Langevin Simulation of Nonlocal Ginzburg-Landau Model for Superconductors in a Magnetic Field

We numerically investigate the phenomenological nonlocal Ginzburg-Landau Hamiltonian for two-dimensional superconductors in a strong magnetic field by Langevin equation. We calculate the structure factor and Abrikosov factor at various temperatures. We also evaluate the specific heat and obtain a cusp which indicates the melting of the vortex lattice. PACS numbers: 74.20.De, 74.60.Ge Typeset using REVTEX 1 The study of the high Tc superconductor (HTSC) in a magnetic field has renewed the interests in the mixed state. The most characteristic behavior of this unconventional superconductor is the large superconducting fluctuation, and it is known that the vortex liquid phase appears below the Hc2 line in the H − T phase diagram. The melting transition of the vortex lattice has been studied theoretically ever since the discovery of the HTSC. [1,2] For the two-dimensional case, there has been a large amount of studies which investigate the vortex lattice melting transition of the Kosterlitz-Thouless type which is accompanied by the dissociation of the dislocation pairs. [3,4] Experimentally, the vortex lattice melting transition is observed as the first order phase transition by revealing a sharp step of the magnetization in both Bi2Sr2CaCu2O8+δ and Y Ba2Cu3O7−δ. [5,6] The first order melting transition of the vortex lattice is already predicted theoretically from the renormalization group analysis based on Ginzburg-Landau model for three dimensional system. [7] Although it is not yet certainly clear whether the first order melting transition survive in the strong magnetic field phase where the fluctuation effect is very large and the system behaves more like two-dimensional. On the other hand, the oxide high Tc superconductor has a layered structure and it is considered as a quasi-two-dimensional system. It is pointed out that the two-dimensional square lattice structure of CuO2 plane has an effect not only on the symmetry of the order parameter but also on the vortex lattice configuration. [8] Recently many experimental evidences which support the notion that the order parameter of HTSC has dx2−y2 symmetry are reported. The expansion of the GL model into a form which deals with the d-wave superconductors are thus necessary in order to investigate the vortex state in high-Tc cuprates. In Ref. [9], we have previously investigated a nonlocal GL model starting with following two motivations. One is that the nonlocal GL model has a remarkable similarity to the ncomponent GL model. We find that a parameter of the characteristic range of the nonlocal interaction plays a role of n for the n-component GL model. The other one is that because of the shortness of the coherence length of HTSC, modification of the quartic interaction of the order parameter into a nonlocal form is considered to be reasonable. 2 We introduce a phenomenological nonlocal Hamiltonian, H[ψ] = ∫