Triple point of the lock-in and onset transitions of charge-density waves

Abstract

We have studied a Landau theory for charge-density waves keeping terms up to sixth order in the free-energy expansion. Even with amplitude modulation, the lock-in transition remains continuous. We construct a phase diagram with three phases: normal, incommensurate, and commensurate meeting at a triple point. With no sixth-order term in the free energy there is a degeneracy of a tricritical point with the triple point, and the first-order phase transition line ends abruptly at the triple point. This is explained in terms of the singular behavior of the derivatives of the entropy in the incommensurate phase near the triple point. With the inclusion of a sixth-order term, the degeneracy of the tricritical point with the triple point is removed and the abrupt ending of the first-order line no longer occurs.