Effects of high pressure, magnetic fields, and substitutions - ICMR ...

Symmetry and Landau theory (Kenzelmann, Mostovoy)
Lowest order coupling Φ em
(P,M) between FE polarization, P, and magnetic
modulation, M, that is invariant upon time reversal and spatial inversion:
t -> -t : P -> P and M -> -M Φ em
quadratic in M
x -> -x: P -> -P and M -> M Φ em
~ P and ∇M (Lifshits invariant), P uniform
for cubic symmetry: Φ em
= const P* [ M(∇*M) – (M*∇)M ]
Φ e
= P 2 /(2χ e
) =>
P = const χ e
[ M(∇*M) – (M* ∇)M ]
Spiral magnetic order: M = m 1
e 1
cos(Qx) + m 2
e 2
sin(Qx) + m 3
e 3
Average polarization: = const χ e
m 1
m 2
[ e 3
× Q ]
Sinusoidal magnetic order: m 1
or m 2
=0 => = 0