Spin spirals

Generalized Bloch condition

Spin spirals may be conveniently modeled using a generalization of the Bloch condition (set LNONCOLLINEAR=.TRUE. and LSPIRAL=.TRUE.):

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

i.e., from one unit cell to the next the up- 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, where R is a lattice vector of the crystalline lattice, and q is the so-called spin-spiral propagation vector.

The spin-spiral propagation vector is commonly chosen to lie within the first Brillouin zone of the reciprocal space lattice, and has to be specified by means of the QSPIRAL-tag.

The generalized Bloch condition above gives rise to the following behavior 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 spin-spiral propagation vector q.

Basis set considerations

redefining the Bloch functions \[ \Psi^{\uparrow}_{\bf k}(\bf r) = \sum _{\bf G} \rm C^{\uparrow}_{\bf k \bf G} e^{i(\bf k + \bf G -\frac{\bf q}{2})\cdot \bf r} \hspace{0.5cm} and \hspace{0.5cm} \Psi^{\downarrow}_{\bf k}(\bf r) = \sum _{\bf G} \rm C^{\downarrow}_{\bf k \bf G} e^{i(\bf k + \bf G +\frac{\bf q}{2})\cdot \bf r} \] %\[ %\left( \begin{array}{c} \mid \Psi^{\uparrow} \rangle \\ \mid \Psi^{\downarrow} \rangle \end{array} \right) %\rightarrow %\left( \begin{array}{c} e^{-i\bf q \cdot \bf r / 2} \mid \Psi^{\uparrow} \rangle \\ e^{+i\bf q \cdot \bf r / 2}\mid \Psi^{\downarrow} \rangle \end{array} \right) %\]

the Hamiltonian changes only minimally \[ \left( \begin{array}{cc} H^{\alpha\alpha} & V^{\alpha\beta}_{\rm xc} \\ V^{\beta\alpha}_{\rm xc} & H^{\beta\beta} \end{array}\right) \rightarrow \left( \begin{array}{cc} H^{\alpha\alpha} & V^{\alpha\beta}_{\rm xc} e^{-i\bf q \cdot \bf r} \\ V^{\beta\alpha}_{\rm xc}e^{+i\bf q \cdot \bf r} & H^{\beta\beta} \end{array}\right) \]

where in $H^{\alpha\alpha}$ and $H^{\beta\beta}$ the kinetic energy of a plane wave component changes to

$H^{}:\qquad |{\bf {k}}+{\bf {G}}|^{2}\rightarrow |{\bf {k}}+{\bf {G}}-{\bf {q}}/2|^{2}$

$H^{}:\qquad |{\bf {k}}+{\bf {G}}|^{2}\rightarrow |{\bf {k}}+{\bf {G}}+{\bf {q}}/2|^{2}$