We present an analytical study of a two-dimensional photonic lattice that simultaneously incorporates fractional coupling, parity-time ($\mathcal{PT}$) symmetry, and nonreciprocal interactions. The system features a checkerboard arrangement of alternating gain and loss sites, with asymmetric coupling amplitudes between sublattices. The dynamics are governed by a fractional discrete Laplacian as introduced by [O. Ciaurri et al., arXiv:1507.04986; O. Ciaurri et al., Adv. Math. 330, 688 (2018)], capturing long-range interactions. We derive a closed-form dispersion relation that explicitly depends on the product of forward and backward coupling strengths, as well as the fractional exponent. Our analysis reveals that increasing nonreciprocity or fractionality can significantly enhance the stability of the $\mathcal{PT}$-symmetric phase, pushing the threshold for $\mathcal{PT}$ breaking to higher gain or loss values. We also compute the participation ratio analytically, showing that nonreciprocity reduces mode delocalization. These results suggest a mechanism for tuning light stability and localization in photonic systems, with potential applications in the design of nonreciprocal and robust optical devices.

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• Complex systems
• Long-range interactions
• Mathematical physics
• Metamaterials
• Non-reciprocal propagation
• Photonics
• PT-symmetry
• Nonperturbative methods
• Schroedinger equation