu)\end{split}\]
Group velocity
\[ \nabla_{q_{\alpha}} \omega_{\mathbf{q}}^{2}
= \sum_{ \kappa \mu }\sum_{ \kappa' \mu' }
f_{\mu}(\kappa; \mathbf{q}\omega_{\mathbf{q}}\nu)^{\ast}
\left( \nabla_{q_{\alpha}} \Phi_{\mu\mu'}(\kappa\kappa'; \mathbf{q}) \right)
f_{\mu'}(\kappa'; \mathbf{q}\omega_{\mathbf{q}}\nu)\]
If the group velocity has scalar component, that is
(1)¶\[\left\langle
\Gamma^{ (\mathbf{q} \omega_{\mathbf{q}}) \ast} \otimes \Gamma^{(\mathbf{\nabla})} \otimes \Gamma^{ (\mathbf{q} \omega_{\mathbf{q}}) }
\mid
\Gamma^{ (\mathrm{id}) }
\right\rangle
\geq 1,\]
it should be zero at \(\mathbf{q}\).
The gradient operator \(\mathbf{\nabla}_{\mathbf{q}}\) behaves as covariant vector:
\[\begin{split} \mathbf{q}' &= \mathbf{R}^{\top}\mathbf{q} \quad (\mathbf{R} \in O(3)) \\
\nabla_{q'_{\alpha}}
&= \sum_{\beta} \frac{ \partial q_{\beta} }{ \partial q'_{\alpha} } \nabla_{q_{\beta}}
= \sum_{\beta} R_{\alpha\beta} \nabla_{q_{\beta}}\end{split}\]
The left hand side of Eq. (1):
\[ \sum_{g \in \mathcal{G}^{\mathbf{q}} / \mathcal{T} }
| \chi^{(\mathbf{q} \omega_{\mathbf{q}})}(g) |^{2} \chi^{(\mathbf{\nabla})}(\mathbf{R}_{g})\]
References¶
- BIR74
Joseph L BIRMAN. A quick trip through group theory and lattice dynamics. Dynamical Properties of Solids: Crystalline solids, fundamentals, 1:83, 1974.