Polar Metals Taxonomy for Materials Classification and Discovery

Over the past decade, materials that combine broken inversion symmetry with metallic conductivity have gone from a thought experiment to one of the fastest growing research topics. In 2013, the observation of the first uncontested polar transition in a metal, LiOsO$_3$, inspired a surge of theoretical and experimental work on the subject, uncovering a host of materials which combine properties previously thought to be contraindicated [Nat. Mater. \textbf{12}, 1024 (2013)]. As is often the case in a nascent field, the sudden rise in interest has been accompanied by diverse (and sometimes conflicting) terminology. Although ``ferroelectric-like" metals are well-defined in theory, i.e., materials that undergo a symmetry-lowering transition to a polar phase while exhibiting metallic electron transport, real materials find a myriad of ways to push the boundaries of this definition. Here, we review and explore the burgeoning polar metal frontier from the perspectives of theory, simulation, and experiment while introducing a unified taxonomy. The framework allows one to describe, identify, and classify polar metals; we also use it to discuss some of the fundamental tensions between theory and models of reality inherent in the terms ``ferroelectric" and ``metals.'' In addition, we highlight shortcomings of electrostatic doping simulations in modeling different subclasses of polar metals, noting how the assumptions of this approach depart from experiment. We include a survey of known materials that combine polar symmetry with metallic conductivity, classified according to the mechanisms used to harmonize those two orders and their resulting properties. We conclude by describing opportunities for the discovery of novel polar metals by utilizing our taxonomy.