Real-Space Integrated Quantum Metric

• Real-space integrated quantum metric is a set of constructions linking the quantum geometric tensor to real-space observables, thus quantifying localization and state spread.
• It connects band geometry with concrete measures like Wannier spread via projector, plaquette, and fidelity-marker techniques to assess phase transitions.
• The approach provides actionable diagnostics in disordered and non-periodic systems, influencing transport properties and topological responses.

Real-space integrated quantum metric denotes a set of closely related constructions that connect the quantum metric to real-space formulas, real-space observables, or explicitly real-space notions of distance. In band geometry, the quantum metric is the real part of the quantum geometric tensor, and its Brillouin-zone integral yields an integrated quantity tied to Wannier spread and to the second cumulant of position (Matsuura et al., 2010). In disordered and non-periodic systems, the same geometric content can be written directly in real space with projector commutators, resolved into local markers, or extracted from plaquette-operator and fidelity-marker constructions (Romeral et al., 2024, Sousa et al., 2023, Chatterjee et al., 6 Apr 2026). A distinct but related line of work defines a genuine real-space quantum metric from overlaps of projected local states, while another metric-space formulation treats wave functions and densities themselves as points in a metric space with distances obtained by integrating local differences over space (Oliveira et al., 2024, D'Amico et al., 2011). These usages differ in emphasis, but all treat quantum geometry as a physically meaningful measure of localization, distinguishability, and transport structure in real space.

1. Band-geometric core and the meaning of “integrated”

The band-geometric starting point is the quantum geometric tensor. In one common notation,

$$q_{\alpha\beta}(\mathbf{k}) = g_{\alpha\beta}(\mathbf{k}) + \mathrm{i}\,\Omega_{\alpha\beta}(\mathbf{k})/2,$$

where $g_{\alpha\beta}$ is the quantum metric and $\Omega_{\alpha\beta}$ is the Berry curvature (Romeral et al., 2024). For two-level Hamiltonians written as $$H=d_0(\mathbf{k})I+\mathbf d(\mathbf k)\cdot\boldsymbol\sigma$$, the metric is often written as

$$g_{\mu\nu}^{\pm}(\mathbf k)=\frac{1}{4}\,\partial_{k_\mu}\hat{\mathbf d}\cdot \partial_{k_\nu}\hat{\mathbf d},$$

with $\hat{\mathbf d}=\mathbf d/|\mathbf d|$ (Iskin, 2018, Narayan, 9 Mar 2026). This makes explicit that the metric measures how rapidly eigenstates change in parameter space.

The integrated object is obtained by Brillouin-zone integration: $$\mathcal{Q}_{\alpha\beta}=\int_{\mathrm{BZ}} q_{\alpha\beta}(\mathbf{k})\, d\mathbf{k}, \qquad \mathcal{G}_{\alpha\beta}=\frac{1}{2}\Re\{\mathcal{Q}_{\alpha\beta}\},$$ while the imaginary part yields the Chern number in two dimensions (Romeral et al., 2024). In this formulation, the integrated quantum metric is not a local invariant at a single $\mathbf k$-point; it is an aggregate of the full occupied-state geometry over the band manifold.

The real-space significance of this integral was stated early in the momentum-space geometry literature. For gapped band insulators, the Brillouin-zone integral of the quantum metric gives the second cumulant of the position operator,

$$\langle x_\mu x_\nu\rangle_c = \frac{v}{N_f(2\pi)^d}\int_{\mathrm{BZ}} d\mathbf k\, g_{\mu\nu}(\mathbf k) \equiv \frac{v}{N_f}\Omega_{I,\mu\nu},$$

so the integrated metric is a real-space fluctuation measure rather than a purely formal momentum-space tensor (Matsuura et al., 2010). This relation underlies much of the later literature: “integrated” refers mathematically to momentum-space integration, but the result characterizes real-space spread, localization, or effective dynamics.

