Quantum Geometry and Transport Phenomena

• Quantum Geometry and Transport Phenomena is defined by the quantum geometric tensor, combining the quantum metric (measuring state distances) and Berry curvature (encoding phase information) to control response properties.
• Nonlinear transport arises through symmetry-dependent contributions, including nonlinear Drude, Berry curvature dipole, and quantum metric dipole terms, which manifest in optical, thermal, and electronic measurements.
• Recent studies extend quantum geometry to many-body and real-space frameworks, offering innovative experimental probes and theoretical models for understanding correlated phases and metal-insulator transitions.

Quantum geometry refers to the intrinsic geometry of the Hilbert space of quantum states, with direct consequences for the dynamics and transport properties of both noninteracting and correlated quantum systems. In condensed matter, quantum geometry is encoded via the quantum geometric tensor (QGT), whose real part is the quantum metric (a measure of “distance” between nearby quantum states in momentum space) and whose imaginary part is proportional to the Berry curvature (the “phase” gauge field underlying many topological effects). These quantum-geometric quantities fundamentally control linear and nonlinear electronic, optical, and thermal transport phenomena, both at the single-particle (band theory) and many-body levels.

1. Quantum-Geometric Tensors: Structure and Physical Meaning

The quantum geometric tensor for an isolated band $|u_n(\mathbf{k})\rangle$ at crystal momentum $\mathbf{k}$ is defined as

$$Q_{n}^{ab}(\mathbf{k}) = \langle \partial_{k_a} u_n | (1 - |u_n\rangle\langle u_n|) | \partial_{k_b} u_n \rangle = g_{n}^{ab}(\mathbf{k}) - \frac{i}{2} \Omega_{n}^{ab}(\mathbf{k}),$$

where

• $$g_{n}^{ab}(\mathbf{k}) = \operatorname{Re}\, Q_{n}^{ab}(\mathbf{k})$$ is the quantum metric, quantifying infinitesimal distances in Hilbert space,
• $$\Omega_{n}^{ab}(\mathbf{k}) = -2 \operatorname{Im}\, Q_{n}^{ab}(\mathbf{k})$$ is the Berry curvature tensor, encoding gauge-invariant phase information.

The Berry connection $$A_n^a(\mathbf{k}) = i\langle u_n(\mathbf{k}) | \partial_{k_a} u_n(\mathbf{k})\rangle$$ and the quantum metric are responsible, respectively, for anomalous velocity (topological transport) and the geometry-driven enhancement or suppression of response functions. The quantum metric's integral over the Brillouin zone determines the spread of Wannier orbitals and the scale of geometric dipole fluctuations, which play a dominant role in various response coefficients (Jiang et al., 6 Mar 2025, Gao et al., 1 Aug 2025, Liu et al., 2024).

2. Nonlinear Transport: Role of Quantum Metric and Berry Curvature Dipoles

In metallic or doped semiconductors, quantum geometry manifests distinctly in nonlinear (quadratic and higher-order) transport. The leading zero-frequency ($\omega \to 0$) second-order current response is

$J^c = \chi^{ab;c} E^a E^b,$

with a decomposition

$$\chi^{ab;c} = \chi_\mathrm{NLD}^{ab;c} + \chi_\mathrm{BCD}^{ab;c} + \chi_\mathrm{QMD}^{ab;c},$$

where

• $\chi_\mathrm{NLD}^{ab;c} \propto \tau^2$: Nonlinear Drude term (skewness of the Fermi velocity),
• $\mathbf{k}$0: Berry curvature dipole term,
• $\mathbf{k}$1: Quantum metric dipole term.

Explicitly, the Berry curvature dipole

$\mathbf{k}$2

drives the nonlinear Hall effect in time-reversal symmetric, inversion-broken metals. The quantum metric dipole,

$\mathbf{k}$3

generates nonlinear longitudinal and transverse signals in systems with PT symmetry, where Berry curvature effects are symmetry-forbidden (Jiang et al., 6 Mar 2025, Fang et al., 22 May 2025, Liu et al., 2024).

