We propose a scheme to achieve controllable nonreciprocal Goos-Hänchen shift (NGHS) in a cavity optomagnonic system consisting of a spinning yttrium iron garnet (YIG) sphere and a degenerate optical parametric amplifier, where the physical mechanism originates from the asymmetry of the large enough anharmonicity in the energy spectrum induced by the Barnett effect. By increasing the photon-magnon coupling strength, transparency due to normal-mode splitting appears and significantly reduces the absorption of the probe field in the system. Simultaneously, when the YIG sphere rotates at a fixed angular velocity, opposite Barnett shifts can be induced by reversing the direction of the static magnetic field, which alters the dispersion of the probe field within the system. Consequently, we investigate the GHS of the reflected probe beam. The results demonstrate that when the probe field is incident at a fixed angle, the GHS in the cavity optomagnonic system exhibits different responses in different directions of the static magnetic field, indicating the occurrence of NGHS. Our work may have potential applications in designing magnetic-controlled optical switches and in measuring the Barnett field.

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• Non-reciprocal transmission
• Rotating wave approximation