Frenkel-Kontorova model: Crossover from the classical to the quantum mechanical

The Frenkel-Kontorova (FK) model describes a chain of atoms connected by springs subject to an external potential. This simple classical model exhibits a wealth of complex behavior. It has also found applications in many condensed matter systems such as charge density waves, magnetic spirals, modulated phases and tribology. However, an in-depth understanding of some of these problems, for example, tribology in the nano-regime, demands an understanding of its quantum mechanical behavior. To achieve this goal, we use a squeezed-state approach first used in quantum optics. We found that quantum fluctuations renormalize the standard map, which governs the classical behavior of the FK model, to a sawtooth map. This result is borne out by Monte-Carlo simulations. We also found that the ground state wave function changes from an extended state to a localized state when the coupling constant increases. Although quantum fluctuations largely smear the transition by breaking of analyticity observed in the classical case, the remnant of this transition is still discernible.