Displacement IGM Current

• Displacement IGM current is the Fermi-sea component of the intrinsic gyrotropic magnetic response driven solely by Zeeman quantum metric contributions at magnetic topological insulator–unconventional magnet interfaces.
• It exhibits distinct 2n-fold angular harmonics that reveal the orbital symmetry and parity of the underlying unconventional magnetic order.
• Experimental estimates demonstrate that measurable voltage and current signatures make it a viable non-invasive probe for mapping emergent magnetic orders in heterostructures.

Displacement IGM current is the Fermi-sea component of the intrinsic gyrotropic magnetic (IGM) response at magnetic topological insulator–unconventional magnet interfaces, manifesting purely from Zeeman quantum metric contributions. In these heterostructures—where the magnetic layer is insulating—the displacement IGM current arises entirely from the proximity-induced magnetic exchange field without any influence from itinerant carriers. It encodes symmetry fingerprints of unconventional magnetic orders, directly tracing momentum-dependent structure of spin-split bands onto measureable transport harmonics and parity-sensitive sign-reversal patterns (Chakraborti et al., 21 Dec 2025).

1. Linear IGM Response in Magnetic Topological Insulator–Unconventional Magnet Interfaces

Under a weak, time-dependent magnetic field $\mathbf{B}(\omega)$ applied to the interface between a magnetic topological insulator (MTI) and an unconventional magnetic insulator (UMI), the linear intrinsic gyrotropic magnetic (IGM) current decomposes as

$$j_a^{\rm IGM} = \chi_{ab} B_b = j^{\rm cond}_a + j^{\rm disp}_a.$$

The conduction IGM current $j^{\rm cond}_a$ is governed by Zeeman Berry curvature and resides on the Fermi surface, while the displacement IGM current $j^{\rm disp}_a$ is governed by the Zeeman quantum metric (ZQM) and involves a Fermi-sea ($\mathbf{k}$-space) integral. In frequency space: $$j_a^{\rm cond}(\omega) = \sigma^C_{ab}(\omega) B_b(\omega), \qquad j_a^{\rm disp}(\omega) = \sigma^D_{ab}(\omega) B_b(\omega).$$ In the adiabatic regime, $\hbar\omega \ll \Delta$, the displacement conductivity simplifies to

$$\sigma^D_{ab} \propto \sum_{m \neq p} \int \frac{d^dk}{(2\pi)^d} \frac{2\hbar \omega}{\epsilon_{pm}}\, \mathcal{Q}^{ab}_{mp}(k),$$

where $\mathcal{Q}^{ab}_{mp}$ is the real ZQM coupling bands $m$ and $p$.

2. Theoretical Framework and Derivation of Displacement IGM Current

Employing Kubo formalism, the displacement IGM conductivity is expressed as

$$\sigma^D_{ab} = \frac{2\hbar\omega\, e^2}{\hbar^2} \sum_{m\neq p} \int \frac{d^dk}{(2\pi)^d}\, \frac{f_m}{\epsilon_{pm}} \mathcal{Q}^{ab}_{mp},$$

with $f_m$ as the Fermi–Dirac factor and $\epsilon_{pm} = \epsilon_p - \epsilon_m$. In the $\omega \rightarrow 0$ limit, one defines the polarization: $$P_a = -\frac{e}{\hbar} \sum_n \int \frac{d^dk}{(2\pi)^d} g^{(n)}_{ab}(\mathbf{k}) B_b,$$ yielding the displacement current $j^{\rm disp}_a = i\omega P_a$. Integration by parts (over the periodic Brillouin zone) renders: $$j^{\rm disp}_a = -\frac{e}{\hbar} \sum_n \int \frac{d^dk}{(2\pi)^d} \partial_{k_b}\left[ g^{(n)}_{ab}(\mathbf{k}) \right] B_b.$$ Here, $g^{(n)}_{ab}(\mathbf{k})$ is the band-resolved ZQM, quantifying the infinitesimal response of the local Bloch spinor under combined momentum shift and Zeeman-induced spin rotation. Its gradient, $\partial_{k_b}g^{(n)}_{ab}$, captures variations of local quantum geometry across $\mathbf{k}$-space, driving bound (displacement) charge motion under oscillatory Zeeman field.

3. Symmetry Analysis: Angular Harmonics and Orbital Fingerprints

A salient feature of the displacement IGM current is its angular-harmonic decomposition, directly reflecting the form factor symmetry of the underlying unconventional magnetic order. For a form factor $g_{\mathbf{k}} \propto k^n\cos(n\phi)$ or $\sin(n\phi)$, the transverse displacement conductivity adopts a pure $2n$-fold harmonic in the field orientation angle $\theta$: $$\sigma^D_{xy}(\theta) \propto \sin(2n\theta), \quad n = 1,\ldots,6 \;\;(p, d, f, g, i\text{-wave}).$$ Correspondingly:

• $p$-wave ($n=1$): $\sin(2\theta)$ (2 nodes)
• $d$-wave ($n=2$): $\sin(4\theta)$ (4 nodes)
• $f$-wave ($n=3$): $\sin(6\theta)$
• $g$-wave ($n=4$): $\sin(8\theta)$
• $i$-wave ($n=6$): $\sin(12\theta)$

The emergence of these “$2n$-fold” harmonics (Editor’s term) is attributed to the sign-alternating structure of the proximity-induced Dirac mass $g_{\mathbf{k}}$ in the magnetically proximitized surface states.