The symmetry of the underlying crystal dictates which tensorial components survive. For example, BCD vanishes under PT symmetry; QMD vanishes if the system is invariant under time-reversal alone; rotation symmetries further constrain measurable components. The scaling with relaxation time $\mathbf{k}$4 allows separation of Drude, BCD, and QMD contributions in transport and optical measurements (Jiang et al., 6 Mar 2025, Liu et al., 2024).

3. Quantum Geometry Beyond Single-Particle Theory

Many-Body Quantum Geometry and Metal-Insulator Transitions

Optimal transport theory extends quantum geometry to many-body correlated states. The quantum distance between two momentum points $\mathbf{k}$5 for a many-body state $\mathbf{k}$6 is

$\mathbf{k}$7

where $\mathbf{k}$8 is the exchange operator. This provides a basis for defining the 2-Wasserstein distance and Wasserstein barycenter over the Brillouin zone, constructing geometric order parameters for detecting metal-insulator transitions, as demonstrated in the 1D $\mathbf{k}$9–$$Q_{n}^{ab}(\mathbf{k}) = \langle \partial_{k_a} u_n | (1 - |u_n\rangle\langle u_n|) | \partial_{k_b} u_n \rangle = g_{n}^{ab}(\mathbf{k}) - \frac{i}{2} \Omega_{n}^{ab}(\mathbf{k}),$$0 model (Hassan et al., 2019). In metals, the many-body spectrum exhibits sharply clustered distances (reflecting Fermi surface structure), while in insulators, quantum distances become uniform, associated with vanishing conductance.

Entropy Production and Thermoelectric Transport

The quantum metric also dictates dissipation rates: the microscopic entropy production in driven Bloch electrons is governed by geometric combinations of the quantum metric tensor. In contrast, Berry curvature controls only dissipationless (Hall-type) responses. In thermoelectric transport, the Hilbert-Schmidt quantum distance controls the scattering rate, and the thermoelectric power factor is enhanced as the maximum quantum distance between states on the Fermi surface increases. For certain quadratic band-touching models, the power factor can double in the maximally geometric regime relative to a trivial geometry (Oh et al., 2024, Xiang et al., 2 Feb 2026).

4. Experimental Manifestations and Probes

The influence of quantum geometry is now established across a wide range of experimental measurements:

• Nonlinear Hall effect (BCD): Observed in WTe$$Q_{n}^{ab}(\mathbf{k}) = \langle \partial_{k_a} u_n | (1 - |u_n\rangle\langle u_n|) | \partial_{k_b} u_n \rangle = g_{n}^{ab}(\mathbf{k}) - \frac{i}{2} \Omega_{n}^{ab}(\mathbf{k}),$$1, MoTe$$Q_{n}^{ab}(\mathbf{k}) = \langle \partial_{k_a} u_n | (1 - |u_n\rangle\langle u_n|) | \partial_{k_b} u_n \rangle = g_{n}^{ab}(\mathbf{k}) - \frac{i}{2} \Omega_{n}^{ab}(\mathbf{k}),$$2, BaMnSb$$Q_{n}^{ab}(\mathbf{k}) = \langle \partial_{k_a} u_n | (1 - |u_n\rangle\langle u_n|) | \partial_{k_b} u_n \rangle = g_{n}^{ab}(\mathbf{k}) - \frac{i}{2} \Omega_{n}^{ab}(\mathbf{k}),$$3, etc.
• Quantum-metric dipole (QMD): Dominates nonlinear responses in antiferromagnets such as MnBi$$Q_{n}^{ab}(\mathbf{k}) = \langle \partial_{k_a} u_n | (1 - |u_n\rangle\langle u_n|) | \partial_{k_b} u_n \rangle = g_{n}^{ab}(\mathbf{k}) - \frac{i}{2} \Omega_{n}^{ab}(\mathbf{k}),$$4Te$$Q_{n}^{ab}(\mathbf{k}) = \langle \partial_{k_a} u_n | (1 - |u_n\rangle\langle u_n|) | \partial_{k_b} u_n \rangle = g_{n}^{ab}(\mathbf{k}) - \frac{i}{2} \Omega_{n}^{ab}(\mathbf{k}),$$5 and in strained or noncoplanar AFMs, independent of spin-orbit interaction (Zhu et al., 2024).
• Nonlinear optical resonances: Injection and shift currents can be decomposed into quantum-geometric contributions—Berry curvature and metric selectivity by light polarization and frequency (Jiang et al., 6 Mar 2025, Gao et al., 1 Aug 2025).
• Thermoelectric and thermal responses: Nonlinear Nernst and Ettingshausen coefficients connected via generalized Mott and Wiedemann–Franz relations to the underlying quantum-geometry dipoles (Fang et al., 22 May 2025).
• Superfluid stiffness in flat bands: The geometric contribution, proportional to the integrated quantum metric, explains supercurrent in systems with vanishing group velocity, as realized in twisted bilayer graphene (Liu et al., 2024, Gao et al., 1 Aug 2025).

