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.