Odd Parity Magnetoresistance

• OMR is defined as the antisymmetric component of magnetoresistance extracted from R(B) - R(–B), appearing only when time-reversal symmetry is broken by intrinsic or proximity-induced magnetization.
• Experimental platforms range from domain-controlled antiferromagnets to proximity-induced topological heterostructures, with some reports showing over 1000% odd resistance change under low magnetic fields.
• Underlying mechanisms include spin-dependent scattering and Berry curvature effects, with tensorial symmetry analysis distinguishing OMR from ordinary magnetoresistive and anomalous Hall responses.

Searching arXiv for papers on odd-parity magnetoresistance and closely related odd-parity magnetoconductivity to ground the article in the cited literature. Odd parity magnetoresistance (OMR) is a longitudinal magnetotransport response in which the resistance or resistivity acquires a component that is antisymmetric under magnetic-field reversal. In conventional magnetoresistance, Onsager reciprocity enforces an even-in-field response when time-reversal symmetry (TRS) is unbroken; OMR appears only when TRS is broken, whether by intrinsic magnetization, a field-selected magnetic domain, or magnetic proximity. Across the literature, OMR has evolved from a symmetry-allowed odd-in-$B$ correction in pyrochlore iridates and ferromagnets to a gate-tunable response in proximitized semiconductor channels and magnetized bilayer graphene, culminating in a proximity-induced topological heterostructure, $\alpha$-Sn/(In,Fe)Sb, where the odd component reaches $1{,}150\%$ at $1\,\mathrm{T}$ (Fujita et al., 2015, Takiguchi et al., 2020, Sahani et al., 2024, Hotta et al., 31 Jul 2025).

1. Definition and reciprocity

Magnetoresistance is commonly written as

$MR(B)=\frac{R(B)-R(0)}{R(0)}.$

With unbroken TRS, Onsager reciprocity requires $R(B)=R(-B)$, so $MR(B)$ is an even function of $B$. The odd component is extracted as

$MR_{\mathrm{odd}}(B)=\frac{MR(B)-MR(-B)}{2},$

or, equivalently in resistivity notation,

$$\rho_{xx}^{\mathrm{odd}}(B)=\frac{\rho_{xx}(B)-\rho_{xx}(-B)}{2}.$$

By construction, $\alpha$0 and $\alpha$1, whereas the conventional magnetoresistive contribution is the symmetric part (Fujita et al., 2015, Sahani et al., 2024, Hotta et al., 31 Jul 2025).

A more general reciprocity statement in magnetic matter is

$\alpha$2

so an odd longitudinal response is allowed only when the material possesses an intrinsic order parameter $\alpha$3 that is itself odd under TRS. This distinguishes OMR from ordinary even-parity magnetoresistance and from the anomalous Hall effect (AHE): OMR is diagonal in the transport tensor and odd in the applied field, whereas the AHE is off-diagonal (Wang et al., 2019, Das et al., 16 Sep 2025).

In several experimentally relevant models, TRS breaking is necessary but not always sufficient. In one-dimensional InAs edge channels magnetized by a proximity effect from (Ga,Fe)Sb, simultaneous breaking of TRS and spatial inversion symmetry (SIS) is required to prevent cancellation of the odd contribution in the Boltzmann transport integrals (Takiguchi et al., 2020).

2. Symmetry structure and tensor formulations

In ferromagnets with a fixed intrinsic magnetization $\alpha$4, the resistivity tensor can be expanded to linear order in magnetic field as

$\alpha$5

Within this structure, the leading odd-parity longitudinal terms arise from $\alpha$6 and from the $\alpha$7 contribution associated with $\alpha$8. In the canonical geometry with $\alpha$9, $1{,}150\%$0, and $1{,}150\%$1 rotated in the $1{,}150\%$2–$1{,}150\%$3 plane, the observed longitudinal and transverse odd responses take the forms

$1{,}150\%$4

linking OMR and the odd-parity planar Hall effect within a common tensorial framework (Wang et al., 2019).

