Integrated Quantum Metric (IQM)

• Integrated Quantum Metric (IQM) is defined by integrating the quantum metric tensor over the Brillouin zone or its real-space analogue, providing a gauge-invariant measure of Wannier function spread.
• IQM establishes bounds between topology and geometry, influencing localization properties and linear response coefficients such as optical conductivity and superfluid weight.
• Efficient numerical methods like the Kernel Polynomial Method and Bott metric enable IQM evaluation in periodic, disordered, and non-Hermitian systems, bridging theory and experiment.

The integrated quantum metric (IQM) captures the global quantum geometric properties of Bloch bands and their real-space analogues, encompassing both crystalline and non-periodic states, with wide-ranging significance for topology, localization, and linear response in quantum materials. Defined as the Brillouin-zone integral (or its appropriate real-space counterpart) of the quantum metric tensor—the symmetric part of the quantum geometric tensor—IQM encodes the gauge-invariant spread of Wannier functions, imposes bounds relating topology and geometry, and underpins several experimentally relevant observables. The extension of IQM to disordered, amorphous, and non-Hermitian systems, along with efficient numerical and experimental protocols, underlines its central role in contemporary quantum materials research.

1. Mathematical Definition and Notation

The quantum geometric tensor (QGT), $q_{\alpha\beta}(k)$, for single-particle bands in periodic systems is

$$q_{\alpha\beta}(k) = \frac{1}{\pi} \sum_{i,j} f_{ik} [1 - f_{jk}] \frac{\langle \psi_{ik} | \partial_{k_\alpha} H(k) | \psi_{jk} \rangle \langle \psi_{jk} | \partial_{k_\beta} H(k) | \psi_{ik} \rangle}{(E_{ik} - E_{jk})^2},$$

where $H(k)$ is the Bloch Hamiltonian, $|\psi_{ik}\rangle$ its $i$th eigenstate, $E_{ik}$ its eigenvalue, and $f_{ik}$ the band occupation.

The QGT decomposes as

$$q_{\alpha\beta}(k) = g_{\alpha\beta}(k) + \frac{i}{2} \Omega_{\alpha\beta}(k)$$

where $g_{\alpha\beta}$ is the real symmetric quantum metric and $\Omega_{\alpha\beta}$ is the Berry curvature.

By integrating over the Brillouin zone ($$q_{\alpha\beta}(k) = \frac{1}{\pi} \sum_{i,j} f_{ik} [1 - f_{jk}] \frac{\langle \psi_{ik} | \partial_{k_\alpha} H(k) | \psi_{jk} \rangle \langle \psi_{jk} | \partial_{k_\beta} H(k) | \psi_{ik} \rangle}{(E_{ik} - E_{jk})^2},$$0-dimensions: $$q_{\alpha\beta}(k) = \frac{1}{\pi} \sum_{i,j} f_{ik} [1 - f_{jk}] \frac{\langle \psi_{ik} | \partial_{k_\alpha} H(k) | \psi_{jk} \rangle \langle \psi_{jk} | \partial_{k_\beta} H(k) | \psi_{ik} \rangle}{(E_{ik} - E_{jk})^2},$$1 BZ), the IQGT is

$$q_{\alpha\beta}(k) = \frac{1}{\pi} \sum_{i,j} f_{ik} [1 - f_{jk}] \frac{\langle \psi_{ik} | \partial_{k_\alpha} H(k) | \psi_{jk} \rangle \langle \psi_{jk} | \partial_{k_\beta} H(k) | \psi_{ik} \rangle}{(E_{ik} - E_{jk})^2},$$2

The integrated quantum metric (IQM) is its real part, $$q_{\alpha\beta}(k) = \frac{1}{\pi} \sum_{i,j} f_{ik} [1 - f_{jk}] \frac{\langle \psi_{ik} | \partial_{k_\alpha} H(k) | \psi_{jk} \rangle \langle \psi_{jk} | \partial_{k_\beta} H(k) | \psi_{ik} \rangle}{(E_{ik} - E_{jk})^2},$$3, while the Chern number arises from its imaginary part, $$q_{\alpha\beta}(k) = \frac{1}{\pi} \sum_{i,j} f_{ik} [1 - f_{jk}] \frac{\langle \psi_{ik} | \partial_{k_\alpha} H(k) | \psi_{jk} \rangle \langle \psi_{jk} | \partial_{k_\beta} H(k) | \psi_{ik} \rangle}{(E_{ik} - E_{jk})^2},$$4 in 2D.

