Anomalous scaling dimensions and critical points in type-II superconductors

The existence of a stable critical point, separate from the Gaussian and XY critical points, of the Ginzburg-Landau theory for superconductors, is demonstrated by direct extraction via Monte-Carlo simulations, of a negative anomalous dimension ηφ of a complex scalar field φ forming a dual description of a neutral superfluid. The dual of the neutral superfluid is isomorphic to a charged superfluid coupled to a massless gauge-field. The anomalous scaling dimension of the superfluid order-field is positive, while we find that the anomalous dimension of the dual field is negative. The dual gauge-field does not decouple from the dual complex matter-field at the critical point. These two critical theories represent separate fixed points. The physical meaning of a negative ηφ is that the vortex-loop tangle of the superfluid at the critical point fills space more efficiently than random walkers, without collapsing. This is due to the presence of the massless dual gauge-field, and the resulting long-ranged vectorial Biot-Savart interaction between vortex-loop segments, which is a relevant perturbation to the steric |ψ| repulsion term. Hence, the critical dual theory is not in the universality class of the |ψ|-theory. A charged superfluid (spin-singlet superconductor) is described by the Ginzburg-Landau theory of a complex matter field ψ coupled to a massless gauge-field A, with coupling constant 2e and a local gauge-symmetry; here e is the electron charge. In 3D, the dual of this theory is given by a dual complex matter-field φ coupled to a massive dual gauge-fieldhwith mass e, exhibiting a globalU(1)symmetry due to the mass of h [1]. It renders the critical point of the theory in the universality class of the |φ|-theory. In renormalization group sense, the short-range interaction mediated by the massive h, is an irrelevant perturbation to the steric repulsion stemming from the |ψ|-term [2]. How1 Corresponding author. FAX: +47 73 59 77 10. E-mail: [email protected] ever, in the limit e → 0, the dual gauge-field becomes massless, and the symmetry of the problem changes to a local gauge-symmetry. This profoundly changes the physics. In this case, the |ψ|-theory of the 3D superfluid, Hψ = |∂iψ| 2+m2ψ|ψ| 2+(uψ/2)|ψ| 4 has a dual theory given by Hφ,h = |(∂i − ighi)φ| 2 + m2φ|φ| 2 + (uφ/2)|φ| 4 + (1/2)(∇×h) [1]. The coupling constant g is given by g = 2πmh/e, where mh = eω, where ω is the amplitude of the order-field ψ. Thus, the dual of the neutral superfluid is isomorphic to the Ginzburg-Landau theory of a charged superfluid coupled to a massless (dual) gauge-field h. In Ref. [3] it is argued that as T → T− c , ω → 0. This would imply that at and above Tc, h decouples from φ. Thus, the critical dual theory would Preprint submitted to Physica B 1 February 2008 be that of a |φ|-theory, i.e. the superfluid and its dual theory would be self-dual. This is a mean-field argument. At the true Tc, ω > 0, and hence the dual gauge-field does not decouple from φ as T → T− c . At the true critical point, the dual of the |ψ|-theory is not a |φ|theory, in particular the symmetries of Hψ and Hφ,h, are different. We thus expect the critical exponents of these two theories to be different, describing two separate critical points. This has been checked by large-scaleMonte-Carlo simulations [4]. The main point of this short communication is the following. Direct evaluation of the anomalous dimension of the matter field ψ, yields ηψ = 0.038 [5]. In contrast, the anomalous dimension of the dual field φ, is given by ηφ = −0.18 ± 0.07 [4]. The latter result is obtained in our Monte-Carlo simulations by analysing the statistics of vortexloops in the 3DXY -model, using a mapping to the dual theory. Therefore, we demonstrate that the critical behavior of the superfluid, which is a |ψ|theory, is in a different universality class than the critical behavior of the dual theory. The results of Ref. [6] thus rest on a firm theoretical footing, in that the existence of a novel charged fixed point need not be assumed, it is demonstrated. The results for the vortex-loop distribution function D(p) as a function of vortex-loop perimeter p are shown in Fig. 1 [4]. We fit D(p) to the form D(p) = Ap exp(−ε(T )/kBT ) where T is temperature, α ≈ 2.5, and ε(T ) is the line tension of the thermally generated vortex-loops, the topological defects destroying superfluid order [4]. We fit T -dependence of ε(T ) to the form ε(T ) ∼ |T − Tc| γ [4]; γ is identified as the susceptibility exponent γφ of the dual field φ [4]. By the scaling law γφ = νφ(2−ηφ) and the observation that νφ = νψ [6,4], where νψ = 0.672 [5], we extract ηφ from the line-tension of the vortex-loops of the 3DXY model. We find γφ = 1.45± 0.05, and hence ηφ = −0.18 ± 0.07. This should be compared to ηψ = 0.038 [5]. Thus, the interaction mediated by the massless gauge-field h is a relevant perturbation to the non-trivial fixed-point of the |φ|-theory, in renormalization group sense, 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 10 10