# The B = 8h Skyrmion's vibrational modes

Each mode's symmetry is denoted as dI, it has a degeneracy d, and transforms as an irrep I of the Skyrmion's symmetry group. We use the notation of Cotton. Character tables may be found here.

The zero modes are translations, rotations and isorotations. These decompose as 1B2 + 2E1, 1A2 + 2E5 and 1A2 + 2E2 respectively.

The B = 8h Skyrmion has a torus on it's top and bottom. It is often useful to describe the vibrations in terms of deformations of these tori.

Freq. Symmetry Description/Notes Visualization
0.13 2E3
The top and bottom tori slide back and forth, out of phase, yielding a shear mode.
0.21 2E5
The top and bottom tori rotate around an axis perpendicular to their symmetry axis, out of phase.
0.24 2E4
The Skyrmion begins to split in half, along the x-axis and then the y-axis.
0.28 2E2
Two opposite vertices of the equatorial hexagon move upwards while the edges which lie between them move downwards.
0.33 1B1
The top and bottom tori rotate, out of phase.
0.40 2E1
A single Skyrmion lying on a vertex of the equatorial hexagon pulls away from the center. The remaining Skyrmion deforms to compensate for this motion.
0.45 1A1
The top and bottom tori move away from the center at the same time.
0.45 1B2
The top and bottom tori move away from, and then towards the center, out of phase.
0.45 2E2
The top and bottom tori split into individual Skyrmions, out of phase.
0.49 2E3
The tori move from side to side in phase, while the equator moves in the opposite direction to keep the center of mass constant.
0.5 2E2
The top and bottom tori split into individual Skyrmions, in phase. The equator deforms significantly to accommodate the splitting.
0.57 2E1
Two sides of the hexagonal equator breathe, out of phase.
0.69 1B2
The top and bottom tori breathe, out of phase.
0.70 1A1
The entire Skyrmion breathes.
0.75 2E5
Two opposite quarters of the Skyrmion inflate, pushing the torus closest to them away.
0.76 2E1
One half of the Skyrmion inflates, pushing both tori away from the center.
0.87 2E5
Two holes which lie on an axis stay stationary. The neighboring holes move towards one and away from the other.
0.88 2E4
The top and bottom tori split into individual Skyrmions, out of phase. Unlike the 0.50 mode, the equator does not deform.
0.89 1A1
The top and bottom tori breathe, in phase.
0.89 2E4
The equator inflates along one axis and deflates along the perpendicular one.
0.90 2E1
Similar to the 0.40 mode but physically due to isorotation.
0.96 2E2
Two neighboring faces move towards, then away, from one another.
1.01 2E4
Similar to the 0.24 mode but physically due to isorotation.
1.02 2E5
Similar to the 0.21 mode but physically due to isorotation.
1.03 2E3
A tri-axially symmetric breathing motion.

## Notes

Hover over an image (or if you're on a tablet/phone: tap on an image) to make it come to life. The hover-over text tells you the Cartesian realization of the element of the irrep you are looking at.