Resolving the quantum criticality paradox in O-18 isotopic SrTiO(3)

Recently there has been considerable interest in the displacive ferroelectric phase transition near T = 28 K in O-18 isotopic strontium titanate. Special efforts have been made to combine the quantum criticality exponents α = -2 (2D) or -3 (3D), δ = 3, and γ = 2 with the thermodynamic inequalities of Rushbrooke, Griffiths, Widom et al, which become exact equalities under the hypothesis of scaling. In particular, these have led others to the inference that γ = 2.0 and β = 1.2 in SrTiO(3). First we show that this is mathematically incorrect and explain why (quantum criticality is exact only at T = 0, whereas the thermodynamic (in)equalities are valid everywhere except T = 0). Second, we show that the inferred values strongly violate a new equality, γ-2β = ν(4-d-2η)>0, we derive from hyperscaling. Third, we show that the existing soft mode frequency data ω(T) from Takesada et al (2006 Phys. Rev. Lett. at press) yield above T(c) (from the Lyddane-Sachs-Teller relationship) γ = 1.0. Fourth, we remeasure β from the polarization P(T) and find β = 0.50 ± 0.02. Fifth, we remeasure the electric susceptibility and find that it perfectly satisfies the Salje-Wruck-Thomas equation, which requires γ = 1.0. The important conclusions are: (a) O-18 SrTiO(3) near T(c) is mean-field; (b) the thermodynamic scaling equalities of Rushbrooke, Griffiths et al are mathematically incompatible with quantum criticality theory; (c) a new hyperscaling relationship makes β = 1.2 and β>γ/2 impossible.