Intraband Quantum Metric Dipole (intraQMD)

• Intraband QMD is a gauge-invariant property defined as the momentum derivative of the quantum metric, capturing anisotropy in the Fermi surface.
• It plays a central role in nonlinear Hall effects, nonreciprocal optics, and plasmonic responses by breaking individual inversion and time-reversal symmetries.
• Its evaluation via k-space differentiation and experimental second-harmonic Hall measurements offers predictive insights for topological magnets and antiferromagnets.

The intraband quantum metric dipole (intraQMD) is a gauge-invariant, quantum-geometric Fermi surface quantity encapsulating the momentum-space gradient of the quantum metric within a single electronic band. It plays a central role in various nonlinear and nonreciprocal transport phenomena—most notably, as a key geometric contribution to the intrinsic nonlinear Hall effect in topological magnets and antiferromagnets, the nonreciprocal directional dichroism in optical responses, the intrinsic nonreciprocity of bulk plasmons, and the energy scale and topological robustness of collective excitations in flat-band ferromagnets. The intraQMD emerges wherever the momentum-resolved quantum metric $g_{ab}(\mathbf{k})$ of Bloch states exhibits anistropy or skewness across the Fermi surface and survives in situations where both Berry curvature dipoles and interband quantum metric dipoles vanish, specifically when combined spatial inversion $\mathcal{P}$ and time-reversal $\mathcal{T}$ symmetries are broken individually but preserved jointly.

1. Definition and Mathematical Formulation

For a single Bloch band with periodic Bloch state $|u_n(\mathbf{k})\rangle$, the quantum metric is defined as

$$g_n^{ab}(\mathbf{k}) = \mathrm{Re}\left[ \langle \partial_{k_a} u_n(\mathbf{k}) | (1 - |u_n(\mathbf{k})\rangle\langle u_n(\mathbf{k})| ) | \partial_{k_b} u_n(\mathbf{k}) \rangle \right]$$

or, equivalently, using matrix projectors,

$$g_n^{ab}(\mathbf{k}) = \frac{1}{2} \mathrm{Tr}[ \partial_{k_a} \hat{P}_n(\mathbf{k}) \, \partial_{k_b} \hat{P}_n(\mathbf{k}) ]$$

where $\hat{P}_n(\mathbf{k})$ is the projector onto band $n$ at momentum $\mathbf{k}$.

The intraband quantum metric dipole is the first momentum derivative of the quantum metric: $$D_n^{abc}(\mathbf{k}) = \partial_{k_c} g_n^{ab}(\mathbf{k})$$ A Fermi sea or Fermi surface average yields the physical intraQMD: $$D_{ab,c} = \int \frac{d^d k}{(2\pi)^d} \sum_n \partial_{k_c} g_{ab}^{(n)}(\mathbf{k}) \, f_0[ E_n(\mathbf{k}) ]$$ where $f_0$ is the Fermi–Dirac distribution. The intraQMD tensor is generically anisotropic and transforms naturally under crystal symmetries (Ulrich et al., 20 Jun 2025, Zhao et al., 10 Aug 2025, Gao et al., 2018, Dutta et al., 2022).

2. Role in Nonlinear Hall and Nonreciprocal Transport

The intraQMD is one of three quantum geometric sources of the intrinsic second-order (nonlinear) Hall effect (NLHE) in clean topological metals and semimetals. The full second-order conductivity at zero temperature decomposes as

$$\sigma^{a;bb} = \int \frac{d^d k}{(2\pi)^d} \sum_n \delta(E_n - \mu) \left[ \sigma^{a;bb}_{\mathrm{NLD}} + \sigma^{a;bb}_{\mathrm{BCD}} + \sigma^{a;bb}_{\mathrm{interQMD}} + \sigma^{a;bb}_{\mathrm{intraQMD}} \right]$$

where the intraQMD term is

$$\sigma^{a;bb}_{\rm intraQMD} = -\frac{e^3}{2\hbar} \int \frac{d^d k}{(2\pi)^d} \sum_n \delta(E_n - \mu) \, D_n^{bba}(\mathbf{k})$$

Notably, unlike the Berry curvature dipole (BCD) or the interband QMD (interQMD), the intraQMD does not involve an inverse band gap denominator and therefore survives in degenerate or nearly degenerate bands, as in topological antiferromagnets or at nodal structures (Ulrich et al., 20 Jun 2025).

In the context of nonlinear planar Hall effect induced by an external magnetic field in Dirac semimetals (e.g., Cd$_3$As$_2$), the field-induced breaking of symmetries enables a nonvanishing intraQMD contribution, which can be experimentally extracted by scaling analysis of second-harmonic Hall voltages (Zhao et al., 10 Aug 2025).

