High-order harmonics and the reverse of the squaring up process in the triangular-lattice magnet ${\mathrm{HoPdAl}}_4{\mathrm{Ge}}_2$

We uncover high-order harmonics and the reverse of the squaring-up process, in terms of analyzing the evolution of magnetic orders in a centrosymmetric layered triangular-lattice magnet ${\mathrm{HoPdAl}}_4{\mathrm{Ge}}_2$, based on a detailed study of the crystal structure, magnetic susceptibility, magnetization, heat capacity, and magnetic structure. Temperature dependencies of magnetic susceptibility and heat capacity show two magnetic transitions at $T_N=10.5$ K and $T_t=5.5$ K. Below $T_N, {\mathrm{Ho}}^{3+}$ spins order antiferromagnetically as a transverse spin-density wave with the propagation vector $k_1=(001.5-\delta )$ with $\delta \approx 0.18$. Upon further cooling through $T_t$, the high-order harmonics with $k_n=(001.5-n\delta ), n=3$, 5, 7 develop, suggesting a squaring-up process. It is surprising that the squaring-up process does not continue down to 0 K but reverses the trend below 3 K. Magnetic-field-induced metastable transitions were observed in $M\left(H\right)$ curves with the fields applied both parallel and perpendicular to the triangular-lattice plane. Neutron-diffraction results suggest that the magnetization process in ${\mathrm{HoPdAl}}_4{\mathrm{Ge}}_2$ involves the conversion of $k_n=(001.5-n\delta )$ with $n=1$, 3, 5, 7, to a ferromagnetic $k_F=\left(000\right)$ component. It is worth noting that the already weak seventh harmonic magnetic peak is enhanced by applying a small magnetic field in the $ab$ plane at 1.5 K or warming up to 3 K, accompanied by a slight decrease of $\delta$.