Quantum Geometric Contribution

• Quantum geometric contribution is the influence of the quantum state manifold on observables, encoded via metrics like the quantum geometric tensor, quantum metric, and Berry curvature.
• It enables a precise partitioning of phase evolution into geometric and dynamical components, underpinning robust quantum control and high-fidelity quantum gate designs.
• This contribution plays a crucial role in phenomena such as enhanced superfluidity in superconductors and novel transport effects in topological and strongly-correlated systems.

Quantum geometric contribution refers to the part of an observable or response function in quantum systems that can be attributed to the geometry of the quantum state manifold, specifically as encoded by geometric quantities such as the quantum geometric tensor (QGT), quantum metric, and Berry curvature. These contributions are universal and often nontrivial in multi-band, topological, or strongly-correlated systems. Quantum geometry is essential for understanding phase evolution, superfluid properties, nonlinear responses, quantum transport, and even entanglement diagnostics.

1. Quantum Geometric Phase–Dynamical Phase Partition Law

The geometric contribution in quantum evolution can be precisely separated from the dynamical part using the universal phase-partition law established for pure-state quantum dynamics. For an evolution $$|\psi(0)\rangle \to |\psi(t)\rangle = U(t)|\psi(0)\rangle$$ with total (Pancharatnam) phase $\Phi_{\mathrm{tot}}(t)$, the infinitesimal geometric and dynamical components obey

$$d\Phi_{\mathrm{geom}} = \sin^2\left[\mathcal{L}(t)/2\right]\, d\Phi_{\mathrm{tot}}, \qquad d\Phi_{\mathrm{dyn}} = \cos^2\left[\mathcal{L}(t)/2\right]\, d\Phi_{\mathrm{tot}},$$

where $$\mathcal{L}(t) = \arccos |\langle\psi(0)|\psi(t)\rangle|$$ is the Bargmann (Bures/Fubini–Study) angle. The fraction of the total phase that is geometric is $f_{\mathrm{geom}}(t) = \sin^2[\mathcal{L}(t)/2]$ and the dynamical fraction is $f_{\mathrm{dyn}}(t) = \cos^2[\mathcal{L}(t)/2]$, with $f_{\mathrm{geom}} + f_{\mathrm{dyn}} = 1$ (Pati et al., 17 Nov 2025). This decomposition holds universally in any finite-dimensional Hilbert space and can be exploited to design high-fidelity geometric gates and robust quantum control protocols.

2. Quantum Geometric Tensor: Structure and Measurement

The quantum geometric tensor for a Bloch band $|u_n(\mathbf{k})\rangle$ is

$$\mathcal{G}_{ij}(\mathbf{k}) = \langle \partial_{k_i}u_n | [1 - |u_n\rangle\langle u_n| ] |\partial_{k_j}u_n \rangle.$$

It decomposes into the quantum metric ($g_{ij} = \mathrm{Re}\,\mathcal{G}_{ij}$, symmetric) and the Berry curvature ($\Omega_{ij} = -2\,\mathrm{Im}\,\mathcal{G}_{ij}$, antisymmetric). The metric quantifies the quantum distance (fidelity susceptibility) between infinitesimally close states, while the Berry curvature captures the local "twist" (gauge structure) of the bundle (Kang et al., 23 Dec 2024).

Experimentally, the QGT has been mapped in bulk materials using momentum-resolved photoemission protocols. The "quasi-QGT" $$q_{ij}(\mathbf{k}) = \langle\partial_i u_n| (H-E_n) | \partial_j u_n \rangle$$ is accessible through band Drude weight (real part) and orbital angular momentum (imaginary part) measurements, e.g., via energy-resolved ARPES and spin-resolved circular dichroism. In two-band systems, $$G_{ij}(\mathbf{k}) = q_{ij}(\mathbf{k})/\Delta E(\mathbf{k})$$, with $\Delta E = E_m-E_n$ (Kang et al., 23 Dec 2024).

3. Quantum Geometric Contribution in Superconductivity and Superfluidity

Quantum geometry leads to nontrivial contributions to the superfluid weight (SW) and effective mass in multiband, spin-orbit coupled, and flat band systems:

• In the BCS-BEC crossover for spin-orbit coupled Fermi superfluids, the pair (Cooper-pair) effective mass receives a geometric contribution $c_{ij}^{\mathrm{inter}}$ directly proportional to the quantum metric (Iskin, 2018). Near unitarity, this geometric component can account for $20$–$25\%$ of the inverse pair mass, with significant implications for sound velocity, superfluid density, and critical fields.
• The time-dependent Ginzburg–Landau theory in multiband Hubbard models identifies a kinetic coefficient $$c_{ij} = c_{ij}^{\mathrm{intra}} + c_{ij}^{\mathrm{inter}}$$, with the latter entirely geometric and proportional to the quantum-metric tensor $g_{ij}^{nm}$ (Iskin, 2023). In flat-band limits, $c_{ij}^{\mathrm{inter}}$ can dominate, enabling superfluidity where band-dispersion is negligible.
• For flat-band superconductors, the sole source of superfluid weight is the quantum metric: $D_{ij} \propto g_{ij}$ (Iskin, 2022, Hu et al., 18 Sep 2024). Geometric SW contributions can result from both effective Josephson coupling and virtual inter-band tunneling, and can even drive negative SW, leading to pair-density-wave phases (Hu et al., 18 Sep 2024).
• The same interband, metric-driven processes control the stiffness and sound velocity of magnon and Goldstone modes in both Bose and Fermi systems (Subasi et al., 2021, Iskin, 2019).

