Universal Scaling Functions of the Gr{\"u}neisen Ratio near Quantum Critical Points

The Gr\"uneisen ratio, defined as $\Gamma_g \equiv (1/T) (\partial T/\partial g)_S$, serves as a highly sensitive probe for detecting quantum critical points (QCPs) driven by an external feild $g$ and for characterizing the magnetocaloric effect (MCE). Near a QCP, the Gr\"uneisen ratio displays a universal divergence which is governed by a universality-class-dependent scaling function stemming from the scale invariance. In this work, we systematically investigate the universal scaling functions of Gr\"uneisen ratio in both one-dimensional (1D) and two-dimensional (2D) quantum spin systems, including the transverse-field Ising model, the spin-1/2 Heisenberg model, the quantum $q$-state Potts model ($q=3,4$) and the $J_1$-$J_2$ columnar dimer model. Our approach employs the thermal tensor-network method for infinite-size 1D systems and the stochastic series expansion quantum Monte Carlo (SSE QMC) simulations for 2D systems, enabling precise calculations of the Gr\"uneisen ratio near QCPs. Through data collapse analysis, we extract the corresponding scaling functions, which establish quantitative frameworks to interpret magnetocaloric experiments and guide the development of ultralow-temperature refrigeration.