  As the surface Andreev bound state (ABS) forms at the open ($1,1$) edge of a
$d_{x^2-y^2}$-wave superconductor, the local density of states (LDOS)
increases. Therefore, a strong electron correlation and drastic phenomena may
occur. However, a theoretical study on the effects of the ABS on the electron
correlation has not been performed yet. To understand these effects, we study
large cluster Hubbard model with an open ($1,1$) edge in the presence of a bulk
$d$-wave gap. We calculate the site-dependent spin susceptibility by performing
random-phase-approximation (RPA) and modified fluctuation-exchange (FLEX)
approximation in the real space. We find that near the ($1,1$) edge, drastic
ferromagnetic (FM) fluctuations occur owing to the ABS. In addition, as the
temperature decreases, the system rapidly approaches a magnetic-order phase
slightly below the transition temperature of the bulk $d$-wave
superconductivity (SC). In this case, the FM fluctuations are expected to
induce interesting phenomena such as edge-induced triplet SC and quantum
critical phenomena.