In all-in/all-out antiferromagnets, the symmetry description is naturally expressed through a rank-3 tensor $1{,}150\%$5 coupling $1{,}150\%$6 to $1{,}150\%$7:

$1{,}150\%$8

For Eu$1{,}150\%$9Ir$1\,\mathrm{T}$0O$1\,\mathrm{T}$1 thin films, this permits an odd-in-$1\,\mathrm{T}$2 longitudinal contribution once the all-in/all-out order breaks TRS while preserving inversion and cubic symmetries on average. Projected onto the longitudinal channel, the result is $1\,\mathrm{T}$3, consistent with a linear odd magnetoresistance below the magnetic transition (Fujita et al., 2015).

A later general symmetry analysis formulated the same issue in terms of odd-parity magnetoconductivity (OMC). In that treatment, the conductivity tensor is expanded as

$1\,\mathrm{T}$4

Longitudinal OMC follows the same point-group constraints as the anomalous Hall conductivity, whereas symmetric transverse OMC obeys distinct rules. This places OMR within a broader band-geometric transport classification rather than treating it only as a resistive anomaly (Das et al., 16 Sep 2025).

3. Experimental platforms and phenomenology

OMR has been realized in several distinct material classes, with widely different magnitudes, field scales, and control parameters. Representative cases are summarized below (Fujita et al., 2015, Wang et al., 2019, Takiguchi et al., 2020, Sahani et al., 2024, Hotta et al., 31 Jul 2025).

Platform	Control or geometry	Reported odd response
Eu$1\,\mathrm{T}$5Ir$1\,\mathrm{T}$6O$1\,\mathrm{T}$7 thin films	Cooling-field selection of all-in/all-out domains	$1\,\mathrm{T}$8 at $1\,\mathrm{T}$9
SmCo$MR(B)=\frac{R(B)-R(0)}{R(0)}.$0	Fixed intrinsic magnetization	$MR(B)=\frac{R(B)-R(0)}{R(0)}.$1 at $MR(B)=\frac{R(B)-R(0)}{R(0)}.$2
InAs/(Ga,Fe)Sb	Gate control of magnetic proximity and edge asymmetry	$MR(B)=\frac{R(B)-R(0)}{R(0)}.$3 at $MR(B)=\frac{R(B)-R(0)}{R(0)}.$4; $MR(B)=\frac{R(B)-R(0)}{R(0)}.$5 upon full field reversal
Magnetized bilayer graphene	Electrostatic tuning near band edges	$MR(B)=\frac{R(B)-R(0)}{R(0)}.$6 at $MR(B)=\frac{R(B)-R(0)}{R(0)}.$7
$MR(B)=\frac{R(B)-R(0)}{R(0)}.$8-Sn/(In,Fe)Sb	In-plane $MR(B)=\frac{R(B)-R(0)}{R(0)}.$9 in a proximitized ultrathin film	$R(B)=R(-B)$0 at $R(B)=R(-B)$1, $R(B)=R(-B)$2

The Eu$R(B)=R(-B)$3Ir$R(B)=R(-B)$4O$R(B)=R(-B)$5 case established a domain-controlled odd linear term: after zero-field cooling, $R(B)=R(-B)$6 is symmetric; after cooling in $R(B)=R(-B)$7, a clear asymmetric term appears, $R(B)=R(-B)$8, with the sign of $R(B)=R(-B)$9 determined by the A- or B-domain selected during cooling. The coefficient saturates once $MR(B)$0 and vanishes above $MR(B)$1, confirming the direct relation to all-in/all-out order (Fujita et al., 2015).

