- #1

dRic2

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Generally speaking, if the Hamiltonian has a specific symmetry defined by an operator M, that is ##[H,M]=0##, when I apply such symmetry operator to a Bloch state I would expect the state to be left unchanged up to a phase:

$$M\ket{ψ_{\mathbf k}}=e^{iϕ(\mathbf k)}\ket{ψ_{\mathbf k}}$$

For the simple case of a translation of a lattice vector R the phase it gets is just ##e^{−i \mathbf k \cdot \mathbf R}##.

What about more complicated symmetries? In my case, I would like to consider a 2D periodic lattice where there is also an additional symmetry called "glide mirror symmetry". This symmetry is composed by a reflection along the x axis (##y \rightarrow −y##) and a translation in the

If I apply this symmetry to a Bloch state then what is the resulting phase? I get stuck at the very beginning:

$$M \psi_{\mathbf k}(\mathbf x) = e^{-i[k_x(x-\frac a 2)-k_yy]} u(x-a/2, -y)$$

Thanks

(Let's consider spinless electrons, i.e. scalar Bloch states)

$$M\ket{ψ_{\mathbf k}}=e^{iϕ(\mathbf k)}\ket{ψ_{\mathbf k}}$$

For the simple case of a translation of a lattice vector R the phase it gets is just ##e^{−i \mathbf k \cdot \mathbf R}##.

What about more complicated symmetries? In my case, I would like to consider a 2D periodic lattice where there is also an additional symmetry called "glide mirror symmetry". This symmetry is composed by a reflection along the x axis (##y \rightarrow −y##) and a translation in the

*x-*direction of half of a lattice vector (##x \rightarrow x - \frac a 2##), where ##a## a is the lattice vector).If I apply this symmetry to a Bloch state then what is the resulting phase? I get stuck at the very beginning:

$$M \psi_{\mathbf k}(\mathbf x) = e^{-i[k_x(x-\frac a 2)-k_yy]} u(x-a/2, -y)$$

Thanks

(Let's consider spinless electrons, i.e. scalar Bloch states)

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