Quantum Metric Quadrupole in Nonlinear Transport

• Quantum metric quadrupole is a higher-order momentum-space structure derived from the real part of the quantum geometric tensor, quantifying the second moments of the quantum metric in Bloch states.
• It underpins a third-order nonlinear longitudinal electrical response, with experimental signatures such as cubic current scaling and symmetry-enforced signal separation.
• Experimental studies in bulk MnBi₂Te₄ and few-layer WTe₂ use angular dependence, magnetic field symmetry, and temperature scaling to isolate its distinct transport effects from Berry curvature contributions.

Searching arXiv for papers on quantum metric quadrupole and closely related quantum-geometry transport work. Quantum metric quadrupole denotes the quadrupole moment of the real part of the quantum geometric tensor of Bloch states. In the condensed-matter setting, the quantum geometric tensor encodes the geometry of electronic bands in momentum space: its imaginary part gives the Berry curvature, while its real part gives the quantum metric. The quantum metric quadrupole is therefore a higher-order momentum-space distribution of the quantum metric, and recent work links it directly to third-order nonlinear electrical transport, especially third-harmonic longitudinal response. Experiments in bulk MnBi$_2$Te$_4$ and few-layer WTe$_2$ have made this notion operational by separating quantum metric quadrupole effects from Berry-curvature-quadrupole effects through symmetry, angular dependence, and scaling analyses (Li et al., 2023, Liu et al., 22 Jan 2025).

1. Quantum-geometric definition

The quantum geometric tensor of Bloch states is the basic object from which the quantum metric quadrupole is constructed. One formulation writes

$$\mathcal{T}_{ab} = G_{ab} - \dfrac{i}{2}\Omega_{ab},$$

with $G_{ab}$ the quantum metric tensor and $\Omega_{ab}$ the Berry curvature. In equivalent notation,

$$\Omega_{\alpha\beta} = 2\, \mathrm{Im}(T_{\alpha\beta}(\mathbf{k})), \qquad g_{\alpha\beta} = \mathrm{Re}(T_{\alpha\beta}(\mathbf{k})).$$

The Berry curvature has been well studied in topological and Hall phenomena, whereas the experimental investigation of quantum metric effects has been at an early stage (Li et al., 2023, Liu et al., 22 Jan 2025).

A quadrupole refers to the second moment, or spatial distribution, of a quantity. Applied to quantum geometry, this means a higher-order momentum-space structure of either the quantum metric or the Berry curvature. In the transport literature summarized here, the quantum metric quadrupole is represented through second derivatives of the quantum metric with respect to crystal momentum; for longitudinal third-order conductivity along $x$, the directly relevant object is $\partial_{k_x}\partial_{k_x} G_{xx}$ (Liu et al., 22 Jan 2025).

This usage is narrower than a generic statement about “quantum geometry.” It does not refer merely to the existence of a nonzero quantum metric; it refers to a particular higher moment of that metric distribution in the Brillouin zone. That distinction is central because the transport signatures discussed below are not attributed to the quantum metric in the abstract, but to its quadrupolar structure.

2. Third-order nonlinear transport mechanism

The principal transport statement is that the quantum metric quadrupole induces a third-order nonlinear longitudinal electrical response. In semiclassical form, the third-order current is written as

$j_a^{(3)} = \chi_{abcd}^{(3)} E_b E_c E_d,$

and the intrinsic quantum-metric contribution to $_4$0 involves Brillouin-zone integrals of second derivatives of $_4$1. The corresponding Berry-curvature-quadrupole contribution generates a transverse third-order Hall-like response (Liu et al., 22 Jan 2025).

Symmetry is decisive. In materials with inversion symmetry, second-order nonlinear effects are forbidden, so third-order responses become the leading nonlinear channel. This is the logic explicitly used in bulk MnBi$_4$2Te$_4$3, where second-order nonlinear responses are negligible as required by inversion symmetry and finite third-order nonlinear responses are observed (Li et al., 2023). The same general framework also clarifies why the quadrupole moment of the quantum geometric tensor is associated with higher-order quantum nonlinearity, whereas the Berry curvature dipole in nonmagnetic materials and the quantum metric dipole in antiferromagnets have been explored through the second-order nonlinear Hall effect (Liu et al., 22 Jan 2025).