In InAs/(Ga,Fe)Sb, the odd component originates in one-dimensional edge channels rather than the bulk two-dimensional channel. Four-terminal measurements show a large $MR(B)$2 linear up to $MR(B)$3, persisting to room temperature. Opposite edges yield equal-magnitude OMR of opposite sign, while a two-terminal geometry across the full width produces nearly zero $MR(B)$4 because the two edge contributions cancel. Gate voltages can turn the effect on and off, and can reverse the sign of the uncompensated response (Takiguchi et al., 2020).

In bilayer graphene proximitized by Cr$MR(B)$5Te$MR(B)$6Ge$MR(B)$7, the longitudinal antisymmetric response is absent in a BLG/hBN control device but substantial in BLG/CGT. The odd part is linear in $MR(B)$8 up to quantum oscillations, peaks near the band edges, is reduced at higher carrier density, remains essentially unchanged up to $MR(B)$9, and vanishes near $B$0 with a fitted critical form $B$1, $B$2 (Sahani et al., 2024).

4. Microscopic mechanisms

Two microscopic mechanisms were isolated in ferromagnetic systems with pinned magnetization. One is spin-polarization-dependent scattering on a Zeeman-shifted Fermi surface: unequal relaxation times for the two spin populations imply that a small field-induced change in spin polarization generates a longitudinal term proportional to $B$3. The other is an anomalous-velocity mechanism usually associated with Berry-curvature physics. Starting from

$B$4

iteration to first order in $B$5 produces a term proportional to $B$6, which contributes both to odd longitudinal magnetoresistance and to the odd planar Hall effect (Wang et al., 2019).

In one-dimensional InAs edge channels, the effective Hamiltonian is

$B$7

Here $B$8 and $B$9 encode Rashba terms from side-wall and top-surface electric fields, while $MR_{\mathrm{odd}}(B)=\frac{MR(B)-MR(-B)}{2},$0 is the Zeeman term. Under the assumption $MR_{\mathrm{odd}}(B)=\frac{MR(B)-MR(-B)}{2},$1, solving the linearized Boltzmann equation with distinct relaxation times $MR_{\mathrm{odd}}(B)=\frac{MR(B)-MR(-B)}{2},$2 for the two Rashba branches yields an odd term in $MR_{\mathrm{odd}}(B)=\frac{MR(B)-MR(-B)}{2},$3 proportional to $MR_{\mathrm{odd}}(B)=\frac{MR(B)-MR(-B)}{2},$4, with $MR_{\mathrm{odd}}(B)=\frac{MR(B)-MR(-B)}{2},$5. In this framework, giant OMR requires the simultaneous presence of magnetic proximity, edge-induced inversion breaking, and asymmetric scattering between the Rashba branches (Takiguchi et al., 2020).

Berry curvature and orbital magnetic moment provide a more general band-geometric description. In magnetized bilayer graphene, the leading odd contribution to the longitudinal conductivity is

$MR_{\mathrm{odd}}(B)=\frac{MR(B)-MR(-B)}{2},$6

so the response is finite only when exchange coupling breaks TRS and generates net $MR_{\mathrm{odd}}(B)=\frac{MR(B)-MR(-B)}{2},$7 and $MR_{\mathrm{odd}}(B)=\frac{MR(B)-MR(-B)}{2},$8 (Sahani et al., 2024). The later general OMC theory extends this result by showing that $MR_{\mathrm{odd}}(B)=\frac{MR(B)-MR(-B)}{2},$9 contains three distinct band-geometric contributions: terms involving $$\rho_{xx}^{\mathrm{odd}}(B)=\frac{\rho_{xx}(B)-\rho_{xx}(-B)}{2}.$$0, a term containing $$\rho_{xx}^{\mathrm{odd}}(B)=\frac{\rho_{xx}(B)-\rho_{xx}(-B)}{2}.$$1, and a Berry-curvature term proportional to $$\rho_{xx}^{\mathrm{odd}}(B)=\frac{\rho_{xx}(B)-\rho_{xx}(-B)}{2}.$$2. Inverting conductivity to resistivity gives

$$\rho_{xx}^{\mathrm{odd}}(B)=\frac{\rho_{xx}(B)-\rho_{xx}(-B)}{2}.$$3

which makes explicit the conductivity origin of the odd resistive signal (Das et al., 16 Sep 2025).