Orbital symmetry	Form factor $g_{\mathbf{k}}$	Transverse harmonic $\sigma^D_{xy}(\theta)$
$p$	$k\cos\phi$/$\sin\phi$	$\sin(2\theta)$
$d$	$k^2\cos(2\phi)/\sin(2\phi)$	$\sin(4\theta)$
$f$	$k^3\cos(3\phi)/\sin(3\phi)$	$\sin(6\theta)$
$g$	$k^4\cos(4\phi)/\sin(4\phi)$	$\sin(8\theta)$
$i$	$k^6\cos(6\phi)/\sin(6\phi)$	$\sin(12\theta)$

4. Parity and Longitudinal Displacement Response: Sign-Reversal Patterns

Parity symmetry underpins the longitudinal displacement IGM conductivity pattern. For $\sigma^D_{xx}(\theta)$:

• Odd-parity ($p, f$): Two sign reversals (nodes at $\theta=0, \pi$), harmonic $\cos(2\theta)$.
• Even-parity ($d, g, i$): Four sign reversals (nodes at $\theta = n\pi/2$, $n=1,2,3,4$), harmonic $\cos(4\theta)$.

Consequently, the longitudinal “sign-reversal count” (Editor’s term) serves as a parity test for the unconventional magnetic order, cleanly differentiating between odd- and even-parity form factors via distinct nodal patterns in the $[0,2\pi)$ interval.

5. Experimental Realizability and Magnitude Estimates

The displacement IGM current yields voltages and current densities within experimentally accessible ranges in realistic topological insulator–magnet heterostructures. For instance, Cr-doped Bi$_2$Se$_3$ ($5\%$ Cr: $\Delta_{\rm ex} \approx 0.2$ eV) interfaced with a $d_{x^2-y^2}$-wave altermagnet (RuO$_2$, $J_0 \sim 1$ eV) under

• DC field $B_{\rm dc} \sim 0.2$ T,
• AC probing field $B_{\rm ac} \sim 10$ mT at $\omega = 10^{13}$ Hz,
• TI thickness $L \sim 100~\mu$m, sheet resistance $R \sim 10^2~\Omega$,

results in

$$V_\perp \approx 1.1~\mathrm{mV}, \qquad V_\parallel \approx 0.11~\mathrm{mV},$$

well above lock-in amplifier detection thresholds. Using the scaling

$$j^{\rm disp} \sim \frac{e J_0}{\hbar} \left( \frac{\Delta_{\rm ex}}{v_F} \right)^n B_{\rm ac}$$

(with $v_F \sim 10^5$ m/s), displacement currents span $\mu$A–mA per meter width, supporting feasibility for device-scale gyrotropic transport measurement protocols.

6. Displacement IGM Current as a Symmetry Fingerprint

Distinct from conduction IGM current, which is governed by Berry curvature and displays minimal sensitivity to underlying magnetic symmetry, the displacement IGM current is a unique, high-fidelity fingerprint for unconventional magnetic order. Its salient features are:

• Exclusivity to ZQM: The response is determined entirely by the real (metric) part of band quantum geometry under combined momentum and Zeeman field rotation.
• Orbital resolution via harmonics: The transverse $2n$-fold harmonic directly identifies the orbital character ($p$, $d$, $f$, $g$, $i$) of the magnetically proximitized mass.
• Parity discrimination: The longitudinal sign-reversal pattern efficiently separates odd- and even-parity magnetic orders.
• Absence in uniform magnets: For uniform (ferro-)magnets, the ZQM vanishes, meaning no displacement IGM current is generated.

These properties establish the displacement IGM current as a powerful gyrotropic fingerprint, enabling direct, symmetry-enforced mapping from the band topology and magnetic symmetry content of unconventional magnetic insulators to transport responses measurable via standard experimental techniques.

7. Implications and Future Prospects

The rigorous symmetry selectivity and experimental accessibility position displacement IGM current detection as a generalized protocol for characterizing momentum-dependent orders in insulating magnets. Its pure dependence on quantum metric phenomena expands the toolbox for non-invasive, transport-based identification of emergent symmetry and topological properties in magnetic heterostructures. A plausible implication is that ongoing advances in lock-in and magneto-optical detection methods will further amplify the sensitivity to these fingerprint signals, accelerating identification and engineering of exotic magnetic orders in topological matter (Chakraborti et al., 21 Dec 2025).