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Renormalization and Quantum Scaling of Frenkel–Kontorova Models

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Abstract

We generalise the classical Transition by Breaking of Analyticity for the class of Frenkel–Kontorova models studied by Aubry and others to non-zero Planck’s constant and temperature. This analysis is based on the study of a renormalization operator for the case of irrational mean spacing using Feynman’s functional integral approach. We show how existing classical results extend to the quantum regime. In particular we extend MacKay’s renormalization approach for the classical statistical mechanics to deduce scaling of low frequency effects and quantum effects. Our approach extends the phenomenon of hierarchical melting studied by Vallet, Schilling and Aubry to the quantum regime

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Author notes

  1. Nuno R. Catarino

    Present address: School of Mathematical and Computer Sciences, Colin Maclaurin Building, Heriot-Watt University, Edinburgh, EH14 4AS, U.K

Authors and Affiliations

  1. Mathematics Institute, University of Warwick, Gibbet Hill Road, Coventry, CV4 7AL, U.K

    Nuno R. Catarino & Robert S. MacKay

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  1. Nuno R. Catarino
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  2. Robert S. MacKay
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Correspondence to Robert S. MacKay.

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Catarino, N.R., MacKay, R.S. Renormalization and Quantum Scaling of Frenkel–Kontorova Models. J Stat Phys 121, 995–1014 (2005). https://doi.org/10.1007/s10955-005-8671-9

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  • DOI: https://doi.org/10.1007/s10955-005-8671-9

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