Modified transverse Ising model for the dielectric properties of SrTiO$_3$ films and interfaces

The transverse Ising model (TIM), with pseudospins representing the lattice polarization, is often used as a simple description of ferroelectric materials. However, we demonstrate that the TIM, as it is usually formulated, provides an incorrect description of SrTiO$_{3}$ films and interfaces because of its inadequate treatment of spatial inhomogeneity. We correct this deficiency by adding a pseudospin anisotropy to the model. We demonstrate the physical need for this term by comparison of the TIM to a typical Landau-Ginzburg-Devonshire model. We then demonstrate the physical consequences of the modification for two model systems: a ferroelectric thin film, and a metallic LaAlO$_{3}$/SrTiO$_{3}$ interface. We show that, in both cases, the modified TIM has a substantially different polarization profile than the conventional TIM. In particular, at low temperatures the formation of quantized states at LaAlO$_{3}$/SrTiO$_{3}$ interfaces only occurs in the modified TIM.

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