Anharmonic corrections to the free energy and to the spectrum of collective modes in the 2d Wigner crystal

In this paper we study the effect of anharmonicity on equilibrium properties of the two-dimensional Wigner crystal. The leading-order perturbation theory is used restricted to high- and zero-temperature limits. The correction to the harmonic free energy of a crystal is found in both regimes. The correction to the Dulong-Petit law is found, which appears to be positive. The shifts to the spectrum of the harmonic phonon frequencies are calculated. In the high-temperature limit, it is shown that the anharmonicity softens all the spectrum of phonon frequencies, and the shifts are rather small. In the quantum limit, the anharmonicity softens only the lower transverse branch, whereas for the upper longitudinal branch the corrections are multidirectional. The relative shifts even close to quantum melting are less than 15%. The phonon lifetimes are also calculated for both limits. We find Wigner crystal to be rather harmonic, at least in the limits considered. Our findings can be used for the experiments on equilibrium properties of Wigner crystals as well as for the theoretical investigation of the phase diagram of interacting electrons.