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

The Frenkel-Kontorova (FK) model describes a chain of atoms connected by sp rings subject to an external potential. This simple classical model exhibit s a wealth of complex behavior. It has also found applications in many cond ensed matter systems such as charge density waves, magnetic spirals, modula ted phases and tribology. However, an in-depth understanding of some of the se problems, for example, tribology in the nano-regime, demands an understa nding of its quantum mechanical behavior. To achieve this goal, we use a sq ueezed-state approach first used in quantum optics. We found that quantum f luctuations renormalize the standard map, which governs the classical behav ior of the FK model, to a sawtooth map. This result is borne out by Monte-C arlo simulations. We also found that the ground state wave function changes from an extended state to a localized state when the coupling constant inc reases, Although quantum fluctuations largely smear the transition by break ing of analyticity observed in the classical case, the remnant of this tran sition is still discernible.