Direct measurement protocols for the quantum metric and Berry curvature in tunneling experiments, cold atom platforms, and photonic lattices are under development. Quantum geometry can be extracted from resonance-enhanced conductance changes under weak parameter drives, allowing full reconstruction of the local quantum-geometric tensor (Klees et al., 11 Aug 2025).

5. Theoretical Formalism and Extension to Real-Space Geometry

Recent theoretical advances provide a wave-packet and kinetic equation framework in which real-space and momentum-space quantum geometries are treated on equal footing. The phase-space quantum metric introduces corrections to wave-packet energy, Berry connection, and the density of states, yielding geometric contributions to polarization and Hall-like signals rooted in gradients of the quantum metric, even in systems where Berry curvature vanishes (Maranzana et al., 22 Mar 2026). In manifold-constrained quantum systems (e.g., quantum particles on corrugated surfaces), the interplay of local metric, global topology, and Berry-phase structure modifies band formation and transport conductance, introducing geometry-induced transport features (Schwager et al., 18 Dec 2025).

6. Higher-Order Nonlinear Responses and Magnetic Quantum Geometry

Quantum geometry extends beyond quadratic response. In systems such as planar altermagnets with $$Q_{n}^{ab}(\mathbf{k}) = \langle \partial_{k_a} u_n | (1 - |u_n\rangle\langle u_n|) | \partial_{k_b} u_n \rangle = g_{n}^{ab}(\mathbf{k}) - \frac{i}{2} \Omega_{n}^{ab}(\mathbf{k}),$$6 symmetry, symmetry constraints suppress second-order transport but permit leading third-order (cubic) nonlinearities. Here, quantum metric quadrupole (QMQ) and Berry curvature quadrupole (BCQ) contributions appear separately in longitudinal and transverse signals, respectively. Magnetic ordering (even independent of spin-orbit interaction) induces robust quantum-metric and Berry-dipole features, enabling large nonlinear transport in broad classes of antiferromagnets and spintronic materials (Mandal et al., 2023, Fang et al., 2023, Zhu et al., 2024).

7. Outlook and Open Directions

Quantum geometry is now recognized as an essential ingredient in modern condensed matter theory, dictating the structure of transport coefficients, thermoelectric figures of merit, nonlinear optical selection rules, and the formation of correlated phases. Open theoretical challenges include:

• Constructing basis- and gauge-invariant definitions of the quantum metric in complicated tight-binding systems,
• Taming singular behavior at nodal points and band touchings,
• Formulating quantum geometry for interacting many-body states,
• Implementing direct “smoking-gun” experimental probes of the metric tensor at the Brillouin-zone and real-space level.

Developments in materials with engineered band geometry, ultracold atom and photonic simulators, and quantum-transport measurements are expected to further elucidate the central role of quantum geometry in shaping transport and response phenomena across quantum matter (Jiang et al., 6 Mar 2025, Gao et al., 1 Aug 2025, Liu et al., 2024, Schwager et al., 18 Dec 2025, Mandal et al., 2023, Fang et al., 2023).