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

TL;DR

This work establishes odd-parity longitudinal magnetoconductivity (OMC) as a robust transport signature of intrinsic time-reversal symmetry breaking in metals. By integrating Berry curvature and orbital magnetic moment effects into a semiclassical transport framework, it derives explicit expressions for the OMC and related conductivities, and clarifies their symmetry constraints. The authors demonstrate, in valley-polarized gapped graphene, that a leading $B$-linear OMC emerges only when TRS is broken and valley polarization is present, with peaks near band edges and vanishing signals inside the gap; they also show OMC tracks the magnetic order parameter with temperature. Extending to the quantum oscillation regime, they reveal both odd- and even-$B$ contributions, including a universal $B$-linear term from the zeroth Landau level in the ultra-quantum limit (Abrikosov linear magnetoresistance). Collectively, OMC and its resistive counterpart offer a direct, symmetry-protected route to identify intrinsic TRS breaking and topological magnetic phases, complementing anomalous Hall measurements.

Abstract

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.

Figures (4)

• Figure 1: Schematic of longitudinal odd-parity magnetoconductivity (OMC) in the low-field limit. In time-reversal symmetric (TRS) materials, Onsager reciprocity forbids OMC, leaving only even-$B$ responses. In contrast, intrinsic TRS-breaking in magnetic systems produces a leading $B$-linear longitudinal response (OMC). Thus, a finite OMC serves as a direct transport signature of intrinsic TRS breaking.
• Figure 2: (a) Band dispersion of gapped Dirac Hamiltonian [Eq. \ref{['Ham_dirac']}] for the $\xi=+1$ valley with $v_F=2\times10^5$ m/s, and $\Delta=0.01$ eV. (b) Berry curvature distribution in the $k_x$-$k_y$ plane for the conduction band of the $\xi=+1$ valley, showing pronounced peaks near the band edges. The orbital magnetic moment shows a similar distribution.
• Figure 3: (a) $B$-linear (orange) and the $B$-quadratic (green) longitudinal magnetoconductivities versus chemical potential. (b) $B$-linear conductivity $\sigma_{yx}(B) = \sigma_{yx}^{\rm O}(B) + \sigma_{yx}^{\rm L}(B)$ (blue) and $B$-quadratic conductivity $\sigma_{yx}(B^2)$ (red) versus chemical potential. Both panels are for valley-polarized graphene with $\Delta_0=0.01$ eV, and $B=1$ T. In both (a) and (b), the solid line represents the numerically evaluated result, whereas the open circles denote the analytical results. (c) Numerically calculated temperature dependence of OMC%, assuming valley polarization of Eq. \ref{['delta_T_form']}. Here, $\sigma^{\rm even}_{xx}$ includes both the Drude and $\sigma_{xx}(B^2)$ contributions. The inset shows the magnetic order parameter $\Delta(T)$ in eV of Eq. \ref{['delta_T_form']} for $\xi = +1$. We have used $\beta=0.20$, $\mu=15$ meV, and $T_c=25$ K. The OMC% follows $\Delta(T)$ and vanishes at $T_c$, thereby serving as an effective order parameter for TRS breaking in transport experiments.
• Figure 4: (a) Landau levels of Hamiltonian \ref{['Ham_B']} for the $\xi=+1$ valley. The zeroth Landau level lies at $-\Delta$ for this valley, and it is independent of $B$. (b) Even-parity ($\rho^{\rm even}_{xx}$) and odd-parity ($\rho_{xx}^{\rm odd}$) resistivities for valley polarized gapped Dirac fermions, showing the symmetric and antisymmetric dependence on $B$. (c) Odd-parity magnetoresistivity (OMR%) as a function of $B$ for chemical potential $\mu=0.025$ eV. We have taken $k_s=10^8~\rm m^{-1}$, $n_{\rm im}=10^{13}~ \rm m^{-2}$, $\Gamma_0=1$ meV, and $\Delta=0.01$ eV.