Dieter Belitz: Skyrmion Flux Lattices

Dieter Belitz presented a theory for the skyrmion flux lattice in
triplet (p-wave) superconductors.

Dieter started out by noting that in singlet (s-wave)
superconductors, the superconducting order parameter possesses an
SO(2) symmetry, in which case the topological excitations are given
by (conventional) vortices. The energy per length of the vortex is
E_vortex=Phi^2 * ln(R)/\lambda^2, where R=lambda/xi, lambda
is the Kondo penetration depth, xi is the superconducting coherence
length, and phi is the flux quantum. In an applied magnetic field,
the vortices form an Abrikosov flux lattice with one flux quantum
per vortex.

Dieter then pointed out that in a triplet superconductor, the spin
sector forms an SO(3) subgroup, which allows two different types of
topological excitations: vortices and skyrmions.

Dirk Morr asked whether Dieter considers a particular spin state, as
represented by the d-vector in a triplet superconductor, and Dieter
replied that he consider the non-unitary spin state described by
d=(1,i,0), representing |up,up> - pairing. Diete then drew a picture
of a skyrmion, in which the spin part of the superconducting order
parameter rotates from |up,up> to |down,down> as one moves radially
outward from the center of the skyrmion. Diete noted that there is
no singularity at the center of the skyrmion, in contrast to a
vortex. Dieter showed that the energy per unit length of the
skyrmion is E_s= Phi^2 /\lambda^2 which is smaller than the vortex
energy E_vortex for R>>1 (Dieter noted in passing that this result
was obtained in a purely classical theory). The skyrmion lattice
contains two flux quanta per skyrmion.

Dieter then described a perturbative result (in 1/R) for the energy
of a skyrmion as a function of the skyrmion radius, which is given
by E(R)= Phi^2 /\lambda^2 *(1 + 1/R - ln(R)/R^2 - 1/R^2 + ...) (R
is given in units of lambda). Dieter noted that this result agrees
very well with a numerical solution of the problem by Rosenstein.
The resulting skyrmion potential is then given by V ~ 1/R, in
contrast to the vortex potential that is given by V ~ exp(-R)/R.

This long-range interaction leads to some distinct differences betweenvortex
flux lattices and skyrmion flux lattices. In particular, they have qualitatively
different melting curves. Dieter sketched a phase diagram for a vortex flux
lattice, which always melts if one gets sufficiently close to the lower critical
field H_c1, and one for a skyrmion flux lattice, which never melts close to H_c1.