Abstract
The half-quantized Hall phase represents a unique metallic or semi-metallic state of matter characterized by a fractional quantum Hall conductance, precisely half of an integer ν multiple of e²/h. Here we demonstrate the existence of a novel 1/2Z topological invariant that sets the half-quantized Hall phase apart from two-dimensional ordinary metallic ferromagnets. The 1/2Z classification is determined by the line integral of the intrinsic anomalous Hall conductance, which is safeguarded by two distinct categories of local unitary and anti-unitary symmetries in proximity to the Fermi surface of electron states. We further validate the 1/2Z topological order in the context of the quantized Hall phase by examining semi-magnetic topological insulator Bi₂Te₃ and Bi₂Se₃ film for ν=1 and topological crystalline insulator SnTe films for ν=2 or 4. Our findings pave the way for future exploration and understanding of topological metals and their unique properties.