Berry Curvature Quadrupole in Topological Systems

• Berry curvature quadrupole is a higher-order multipole moment that quantifies the spatial inhomogeneity of Berry curvature in momentum space.
• It emerges under specific symmetry conditions in crystals, driving third-order nonlinear Hall and thermal responses in materials like spin-canted FeSn and monolayer SrMnBi₂.
• Quantitative insights from first-principles calculations and experiments underpin its role in device applications such as high-frequency rectification and topological photonic circuitry.

The Berry curvature quadrupole constitutes a higher-order multipole moment of the Berry curvature field in momentum space, generalizing the concept of the Berry curvature monopole (responsible for the linear anomalous Hall effect) and dipole (driving the second-order nonlinear Hall response). Defined as the Brillouin-zone pseudotensor encoding the spatial inhomogeneity of Berry curvature, the quadrupole plays a fundamental role in third-order nonlinear anomalous Hall and thermal transport phenomena, particularly in crystals with specific magnetic symmetry constraints. Notably, the Berry curvature quadrupole manifests experimentally as a leading-order nonlinear transverse response in antiferromagnets and selected topological materials, with direct evidence established by third-harmonic Hall measurements in materials such as spin-canted FeSn and epitaxial monolayer SrMnBi₂. Its symmetry-selective emergence and topological ramifications underpin contemporary research in nonlinear Hall transport, Berryology, and symmetry-driven phenomena in quantum matter (Sankar et al., 2023, Korrapati et al., 2024, Zhang et al., 2020).

1. Mathematical Definition and Multipole Hierarchy

The Berry curvature $\Omega_a(\mathbf{k})$ in crystalline solids is a pseudovector field in the Brillouin zone, with monopole, dipole, quadrupole, and higher multipole moments constructed by successively taking derivatives with respect to momentum. The $n$-th moment is:

$$P_{\alpha_1\cdots\alpha_n\beta} = \int[d\mathbf{k}]\, f_0(\epsilon_{\mathbf{k}}) \partial_{\alpha_1}\cdots\partial_{\alpha_n}\Omega_\beta(\mathbf{k})$$

where $f_0$ is the Fermi-Dirac distribution. The dipole is $$D_{ab} = \int [d\mathbf{k}]\, f_0\, \partial_a\Omega_b$$, and the quadrupole is:

$$Q_{abc} = \int [d\mathbf{k}]\, f_0\, \partial_a\partial_b\Omega_c(\mathbf{k})$$

Alternatively, integrating by parts emphasizes its Fermi-surface character:

$$Q_{abc} = -\int [d\mathbf{k}]\, f_0'(\epsilon_{\mathbf{k}}) (\partial_a\epsilon_{\mathbf{k}})(\partial_b\Omega_c)$$

This construction generalizes to higher-rank Berry curvature moments (hexapole, etc.), each characterized by the corresponding higher-order symmetry of $\Omega_a(\mathbf{k})$'s spatial distribution (Zhang et al., 2020, Korrapati et al., 2024).

2. Symmetry Conditions and Material Constraints

The Berry curvature quadrupole is sharply constrained by crystal and magnetic symmetry. Time-reversal ($\mathcal{T}$) symmetry inverts the Berry curvature, causing both monopole and quadrupole integrals to vanish unless $\mathcal{T}$ is broken. The quadrupole, being parity-even, can survive even if inversion remains, provided $n$0 is broken. Additional crystalline constraints—mirrors, rotations—select which tensor components can be nonzero and, in many point groups, symmetry forces both the monopole and the dipole to vanish while permitting a finite quadrupole.

Comprehensive group-theoretical analyses reveal that out of 122 three-dimensional magnetic point groups, exactly 66 admit nonzero Berry curvature quadrupoles, and 15 possess symmetry such that the quadrupole is the lowest-allowed (leading) Berry curvature moment (Zhang et al., 2020). In 2D, out of 31 magnetic point groups, three (2mm, 4′, 4′m′m) admit a leading quadrupole (Zhang et al., 2020, Korrapati et al., 2024). The paradigmatic experimental realization is spin-canted FeSn, where an out-of-plane magnetic field reduces the symmetry from $n$1 (inhibiting all but the monopole) to $n$2 ($n$3), allowing a finite $n$4 and thus a third-order nonlinear Hall effect while still forbidding a second-order dipolar response (Sankar et al., 2023).

