Scalar Domination and Normal Fluctuations in N-Vector Quantum Anharmonic Crystals

Abstract

A lattice model of N-dimensional quantum anharmonic oscillators with a polynomial anharmonicity and a ‘ferroelectric’ pair interaction is considered. For arbitrary N ε$\mathbb{N}$, correlation inequalities, showing that the temperature Green functions of this model are dominated by the corresponding Green functions of the scalar (N=1) model, are proven. These inequalities are then used to prove that the fluctuations of displacements of particles remain normal at all temperatures provided the model parameters obey a certain condition. In particular this means that the smallest distance between the energy levels of the corresponding one-dimensional isolated oscillator should be large enough or its mass should be small enough.