Magnetic neutron scattering resonance of high-$T_c$ superconductors in external magnetic fields: An SO(5) study

The magnetic resonance at 41 meV observed in neutron scattering studies of ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_7$ holds a key position in the understanding of high-$T_c$ superconductivity. Within the SO(5) model for superconductivity and antiferromagnetism, we have calculated the effect of an applied magnetic field on the neutron scattering cross section of the magnetic resonance. In the presence of Abrikosov vortices, the neutron scattering cross section shows clear signatures of not only the fluctuations in the superconducting order parameter $\psi ,$ but also the modulation of the phase of $\psi$ due to vortices. In reciprocal space we find that (i) the scattering amplitude is zero at $(\pi /a,\pi /a),$ (ii) the resonance peak is split into a ring with radius $\pi /d$ centered at $(\pi /a,\pi /a),$ d being the vortex lattice constant, and consequently, (iii) the splitting $\pi /d$ scales with the magnetic field as $\sqrt{B}.$