3. Nonlinear Transport and Third-Order Anomalous Hall Effect

The Berry curvature quadrupole gives rise to a third-order nonlinear anomalous Hall effect (NLAHE), leading to a transverse voltage response at the third harmonic of an applied a.c. bias. Within the semiclassical Boltzmann approach, the third-order current response is:

$n$5

For the Hall geometry with electric field along $n$6, the relevant component is

$n$7

in the dc limit ($n$8 is the relaxation time, $n$9 the effective mass). In practical Hall bar geometries, the third-harmonic transverse voltage $$P_{\alpha_1\cdots\alpha_n\beta} = \int[d\mathbf{k}]\, f_0(\epsilon_{\mathbf{k}}) \partial_{\alpha_1}\cdots\partial_{\alpha_n}\Omega_\beta(\mathbf{k})$$0 scales as $$P_{\alpha_1\cdots\alpha_n\beta} = \int[d\mathbf{k}]\, f_0(\epsilon_{\mathbf{k}}) \partial_{\alpha_1}\cdots\partial_{\alpha_n}\Omega_\beta(\mathbf{k})$$1, and is distinguished by its cubic current dependence, B-field parity, vanishing as $$P_{\alpha_1\cdots\alpha_n\beta} = \int[d\mathbf{k}]\, f_0(\epsilon_{\mathbf{k}}) \partial_{\alpha_1}\cdots\partial_{\alpha_n}\Omega_\beta(\mathbf{k})$$2, and sign-reversal at magnetic phase transitions (Sankar et al., 2023). Scaling analysis shows that, at high temperature, the intrinsic contribution (proportional to $$P_{\alpha_1\cdots\alpha_n\beta} = \int[d\mathbf{k}]\, f_0(\epsilon_{\mathbf{k}}) \partial_{\alpha_1}\cdots\partial_{\alpha_n}\Omega_\beta(\mathbf{k})$$3) dominates over extrinsic skew-scattering effects (Sankar et al., 2023, Zhang et al., 2020).

4. Photonic and Topological Realizations

The Berry curvature quadrupole also features in engineered photonic and electronic systems. In D$$P_{\alpha_1\cdots\alpha_n\beta} = \int[d\mathbf{k}]\, f_0(\epsilon_{\mathbf{k}}) \partial_{\alpha_1}\cdots\partial_{\alpha_n}\Omega_\beta(\mathbf{k})$$4 photonic metacrystals with roto-inversion-time ($$P_{\alpha_1\cdots\alpha_n\beta} = \int[d\mathbf{k}]\, f_0(\epsilon_{\mathbf{k}}) \partial_{\alpha_1}\cdots\partial_{\alpha_n}\Omega_\beta(\mathbf{k})$$5) symmetry, the quadratic-in-momentum Hamiltonian near straight nodal lines generates a Berry curvature with a quadrupolar distribution. The dominant nonzero tensor elements $$P_{\alpha_1\cdots\alpha_n\beta} = \int[d\mathbf{k}]\, f_0(\epsilon_{\mathbf{k}}) \partial_{\alpha_1}\cdots\partial_{\alpha_n}\Omega_\beta(\mathbf{k})$$6 reflect a characteristic four-lobe pattern. The quadrupole moment may be continuously tuned by geometry, allowing experimental control (Wang et al., 2021). The quadrupole gives rise to $$P_{\alpha_1\cdots\alpha_n\beta} = \int[d\mathbf{k}]\, f_0(\epsilon_{\mathbf{k}}) \partial_{\alpha_1}\cdots\partial_{\alpha_n}\Omega_\beta(\mathbf{k})$$7-quantized Zak phases along all crystallographic directions, which, through bulk-boundary correspondence, guarantee robust surface “drumhead” states on every facet—manifesting as topological “photon Fermi arcs.” The evolution of these surface state equifrequency contours with excitation energy directly visualizes the underlying Berry curvature quadrupole and supports photonic super-imaging (Wang et al., 2021).

