Spin spirals

Generalized Bloch condition

Spin spirals may be conveniently modeled using a generalisation of the Bloch condition:

[math]\displaystyle{ \left[ \begin{array}{c} \Psi^{\uparrow}_{\bf k}(\bf r) \\ \Psi^{\downarrow}_{\bf k}(\bf r) \end{array} \right] = \left( \begin{array}{cc} e^{-i\bf q \cdot \bf R / 2} & 0\\ 0 & e^{+i\bf q \cdot \bf R / 2} \end{array}\right) \left[ \begin{array}{c} \Psi^{\uparrow}_{\bf k}(\bf r-R) \\ \Psi^{\downarrow}_{\bf k}(\bf r-R) \end{array} \right], }[/math]

i.e., from one unit cell to the next the up-spinor and down-spinors pick up an additional phase factor of [math]\displaystyle{ \exp(-i{\bf q}\cdot {\bf R}/2) }[/math] and [math]\displaystyle{ \exp(-i{\bf q}\cdot {\bf R}/2) }[/math], respectively.

This condition gives rise to the following behaviour of the magnetization density:

[math]\displaystyle{ {\bf m} ({\bf r} + {\bf R})= \left( \begin{array}{c} m_x({\bf r}) \cos({\bf q} \cdot {\bf R}) - m_y({\bf r}) \sin({\bf q} \cdot {\bf R}) \\ m_x({\bf r}) \sin({\bf q} \cdot {\bf R}) + m_y({\bf r}) \cos({\bf q} \cdot {\bf R}) \\ m_z({\bf r}) \end{array} \right) }[/math]

This is schematically depicted in the figure at the top of this page: the components of the magnization in the xy-plane rotate about the so-called spin-spiral propagation vector q.