Resolving the quantum criticality paradox in O-18 isotopic SrTiO3

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 alpha = -2 (2D) or -3 (3D), delta = 3, and gamma = 2 with the thermodynamic inequalities of Rushbrooke, Griffiths, Widom el al, which become exact equalities under the hypothesis of scaling. In particular, these have led others to the inference that gamma = 2.0 and = 1.2 in SrTiO3. 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, gamma - 2 beta = v(4 - d - 2 eta) > 0, we derive from hyperscaling. Third, we show that the existing soft mode frequency data omega(T) from Takesada et at (2006 Phys. Rev. Lett. at press) yield above T-c (from the Lyddane-Sachs-Teller relationship) gamma = 1.0. Fourth, we remeasure 8 from the polarization P(T) and find = 0.50 +/- 0.02. Fifth, we remeasure the electric susceptibility and find that it perfectly satisfies the Salje-Wruck-Thomas equation, which requires gamma = 1.0. The important conclusions are: (a) O-18 SrTiO3 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 8 = 1.2 and beta > gamma/2 impossible.