Lro in Lattice Systems of Linear Classical and Quantum Oscillators. Strong N-N Pair Quadratic Interaction

For systems of one-component interacting oscillators on the d-dimensional lattice, d>1, whose potential energy besides a large nearest-neighbour (n-n) ferromagnetic translation-invariant quadratic term contains small non-nearest-neighbour translation invariant term, an existence of a ferromagnetic long-range order for two valued lattice spins, equal to a sign of oscillator variables, is established for sufficiently large magnitude g of the n-n interaction with the help of the Peierls type contour bound. The Ruelle superstability bound is used for a derivation of the contour bound.