arXiv:cond-mat/9807338v3 [cond-mat.stat-mech] 18 Jan 1999 Vortex Loop Phase Transitions in Liquid Helium, Cosmic Strings, and High-T c Superconductors Gary A. Williams Department of Physics and Astronomy, University of California, Los Angeles, CA 90095 (October 13, 2018) The distribution of thermally excited vortex loops near a superfluid phase transition is calculated from a renormalized theory. The number density of loops with a given perimeter is found to change from exponential decay with increasing perimeter to algebraic decay as Tc is approached, in agree- ment with recent simulations of both cosmic strings and high-Tc superconductors. Predictions of the value of the exponent of the algebraic decay at Tc and of critical behavior in the vortex density are confirmed by the simulations, giving strong support to the vortex-folding model proposed by Shenoy. 64.60.Cn, 67.40.Vs, 11.27.+d, 74.20.-z. The role of thermally excited vortex loops in three- dimensional phase transitions where a U(1) symmetry is broken recently has become a prime topic in cosmology [1,2] and high-T c superconductivity [3–5]. These tran- sitions are in the same universality class as the super- fluid λ-transition in 4 He. In the helium case our orig- inal renormalization theory based on vortex loops [6,7] has been extended to calculations of the specific heat [8] and to the dynamics of the transition [9]. In this theory the Landau-Ginzburg-Wilson Hamiltonian is rewritten to cast it in terms of its elementary excitations, spin waves and vortex loops, providing an alternative method for carrying out the renormalization process, compared to the more traditional perturbation theories which expand the Hamiltonian expressed in terms of the order param- eter. Here we further employ the loop theory to gain insight into recent simulations of the high-T c transition [4,5] and of cosmic-string phase transitions in the early universe [1,2]. The probability of occurrence of a vortex loop with perimeter P is calculated, and in agreement with the simulations we find a crossover from quasi-exponential decay of the probability with perimeter at low temper- atures, to purely algebraic decay precisely at T c . This provides strong support for the phenomenological ”Flory scaling” treatment of the random-walking loops devel- oped by Shenoy and co-workers [10]. In the vortex-loop theory the superfluid density ρ s is reduced by thermally excited loops whose average diam- eter a increases as the temperature is increased, and the density is finally driven to zero at T c by loops of infinite size [6]. Defining a dimensionless superfluid density by K r =¯ h 2 ρ s a o /m 2 k B T , where m is the mass of the 4 He atom and a o the smallest ring diameter, the equation for the renormalized density is given by [8] 1 K r = 1 K o + A o  a ao  a a o  6 exp  - U (a) k B T  da a o . (1) Here A o =4π 3 /3, K o is the ”bare” superfluid density resulting from the spin waves (and is the initial value of K r ), and U(a) is the renormalized energy of a ring, given by U (a)/k B T = π 2  a ao K r (ln  a a c  + 1) da a o + π 2 K o C (2) where C is a nonuniversal constant characterizing the core energy. For helium C and a o are determined from two experimental inputs, T c = 2.172 K and the amplitude of the superfluid density [8], yielding C = 1.03 and a o = 2.5 ˚ A. The effective core size a c in Eq. (2) was suggested by Shenoy and co-workers [7,10] to be a result of the ran- dom walk of the loop giving rise to radial fluctuations of order a c about the average diameter. This folding of the loop occurs because antiparallel vortex segments lower the energy. A simple polymer-type calculation [10] us- ing energy-entropy arguments yields a c /a =(K r a/a o ) θ , where θ = d/(d + 2) = 0.6 in d = 3 dimensions has the same form as the well-known Flory exponent of the self- avoiding walk. Eqs. (1) and (2) then constitute a coupled set of in- tegral equations for the renormalized superfluid density, and can be solved recursively starting from the bare scale a o and iterating to distances greater than the correlation length ξ = a o /K r . In practice these are converted to a set of coupled differential equations similar to the Kosterlitz recursion relations [11] for the two-dimensional case, and are solved using a Runge-Kutta technique [9]. As T is increased (K o decreased) the solution for ρ s falls to zero as (T c -T) ν , with ν = 0.6717 for θ = 0.6. This can be better matched to the most precise experimental value [12] ν = 0.6705 by adjusting to θ = 0.594, which is rea- sonable since it is known that the Flory-type arguments are not exact in three dimensions [13]. The arguments of Ref. 10 also yield a result for the average perimeter of a loop of diameter a, 1