The quantum phase transition and thermodynamics of a periodic Anderson-like polymer chain in a magnetic field are investigated by Green's function theory. The T-h phase diagram is explored, wherein a crossover temperature T-* denoting the gapless phase crossover into quantum critical regimes, smoothly connects near the critical fields to the universal linear line T-* similar to (h - h(c,s)), and ends at h(c,s), providing a new route to capture quantum critical point (QCP). The quantum critical scaling around QCPs is demonstrated by analyzing magnetization, specific heat and Gruneisen parameter Gamma(h), which provide direct access to distill the power-law critical exponents (beta, delta and alpha) obeying the critical scaling relation alpha + beta(1 + delta) = 2, analogous to the quantum spin system. Furthermore, scaling hypothesis equations are proposed to check the scaling analysis, for which all the data collapse onto a single curve or two independent branches for the plot against an appropriate scaling variable, indicating the self-consistency and reliability of the obtained critical exponents. (C) 2017 Elsevier B.V. All rights reserved.