Two dimensional vortices in the presence of a periodic potential

Abstract

The major features of the phase diagram obtained from the simulation have been reproduced qualitatively in the density functional study. Of course, the DFT does not capture the continuous nature of the transition, which is to be expected from a mean-field treatment. But it has also to be pointed out that though the different phases in the temperature–substrate strength plane obtained from the DFT calculation match those seen in the MC simulation in terms of symmetry, the DFT results and the ones from simulation do differ in details. This is most pronounced in the density distribution in the low-temperature partially pinned phase, where the displacement of the density peak from the center of the square is much less than what one sees in the simulation, and the density distribution seems to be unusually flattened. Also, the actual temperatures of the transitions are different in the two cases. There are quite a few possible reasons for these discrepancies. To start with, we are truncating the free energy functional to second order in the density difference. This could bring in some inaccuracies into the analysis. Even within this approximation, we are not using the exact correlation functions, the HNC solution being an approximate one. The HNC approximation usually underestimates the correlations at a given temperature, and hence one can expect a transition to the partially pinned phase at a higher temperature if the correct correlation functions are used. This would bring the transition temperatures obtained in the DFT calculation closer to the values seen in the simulation. Also, the correct reference state about which the free energy should be expanded is the nonuniform modulated liquid, rather than the uniform liquid phase we have used. But this brings in the complication of computing the two-body correlations in the presence of the substrate potential, which renders the system inhomogeneous. This is difficult to treat analytically. One can ideally go back to the simulations, compute the two-body correlations from there (which will now depend on both r and r? rather than |r ? r?|) and use those in the DFT calculations. In practice, however, it would be extremely difficult to carry out such a calculation.