CASINO manual - Theory of Condensed Matter

36 Noncollinear-spin systems
36.1 Wave functions for noncollinear-spin systems
In a noncollinear-spin system, the particles of interest can have spin directions that are not parallel to
the global quantization axis and/or the spin direction can vary with position in space. To treat such
a system, it is not possible to assign a definite spin to each particle. Instead, the full four-dimensional
position–spin coordinates of the particles must be considered.
Casino can perform VMC calculations on noncollinear-spin systems by evaluating the energy expectation
value
〈Ψ(X)|Ĥ|Ψ(X) 〉
∑ ∫
S |Ψ(R, S)| 2 E L (R, S) dR
E =
= ∑ ∫ , (446)
〈Ψ(X)|Ψ(X)〉
S |Ψ(R, S)|2 dR
where X is the 4N-dimensional vector of position and spin coordinates for all the particles in the
system, R is their real-space positions and S is their spin coordinates. This is achieved by extending the
standard Metropolis algorithm, so that there is a spin-flip step which may change the spin coordinate
s in addition to the real-space position r of each particle. This extension is supported for VMC
methods 1 and 3.
casino supports noncollinear calculations for Slater–Jastrow(–backflow) wave functions of the form
ψ 1 (x 1 ) · · · ψ 1 (x N )
Ψ(X) = exp[J(R)]
.
.
. (447)
∣ ψ N (x 1 ) · · · ψ N (x N ) ∣
The Jastrow factor and backflow function can only depend on the real-space positions of the particles,
but the single-particle orbitals depend on both position- and spin-coordinates. Equivalently, we can
say that the single-particle orbitals in the determinant can be arbitrary two-component spinors.
There are no specific keywords in the input files that control whether casino performs a noncollinear
calculation or not. If noncollinear mode is supported for the selected system type and the input data
about the wave function indicates noncollinear orbitals, casino automatically performs a noncollinear
VMC calculation.
DMC does not support noncollinear spins.
36.2 Spiral spin-density waves in the HEG
Currently the only noncollinear-spin system type that can be studied using casino is a spiral spindensity
wave state in the HEG. In such a system, the single-particle orbitals are of the spinor form
ψ k
(r) = √ 1 (
e ik·r cos (
1
2 θ )
k)e −i 1 2 q·r
Ω sin ( 1 2 θ k)e +i 1 2 q·r , (448)
where k is the familiar plane-wave-vector, q is a constant vector (magnetization wave-vector) which
is the same for all orbitals, and the values θ k are independent parameters for each orbital. For each
orbital ψ k
, an electron can also occupy the orbital orthogonal to it, obtained by the replacement
θ k → θ k + π. A determinant of orbitals of the above form gives rise to a static, spiral spin density,
with wave-vector q.
Setting up a calculation of this form is very similar to a standard electron fluid calculation. In addition
to the usual parameters, the input file must contain a definition of the magnetization wavevector
in the free particles block and the correlation.data file must contain a block specific to
the SDW system, giving a definition of the occupied single-particle orbitals. For an example, see
~/CASINO/examples/electron phases/3D fluid sdw.
The calculation of the spin-density matrix and hence magnetization density in a spiral spin density
wave is described in Sec. 33.2.3.
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