Quantum Critical Paraelectrics and the Casimir Effect in Time

We study the quantum paraelectric-ferroelectric transition near a quantum critical point, emphasizing the role of temperature as a "finite size effect" in time. The influence of temperature near quantum criticality may thus be likened to a temporal Casimir effect. The resulting finite-size scaling approach yields $\frac{1}{T^2}$ behavior of the paraelectric susceptibility ($\chi$) and the scaling form $\chi(\omega,T) = \frac{1}{\omega^2} F(\frac{\omega}{T})$, recovering results previously found by more technical methods. We use a Gaussian theory to illustrate how these temperature-dependences emerge from a microscopic approach; we characterize the classical-quantum crossover in $\chi$, and the resulting phase diagram is presented. We also show that coupling to an acoustic phonon at low temperatures ($T$) is relevant and influences the transition line, possibly resulting in a reentrant quantum ferroelectric phase. Observable consequences of our approach for measurements on specific paraelectric materials at low temperatures are discussed.