Spin spirals

Generalized Bloch condition

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

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- 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, 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:

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)

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:

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/":): \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}}}}}}}}}}}}

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/":): \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:

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}{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 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/":): H^{{\uparrow \uparrow }} 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/":): H^{{\downarrow \downarrow }} the kinetic energy of a plane wave component changes to:

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

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

In the case of spin-spiral calculations the cutoff energy of the basis set is specfied by means of the ENINI-tag.