Critical fluctuations in superconductors and the magnetic field penetration depth.

The superconducting transition is studied within the one-loop RG in fixed dimension D = 3 and at the critical point. A tricritical behavior is found and, for κ > κc, an attractive charged fixed point, distinct from that of a neutral superfluid. The critical exponents of the continuous transition are evaluated and it is shown that the anomalous dimension of the gauge-field equals unity. This implies the proportionality of the magnetic field penetration depth and the superconducting correlation length below the transition. The penetration depth exponent is non-classical. We argue that it cannot be extracted from the dual theory in a straigthforward manner since it is not renormalized by fluctuations of the dual field. PACS: 74.40.+k The problem of a charged scalar field coupled to a gauge vector potential arises frequently in theoretical physics. In its original version, it describes formation of Meissner state in superconductors [1] and Higgs mechanism in particle physics [2]. Furthermore, the nematicsmectic-A transition in liquid crystals [3] and, more recently, the transitions between plateaus in the quantum Hall effect [4, 5] and the finite-magnetic-field critical behavior in extreme type-II superconductors [6] have also been related to this problem. In a superconductor, the scalar field represents the fluctuating superconducting order parameter Ψ which, BCS pairs being charged, is coupled to fluctuations in the electromagnetic potential ~ A. At the mean-field level the transition is discontinuous and remains so when fluctuations in Ψ are included via the ǫ(≡ 4 − D)-expansion [1]. Numerical simulations [7, 8] of related lattice models support this scenario for smaller values of the Ginzburg parameter κ. However, for large κ, the results are consistent with a continuous, second-order phase transition [7, 8]. The picture obtained in numerical work is in accordance with the 1/n-expansion [1, 9] and with duality arguments which connect the lattice version of the theory to a dual gas of interacting vortex loops and the “inverted” 3D XY model [7, 10]. In this Letter we study this superconducting-Higgs electrodynamics (SHE) directly in D = 3 within perturbation theory at the critical point corresponding to the charged superfluid. Our results are as follows: We first show that the anomalous dimension of the gauge field, ηA, equals unity to all orders in perturbative expansion. By combining this result with the Josephson relation, we argue that the magnetic field penetration depth, λ, and the superconducting correlation length, ξ, diverge with the same exponent (ν) as the transition is approached from below. These results should be contrasted with ηA = 0 and λ ∝ √ ξ at the unstable critical point for neutral superfluid. We then demonstrate that our one-loop results imply the presence of the tricritical point in SHE. For bare κ < κc the renormalization group (RG) flows are unstable and the transition is likely first order while, for κ > κc, we find a stable fixed point indicating a continuous transition. By selecting the regularization