Magnetic phase diagram of Ba₂FeSi₂O₇: a quasi-two-dimensional square lattice Heisenberg antiferromagnet with a strong uniaxial magnetic anisotropy.

We study the strongly anisotropic quasi-two-dimensional square lattice S = 2 quantum magnet Ba₂FeSi₂O₇ with magnetic field along the anisotropic axis. The compound shows a XY antiferromagnetic (XY AFM) order at T_N∼5.5K at zero field.  We demonstrate from magnetization, magnetocaloric measurement and generalized SU(5) mean field calculation at zero temperature that the XY AFM phase becomes unstable with increasing magnetic field due to level crossings of spin states, and a quantum paramagnetic state appears around 30 T. Another XY AFM state is observed upon further increasing field above 35 T before the magnetization saturates around 45 T. Ultimately, ferroelectricity is also observed in the antiferromagnetic phases by orbital hybridization. We map the temperature/magnetic field phase diagram that consists of two dome shaped AF and multiferroic phases that feature the magnetic Bose-Einstein condensates accompanying ferroelectricity.