Low-Temperature Effects of the Tunneling Integral in Hydrogen-Bonded Ferroelectrics

The coupling of proton tunneling to acoustic phonons in hydrogen-bonded ferroelectrics is studied. The resultant collective modes are obtained. The presence of proton tunneling in renormalized acoustic phonons leads to a $T^4$ dependence of the low-temperature growth of the disorder in the ferroelectric. Also, the acoustic phonons velocities are decreased by coupling to proton tunneling; this suggests that undeuterated ferroelectrics will have a smaller acoustic-phonon velocity than the deuterated ferroelectrics. The low-temperature decrease of the spontaneous polarization is shown to have a leading term, $T^4$, which results from two physical causes: anharmonicity of the lattice forces and the decrease with temperature of the proton order parameter.