The table below summarizes the origin of quantum-geometric contributions in select contexts:

System	Observable	Geometric Contribution
SOC Fermi superfluid	Pair mass, $m_B$	Quantum metric, $g_{ij}$
Multiband GL theory	Pair stiffness, $c_{ij}$	Interband $g_{ij}^{nm}$
Flat-band/2-band BEC/SC	Superfluid weight $D_{ij}$	$g_{ij}(\mathbf{k}_c)$
Bose/Fermi, Goldstone mode	Sound velocity $c$	$Q_{ij}^\mathrm{geom}$

4. Quantum Geometry and Transport Phenomena

Quantum metric and Berry curvature contribute to both linear and nonlinear transport:

• In Dirac and Weyl semimetals, the quantum geometric part of the diffusion constant $D_{ab}$ corresponds to the transverse part of a rank-2 tensor in momentum space. At charge neutrality in 3D, the entire diffusion constant is quantum-geometric in origin, with band velocity contributions vanishing due to accidental cancellation (Burkov, 18 Dec 2025). In 2D, $3/4$ of $D_2$ arises geometrically.
• The intrinsic nonlinear Hall effect can be decomposed into Berry curvature dipole (BCD), interband quantum metric dipole (interQMD), and intraband quantum metric dipole (intraQMD) terms. The intraband metric dipole, involving derivatives of $g_{nn}$, provides dominant corrections in certain topological antiferromagnets (Ulrich et al., 20 Jun 2025).
• Second harmonic generation (SHG) contains quantum metric and connection contributions in injection, shift, anomalous, double-resonant, and higher-order-pole currents (Bhalla et al., 2021). These enable "metric-selective" nonlinear optical diagnostics in materials such as WTe$_2$ and CuMnAs.

5. Quantum Geometric Effects in Dynamics and Entanglement

Quantum geometric contributions appear in non-equilibrium and entanglement contexts:

• In quantum quenches, the interplay between dynamical and geometric phase contributions can lead to non-equilibrium phase transitions, modify defect generation statistics (changing the Kibble–Zurek exponent), and alter transition probabilities in driven many-body systems (Tomka et al., 2011).
• The entanglement entropy in quantum Hall systems carries a universal geometric "corner" correction $S^{\mathrm{corner}}(\theta)$ related to the opening angles of subsystem boundaries, consistent with CFT predictions and robust against edge reconstruction (Ye et al., 2022).
• The geometric part of the Berry phase in non-Hermitian systems induces a path-independent amplification or decay, measurable in terms of Petermann factors (Ozawa et al., 20 Sep 2024).

6. Generalizations and Differential Geometric Framework

A fully general treatment of quantum geometric contributions requires the geometry of vector bundles endowed with connections:

• The quantum geometric tensor can be constructed from a subbundle projector $P$ and an arbitrary covariant derivative, leading to a split into quantum metric, Berry curvature, and an extra curvature term encoding the extrinsic geometry (e.g., in curved backgrounds) (Oancea et al., 21 Mar 2025). The generalized Gauss–Codazzi–Mainardi equations relate these pieces and explain the influence of spatial or spacetime curvature on quantum state transport, response coefficients, and semiclassical dynamics.

7. Experimental Protocols and Outlook

Recent advances enable direct measurement and control of geometric contributions:

• Step-response (relaxation from constrained equilibrium) measurements provide a real-time probe of the quantum metric via the time-dependent quantum geometric tensor; the polarization step at $t=0$ is directly proportional to $g_{\mu\nu}$ (Verma et al., 25 Jun 2024).
• ARPES-based quasi-QGT protocols deliver momentum- and energy-resolved maps of the quantum metric and Berry curvature (Kang et al., 23 Dec 2024).
• Benchmarking in twisted bilayer systems, moiré heterostructures, antiferromagnets, and cold-atom platforms has confirmed quantum-geometric control over transport, optical, and superfluid phenomena.

The geometric contribution is thus a central organizing principle unifying phenomena across quantum dynamics, transport, superconductivity, non-equilibrium processes, and quantum gravity-inspired frameworks. Ongoing developments leverage its predictive power in quantum material design, device engineering, and foundational quantum physics.