Search for a ferromagnetic quantum criticality in YbIn1$_{1-x}$Rh$_{x}$Cu$_{4}$

The geometrically frustrated system YbIn$_{1-x}$Rh$_{x}$Cu$_{4}$ is investigated in order to search for a quantum criticality associated with ferromagnetic correlations. The undoped YbInCu$_{4}$ undergoes a first-order isostructural valance transition at 42K. The valance transition is suppressed with Rh substitution of x$\sim $0.3 and ferromagnetic correlations grow with increasing Rh content as can be seen from the increasing Weiss temperature in magnetic susceptibility. We report low temperature specific heat measurements down to 50mK on YbIn$_{1-x}$Rh$_{x}$Cu$_{4 }$with x around suspected critical concentration x$\sim $0.6 and discuss the relation between the ferromagnetic correlations and non-Fermi liquid behavior.