Ultra Large-scale Exact-diagonalization for Confined Fermion-Hubbard Model on the Earth Simulator : Exploration of Superfluidity in Confined Strongly-Correlated Systems

In order to explore a possibility of superfluidity in confined strongly-correlated fermion systems, e.g., nano-scale cuprate High-Tc superconductors and atomic Fermi gases loaded on optical lattice, we implement an exact diagonalization code for their mathematical model, i.e., a trapped Hubbard model on the Earth Simulator. We compare two diagonalization algorithms, the traditional Lanczos method and a new algorithm, the preconditioned conjugate gradient (PCG) method, and find that when using the PCG the total CPU time can be reduced to 1/3 ~ 1/5 compared to the former one since the convergence can be dramatically improved by choosing a good preconditioner and the communication overhead is much more efficiently concealed in the PCG method. Consequently, such a performance improvement enables us to do systematic studies for several parameters. Numerical simulation results reveal that an unconventional type of pairing specific to the confined system, which may cause superfluidity, develops under a strong repulsive interaction.