I have heard that applying uniform torsion to a system of Bloch electrons will induce Landau levels. Essentially, if I apply a uniform torsion about the $z$-direction, then a pseudo-magnetic field should emerge. My question is: how can I derive this by looking at local Bloch states? When I try to do it, I find that the magnetic field is perpendicular to the $z$-axis. Am I wrong in supposing that torsion about the $z$-axis induces a pseudo magnetic field also along the $z$-axis, or is my derivation flawed?
DERIVATION: Consider a system of non-interacting electrons in a periodic (Bloch) potential. The wave function can be expressed as $$\psi_{n{\bf k}}({\bf r}) = u_{n{\bf k}}({\bf r}) e^{i{\bf k\cdot r}},$$ where $n\in\mathbb N$ indexes the bands and ${\bf k}$ is the lattice momentum, belonging to the first Brillouin zone. Moreover, for lattice vectors ${\bf R}$, we have $u_{n{\bf k}}({\bf r})=u_{n{\bf k}}({\bf} r+{\bf R})$. Now suppose I apply torsion to the system along the $z$-direction; then the wave function no longer takes the Bloch form. If the torsion is gentle, however, the Bloch form should still hold approximately in local patches. Let $\theta(z)$ represent the angle by which the local lattice vectors have been twisted in the $x,y$-plane at height $z$. Then I would expect the pseudo-gauge connection to take the form $${\bf A} = \langle u_{n\bf k}({\theta(z) })| {\bf\nabla} | u_{n\bf k}({\theta(z) }\rangle .$$ But with this definition, it follows that ${\bf A} \propto {\bf e}_z$. But if ${\bf A}$ is directed along the $z$-axis, then the pseudo-magnetic field must be perpendicular to the $z$-direction.
