Ising model in a transverse tunneling field and proton-lattice interaction in H-bonded ferroelectrics

Abstract

The addition of a ${B}_{\mathrm{ij}}{S}_{i}^{x}{S}_{j}^{x}$-type coupling between the tunneling motion of one proton and the tunneling motion of another to the Ising model in a transverse-field Hamiltonian, or the addition of the probably larger ${S}_{i}^{x}{F}_{i}^{x}{Q}_{i}$-type pseudospin phonon coupling (describing the modulation of the distance between the two equilibrium sites in an O-H \ifmmode\cdot\else\textperiodcentered\fi{} \ifmmode\cdot\else\textperiodcentered\fi{} \ifmmode\cdot\else\textperiodcentered\fi{} O bond by nonpolar phonons), results in a temperature-dependent renormalization of the proton tunneling integral. This is important close to ${T}_{C}$, where the soft-mode frequency vanishes, ${\ensuremath{\omega}}_{\mathrm{crit}}\ensuremath{\rightarrow}0$. This may lead to large isotope shifts in ${T}_{C}$ on deuteration even for small values of the tunneling intergal and may explain some phenomena recently observed in PbHP${\mathrm{O}}_{4}$ and squaric acid as well as the dependence of the effective proton-lattice interaction constant on hydrostatic pressure in K${\mathrm{H}}_{2}$P${\mathrm{O}}_{4}$-type systems. This last effect may be also due to the presence of a ${S}_{i}^{x}{D}_{i}^{x}{{Q}_{i}}^{2}$ term in addition to the Kobayashi ${S}_{i}^{z}{F}_{i}^{z}{Q}_{i}$ term when coupling with polar optic phonons is taken into account. When the lattice motion is so anharmonic that the lattice ions move in a double-well potential, and the proton-lattice coupling so strong that the protons can tunnel in only one out of the two possible lattice configurations, two Curie temperatures may appear.