We studied the phase diagram for a two-dimensional d-wave superconducting system under an in-plane magnetic field or an exchange field. According to the spatial configuration of the order parameter, we show that there exists quantum phase transitions in which the uniform phase transforms to the one-dimensional Fulde-FerrellLarkin-Ovchinnikov (FFLO) state, and then to two-dimensional FFLO state upon increasing the exchange field. The local density of states are calculated and suggested to be signatures to distinguish these phases.The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state was predicted several decades ago by Fulde and Ferrell [1], and Larkin and Ovchinnikov [2] for the superconductor in a strong magnetic field, where the superconducting (SC) order parameter varies periodically in space. While the occurance of the FFLO state requires very stringent conditions on the SC materials, namely, the Pauli paramagnetism effect should dominate over the orbital effect [3], and the material needs to be very clean [4]. As a result, this long thought of inhomogeneous SC state has never been observed in conventional superconductors.For layered systems with an exchange field or a magnetic field parallel to the SC plane, the orbital effect will be suppressed strongly due to the low dimensionality. Thus they could be strong candidates to look for the FFLO state. Actually, in the past decade, indications for possible FFLO state have been reported in the heavy fermion materials CeCoIn 5 [5,6], organic superconductors λ-(BETS) 2 GaCl 4 [7],λ-(BETS) 2 FeCl 4 [8, 9] and κ-(BEDT-TTF) 2 Cu(NCS) 2 [10,11]. All of them are quasi two dimensional (2D) layered compounds. The experimental developments have attracted renewed interest on the property of the FFLO state. Theoretically, the existence and the character of the FFLO state can be investigated through analyzing the Ginzburg-Landau (GL) free-energy function, which is valid at temperatures not too below the superconducting transition temperature. Another effective method is the Bogoliubovde-Gennes (BdG) technique and it has been proven to be a powerful tool to study the inhomogeneous state and the local density of states (LDOS) self-consistently in the lowdimensional system. In fact, in the past, the FFLO state has been studied intensively based on the above two techniques [12,13,14,15,16,17,18]. For a 2D system, it is somewhat established [14,15]that the order parameter has a 2D checkerboard pattern for a superconductor with d-wave pairing symmetry, and a 1D stripe-like pattern for s-wave pairing symmetry [15,16]. On the other hand, it has been shown that in the presence of dilute impurities, the pattern of the FFLO state becomes 1D stripe-like in a 2D d-wave superconductor [17,18]. This implies that the 2D and 1D FFLO states may be present in the system in different parameter region. It was also argued without a calculation that the H −T phase diagram for the 2D isotropic systems, regardless it is s or d-wave pairing, should include both 1D FFLO state and 2D FFLO states with squar...