Journal of Statistical Physics

, Volume 121, Issue 5–6, pp 995–1014 | Cite as

Renormalization and Quantum Scaling of Frenkel–Kontorova Models

  • Nuno R. Catarino
  • Robert S. MacKayEmail author


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


Transition by breaking of analyticity renormalization quantum scaling specific heat 


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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryU.K
  2. 2.School of Mathematical and Computer Sciences, Colin Maclaurin BuildingHeriot-Watt UniversityEdinburghU.K

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