ANNALS OF PHYSICS 139, 343-418 (1982) Critical Behavior of the One-Dimensional Commensurate-Incommensurate Transition L. A. TURKEVICH* Department of Physics, Stanford University, Stanford, California 94305, and Department of Physics, University of California, Los Angeles, California 90024 AND S. DONIACH Department of Applied Physics, Stanford University, Stanford, California 94305 Received August 14, 1981 We study the low-temperature critical behavior of one-dimensional charge-density-waves coupled to an underlying lattice, using the McMillan free energy. For weak coupling, the incommensurate CDW orders at T= 0 as a lattice of phase slip solitions with XY critical behavior. For strong coupling, the commensurate CDW orders at T = 0 with Ising critical behavior. Analytic expressions for the low-temperature inverse correlation length and average phase (change are obtained for all values of the coupling to the lattice. The possibility in systems of reduced dimensionality of coupling a charge-density- wave (CDW) distortion to an underlying lattice dates back to Peierls [ 11. The dimerized (commensurate CDW) state had previously been invoked in the problem of bond alternation in long chain molecules by Kuhn [2], and the commensurate Peierls instability per se was then treated quantitatively by the chemists [3]. The omnipresence of CDWs in systems of reduced dimensionality, and the existence in several such systems (TTF-TCNQ [4], TaSe, [5], NbSe, [6]) of transitions where an incommensurate CDW “locks onto” an underlying lattice prompts renewed interest in this problem. The effect of lattice coupling also drastically alters the transport properties [ 1, 7, 81. In this paper, we examine the equilibrium thermodynamics of one-dimensional CDW systems where there is coupling to an underlying lattice. In particular., associated with the transition from the incommensurate to the commen- surate CDW state is a cross-over from XY to Ising critical behavior [9]. The purpose of this paper is twofold. We are interested in the thermodynamics of * Present address: Corporate Research-Science Laboratories, Exxon Research and Engineering Company, Linden, New Jersey 07036. 343 0003-49 16/82/040343-76$05,00/O Copyright ‘C 1982 by Academic Press, Inc. All rights of reproduction in any form reserved.