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

The generalized Bloch condition redefines the Bloch functions as follows:

$\Psi _{\bf {k}}^{\uparrow }({\bf {r)=\sum _{\bf {G}}{\rm {C_{\bf {k{\bf {G}}}}^{\uparrow }e^{i({\bf {k+{\bf {G-{\frac {\bf {q}}{2}})\cdot {\bf {r}}}}}}}}}}}$

$\Psi _{\bf {k}}^{\downarrow }({\bf {r)=\sum _{\bf {G}}{\rm {C_{\bf {k{\bf {G}}}}^{\downarrow }e^{i({\bf {k+{\bf {G+{\frac {\bf {q}}{2}})\cdot {\bf {r}}}}}}}}}}}$

This changes the Hamiltonian only minimally:

$\left({\begin{array}{cc}H^{\uparrow \uparrow }&V_{\rm {xc}}^{\uparrow \downarrow }\\V_{\rm {xc}}^{\downarrow \uparrow }&H^{\downarrow \downarrow }\end{array}}\right)\rightarrow \left({\begin{array}{cc}H^{\uparrow \uparrow }&V_{\rm {xc}}^{\uparrow \downarrow }e^{-i{\bf {q\cdot {\bf {r}}}}}\\V_{\rm {xc}}^{\downarrow \uparrow }e^{+i{\bf {q\cdot {\bf {r}}}}}&H^{\downarrow \downarrow }\end{array}}\right),$

where in $H^{\uparrow \uparrow }$ and $H^{\downarrow \downarrow }$ the kinetic energy of a plane wave component changes to:

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

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