Absence of Critical Points for a Class of Quantum Hierarchical Models

Abstract:

Hierarchical models of quantum anharmonic oscillators with a polynomial anharmonicity and interaction decaying as (distance)^-1-λ are considered. For a class of such models (including ϕ⁴-type anharmonicity ones), it is shown that the critical fluctuations of the position operator are absent, for all λ > 0 and all temperatures, provided the oscillators mass in less than some threshold value depending on the anharmonicity parameters. This result may be interpreted as a rigorous mathematical justification of physical arguments showing that quantum fluctuations can damp phase transitions.