Analysis of the perturbation series for the specific heat of a thin-film superconductor nearHc2

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Abstract
The asymptotic perturbation series developed over many years from the Landau-Ginzburg theory is used to study the specific heat of a thin-film superconductor in a magnetic field. It is found that rewriting the series as an expansion in the entropy improves the self-consistency of its Padé or Padé-Borel resummations at low temperatures. However, there is a discrepancy between Monte Carlo data and the series resummations, possibly due to saddle-point contributions that are only a finite energy away from the point about which the perturbation expansion is performed. Our expansion is also used on the exactly soluble zero-dimensional Landau-Ginzburg model, our toy model. The results of the Padé and Padé-Borel resummations are more accurate than previous methods, but still do not converge to the exact low-temperature limit. The δ expansion and Stevenson transformation are tried on the toy model but the former is discarded because its convergence is too slow to be practical and the latter because results are very poor in the temperature region where the Borel sum fails to converge to the exact result. We conclude that no satisfactory resummation procedure exists for this problem. Finally Monte Carlo data are compared with recent experimental results and the agreement is found to be poor at low temperatures. The extension of our analysis to the series for bulk systems has been made. No strong evidence for a transition could be found.