Elasticity, fluctuations, and vortex pinning in ferromagnetic superconductors: A columnar elastic glass

We study the elasticity, fluctuations, and pinning of a putative spontaneous vortex solid in ferromagnetic superconductors. Using a rigorous thermodynamic argument, we show that in the idealized case of vanishing crystalline pinning anisotropy the long-wavelength tilt modulus of such a vortex solid vanishes identically, as guaranteed by the underlying rotational invariance. The vanishing of the tilt modulus means that, to lowest order, the associated tension elasticity is replaced by the softer, curvature elasticity. The effect of this is to make the spontaneous vortex solid qualitatively more susceptible to the disordering effects of thermal fluctuations and random pinning. We study these effects, taking into account the nonlinear elasticity, that, in three dimensions, is important at sufficiently long length scales, and showing that a “columnar elastic glass” phase of vortices results. This phase is controlled by a previously unstudied zero-temperature fixed point, and it is characterized by elastic moduli that have universal strong wave-vector dependence out to arbitrarily long length scales, leading to non-Hookean elasticity. We argue that, although translationally disordered for weak disorder, the columnar elastic glass is stable against the proliferation of dislocations and is, therefore, a topologically ordered elastic glass. As a result, the phenomenology of the spontaneous vortex state of isotropic magnetic superconductors differs qualitatively from a conventional, external-field-induced mixed state. For example, for weak external fields $H$, the magnetic induction scales universally like $B\left(H\right)\sim B\left(0\right)+c{H}^{\alpha }$, with $\alpha \approx 0.72$.

Figure 1

The nonlinear and universal power-law $B\left(H\right)$ scaling, which at weakest fields $H<H_{cr}$ and strongest fields $H>H_{NL}$ is cut off by the crystal symmetry breaking anisotropy and ${\xi }^{NL}$, respectively.

Figure 2

Schematic representation of the mechanism of random tilt pinning. Randomly oriented anisotropic clusters of point-pins exert random torques on vortex lines.

Figure 3

Renormalization group flow of the ratio of the Lamé coefficients $x=\lambda ∕\mu$. The fixed point at $x_2=-0.033$ controls the long-scale properties (anomalous elasticity and associated phenomenology) of the pinned spontaneous vortex solid. The critical point at $x_1=-4∕5$ controls the mechanical instability of the spontaneous vortex solid.

Figure 4

A plot of the function $\mathbf{(}{F}_D\left(x\right)-F_{\mu }\left(x\right)\mathbf{)}$, with its zeros, $\mathbf{(}{F}_D\left(x\right)-F_{\mu }\left(x\right)\mathbf{)}=0$ at $x_1=-4∕5$ and $x_2\simeq -0.03272$ determining the fixed-point solutions for the Lamé coefficients ratio, $x_{*}={\lambda }^{*}∕{\mu }^{*}.$

Figure 5

A square spontaneous vortex with an extra row of flux lines introduced from the right, along the $x$ axis. The core of the dislocation comes out of the page.

Figure 6

The Feynman diagrams that dominate the renormalization of the reduced dual temperature $t$.

Figure 7

The region of ${\eta }_{\kappa }$ and ${\eta }_{\mu }$ that satisfy the criteria Eqs. (7.52a, 7.52b, 7.52c) for a long-range orientationally ordered, stable (at sufficiently low disorder strength) columnar elastic-glass phase.