Novel quantum spin liquid ground state in the trimer rhodate Ba$_4$NbRh$_3$O$_{12}$

Frustrated magnets offer a plethora of exotic magnetic ground states, including quantum spin liquids (QSLs), in which enhanced quantum fluctuations prevent a long-range magnetic ordering of the strongly correlated spins down to lowest temperature. Here we have investigated the trimer based mixed valence hexagonal rhodate Ba$_4$NbRh$_3$O$_{12}$ using a combination of dc and ac magnetization, electrical resistivity, specific heat, and muon spin rotation/relaxation ($\mu$SR) measurements. Despite the substantial antiferromagnetic exchange interactions, as evident from the Weiss temperature ($\theta_{\mathrm{W}}\sim -35$ to -45 K), among the Rh-local moments, neither long-range magnetic ordering nor spin-freezing is observed down to at least 50 mK, in ac-susceptibility, specific heat and ZF-$\mu$SR measurements (down to 0.26 K). We ascribe the absence of any magnetic transition to enhanced quantum fluctuations as a result of geometrical frustration arising out of the edge-sharing equilateral Rh-triangular network in the structure. Our longitudinal-field $\mu$SR result evidences persistent spin fluctuations down to 0.26~K, thus stabilizing a dynamic QSL ground state in Ba$_4$NbRh$_3$O$_{12}$. Furthermore, the magnetic specific heat ($C_{\mathrm{m}}$) data at low-$T$ reveal a significant $T$-linear contribution plus a quadratic $T$-dependence. A $T$-linear behavior is evocative of gapless spin excitations, while the $T^2$-term of $C_{\mathrm{m}}$ may indicate the Dirac QSL phenomenology of the spinon excitations with a linear dispersion.