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Odd-parity longitudinal magnetoconductivity in time-reversal symmetry broken materials

Magnetotransport measurements are a sensitive probe of symmetry and electronic structure in quantum materials. While conventional metals exhibit longitudinal magnetoconductivity that is even in a magnetic field ($B$) for small $B$, we show that magnetic materials which intrinsically break time-reversal symmetry (TRS) show an {\it odd-parity magnetoconductivity} (OMC), with a leading linear-$B$ response. Using semiclassical transport theory, we derive explicit expressions for the longitudinal and transverse conductivities and identify their origin in Berry curvature and orbital magnetic moment. Crystalline symmetry analysis shows that longitudinal OMC follows the same point-group constraints as the anomalous Hall effect, while transverse OMC obeys distinct rules, providing an independent probe of TRS breaking. In the large $B$ quantum oscillation regime, we uncover both odd- and even-$B$ contributions, demonstrating OMC beyond the semiclassical picture. Explicit calculations in valley-polarized gapped graphene show that OMC peaks near the band edges, vanish in the band gap and follow the temperature dependence of the magnetic order parameter. Our results explain the odd-parity magnetoresistance recently observed in magnetized graphene and establish OMC as a robust transport signature of intrinsic TRS breaking in metals.