“Soft” Anharmonic Vortex Glass in Ferromagnetic Superconductors

Ferromagnetic order in superconductors can induce a spontaneous vortex (SV) state. For external field $\mathbf{H}=0\mathrm{}$, rotational symmetry guarantees a vanishing tilt modulus of the SV solid, leading to drastically different behavior than that of a conventional, external-field-induced vortex solid. We show that quenched disorder and anharmoinc effects lead to elastic moduli that are wave-vector dependent out to arbitrarily long length scales, and non-Hookean elasticity. The latter implies that for weak external fields $H$, the magnetic induction scales universally like $B\left(H\right)\sim B\left(0\right){+cH}^{\alpha }$, with $\alpha \approx 0.72$. For weak disorder, we predict the SV solid is a topologically ordered glass, in the “columnar elastic glass” universality class.