Ferromagnetic order in superconductors can induce a spontaneous vortex (SV) state. For external field H = 0, 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 anharmonic effects lead to elastic moduli that are wavevectordependent 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(H) ∼ B(0) + cH α , with α ≈ 0.72. For weak disorder, we predict the SV solid is a topologically ordered vortex glass, in the "columnar elastic glass" universality class. 64.60Fr,05.40,82.65Dp Rare-earth borocarbide materials exhibit a rich phase diagram that includes superconductivity, antiferromagnetism, ferromagnetism and spiral magnetic order. [1][2][3] In particular, there is now ample experimental evidence that, at low temperatures, superconductivity and ferromagnetism competitively coexist in ErNi 2 B 2 C compounds. Other possible examples of such ferromagnetic superconductors (FS) are the recently discovered high temperature superconductor Sr 2 Y Ru 1−x Cu x O 6 and the putative p-wave triplet strontium ruthenate superconductor, Sr 2 Ru O 4 , which spontaneously breaks time reversal symmetry. For sufficiently strong ferromagnetism, such FS's have been predicted [3] to exhibit a spontaneous vortex (SV) state driven by the spontaneous magnetization, rather than by an external magnetic field H. The novel phenomenology of the associated SV solid is the subject of this Letter.Here we will show that for H = 0, the elastic properties of the resulting SV solid differ dramatically and qualitatively from those of a conventional Abrikosov lattice. The key underlying difference is the vanishing of the tilt modulus, which is guaranteed by the underlying rotational invariance (but see below). Although this invariance is broken by the magnetization, M, the tilt modulus remains zero because this breaking is spontaneous. This contrasts strongly with a conventional vortex solid, where the rotational symmetry is explicitly broken by the applied field H. All of our conclusions, e.g., the unusual B(H) relation, illustrated in Fig.1 are a direct consequence of this important observation.In particular, we find that this "softness" (i.e., vanishing tilt modulus) of the SV lattice drastically enhances the effects of quenched disorder. As in conventional vortex lattices [4], any amount of disorder ∆ V , however weak, is sufficient to destroy translational order in SV lattices. Here the finite ordered domains are divergently anisotropic, with dimensions ξ

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We study the collective states formed by two-dimensional electrons in Landau levels of index n > or = near half filling. By numerically solving the self-consistent Hartree-Fock (HF) equations for a set of oblique two-dimensional lattices, we find that the stripe state is an anisotropic Wigner crystal (AWC), and determine its precise structure for varying values of the filling factor. Calculating the elastic energy, we find that the shear modulus of the AWC is small but finite (nonzero) within the HF approximation. This implies, in particular, that the long-wavelength magnetophonon mode in the stripe state vanishes q(3/2) like as in an ordinary Wigner crystal, and not like q(5/2) as was found in previous studies where the energy of shear deformations was neglected.

We study the cohesive energy and elastic properties as well as normal modes of the Wigner and bubble crystals of the two-dimensional electron system (2DES) in higher Landau levels. Using a simple Hartree-Fock approach, we show that the shear moduli (c66's) of these electronic crystals show a non-monotonic behavior as a function of the partial filling factor ν * at any given Landau level, with c66 increasing for small values of ν * , before reaching a maximum at some intermediate filling factor ν * m , and monotonically decreasing for ν * > ν * m . We also go beyond previous treatments, and study how the phase diagram and elastic properties of electron solids are changed by the effects of screening by electrons in lower Landau levels, and by a finite thickness of the experimental sample. The implications of these results on microwave resonance experiments are briefly discussed.

A review of the theory of the lock-in transition of vortices onto the layer direction in type II layered superconductors is presented. This phenomenon originates from the lowering of free energy when vortex cores are centered between adjacent layers. The approaches for the low-temperature “quasi-2D" as well as the 3D anisotropic regimes are reviewed. variational method. The phenomenological model with a modulated vortex core is shown In particular, the temperature variation of the core energy modulation is obtained by a to work in all the temperature range. The behaviour of the reversible torque is described as a function of the various parameters: field, temperature, anisotropy, and the existence of the London maximum is discussed. A simple interpretation of the transition is given as well as the influence of demagnetizing fields.