Casimir interactions between magnetic flux tubes in a dense lattice

Abstract

We use the worldline numerics technique to study a cylindrically symmetric model of magnetic flux tubes in a dense lattice and the nonlocal Casimir forces acting between regions of magnetic flux. Within a superconductor the magnetic field is constrained within magnetic flux tubes and if the background magnetic field is on the order the quantum critical field strength, ${B}_{k}=\frac{{m}^{2}}{e}=4.4\ifmmode\times\else\texttimes\fi{}{10}^{13}$ Gauss, the magnetic field is likely to vary rapidly on the scales where QED effects are important. In this paper, we construct a cylindrically symmetric toy model of a flux tube lattice in which the nonlocal influence of QED on neighboring flux tubes is taken into account. We compute the effective action densities using the worldline numerics technique. The numerics predict a greater effective energy density in the region of the flux tube, but a smaller energy density in the regions between the flux tubes compared to a locally constant-field approximation. We also compute the interaction energy between a flux tube and its neighbors as the lattice spacing is reduced from infinity. Because our flux tubes exhibit compact support, this energy is entirely nonlocal and predicted to be zero in local approximations such as the derivative expansion. This Casimir-Polder energy can take positive or negative values depending on the distance between the flux tubes, and it may cause the flux tubes in neutron stars to form bunches. In addition to the above results we also discuss two important subtleties of determining the statistical uncertainties within the worldline numerics technique. Firstly, the distributions generated by the worldline ensembles are highly non-Gaussian, and so the standard error in the mean is not a good measure of the statistical uncertainty. Secondly, because the same ensemble of worldlines is used to compute the Wilson loops at different values of $T$ and ${x}_{\mathrm{cm}}$, the uncertainties associated with each computed value of the integrand are strongly correlated. We recommend a form of jackknife analysis which deals with both of these problems.