3. Optical and Plasmonic Manifestations

The intraQMD yields a direct, symmetry-constrained geometric origin for nonreciprocal directional dichroism—the differential response of the refractive index upon reversal of light propagation. In the static, quadrupolar limit, the associated transport current takes the form

$$J_i^{\text{tr}} = \gamma_{ijk} \, \partial_k E_j \qquad\mathrm{with}\qquad \gamma_{ijk} = e^2 \sum_n \int \frac{d^3 k}{(2\pi)^3} v_{i,n}(\mathbf{k}) g_{jk,n}(\mathbf{k}) \frac{\partial f_n}{\partial \varepsilon_n}$$

The nonreciprocal part is intrinsically geometric and vanishes when either inversion or time-reversal symmetry is preserved (Gao et al., 2018).

In plasmonic response, the intraQMD appears at $\mathcal{O}(q^3)$ in the small wavevector expansion of the intraband polarization function. The resulting plasmon dispersion acquires a term: $$\omega_p^{\rm intra}(\mathbf{q}) \approx \omega_p^0 + \frac{1}{2} V_q q_a q_b q_c {\cal Q}_{abc} + ...$$ where ${\cal Q}_{abc}$ is the intraQMD tensor and this term is responsible for intrinsic nonreciprocity in plasmon propagation in noncentrosymmetric magnetic metals and moiré systems (Dutta et al., 2022).

4. Topological Flat Bands and Collective Excitations

In flat-band ferromagnets, the intraQMD determines the spatial separation of particle-hole (magnon) excitations, controlling both excitation energy and spin stiffness. In the single-mode approximation, the magnon gap and spin stiffness are directly set by the intraQMD: $$\rho_s = U(\bar{g}) \langle \Omega_n^2 \rangle_{\text{BZ}} \geq U(\bar{g}) C^2$$ where $U(\bar{g})$ is an effective interaction set by the quantum metric, $\Omega_n$ is the Berry curvature, and $C$ the Chern number. Topological constraints enforce a nonzero intrinsic magnon gap in Chern bands, as the quantized topology mandates vortices in the magnon wavefunction, and the intraQMD yields a lower bound on the gap once SU(2) symmetry is reduced to U(1) (Chen et al., 27 Jun 2025).

5. Symmetry Criteria and Material Realizations

A nonzero intraQMD requires the breaking of both inversion ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetries separately but is allowed when combined $\mathcal{PT}$ symmetry is preserved. It is generically forbidden by the presence of either symmetry or rotation/combinations that symmetrize the Fermi surface. This exclusion makes the intraQMD the only geometric NLHE term remaining in certain topological antiferromagnets and nodal-plane materials such as CuMnAs, Yb$_3$Pt$_4$, MnNb$_3$S$_6$, and CoNb$_3$S$_6$ (Ulrich et al., 20 Jun 2025, Dutta et al., 2022).

Additionally, in nonmagnetic Dirac semimetals with externally broken time-reversal via magnetic field, the intraQMD is nonzero and field-tunable, offering pathways for magnetic-field-controlled nonlinear transport (Zhao et al., 10 Aug 2025).

6. Computational and Experimental Extraction

The intraQMD can be cast in projector language in terms of derivatives of the Hamiltonian, facilitating practical calculations: $$D_n^{abc}(\mathbf{k}) = \partial_{k_c} g_n^{ab}(\mathbf{k})$$ with $g_n^{ab}(\mathbf{k})$ expressible as traces over derivatives of $\hat{H}(\mathbf{k})$. Techniques such as numerical $k$-space differentiation or direct computation via Kubo or semiclassical formulas are used to evaluate the tensor components.

Experimentally, extraction involves second-harmonic Hall measurements and scaling fits in nonlinear transport (e.g., in Cd$_3$As$_2$), or comparison against exact and approximate theoretical gaps in magnon branches of flat-band models and twisted moiré systems (Zhao et al., 10 Aug 2025, Chen et al., 27 Jun 2025).

7. Physical Significance and Impact

The concept of the intraQMD generalizes the role of quantum geometry in condensed matter, providing a geometric underpinning for various nontrivial electromagnetic responses that cannot be interpreted in terms of Berry curvature alone. It bridges the understanding of nonlinear electronic transport, nonreciprocal optics, and topologically protected collective excitations. As a Fermi-surface property that responds to symmetry and topology, the intraQMD offers a powerful predictive tool for material discovery and for the engineering of quantum-geometric responses in both magnetic and nonmagnetic multiband systems (Ulrich et al., 20 Jun 2025, Gao et al., 2018, Dutta et al., 2022, Chen et al., 27 Jun 2025).