A distinct antecedent framework was developed for hopping transport through non-zero-angular-momentum impurity orbitals in organic semiconductors and HOPG graphite. In that theory, TRS preserves the degeneracy between $$\rho_{xx}^{\mathrm{odd}}(B)=\frac{\rho_{xx}(B)-\rho_{xx}(-B)}{2}.$$4 and $$\rho_{xx}^{\mathrm{odd}}(B)=\frac{\rho_{xx}(B)-\rho_{xx}(-B)}{2}.$$5 orbitals, so the leading low-field correction is quadratic. Broken TRS lifts the $$\rho_{xx}^{\mathrm{odd}}(B)=\frac{\rho_{xx}(B)-\rho_{xx}(-B)}{2}.$$6 degeneracy and produces a linear negative hopping contribution, written as

$$\rho_{xx}^{\mathrm{odd}}(B)=\frac{\rho_{xx}(B)-\rho_{xx}(-B)}{2}.$$7

with the linear term interpreted as a signature of TRS breaking in hopping-mediated transport (Alexandrov et al., 2012).

5. Proximity-induced topological states in $$\rho_{xx}^{\mathrm{odd}}(B)=\frac{\rho_{xx}(B)-\rho_{xx}(-B)}{2}.$$8-Sn/(In,Fe)Sb

The $$\rho_{xx}^{\mathrm{odd}}(B)=\frac{\rho_{xx}(B)-\rho_{xx}(-B)}{2}.$$9-Sn/(In,Fe)Sb heterostructure provides the largest reported OMR in the supplied literature. The system is grown on InSb(001), with a $\alpha$00 ferromagnetic semiconductor (In,Fe)Sb bottom layer, $\alpha$01, and a $\alpha$02 ($\alpha$03 monolayers) $\alpha$04-Sn top layer grown by MBE. Standard Hall bars of $\alpha$05 are used, with a dc current $\alpha$06 along $\alpha$07 and an in-plane magnetic field rotated parallel or antiparallel to the current so that the ordinary Hall contribution is nulled. At $\alpha$08 and $\alpha$09, the odd magnetoresistance reaches up to $\alpha$10 at $\alpha$11, grows roughly linearly with field with slight nonlinearity at higher $\alpha$12, follows the projection of $\alpha$13 onto the current direction with a $\alpha$14 law, vanishes when $\alpha$15, and disappears above approximately $\alpha$16 (Hotta et al., 31 Jul 2025).

The striking magnitude is not attributed to ordinary size effects in a trivial ultrathin semiconductor. Although $\alpha$17-Sn at this thickness is a trivial narrow-gap semiconductor, perpendicular-field measurements reveal large even magnetoresistance, about $\alpha$18 at $\alpha$19, together with Shubnikov–de Haas oscillations. The FFT of $\alpha$20 versus $\alpha$21 yields a single frequency $\alpha$22, implying a small two-dimensional Fermi pocket; the temperature dependence gives $\alpha$23; the Landau-fan intercept gives a Berry phase $\alpha$24; and the angle dependence follows $\alpha$25 up to $\alpha$26, confirming the two-dimensional character of the pocket before a secondary three-dimensional InSb pocket appears at larger tilt. These observations were interpreted as evidence for linear Dirac/Weyl-type dispersion localized at the interface (Hotta et al., 31 Jul 2025).

First-principles calculations connect these oscillation data to magnetic proximity. Bulk $\alpha$27-Sn under small biaxial compressive strain is a Dirac semimetal with inverted $\alpha$28 ordering, but an $\alpha$29-ML free-standing film becomes trivial because quantum confinement reverses the inversion and opens a small gap. In a slab model with Fe atoms on one surface, Fe $\alpha$30–Sn $\alpha$31 hybridization closes the gap and restores linear dispersion around $\alpha$32. The resulting bands along $\alpha$33–X and $\alpha$34–Y are inequivalent, and each Dirac or Weyl cone is slightly tilted in the film plane, with $\alpha$35 (Hotta et al., 31 Jul 2025).

