Quantum ferroelectrics: A renormalization-group study

Abstract

A renormalization-group transformation for quantum statistics is developed and applied to the ${\ensuremath{\varphi}}^{4}$ model. We find that quantum fluctuations at $T=0$ and thermal fluctuations at $T\ensuremath{\ne}0$ restore the symmetry giving rise to a ferroelectric-paraelectric transition. The renormalized mass (the inverse dielectric susceptibility) and the coupling constant become temperature dependent. The renormalization constants and the Wilson functions are given by the calculation at $T=0$. The inverse susceptibility for $n=1$ and $d=3$ ($n$ being the number of components of the order parameter and $d$ the dimension) is given by ${\ensuremath{\chi}}^{\ensuremath{-}1}\ensuremath{\sim}{\ensuremath{\chi}}_{\mathrm{qmf}}^{\ensuremath{-}1}{|log{\ensuremath{\chi}}_{\mathrm{qmf}}^{\ensuremath{-}1}|}^{\ensuremath{-}\frac{1}{3}}$ (qmf refers to the quantum-mean-field susceptibility in the paraelectric phase). For materials with ${T}_{c}=0$ we find ${\ensuremath{\chi}}_{\mathrm{qmf}}^{\ensuremath{-}1}\ensuremath{\sim}{T}^{2}$ and ${\ensuremath{\chi}}^{\ensuremath{-}1}\ensuremath{\sim}{T}^{2}{|log{T}^{2}|}^{\ensuremath{-}\frac{1}{3}}$.