Finite-temperature effects on commensurate-incommensurate phase transitions in one and two dimensions

We study the effects of finite temperatures on the commensurate-incommensurate phase transition of one dimensionally modulated systems in one and two dimensions. In the one-dimensional case, as expected, the system is always in an incommensurate phase at any finite temperature. On the other hand, in two dimensions, the commensurate phase is still stable below a "critical pressure" ${\delta }_c\mathrm{}\left(T\right)\mathrm{}$ which decreases linearly with increasing temperature. However, above this critical pressure the system is in an incommensurate phase.