Conformal aspect of charge density waves: Theory and experiment

The variety of two-dimensional (2D) discommensurate charge density wave (DC-CDW) phases has been explained by a phenomenological CDW free-energy theory, where the wave vector of different DC-CDW phases corresponds to different local minima of a multivalley free-energy landscapes. Here, we discover that experimental 2D DC-CDW structures in transition-metal dichalcogenides can be understood elegantly by a conformal transformation with complex numbers. Specifically, we represent a 2D CDW wave vector Q by a complex number Q and find that by minimizing the CDW free energy with respect to Q, different wave vectors Q and Q^' are connected by a conformal transformation. Consequently, √n×√n4pt0ex(n=7,9,13) honeycomb and (quasi-)stripe DC-CDWs can be understood using simple conformal groups, which link DC-CDW wave vectors (multivalley landscape) to the incommensurate and commensurate wave vectors.