Helicity modulus and Meissner effect in a fluctuating type-II superconductor.

The helicity modulus for a fluctuating type II superconductor is computed within the elastic medium approximation, as a probe of superconducting phase coherence and the Meissner effect in the mixed state. We argue that at the vortex line lattice melting transition, there remains superconducting coherence parallel to the applied magnetic field, provided the vortex line liquid retains a finite shear modulus at finite wavevector. 74.60.Ge, 64.60-i, 74.40.+k Typeset using REVTEX 1 In the high temperature superconductors, fluctuation effects are believed to be important over a wide region of the H − T phase diagram [1–5]. Recently there has been much controversy concerning the effect of vortex line fluctuations on long range order of the superconducting wavefunction in the mixed state [4–8]. Part of this controversy has concerned the definition of the proper gauge invariant correlation function. In this work we reconsider the question of superconducting coherence by considering instead the behavior of the helicity modulus [9] of the superconductor, which has proven a valuable criteria for coherence in related superfluid [10] and Josephson array [11] models. As will be seen below, it is an intrinsically gauge invariant quantity. Furthermore, the helicity modulus is equivalent to the linear response coefficient between an applied perturbation in magnetic field and the resulting supercurrent. Hence it more directly probes one of the most characteristic properties of a superconductor, the partial Meissner effect of the mixed state. Working in the elastic medium approximation, we compute the helicity modulus to lowest order in the fluctuation of vortex lines. We show that vortex line fluctuations have a dramatically different effect on the helicity modulus parallel versus transverse to the applied magnetic field. For the parallel case, the helicity modulus gives total screening as in the Meissner effect. We argue that this total screening is unaffected by the vortex line lattice melting transition, provided the vortex liquid retains a finite shear modulus at finite wavevector. In the limit of an extreme type II superconductor with λ → ∞, our results explain recent numerical simulations by Li and Teitel [12] which show clearly the persistence of phase coherence parallel to the applied field, well into the vortex liquid state. Similar results were obtained by Feigel’man et al. [13], working with a related 2d boson model. We show a simple relation between their results and ours, which can be expressed in terms of the “winding number” of vortex lines. For simplicity, we carry out our calculation for an isotropic superconductor; the extension to the uniaxial anisotropic case is straightforward. We work within the London approximation, which should be valid provided one is not too close to Hc2. The Landau-Ginzburg Helmholtz free energy [14] for an isotropic uniform superconduc2 tor, within the London approximation of constant wavefunction amplitude, can be written as,