# The B = 8u 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 two ways to describe the B=8 which are useful here. One is to view it as two B=4 cubes; the other is as four B=2 tori.

Freq. Symmetry Description/Notes Visualization
0.12i 1A2g
Each cube isorotates around the red isospin axis, in opposite directions. This mode connects the B=8u Skyrmion to the lower energy B=8t Skyrmion, hence its imaginary frequency.
0.01 1A1u
The cubes rotate around their common C4 symmetry axis, out of phase.
0.21 2Eu
The cubes rotate towards each other, making a bent-arm shape.
0.22 2Eg
The cubes rotate away from each other, yielding a shear mode.
0.29 1B2u
The cubes isorotate about the white/black axis out of phase.
0.30 1A1g
The cubes move away from each other.
0.34 1B2g
The outer tori vibrate like the B=2, 0.37 mode in phase.
0.40 1A1g
Each cube vibrates like the B=4, 0.46 mode. The individual tori come out along the C4 symmetry axis.
0.45 1B1g
Each cube vibrates like the B=4, 0.46 mode in phase. The tori come out along the axes perpendicular to the C4 symmetry axis.
0.47 1B1u
The outer tori vibrate like the B=2, 0.37 mode out of phase.
0.47 2Eg
Each cube vibrates like the B=4, 0.48 mode in phase.
0.48 1A2u
The two cubes vibrate like the B=4, 0.46 mode out of phase.
0.49 2Eu
Each cube vibrates like the B=4, 0.48 mode out of phase.
0.51 1B1u
The central cube vibrates like the B=4, 0.52 mode.
0.52 1B2u
Each cube vibrates like the B=4, 0.46 mode out of phase.
0.53 1B2g
Each cube vibrates like the B=4, 0.52 mode out of phase.
0.57 1B1g
Each cube vibrates like the B=4, 0.62 mode such that one cube is a mirror image of the other.
0.57 1B2g
Each cube vibrates like the B=4, 0.48 mode in phase.
0.68 2Eu
Each cube vibrates like the B=4, 0.62 mode such that one cube is a mirror image of the other.
0.70 2Eg
Each cube vibrates like the B=4, 0.62 mode out of phase.
0.78 1A1g
The central cube breathes.
0.85 1A2u
Similar to the 0.40 mode but physically due to a breathing motion.
0.86 2Eg
Each cube vibrates like the B=4, 0.87 3T1u mode out of phase.
0.86 2Eu
The central cube vibrates like the B=4, 0.87 3T1u mode.
0.88 1B1u
Each cube vibrates like the B=4, 0.94 mode out of phase.
0.91 1A1g
Each cube breathes, in phase.
0.92 2Eu
Similar to the 0.68 mode but physically due to an isorotation.
0.93 1B2u
The cubes vibrate like the B=4, 0.94 mode in phase.
0.96 1B1g
Similar to the 0.45 mode but physically due to a breathing motion.
0.96 1A2u
Similar to the 0.48 mode but physically due to a breathing motion.
0.97 2Eg
Each cube vibrates like the B=4, 0.94 mode in phase.
1.04 1B2g
Similar to the 0.53 mode but physically due to a breathing motion.
1.09 1B2u
The central cube vibrates like the B=4, 0.94 mode.
1.13 2Eu
Each cube vibrates 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.