The effect of disorder on quantum phase transition in the double layered   ruthenates (Sr1-xCax)3Ru2O7
  (Sr1-xCax)3Ru2O7 is characterized by complex magnetic states, spanning from a long-range antiferromagnetically ordered state over an unusual heavy-mass nearly ferromagnetic (NFM) state to an itinerant metamagnetic (IMM) state. The NFM state, which occurs in the 0.4 > x > 0.08 composition range, freezes into a cluster-spin-glass (CSG) phase at low temperatures [Z. Qu et al., Phys. Rev. B 78, 180407(R) (2008)]. In this article, we present the scaling analyses of magnetization and the specific heat for (Sr1-xCax)3Ru2O7 in the 0.4 > x > 0.08 composition range. We find that in a temperature region immediately above the spin freezing temperature T$_f$, the isothermal magnetization M(H) and the temperature dependence of electronic specific heat C_e(T) exhibit anomalous power-law singularities; both quantities are controlled by a single exponent. The temperature dependence of magnetization M(T) also displays a power-law behavior, but its exponent differs remarkably from that derived from M(H) and C_e(T). Our analyses further reveal that the magnetization data M(H,T) obey a phenomenological scaling law of M(H,T) \propto H^\alpha f(H/T^\delta) in a temperature region between the spin freezing temperature T_f and the scaling temperature T_scaling. T_scaling systematically decreases with the decease of Ca content. This scaling law breaks down near the critical concentration x = 0.1 where a CSG-to-IMM phase transition occurs. We discussed these behaviors in term of the effect of disorder on the quantum phase transition. 
Zhe Qu, Jin Peng, Tijiang Liu, David Fobes, Vlad Dobrosavljevic,   Leonard Spinu, and Zhiqiang Mao
cond-mat.str-el cond-mat.mtrl-sci