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Low-temperature specific-heat studies on two square-kagome antiferromagnets

We studied the low-temperature specific heats of two antiferromagnets with the two-dimensional square-kagome structure, i.e., KCu$_6$AlBiO$_4$(SO$_4$)$_5$Cl (KCu6) and Na$_6$Cu$_7$BiO$_4$(PO$_4$)$_4$[Cl,(OH)]$_3$ (NaCu7) with the structural differences that there are interlayer Cu$^{2+}$ ions in NaCu7. Both materials show no magnetic ordering down to 50 mK. At zero field, the $C/T$ of KCu6 has a finite value when the temperature is close to zero K. Under the magnetic field, a seemingly $T^2$ dependence appears and its coefficient is progressively suppressed by the field. For NaCu7, the specific heat exhibits the $T^2$ dependence at zero field and under fields. The ratio of the quadratic coefficients of KCu6 and NaCu7 at high fields is inversely proportional to ratio of the squared Weiss temperatures, which indicates these two compounds host the same ground state under fields. Our results suggest that the interlayer Cu$^{2+}$ ions in NaCu7 play a negligible role in determination of its ground state. We discuss the possible quantum-spin-liquid states in these compounds and further directions to pursue based our results.