A compact way to organize the transport assignments established in MnBi$_4$4Te$_4$5 is the following.

Measured signal	Physical origin	Symmetry in $_4$6
$_4$7	Quantum metric quadrupole	Even
$_4$8	Berry curvature quadrupole	Odd
Second-harmonic ($_4$9)	Forbidden in inversion-symmetric bulk MnBi$_2$0Te$_2$1	Negligible

The conceptual significance of this table is that the real and imaginary parts of the quantum geometric tensor can be separated experimentally through third-order transport, not only inferred jointly from a single nonlinear observable.

3. Bulk MnBi$_2$2Te$_2$3: antiferromagnetic topological-insulator realization

Bulk MnBi$_2$4Te$_2$5 provides the first transport platform in which both the quantum metric quadrupole and the Berry curvature quadrupole were revealed through third-order nonlinear measurements. The material is a topological anti-ferromagnet. Hall-bar devices were fabricated from bulk MnBi$_2$6Te$_2$7 flakes with thickness $_2$8 nm, driven by a low-frequency a.c. current with $_2$9 Hz, while an out-of-plane magnetic field was applied perpendicular to the current. Lock-in detection resolved longitudinal and transverse harmonic voltages $$\mathcal{T}_{ab} = G_{ab} - \dfrac{i}{2}\Omega_{ab},$$0 and $$\mathcal{T}_{ab} = G_{ab} - \dfrac{i}{2}\Omega_{ab},$$1 (Li et al., 2023).

The central observations are threefold. First, second-harmonic signals are negligible, consistent with inversion symmetry. Second, both third-harmonic longitudinal and transverse voltages are finite. Third, the measured $$\mathcal{T}_{ab} = G_{ab} - \dfrac{i}{2}\Omega_{ab},$$2 is an even function of magnetic field $$\mathcal{T}_{ab} = G_{ab} - \dfrac{i}{2}\Omega_{ab},$$3, while $$\mathcal{T}_{ab} = G_{ab} - \dfrac{i}{2}\Omega_{ab},$$4 is odd in $$\mathcal{T}_{ab} = G_{ab} - \dfrac{i}{2}\Omega_{ab},$$5. The field dependence suggests that $$\mathcal{T}_{ab} = G_{ab} - \dfrac{i}{2}\Omega_{ab},$$6 is induced by quantum metric quadrupole and $$\mathcal{T}_{ab} = G_{ab} - \dfrac{i}{2}\Omega_{ab},$$7 is induced by Berry curvature quadrupole (Li et al., 2023).

The measured third-order responses track the magnetic phase structure of the material. Their magnitudes change abruptly as MnBi$$\mathcal{T}_{ab} = G_{ab} - \dfrac{i}{2}\Omega_{ab},$$8Te$$\mathcal{T}_{ab} = G_{ab} - \dfrac{i}{2}\Omega_{ab},$$9 flakes go through magnetic transitions from an AFM state to a canted AFM state and to an FM state, and both signals vanish above the AFM transition temperature. Detailed theoretical modeling with an effective four-band Hamiltonian and a tunable magnetic order parameter $G_{ab}$0 quantitatively reproduces the observed magnetic-field and temperature dependence. The work therefore establishes a direct connection between higher-order moments of quantum geometry and nonlinear transport in a centrosymmetric topological magnet, and it is described as the first experimental observation of quantum metric quadrupole effects in transport (Li et al., 2023).

4. Few-layer WTe$G_{ab}$1: room-temperature and angle-resolved regime

Few-layer WTe$G_{ab}$2 realizes a different experimental regime. The samples were 6-layer and 10-layer WTe$G_{ab}$3, non-centrosymmetric with point group $G_{ab}$4, contacted in a circular geometry for angle-dependent measurements. A sinusoidal a.c. current with frequency $G_{ab}$5 Hz was applied, and first-, second-, and third-harmonic signals were measured simultaneously with lock-in amplifiers. The reported effect is a quantum metric quadrupole induced third-order nonlinear longitudinal electrical response persisting up to room temperature (Liu et al., 22 Jan 2025).