The semiclassical interpretation uses oppositely tilted Weyl cones. Near node $\alpha$36,

$\alpha$37

with dispersion

$\alpha$38

A nonzero tilt breaks $\alpha$39 inversion for each cone and, through Berry curvature, generates a current correction linear in $\alpha$40,

$\alpha$41

For $\alpha$42 and $\alpha$43, the total odd component obeys $\alpha$44; because the two cones have opposite chirality and opposite tilt, their contributions add rather than cancel. A more complete fit gives

$\alpha$45

and the full three-dimensional angular dependence at $\alpha$46 is reproduced with $\alpha$47, consistent with the anisotropic tilts extracted from DFT (Hotta et al., 31 Jul 2025).

6. Significance, applications, and unresolved issues

OMR has become a transport probe of intrinsic TRS breaking in systems where a purely even longitudinal response would otherwise be expected. In BLG/CGT, the odd longitudinal signal is presented as a significant probe of TRS breaking in quantum materials in which the crystal symmetries preclude the appearance of anomalous Hall effect, or where the AHE is too small to detect (Sahani et al., 2024). A later crystalline-symmetry analysis states, however, that longitudinal OMC follows the same point-group constraints as the anomalous Hall conductivity, whereas symmetric transverse OMC obeys different rules (Das et al., 16 Sep 2025). This suggests that comparisons across the OMR literature must distinguish carefully between conductivity and resistivity formulations, between longitudinal and transverse odd responses, and between bulk symmetry arguments and device-level measurement geometries.

The device implications are correspondingly diverse. In $\alpha$48-Sn/(In,Fe)Sb, the odd resistance change is both large and linear, so the device behaves as a “Hall-like” sensor. With sensitivity defined as $\alpha$49, the reported value is $\alpha$50, compared with $\alpha$51–$\alpha$52 for the best commercial InSb Hall sensors. The same work identifies ultrasensitive magnetic-field detectors, low-power spintronic read heads, and on-chip magnetometry as plausible applications (Hotta et al., 31 Jul 2025). In InAs/(Ga,Fe)Sb, the ability to tune magnetic proximity and edge asymmetry independently enables ON/OFF switching and polarity reversal of OMR; in BLG/CGT, the response is field- and gate-programmable and remains operative at room temperature (Takiguchi et al., 2020, Sahani et al., 2024).

A recurrent misconception is to equate OMR with any linear magnetoresistance. In the modern transport literature, OMR is specifically the antisymmetric longitudinal component extracted by $\alpha$53 or $\alpha$54, and it should vanish when the relevant TRS-breaking order disappears or when opposite odd contributions cancel. Eu$\alpha$55Ir$\alpha$56O$\alpha$57 is symmetric after zero-field cooling and loses its odd term above $\alpha$58; two-terminal InAs measurements across both edges show nearly zero odd response because the two edge contributions cancel; and a BLG/hBN control sample exhibits a purely even $\alpha$59 (Fujita et al., 2015, Takiguchi et al., 2020, Sahani et al., 2024).

Taken together, the available results identify OMR as a sharply symmetry-constrained yet microscopically diverse transport phenomenon. Its realizations span domain-selected antiferromagnets, high-coercivity ferromagnets, inversion-broken one-dimensional channels, magnetized graphene, and proximity-induced topological states. The most extreme case so far, in $\alpha$60-Sn/(In,Fe)Sb, indicates that a small Fermi surface in a linear band, strong tilt induced by magnetic-proximity-reinvoked topology, and additive odd responses from two cones can push OMR beyond $\alpha$61 at low field, far above the modest values reported previously (Hotta et al., 31 Jul 2025).