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 $Q^{\prime }$ are connected by a conformal transformation. Consequently, $\sqrt{n}\times \sqrt{n}(n=7,9,13)$ 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.