Gruneisen parameter and thermal expansion near magnetic quantum critical points in itinerant electron systems

Complete expressions of the thermal-expansion coefficient alpha and the Grijneisen parameter Gamma 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 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. For sufficiently low temperatures, the thermal-expansion coefficient at the QCP behaves as alpha approximate to alpha(b )similar to 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 C-v and alpha, Grijneisen parameter Gamma = Gamma(a) + Gamma(b) is derived, where Gamma(a) and Gamma(b) contain alpha(a) and alpha(b), respectively. The inverse susceptibility (renormalized by the mode-mode coupling of spin fluctuations) coupled to the volume V in Gamma(b) gives rise to the divergence of Gamma at the QCP for each class even though the characteristic energy scale of spin fluctuation T-0 is finite at the QCP, which gives a finite contribution in Gamma(a) =-v/T-0(partial derivative T-0/partial derivative V)(T=0)For T << T-0 , the Gruneisen parameter at the QCP behaves as Gamma approximate to Gamma(b) similar to T-2/3 /In T (3D FM), T-1/2 /(const. - T-1/2 ) (3D AFM), -T-2/3 ln T (2D FM), and ln(-ln T )/ [T In Tln (- ln T/ln T)] (2D AFM). General properties of a and F 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.