Spin spirals

Generalized Bloch condition

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \exp(-i{{\bf {q}}}\cdot {{\bf {R}}}/2) and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): \exp(-i{{\bf {q}}}\cdot {{\bf {R}}}/2) , respectively.

The above definition gives rise to the following magnetization density:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://www.vasp.at/wiki/restbase/vasp.at/v1/":): {{\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)