## 3 Results and Discussion
The more apparent trends, the decrease in the critical field (Figure 8a-c or Figure 8d,e) and the Neel temperature (Figure 7) with increasing Mn content, can largely be attributed to the growing frustration of the magnetic order, which intensifies as the Mn concentration in the lattice increases. Neutron diffraction measurements support this observation, revealing a corresponding decrease in the ordered magnetic moment with increasing Mn content [33; 34]. Remnant magnetization of the single-phase rhombohedral samples with \(x\) = 0.35 increases as the temperature decreases, reaching about 0.2 emu/g at \(T\) = 2.5 K (Figure 9a,d). Although this value is comparable to the spontaneous magnetization originating from spin canting in the G-type AFM structure with moments aligned perpendicular to the \(c\) axis [49], the character of the temperature dependencies of magnetization (Figure 6a,d) suggests that this increase is at least partially due to the formation of a glassy magnetic state. A dramatic increase (up to a few emu/g) in low-temperature remnant magnetization is observed in the single-phase rhombohedral samples with \(x\) = 0.4 (Figure 9b,e). While the Bi\({}_{0.85}\)Ca\({}_{0.15}\)Fe\({}_{0.6}\)Mn\({}_{0.4}\)O\({}_{3}\) compound still demonstrates metamagnetic behavior at \(T\) = 2.5 K (Figure 9e), no features of metamagnetic transformation at \(T\) = 2.5 K are seen in the Bi\({}_{0.9}\)Ca\({}_{0.1}\)Fe\({}_{0.6}\)Mn\({}_{0.4}\)O\({}_{3}\) sample (Figure 9b). A further increase in Mn concentration within the Bi\({}_{0.9}\)Ca\({}_{0.1}\)Fe\({}_{1-x}\)Mn\({}_{x}\)O\({}_{3}\) series leads to the disappearance of metamagnetic behavior at a higher temperature of 25 K (Figure 9c), indicating that a decrease in temperature and an increase in magnetic field result in the same effect: reorientation of the AFM coupled spins from the \(c\) to the \(a\) axis, as confirmed by neuron diffraction data [48], with canting of the magnetic moment in the \(xy\) plane [49]. At low temperatures, where magnetization is enhanced due to the glassy component, the field-driven G\({}_{z}\) to G\({}_{x}\) reorientation of the magnetic moments in this partly frustrated system is accompanied by pronounced jumps in magnetization that gradually vanish as the temperature increases (Figure 8a-e,g). In other words, the observed metamagnetic behavior is a manifestation of the spin-reorientation transition (from the collinear to the canted configuration of magnetic moments), which is accompanied by a reorganization of the domain structure and alignment of partially frustrated spins along the field. It is important to emphasize the difference in magnetic properties between Ca/Mn- and widely studied \(RE\)-substituted bismuth ferrites. In the Bi\({}_{1-x}\)RE\({}_{x}\)FeO\({}_{3}\) series [50; 51], chemical substitution does not influence the character of the superexchange interactions; instead, it primarily affects the magnetic anisotropy, thereby reducing the threshold field for the magnetic field-induced transition from cycloidal AFM to canted AFM structure [14; 15; 16]. In contrast, in the Bi\({}_{1-y}\)Ca\({}_{y}\)Fe\({}_{1-x}\)Mn\({}_{y}\)O\({}_{3}\) series, Ca/Mn substitution leads to the emergence of competing AFM and FM exchange interactions, which underlies the strong enhancement of magnetization at low temperatures. The metamagnetic behavior observed in the Bi\({}_{1-y}\)Ca\({}_{y}\)Fe\({}_{1-x}\)Mn\({}_{y}\)O\({}_{3}\) series below room temperature is attributed to the transformation from collinear AFM order to canted AFM order. This transformation, due to the presence of a glassy magnetic component, is accompanied by large magnetization jumps that are not observed in the Bi\({}_{1-x}\)RE\({}_{x}\)FeO\({}_{3}\) series. Given the significant magnetization jumps, where magnetization reaches a few emu/g in fields as low as a few kOe, an effect uniquely characteristic of Ca/Mn-substituted BiFeO\({}_{3}\), these materials hold promising potential for applications involving magnetoelectric coupling.