5. Experimental Evidence and Quantification

Electric transport measurements on the kagome antiferromagnet FeSn provide direct observation of quadrupolar Berry curvature physics: the third-order NLAHE is observed as a strong room-temperature transverse response at 3$$P_{\alpha_1\cdots\alpha_n\beta} = \int[d\mathbf{k}]\, f_0(\epsilon_{\mathbf{k}}) \partial_{\alpha_1}\cdots\partial_{\alpha_n}\Omega_\beta(\mathbf{k})$$8 under an a.c. bias (Sankar et al., 2023). The magnitude of $$P_{\alpha_1\cdots\alpha_n\beta} = \int[d\mathbf{k}]\, f_0(\epsilon_{\mathbf{k}}) \partial_{\alpha_1}\cdots\partial_{\alpha_n}\Omega_\beta(\mathbf{k})$$9, extracted from scaling-law fits, is quantitatively consistent (within a factor of two) with density functional theory + Wannier and classical Monte Carlo estimates, validating theoretical models. The observed signal tracks the magnetic phase diagram, showing maximal response near the Néel temperature due to enhanced spin canting (Sankar et al., 2023).

First-principles calculations predict sizable quadrupole moments and third-harmonic voltages in monolayer SrMnBi$f_0$0 and twisted bilayer graphene near quantum anomalous Hall filling, with predicted $f_0$1V in micron-scale devices (Zhang et al., 2020). In all cases, the sign and magnitude of the experimentally observed third-harmonic Hall signal are determined by the computed $f_0$2 value.

Material	$f_0$3 $f_0$4	$f_0$5 ($f_0$6V)
SrMnBi$f_0$7, monolayer	$f_0$8	$f_0$9
TBG QAH	$$D_{ab} = \int [d\mathbf{k}]\, f_0\, \partial_a\Omega_b$$0	$$D_{ab} = \int [d\mathbf{k}]\, f_0\, \partial_a\Omega_b$$1

6. Extensions: Thermal Hall and Nernst Responses

The Berry curvature quadrupole generalizes its role in electrical transport to thermal transport. In crystals with appropriate symmetry (nonzero quadrupole, vanishing monopole/dipole), the third-order anomalous thermal Hall current and Nernst effect are governed by energy-modulated Berry curvature quadrupole tensors. Specifically, the cubic-in-gradient responses $$D_{ab} = \int [d\mathbf{k}]\, f_0\, \partial_a\Omega_b$$2 (thermal Hall) and $$D_{ab} = \int [d\mathbf{k}]\, f_0\, \partial_a\Omega_b$$3 (Nernst) scale as $$D_{ab} = \int [d\mathbf{k}]\, f_0\, \partial_a\Omega_b$$4, with the dominant tensor element in 2D being $$D_{ab} = \int [d\mathbf{k}]\, f_0\, \partial_a\Omega_b$$5 and $$D_{ab} = \int [d\mathbf{k}]\, f_0\, \partial_a\Omega_b$$6, respectively (Korrapati et al., 2024). At low temperature, the leading behavior is $$D_{ab} = \int [d\mathbf{k}]\, f_0\, \partial_a\Omega_b$$7 and $$D_{ab} = \int [d\mathbf{k}]\, f_0\, \partial_a\Omega_b$$8. Model estimates for materials with confirmed large Berry curvature quadrupoles yield cubic thermal and Nernst currents detectable with present experimental techniques (Korrapati et al., 2024).

7. Outlook: Topological, Spectroscopic, and Device Implications

Berry curvature quadrupole effects are symmetry-allowed in 90 of the magnetic point groups, and their observation provides sensitive probes of hidden or fluctuating magnetic orders, particularly in systems where linear transport is silent. Nonlinear Hall responses at higher harmonics extend to hexapole (fifth harmonic) and beyond, with implications for terahertz NLAHE spectroscopy in lower-symmetry antiferromagnets. The ability to electrically tune multipoles by magnetic order or stacking geometry suggests device applications in high-frequency rectification, frequency upconversion, and spin-optoelectronic devices based on electrically switchable Berry curvature multipoles (Sankar et al., 2023, Korrapati et al., 2024, Zhang et al., 2020).

The quadrupolar structure in photonic systems, with geometry-control over $$D_{ab} = \int [d\mathbf{k}]\, f_0\, \partial_a\Omega_b$$9, supports direct visualization and manipulation of Berryological multipoles, facilitating advances in super-resolution imaging and topological photonic circuitry (Wang et al., 2021). The continued development of nonlinear measurement protocols, symmetry-selective materials design, and first-principles Berryology is anticipated to further expand the landscape of nonlinear and topological quantum transport rooted in Berry curvature quadrupoles.