In real space (including disorder or open systems), the equivalent, gauge-invariant form is

$$q_{\alpha\beta}(k) = \frac{1}{\pi} \sum_{i,j} f_{ik} [1 - f_{jk}] \frac{\langle \psi_{ik} | \partial_{k_\alpha} H(k) | \psi_{jk} \rangle \langle \psi_{jk} | \partial_{k_\beta} H(k) | \psi_{ik} \rangle}{(E_{ik} - E_{jk})^2},$$5

where $$q_{\alpha\beta}(k) = \frac{1}{\pi} \sum_{i,j} f_{ik} [1 - f_{jk}] \frac{\langle \psi_{ik} | \partial_{k_\alpha} H(k) | \psi_{jk} \rangle \langle \psi_{jk} | \partial_{k_\beta} H(k) | \psi_{ik} \rangle}{(E_{ik} - E_{jk})^2},$$6 is the ground-state projector, $$q_{\alpha\beta}(k) = \frac{1}{\pi} \sum_{i,j} f_{ik} [1 - f_{jk}] \frac{\langle \psi_{ik} | \partial_{k_\alpha} H(k) | \psi_{jk} \rangle \langle \psi_{jk} | \partial_{k_\beta} H(k) | \psi_{ik} \rangle}{(E_{ik} - E_{jk})^2},$$7 is the system area, and $$q_{\alpha\beta}(k) = \frac{1}{\pi} \sum_{i,j} f_{ik} [1 - f_{jk}] \frac{\langle \psi_{ik} | \partial_{k_\alpha} H(k) | \psi_{jk} \rangle \langle \psi_{jk} | \partial_{k_\beta} H(k) | \psi_{ik} \rangle}{(E_{ik} - E_{jk})^2},$$8 is the position operator (Romeral et al., 2024, Sun et al., 19 May 2026, Chatterjee et al., 6 Apr 2026).

2. Physical Significance and Bounds

IQM is strictly positive-semidefinite: $$q_{\alpha\beta}(k) = \frac{1}{\pi} \sum_{i,j} f_{ik} [1 - f_{jk}] \frac{\langle \psi_{ik} | \partial_{k_\alpha} H(k) | \psi_{jk} \rangle \langle \psi_{jk} | \partial_{k_\beta} H(k) | \psi_{ik} \rangle}{(E_{ik} - E_{jk})^2},$$9 pointwise, and $H(k)$0 after integration. IQM provides the lower bound on the spread of maximally localized Wannier functions and controls the minimal spatial delocalization of occupied states. In insulators, $H(k)$1 remains finite; in metals, it diverges with system size, reflecting the existence of extended states and their enhanced quantum geometric fluctuations (Romeral et al., 2024).

IQM also plays a direct role in observables such as superfluid weight, dielectric response, and optical spectral weight, where it quantifies the geometric contribution to response coefficients.

3. Real-Space and Non-Hermitian Formulations

IQM has well-defined analogues in real-space and for non-Hermitian systems. For non-periodic systems on (finite) tori, the IQM can be computed via the Bott metric, $H(k)$2, defined from a real-space plaquette operator built from “twist” (coordinate-phase) operators. In the large-system limit, the Bott metric converges to the trace of the IQM tensor: $H(k)$3 This construction is manifestly translational-symmetry-free and thus applies in disordered and amorphous models (Chatterjee et al., 6 Apr 2026).

