Low-temperature excitations, specific heat and hierarchical melting of a one-dimensional incommensurate structure

Abstract

The low-temperature behaviour of an Ising spin model for a one-dimensional incommensurate structure (which is equivalent to the discrete Frenkel-Kontorova model in the non-analytical regime) is analytically investigated. Its specific heat exhibits an infinite series of Schottky anomalies at temperatures Tm going to zero. These anomalies correspond to crossover which can be physically associated with a hierarchical melting of the incommensurate chain. In contrast with early work, the authors show that certain types of phase defect (discommensurations) are the essential low-temperature excitations of the system and in addition that they can be hierarchically classified according to the continued-fraction expansion of the commensurability ratio zeta . Their characteristic energies kBTm and their weights are explicitly calculated at all orders. They present first a physically simple but empirical approach which allows one to predict and to calculate approximately these anomalies. Next, the validity of this approach is confirmed by a more sophisticated and new method of renormalisation with unequal blocks, which becomes exact as T goes to zero. The authors discuss the application of this theory to experimental measurements of the specific heat in K-hollandite.