Zeeman effect in Solids: From field-Induced Topological metal to In-Plane Anomalous Hall Effects

In nonmagnetic centrosymmetric metals, Kramers-degenerate bands carry SU(2) Berry curvature that is traceless, yielding no net anomalous Hall response. An external magnetic field lifts this degeneracy through the Zeeman effect, described by a momentum-dependent g-factor tensor ĝ̂(k) whose orbital part—arising from high-energy bands via Löwdin down-folding—can far exceed the free-electron value g = 2 in materials with strong spin-orbit coupling. We have developed a first-principles method, implemented within the PAW formulation of DFT, to compute ĝ̂(k) and validate it against experiments on Bi and Bi₂Se₃. The k-dependence of ĝ̂ on the Fermi surface generates Zeeman-induced U(1) Berry curvature with three major consequences that have been overlooked for decades: (i) a modified spin-zero effect in quantum oscillations, verified in ZrTe₅; (ii) nonzero Chern numbers on Zeeman-split Fermi surfaces, establishing field-induced topological metals in TaAs₂ and Na₃Bi; and (iii) an in-plane anomalous Hall effect (AHE) driven purely by Berry curvature when the magnetic field lies in the current–voltage plane. For two-dimensional systems without C₂z symmetry, we show analytically and numerically that the in-plane AHE can be quantized or nearly half-quantized. Concrete predictions are made for Sb₂Te₃ thin films, where quantized and half-quantized plateaus emerge under strong in-plane fields, and for Cd₃As₂ thin films, where quantum confinement combined with in-plane Zeeman coupling drives transitions to a 2D Weyl semimetal phase (protected by C₂z𝒯) and, upon breaking this symmetry, to a Chern insulator with quantized in-plane AHE.