Gr\"{u}neisen Parameter and Thermal Expansion near Magnetic Quantum Critical Points in Itinerant Electron Systems

Complete expressions of the thermal-expansion coefficient $\alpha$ and the Gr\"{u}neisen parameter $\Gamma$ are derived on the basis of the self-consistent renormalization (SCR) theory. By considering zero-point as well as thermal spin fluctuation under the stationary condition, the specific heat for each class of the magnetic quantum critical point (QCP) specified by the dynamical exponent $z=3$ (FM) and $z=2$ (AFM) and the spatial dimension ($d=3$ and $2$) is shown to be expressed as $C_{V}=C_a-C_b$, where $C_a$ is dominant at low temperatures, reproducing the past SCR criticality endorsed by the renormalization group theory. Starting from the explicit form of the entropy and using the Maxwell relation, $\alpha=\alpha_a+\alpha_b$ (with $\alpha_a$ and $\alpha_b$ being related to $C_a$ and $C_b$, respectively) is derived, which is proven to be equivalent to $\alpha$ derived from the free energy. The temperature-dependent coefficient found to exist in $\alpha_b$, which is dominant at low temperatures, contributes to the crossover from the quantum-critical regime to the Curie-Weiss regime and even affects the quantum criticality at 2d AFM QCP. Based on these correctly calculated $C_{V}$ and $\alpha$, Gr\"{u}neisen parameter $\Gamma=\Gamma_a+\Gamma_b$ is derived, where $\Gamma_a$ and $\Gamma_b$ contain $\alpha_a$ and $\alpha_b$, respectively. The inverse susceptibility coupled to the volume $V$ in $\Gamma_b$ gives rise to divergence of $\Gamma$ at the QCP for each class even though characteristic energy scale of spin fluctuation $T_0$ is finite at the QCP, which gives a finite contribution in $\Gamma_a=-\frac{V}{T_0}\left(\frac{\partial T_0}{\partial V}\right)_{T=0}$. General properties of $\alpha$ and $\Gamma$ including their signs as well as the relation to $T_0$ and the Kondo temperature in temperature-pressure phase diagrams of Ce- and Yb-based heavy electron systems are discussed.