Quantized intrinsically localized modes of the Fermi–Pasta–Ulam lattice

Abstract

We have examined the quantized n = 2 excitation spectrum of the Fermi–Pasta–Ulam lattice. The spectrum is composed of a resonance in the two-phonon continuum and a branch of infinitely long-lived excitations which splits off from the top of the two-phonon creation continuum. We calculate the zero-temperature limit of the many-body wavefunction and show that this mode corresponds to an intrinsically localized mode (ILM). In one dimension, we find that there is no lower threshold value of the repulsive interaction that must be exceeded if the ILM is to be formed. However, the spatial extent of the ILM wavefunction rapidly increases as the interaction is decreased to zero. The dispersion relation shows that the discrete n = 2 ILM and the resonance hybridize as the center of mass wavevector q is increased towards the zone boundary. The many-body wavefunction shows that as the zone boundary is approached, there is a destructive resonance which occurs between pairs of sites separated by an odd number of lattice spacings. We compare our theoretical results with the recent experimental observation of a discrete ILM in NaI by Manley et al.

Keywords:

• intrinsically localized modes
• discrete breathers
• quantized excitations
• Fermi–Pasta–Ulam model

Acknowledgements

This work was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Materials Science through the award DEFG02-84ER45872. The work is dedicated to David Sherrington (F.R.S.) in honour of his seventieth birthday. One of the authors (PSR) would like to thank A.R. Bishop, S. Flach, M.E. Manley and A.J. Sievers for enlightening conversations.

Notes

Notes

1. The ground state is degenerate under the continuous transformation , and therefore the assumed spontaneous symmetry breaking leads to the occurrence of Goldstone modes.

2. At finite temperatures, this ansatz should be generalized to include terms which involve a product of one creation and one annihilation operator. The ansatz reproduces the exact solutions for the two-boson excitations of models in which the total boson occupation number is a conserved quantity.