Quantum ferroelectrics: A renormalization-group study

A renormalization-group transformation for quantum statistics is developed and applied to the ${\phi }^4$ model. We find that quantum fluctuations at $T=0\mathrm{}$ and thermal fluctuations at $T\ne 0$ 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 $T=0\mathrm{}$. The inverse susceptibility for $n=1\mathrm{}$ and $d=3\mathrm{}$ ($n$ being the number of components of the order parameter and $d$ the dimension) is given by ${\chi }^{-1\mathrm{}}\sim {\chi }_{\mathrm{qmf}}^{-1\mathrm{}}{\left|log{\chi }_{\mathrm{qmf}}^{-1\mathrm{}}\right|}^{-\frac{1}{3}}$ (qmf refers to the quantum-mean-field susceptibility in the paraelectric phase). For materials with $T_c=0\mathrm{}$ we find ${\chi }_{\mathrm{qmf}}^{-1\mathrm{}}\sim T^2$ and ${\chi }^{-1\mathrm{}}\sim T^2{\left|log{T}^2\right|}^{-\frac{1}{3}}$.