Spin spirals

Generalized Bloch condition

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

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

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

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

${\bf {m}}({\bf {r}}+{\bf {R}})=\left({\begin{array}{c}m_{x}({\bf {r}})\cos({\bf {q}}\cdot {\bf {R}})-m_{y}({\bf {r}})\sin({\bf {q}}\cdot {\bf {R}})\\m_{x}({\bf {r}})\sin({\bf {q}}\cdot {\bf {R}})+m_{y}({\bf {r}})\cos({\bf {q}}\cdot {\bf {R}})\\m_{z}({\bf {r}})\end{array}}\right)$

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.