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 T_m 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 k_BT_m 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.