We predict anomalous atomic-scale 0-π transitions in a Josephson junction with a ferromagneticinsulator (FI) barrier. The ground state of such junction alternates between 0-and π-states when thickness of FI is increasing by a single atomic layer. We find that the mechanism of the 0-π transition can be attributed to thickness-dependent phase-shifts between the wave numbers of electrons and holes in FI. Based on these results, we show that stable π-state can be realized in junctions based on high-Tc superconductors with La2BaCuO5 barrier.PACS numbers: 74.50.+r, 03.67.Lx The developing field of superconducting spintronics comprises a plenty of fascinating phenomena that may complement nonsuperconducting spintronics devices [1]. Mesoscopic hybrid structures consisting of superconducting and magnetic materials have attracted considerable attention as devices with novel functionalities [2]. One of most interesting effects is the formation of π-states in superconductor/ferromagnetic-metal/superconductor (S/FM/S) Josephson junctions [3]. Under appropriate conditions a ferromagnet can become a π-phase shifter, providing the phase difference φ = π between two superconductors in the ground state in contrast to φ = 0 in ordinary Josephson junctions. Recently a quiet qubit based on S/FM/S π-junction [4] has been suggested as a promising device to realize quantum computation because the spontaneously generated two-level system in this structure is robust against decoherence due to external fluctuations. However, low energy quasiparticle excitations in a FM provide strong dissipation [5]. Therefore Josephson π junctions with a nonmetallic interlayers are highly desired for qubit applications [6]. Moreover, from the fundamental view point, the Josephson transport through a f erromagnetic insulator (FI) has been studied based on phenomenological models [7] and not yet been explored explicitly.In this Letter, we study theoretically the Josephson effect in superconductor/ferromagneticinsulator/superconductor (S/FI/S) junctions using the tight-binding model. We show that the ground state in such structures alternates between the 0-and π-states when the thickness of a FI (L F ) is increasing by a single atomic layer. This remarkable effect originates from the characteristic band structure of a FI. Quasiparticles in the electron and hole branches acquire different phase shifts while propagating across a FI. We will show that the phase difference is exactly πL F due to the band structure of a FI, thus providing the atomic-scale 0-π transition. This mechanism is in striking contrast to the proximity induced 0-π transition in conventional S/FM/S junctions. On the basis of the obtained results, we predict a stable π-state in a Josephson junction based on high-T c superconductors with a La 2 BaCuO 5 barrier, where electric current flows along the c axis of cuprates.Let us first consider an S/FI/S junction in the twodimensional tight-binding model as shown in Fig. 1(a). The vector r = jx + my points to a lattice site, where