Toroidal loop-current order in kagome AV3Sb5 : Zero-field diode and anomalous Hall

The kagome superconductors AV3Sb5 show a puzzling coexistence of a CDW-driven anomalous Hall effect (AHE) and a zero-field superconducting diode effect (SDE) with normal-state magnetic-field trainability. We propose a unified mechanism via a symmetry-based, gauge-covariant Ginzburg-Landau theory. A composite toroidal loop-current order, generated by the triple-Q CDW, sources an emergent orbital gauge field. The superconducting condensate couples to both electromagnetic and orbital gauge potentials; a Lifshitz invariant enforces a finite pairing momentum. This momentum creates an intrinsic Josephson phase bias, naturally yielding a zero-field SDE. The same structure produces a mixed orbital-electromagnetic response that accounts for the CDW-onset AHE. The framework furnishes quantitative relations used to analyze experiments, whose extracted parameters agree semi-quantitatively with device data. We propose falsifiable, targeted tests, and note that the construction is consistent with the phenomenology of both AV3Sb5 and twisted trilayer graphene.