# The B = 6 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 + 2E3 and 1A2 + 2E2 respectively.

Notes: it is helpful to think of the B=6 Skyrmion as three tori, stacked upon each other. The Cartesian realizations are orientated as if the C4 axis is aligned with the z-axis.

Freq. Symmetry Description/Notes Visualization
0.25 1A1
The outer tori move away from the central torus.
0.28 1B1
The outer tori rotate in opposite directions.
0.33 2E1
The outer tori rotate, deforming the Skyrmion like a hinge.
0.40 2E3
The outer tori shear, out of phase.
0.41 2E2
The outer tori split into two B=1 Skyrmions in the plane perpendicular to their common symmetry axis. The bottom mirrors the motion of the top, but rotated by 45 degrees. In both cases, the B=1's move towards a vertex of the core.
0.46 2E2
A similar motion as the 0.41 mode, but the B=1's move towards a hole of the core.
0.47 1B2
One of the outer tori pull away, leaving a B=4 core.
0.50 2E1
The center torus pulls away from the Skyrmion while the outer tori deform to maintain the center of mass.
0.52 2E2
Viewing the B=6 Skyrmion as two overlapping B=4 cubes, the cubes perform the B=4, 0.52 mode out of phase.
0.76 1A1
The breathing mode.
0.77 2E1
Two of the eight central holes grow while another two shrink.
0.80 2E3
Viewing the B=6 Skyrmion as two overlapping B=4 cubes, the cubes perform the B=4, 0.62 mode out of phase.
0.87 1B2
One half of the Skyrmion deflates as the other inflates. The central torus oscillates between the two outer tori, yielding a Newton's cradle motion of tori.
0.90 2E2
Similar to the 0.46 mode, but physically due to breathing.
0.95 1A1
The Skyrmion elongates then flattens, maintaining D4h symmetry.
0.95 2E3
One face inflates while the diagonally opposite face deflates.
1.01 2E1
Viewing the B=6 Skyrmion as two overlapping B=4 cubes, the cubes perform the B=4, 0.94 mode out of phase.
1.08 2E2
Similar to the 0.41 mode, but physically due to isorotation
1.11 2E3
Viewing the B=6 Skyrmion as two overlapping B=4 cubes, the cubes perform 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.