The electrical signatures are explicit. The first-harmonic voltage scales linearly with current amplitude $G_{ab}$6, while the third-harmonic voltage scales cubically, $G_{ab}$7. The second-harmonic voltage is an order of magnitude weaker and noisy. Angle-resolved third-harmonic current-voltage characteristics are consistent with the intrinsic crystal symmetry of WTe$G_{ab}$8, and longitudinal third-order nonlinearity is observed even when the transverse Hall effect vanishes, namely along the $G_{ab}$9- or $\Omega_{ab}$0-axis (Liu et al., 22 Jan 2025).

The angular dependence is fitted by

$\Omega_{ab}$1

Extracted third-order conductivity tensor components show strong anisotropy, with $\Omega_{ab}$2 much greater than $\Omega_{ab}$3, consistent with model band anisotropy. Through temperature variation and scaling analysis, the study identifies the quantum metric quadrupole as the physical origin of the observed third-order longitudinal nonlinearity. Below 30 K, linear fits separate an intrinsic QMQ contribution $\Omega_{ab}$4, which scales as $\Omega_{ab}$5, from a skew-scattering contribution $\Omega_{ab}$6, which scales as $\Omega_{ab}$7. Frequency tests from 17.777 Hz to 177.77 Hz show no frequency dependence, arguing against a capacitive origin, and the temperature and scaling dependences are inconsistent with pure thermal contributions. The reported QMQ nonlinear conductivity in few-layer WTe$\Omega_{ab}$8 is $\Omega_{ab}$9 larger than in bulk MoTe$$\Omega_{\alpha\beta} = 2\, \mathrm{Im}(T_{\alpha\beta}(\mathbf{k})), \qquad g_{\alpha\beta} = \mathrm{Re}(T_{\alpha\beta}(\mathbf{k})).$$0 or TaIrTe$$\Omega_{\alpha\beta} = 2\, \mathrm{Im}(T_{\alpha\beta}(\mathbf{k})), \qquad g_{\alpha\beta} = \mathrm{Re}(T_{\alpha\beta}(\mathbf{k})).$$1 (Liu et al., 22 Jan 2025).

5. Symmetry diagnostics and experimental interpretation

The experimental literature makes the symmetry logic unusually transparent. In centrosymmetric bulk MnBi$$\Omega_{\alpha\beta} = 2\, \mathrm{Im}(T_{\alpha\beta}(\mathbf{k})), \qquad g_{\alpha\beta} = \mathrm{Re}(T_{\alpha\beta}(\mathbf{k})).$$2Te$$\Omega_{\alpha\beta} = 2\, \mathrm{Im}(T_{\alpha\beta}(\mathbf{k})), \qquad g_{\alpha\beta} = \mathrm{Re}(T_{\alpha\beta}(\mathbf{k})).$$3, inversion symmetry suppresses second-order nonlinear electric responses, so the third-order channel becomes the leading nonlinear signal. The distinction between longitudinal and transverse third-harmonic voltages, together with their even or odd behavior under $$\Omega_{\alpha\beta} = 2\, \mathrm{Im}(T_{\alpha\beta}(\mathbf{k})), \qquad g_{\alpha\beta} = \mathrm{Re}(T_{\alpha\beta}(\mathbf{k})).$$4, is then used to separate the quantum metric quadrupole from the Berry curvature quadrupole (Li et al., 2023).

In few-layer WTe$$\Omega_{\alpha\beta} = 2\, \mathrm{Im}(T_{\alpha\beta}(\mathbf{k})), \qquad g_{\alpha\beta} = \mathrm{Re}(T_{\alpha\beta}(\mathbf{k})).$$5, the emphasis is different. Because the material is non-centrosymmetric and strongly anisotropic, the decisive diagnostics are the crystal-angle dependence, the cubic current scaling, and the temperature-conductivity scaling law that decomposes intrinsic and extrinsic contributions. The paper states that the QMQ mechanism is not symmetry-forbidden in the longitudinal configuration, even for isotropic or time-reversal symmetric bands, and uses this point to explain why a sizable longitudinal third-order nonlinear current can be isolated from more familiar Hall-like Berry-curvature effects (Liu et al., 22 Jan 2025).

A common misconception is to compress all quantum-geometric nonlinear transport into the second-order nonlinear Hall effect. The current literature is more differentiated. Berry curvature dipole in nonmagnetic materials and quantum metric dipole in antiferromagnets have indeed been explored through second-order nonlinear Hall measurements, but the quadrupole moment of the quantum geometric tensor is theoretically predicted to induce higher-order quantum nonlinearity, with the experimentally established signature being third-order longitudinal response (Liu et al., 22 Jan 2025). This suggests that multipolar organization of quantum geometry is not a formal refinement only; it changes the order, tensor structure, and symmetry channel of measurable transport.

