We develop a systematic large-$N$ expansion based on the Schwinger boson representation of SU(4) coherent states of dimers for the paradigmatic spin-$1/2$ bilayer square lattice Heisenberg antiferromagnet. This system exhibits a quantum phase transition between a quantum paramagnetic state and a Néel order state, driven by the coupling constant $g=J^{\prime }/J$, which is defined as the ratio between the interdimer $J^{\prime }$ and intradimer $J$ exchange interactions. We demonstrate that this approach accurately describes static and dynamic properties on both sides of the quantum phase transition. The critical coupling constant $g_c\approx 0.42$ and the dynamic spin structure factor reproduce quantum Monte Carlo results with high precision. Notably, the $1/N$ corrections reveal the longitudinal mode of the magnetically ordered phase along with the overdamping caused by its decay into the two-magnon continuum. The present large-$N$ SU(4) Schwinger boson theory can be extended to more general cases of quantum paramagnets that undergo a quantum phase transition into magnetically ordered states.

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• Magnetic phase transitions
• Magnons
• Spin dynamics
• Triplon
• Large-N expansion in field theory
• Schwinger boson method