# The B = 8t 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 1A2u + 2Eu, 1A2g + 2Eg and 1A1u + 1B2u + 1B2g respectively.

There are several ways to describe the B=8 which are useful here. One is to view it as two B=4 cubes; another is as four B=2 tori. One can also identify it as central cube surrounded by two tori.

Freq. Symmetry Description/Notes Visualization
0.18 1A1u
The cubes rotate around their common C4 symmetry axis, out of phase.
0.19 1A2g
The cubes isorotate around the red iso-axis, out of phase.
0.22 2Eu
The cubes rotate towards each other forming a bent-arm.
0.25 1A1g
The cubes move away from each other.
0.33 1B1g
The cubes isorotate around the white/black iso-axis, out of phase.
0.35 1B1u
Each cube vibrates like the B=4, 0.48 mode out of phase.
0.38 2Eg
Similar to the 0.22 mode, but the rotation is in phase yielding a shear mode of the two cubes.
0.43 1A2u
The cubes each vibrate like the B=4, 0.46 mode out of phase. The outgoing tori lie in the plane perpendicular to the common C4 symmetry axis.
0.44 1A1g
The central cube vibrates like the B=4, 0.46 mode.
0.44 2Eu
Each cube vibrates like the B=4, 0.48 mode out of phase.
0.45 1B2u
Each cube vibrates like the B=4, 0.46 mode out of phase.
0.46 1B1g
Each cube vibrates like the B=4, 0.46 mode in phase.
0.47 1B2g
The outer tori vibrate like the B=2, 0.37 mode in phase.
0.48 1B2g
The central cube vibrates like the B=4, 0.48 mode.
0.48 2Eg
Each cube vibrates like the B=4, 0.48 mode, in phase.
0.52 1B1u
Each cube vibrates like the B=4, 0.52 mode, in phase.
0.59 2Eg
Each cube vibrates like the B=4, 0.62 mode out of phase realized in such a way that the central cube's deformation is small.
0.62 1B2u
Each cube vibrates like the B=4, 0.62 mode in phase realized in such a way that the central cube significantly deforms.
0.72 1A1g
The center cube breathes.
0.72 2Eu
Each cube vibrates like the B=4, 0.94 mode, realized such that two edges of the central cube pull away from each other.
0.81 1B1g
The central cube vibrates like the B=4, 0.87 mode while the top and bottom tori vibrate like the B=2, 0.37 mode. All motion is due to breathing.
0.83 1B2g
Each cube vibrates like the B=4, 0.94 mode in phase.
0.84 1B1u
The central cube vibrates like the B=4, 0.87 mode.
0.85 1A2u
One cube inflates while the other deflates. The central cube moves back and forth between the two tori.
0.86 2Eu
Two neighboring faces inflate while the opposite faces deflate. The faces in between move their positions to compensate.
0.88 2Eg
Each cube vibrates like the B=4, 0.87 mode out of phase.
0.91 1A1g
Both cubes breathe in phase.
0.94 2Eu
Similar to the 0.22 mode but physically due to isorotation.
0.98 1A2u
The top and bottom tori inflate and deflate out of phase.
1.03 2Eg
Each cube vibrates like the B=4, 0.94 mode out of phase.
1.05 1B1g
Similar to the 0.46 mode but physically due to a breathing motion.
1.06 1B1u
Similar to the 0.35 mode but physically due to a breathing motion.
1.08 1B2u
Similar to the 0.45 mode but physically due to a breathing motion.
1.10 2Eu
The two cubes vibrate like the B=4, 1.14 mode in phase.

## 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.