Spin spirals: Difference between revisions
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== Basis set considerations == | == 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 | |||
:<math> | |||
H^{}:\qquad |{\bf k} + {\bf G}|^2 \rightarrow |{\bf k} + {\bf G} - {\bf q} /2|^2 | |||
</math> | |||
:<math> | |||
H^{}:\qquad |{\bf k} + {\bf G}|^2 \rightarrow |{\bf k} + {\bf G} + {\bf q} /2|^2 | |||
</math> |
Revision as of 13:08, 6 July 2018
Generalized Bloch condition
Spin spirals may be conveniently modeled using a generalization of the Bloch condition (set LNONCOLLINEAR=.TRUE. and LSPIRAL=.TRUE.):
i.e., from one unit cell to the next the up- and down-spinors pick up an additional phase factor of and , 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:
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