Electronic Toroidal Metals: Landau Theory and Magnetoelectric Fingerprints

Abstract

Toroidal order is a higher-rank multipolar order whose intrinsic realization in itinerant-electron systems remains unexplored. Here, we develop a generic theory of the electronic toroidal metal (ETM), in which toroidal order emerges spontaneously from electronic degrees of freedom near the Fermi surfaces. Because candidate toroidal bilinears can overlap by symmetry with the uniform charge current, we formulate a projected instability criterion that removes the noncondensable current component. Within this current-orthogonal sector, we identify a soft collective mode in the $\mathcal{P}$-odd and $\mathcal{T}$-odd particle-hole channel and demonstrate that ETM arises as a Fermi-liquid instability. We then define the toroidal moment in an itinerant system through the antisymmetric magnetoelectric response tensor. We further establish an intimate connection between this response and the topology of pseudospin texture: the rearrangement of pseudospin vortices changes the winding structure of the Fermi surfaces and markedly enhances the magnetoelectric response. ETM also exhibits characteristic nonlinear charge transport, including a pronounced enhancement of its interband quantum-geometric contribution. Our results establish ETM as a distinct metallic Landau phase and provide a general framework for understanding toroidal order generated by itinerant electrons.