6. Related quadrupolar concepts and terminological boundaries

The term “quantum metric quadrupole” belongs specifically to Bloch-band quantum geometry and third-order nonlinear transport. Related uses of “quadrupole” in quantum matter are distinct. In incompressible quantum Hall fluids, a new picture treats integer and fractional incompressible quantum Hall fluids as fluids carrying an electric quadrupole, with the local primitive electric quadrupole density $$\Omega_{\alpha\beta} = 2\, \mathrm{Im}(T_{\alpha\beta}(\mathbf{k})), \qquad g_{\alpha\beta} = \mathrm{Re}(T_{\alpha\beta}(\mathbf{k})).$$6 entering a generic expression for Hall viscosity,

$$\Omega_{\alpha\beta} = 2\, \mathrm{Im}(T_{\alpha\beta}(\mathbf{k})), \qquad g_{\alpha\beta} = \mathrm{Re}(T_{\alpha\beta}(\mathbf{k})).$$7

and with continuous rotational symmetry no longer required (Haldane, 2023). This is a many-body electric quadrupole framework rather than a Bloch-band quantum metric quadrupole.

A second related construction is the Cooper pair quadrupole moment. In that setting,

$$\Omega_{\alpha\beta} = 2\, \mathrm{Im}(T_{\alpha\beta}(\mathbf{k})), \qquad g_{\alpha\beta} = \mathrm{Re}(T_{\alpha\beta}(\mathbf{k})).$$8

and the trace of the quadrupole gives the pair size. The framework identifies contributions from amplitude, quantum metric, and phase structure, and shows that when time-reversal symmetry is broken, Berry curvature gives an essential contribution absent from previous quantum-metric theories (Chen et al., 4 May 2026). This suggests a broader quadrupolar language for quantum geometry, but it remains conceptually separate from the transport-defined QMQ of MnBi$$\Omega_{\alpha\beta} = 2\, \mathrm{Im}(T_{\alpha\beta}(\mathbf{k})), \qquad g_{\alpha\beta} = \mathrm{Re}(T_{\alpha\beta}(\mathbf{k})).$$9Te$x$0 and WTe$x$1.

There is also a recurrent terminological ambiguity with general relativity. Papers on the “post-linear quadrupole-quadrupole metric,” the Erez-Rosen metric, or Kerr-like metrics with quadrupole concern spacetime metrics with mass quadrupole moments of gravitating bodies, not the real part of the quantum geometric tensor of Bloch states (Frutos-Alfaro et al., 2015, Frutos-Alfaro, 2015). In the condensed-matter context, “metric” refers to Hilbert-space geometry of electronic states, not to spacetime geometry.

7. Significance and current direction

The present state of the subject is defined by two complementary achievements. Bulk MnBi$x$2Te$x$3 demonstrates that the quadrupoles of both the real and imaginary parts of the quantum geometry tensor can be revealed through third-order nonlinear transport and resolved by magnetic-field symmetry across AFM, canted-AFM, and FM phases (Li et al., 2023). Few-layer WTe$x$4 shows that a quantum metric quadrupole induced third-order nonlinear longitudinal response can be giant, strongly anisotropic, and persistent up to room temperature, with temperature and conductivity scaling used to isolate the intrinsic contribution (Liu et al., 22 Jan 2025).

The resulting picture is that third-order longitudinal nonlinearity has become an efficient method for revealing the quantum metric structure. The available evidence also indicates that higher moment distributions of quantum geometric quantities, rather than only the lowest dipolar moments, can control measurable transport. The MnBi$x$5Te$x$6 study explicitly presents this as a template for probing quantum geometry multipoles, even up to higher-order responses, by leveraging symmetry constraints in material design (Li et al., 2023).

In this sense, the quantum metric quadrupole is both a geometric invariant of band structure and an experimentally addressable transport mechanism. Its importance lies not in replacing Berry-curvature physics, but in placing the real and imaginary parts of the quantum geometric tensor on parallel footing at the level of higher-order nonlinear response.