Grüneisen Parameter and Thermal Expansion near Magnetic Quantum Critical Points in Itinerant Electron Systems

Complete expressions of the thermal-expansion coefficient α and the Grüneisen parameter Γ are derived on the basis of the self-consistent renormalization (SCR) theory. By considering the 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 [feorromagnetism (FM)] and z = 2 [antiferromagnetism (AFM)] and the spatial dimension (d = 3 and 2) is shown to be expressed as CV = Ca − Cb, where Ca 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, α = αa + αb (with αa and αb being related to Ca and Cb, respectively) is derived, which is proven to be equivalent to α derived from the free energy. The temperature-dependent coefficient found to exist in αb, which is dominant at low temperatures, contributes to the crossover from the quantum-critical regime to the Curie-Weiss regime. For sufficiently low temperatures, the thermal-expansion coefficient at the QCP behaves as α ≈ αb ∼ T 1/3 (3D FM), T 1/2 (3D AFM), − ln T (2D FM), and − ln(− ln T )/ ln (− T/ln T ) (2D AFM). Based on these correctly calculated CV and α, Grüneisen parameter Γ = Γa + Γb is derived, where Γa and Γb contain αa and αb, respectively. The inverse susceptibility (renormalized by the mode-mode coupling of spin fluctuations) coupled to the volume V in Γb gives rise to the divergence of Γ at the QCP for each class even though the characteristic energy scale of spin fluctuation T0 is finite at the QCP, which gives a finite contribution in Γa = − V/T0 ( ∂T0/∂V )T =0. For T ≪ T0, the Grüneisen parameter at the QCP behaves as Γ ≈ Γb ∼ −T −2/3/ln T (3D FM), T −1/2/(const. − T 1/2 ) (3D AFM), −T −2/3 ln T (2D FM), and ln(− ln T )/[T ln T ln (− T/ln T )] (2D AFM). General properties of α and Γ including their signs as well as the relation to T0 and the Kondo temperature in temperature-pressure phase diagrams of Ce- and Yb-based heavy electron systems are discussed