For non-Hermitian open-boundary systems, the integrated quantum metric can be defined either in real space,

$H(k)$4

or in non-Bloch (generalized Brillouin zone, GBZ) formalism: $H(k)$5 with $H(k)$6 the left-right quantum metric tensor. An exact equivalence holds between these forms. The non-Bloch integrated quantum metric equals the gauge-invariant spread $H(k)$7 of Wannier functions constructed from non-Bloch states, bounding their localization and encoding the skin effect (Sun et al., 19 May 2026).

4. Experimental Signatures and Measurement Protocols

IQM directly influences various bulk observables, but direct measurement requires isolating the symmetric component of the quantum geometric tensor. Two key protocols have emerged:

• Souza–Wilkens–Martin (SWM) Sum Rule: Isolates the IQM from linear optical conductivity via

$H(k)$8

but requires broadband ($H(k)$9) experimental access (Verma et al., 2024).

• Step-Response Protocol: A sudden switch-off of a constant electric field is used to measure the immediate dipole relaxation. In the high-temperature limit, the initial response is directly proportional to the integrated quantum metric:

$|\psi_{ik}\rangle$0

This circumvents the need for frequency integration with the requirement for ultrafast, time-resolved polarization measurement (Verma et al., 2024).

Both protocols are strictly valid in gapped insulators and require careful experimental control over temperature and timescale.

5. Numerical Approaches and Systematic Studies

Efficient evaluation of IQM in large, disordered, or non-periodic systems exploits several real-space techniques:

• Kernel Polynomial Method (KPM): Expands the ground-state projector using Chebyshev polynomials and evaluates commutators recursively. Stochastic trace approximation ($|\psi_{ik}\rangle$1 scaling) enables simulations of $|\psi_{ik}\rangle$2 sites (Romeral et al., 2024).
• Bott Metric Calculation: Involves only real-space projectors and U(1) twists, requires no translational symmetry, and serves as a numerically efficient proxy for IQM in finite geometries (Chatterjee et al., 6 Apr 2026).

Results in the disordered Haldane model show that IQM tracks topological phase boundaries, remains finite and robust against moderate disorder, and converges in gapped regimes. IQM is more resilient to disorder than the Chern number but collapses at gap closure or in metals, where it diverges with system size (Romeral et al., 2024).

6. Extensions, Physical Implications, and Applications

IQM's scope extends to various systems and observables:

• Topological Insulators, Semimetals, and Correlated Phases: Real-space and KPM-based IQM computation generalizes to 2D $|\psi_{ik}\rangle$3 topological insulators, Weyl semimetals, moiré, and quasicrystalline systems, and to mean-field correlated models such as magic-angle graphene (Romeral et al., 2024).
• Non-Hermitian and Open Systems: IQM distinguishes trivial and topological non-Hermitian phases, diverges at gap closing under open boundary conditions, and tracks localization in the presence of the skin effect (Sun et al., 19 May 2026).
• Localization and Geometry: The connection between IQM and the spread of Wannier functions provides both theoretical and computational access to quantum localization properties, especially in amorphous and disordered crystals (Chatterjee et al., 6 Apr 2026).
• Experimental Probes: Quantum metric signatures have been proposed and partially observed in cold-atom, photonic, and step-response measurement platforms, with IQM offering spatially resolved markers for quantum geometry (Romeral et al., 2024, Verma et al., 2024).

7. Summary Table

Context	IQM Expression/Proxy	Key Insights
Periodic systems (BZ)	$|\psi_{ik}\rangle$4	Fundamental quantum geometry of bands
Disordered/real-space	$|\psi_{ik}\rangle$5	Translational-symmetry-free; KPM/Bott
Non-Hermitian/open systems	$|\psi_{ik}\rangle$6, $|\psi_{ik}\rangle$7 (real-space/GBZ)	Skin effect, spread of non-Bloch Wannier
Bott metric (real-space)	$|\psi_{ik}\rangle$8	Efficient, disorder/amorphous compatible

IQM thus forms a unifying geometric invariant, bridging momentum- and real-space formulations, providing rigorous lower bounds to localization, and underpinning topological and linear-response characterization in a broad class of quantum systems (Romeral et al., 2024, Chatterjee et al., 6 Apr 2026, Sun et al., 19 May 2026, Verma et al., 2024).