We performed AC calorimetry and magnetoresistance measurements under pressure for H || a-axis (easy-magnetization axis) in the novel heavy-fermion superconductor UTe2. Thanks to the thermodynamic information, multiple superconducting phases have been revealed under pressure and magnetic field. The (H,T) phase diagram of superconductivity under pressure displays an abrupt increase of the upper critical field (Hc2) at low temperature and in the high field region, and a strong convex curvature of Hc2 at high temperature. This behavior of Hc2 and the multiple superconducting phases require a state for the superconducting order parameter more complex than a spin-triplet equal spin pairing. Above the superconducting critical pressure, Pc, we find strong indications that the possible magnetic order is closer to antiferromagnetism than to ferromagnetism.
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