Magnetic field dependence of the spin wave excitations in Sr$_2$FeSi$_2$O$_7$

Without field, the multiferroic Sr$_2$FeSi$_2$O$_2$ orders below 4.7 K into a simple square-lattice antiferromagnetic collinear structure. Application of a magnetic field at low temperatures induces four different magnetic phases with spontaneous electric polarization. We report inelastic neutron scattering measurements on a single crystal of Sr$_2$FeSi$_2 $O$_7$ under magnetic fields to investigate how the magnetic fluctuations change with field. In zero field, no dispersion was observed in $L$-direction, indicating that Sr$_2$FeSi$_2 $O$_7$ is a two dimensional magnet. The dispersion relation along ($H$ 0 0.5) shows a Goldstone mode arising from the magnetic Bragg position at (1 0 0.5) with a periodicity of $2 \times2\pi/a$, suggesting that the strongest magnetic interaction is between the nearest neighbor Fe$^{2+}$ ions. Under the field, the magnetic fluctuations become more complex than the simple splitting of the doubly degenerate Goldstone mode into two gapped modes.