Generalized Neumann's Principle as a Unified Framework for Fractional Quantum and Conventional Ferroelectricity

Abstract

Monolayer In$_2$Se$3$ exhibits unexpected in-plane polarization, despite having $C_{3v}$ symmetry, a feature that was traditionally considered forbidden by symmetry. To explain this remarkable behavior, Ji et al. proposed the concept of fractional quantum ferroelectricity (FQFE), in which polarization occurs in fractional multiples of a quantum, and argued that this phenomenon violates {\it conventional} Neumann's principle. In this Letter, we introduce a generalized form of Neumann's principle and demonstrate that both FQFE and conventional ferroelectricity can be consistently described within this unified theoretical framework. We propose a method, based on the generalized Neumann's principle, for the systematic identification of FQFE materials. This approach is straightforward to apply and offers a clear conceptual understanding and deep physical insight for FQFE. Using this method, we determine all symmetry-allowed FQFE cases across the 32 crystallographic point groups. Since practical applications rely on the ability to control polarization, we further show that FQFE can be effectively switched via coupling with conventional polarization. Using HfZnN$_2$ as an illustrative example, we reveal the underlying mechanism of this coupling and outline a strategy to identify other materials with similar switching behavior.