Multistage development of short-range spin correlations and weak magnetic order in the two coupled trillium lattices of K2Fe2(MoO4)(PO4)2

Trillium lattices, where magnetic ions form a chiral network of corner-sharing triangles, offer a three-dimensional magnetic framework that can host fragile classical spin-liquid states. Herein, we report on the magnetization, specific heat, electron spin resonance (ESR), and muon spin relaxation ($\mu$SR) of K${}_2$Fe${}_2$(MoO${}_4$)(PO${}_4$)${}_2$ single crystals. Magnetization measurements reveal strong antiferromagnetic interactions coexisting with weak magnetic order at $T_{\mathrm{N}}=5.2$~K, as evidenced by a $\lambda$-like anomaly observed in the magnetic susceptibility, a critical enhancement of the muon spin relaxation rate and the wipeout of the ESR signal as the temperature approaches $T_{\mathrm{N}}$. Above $T_{\mathrm{N}}$, two distinct developments of short-range spin correlations are identified at $T_{\mathrm{H}}=34$~K and $T_{\mathrm{L}}=10$~K, supported by magnetic specific heat anomalies and the temperature dependence of the ESR linewidth and $g$-factor. Upon cooling below $T_{\mathrm{N}}$, an anomaly appears at $T^{\ast }=3.2$~K in thermodynamic observables and the muon spin relaxation rate, indicative of spin reorientation driven by residual interactions. Despite the presence of magnetic order, $\mu$SR experiments reveal dynamically fluctuating spins persisting even in the ordered state. Moreover, the suppression of $T_{\mathrm{N}}$ under applied magnetic fields (${\mu }_{0}H\ge 2$~T) suggests that K${}_2$Fe${}_2$(MoO${}_4$)(PO${}_4$)${}_2$ constitutes a promising candidate for exploring field-induced spin-liquid behavior in three-dimensionally coupled trillium lattices.