2. Projector, plaquette, and local-state constructions in real space

A major development was the direct real-space rewriting of the integrated quantum geometric tensor. For disordered systems without translational symmetry, one may define

$$\mathcal{Q}_{\alpha\beta} = -\frac{4\pi}{A}\,\mathrm{Tr}\!\left\{ \hat P\,[\hat r_\alpha,\hat P]\,[\hat r_\beta,\hat P] \right\},$$

where $g_{\alpha\beta}$0 is the zero-temperature projector onto occupied states (Romeral et al., 2024). This expression is gauge invariant, equivalent to the Brillouin-zone formula in the thermodynamic limit, and does not require periodicity. Its local trace yields a spatial map whose real part serves as a localization marker and whose imaginary part serves as a local Chern marker.

The same real-space sector can also be accessed through a plaquette-operator construction. The Bott metric is defined from the real part of the trace-log of the plaquette operator,

$g_{\alpha\beta}$1

in direct analogy with the Bott index $g_{\alpha\beta}$2 (Chatterjee et al., 6 Apr 2026). In the thermodynamic limit, under projector-locality assumptions appropriate to spectrally gapped systems and mobility-gap regimes, the Bott metric converges to the trace of the integrated quantum metric,

$g_{\alpha\beta}$3

In this formulation, topology and quantum metric are read from the same plaquette operator: the imaginary part encodes phase winding, while the real part measures amplitude contraction.

A closely related real-space rewrite is the fidelity-marker construction. The Brillouin-zone integral

$g_{\alpha\beta}$4

is recast as

$g_{\alpha\beta}$5

with the local fidelity marker

$g_{\alpha\beta}$6

defined directly from lattice projectors (Sousa et al., 2023). The diagonal components coincide with the localization marker, and the off-diagonal components are the symmetrized combination.

An even more literal real-space formulation constructs a normalized local state by projecting a position eigenstate onto the occupied subspace,

$g_{\alpha\beta}$7

and defines the real-space quantum metric through the overlap of neighboring local states,

$g_{\alpha\beta}$8

(Oliveira et al., 2024). Here the metric lives on real space itself rather than arising as a real-space representation of a momentum-space integral.

3. Localization, Wannier spread, and metric-space interpretations

The integrated quantum metric is directly tied to localization. In the disordered-topological formulation, $g_{\alpha\beta}$9 is described as a measure of the spread of the occupied electronic states and as directly tied to ground-state localization: smaller $\Omega_{\alpha\beta}$0 indicates more localized Wannier-like states, while larger $\Omega_{\alpha\beta}$1 signals delocalization (Romeral et al., 2024). The same work emphasizes the inequality

$\Omega_{\alpha\beta}$2

which follows from positivity of the quantum geometric tensor and states that the quantum metric bounds the topological response from below. In the modern theory of insulators, $\Omega_{\alpha\beta}$3 is finite in insulators but divergent in metals in the thermodynamic limit (Romeral et al., 2024).

The relation to Wannier spread is central across several formalisms. The momentum-space geometry literature states that the quantum metric measures the second cumulant of position, whereas the Berry connection measures the first cumulant or electric polarization (Matsuura et al., 2010). The fidelity-marker formulation identifies the local marker as the local density of the gauge-invariant part of the spread of Wannier functions, and the non-Hermitian extension later showed that the real-space integrated quantum metric equals the gauge-invariant part of the spread functional of localized non-Bloch Wannier functions under open boundary conditions (Sousa et al., 2023, Sun et al., 19 May 2026).

A different, non-band-geometric usage appears in metric-space quantum mechanics. There the density distance

$\Omega_{\alpha\beta}$4

is identified as a real-space integrated quantum metric in the strongest sense, because the separation is obtained by integrating local pointwise differences over space (D'Amico et al., 2011). In that framework, all $\Omega_{\alpha\beta}$5-particle densities lie on a sphere of radius $\Omega_{\alpha\beta}$6, all $\Omega_{\alpha\beta}$7-particle wave functions lie on a sphere of radius $\Omega_{\alpha\beta}$8, and Fock space is stratified into concentric “onion-shell” spheres. The Hohenberg–Kohn map is then represented as a monotonic mapping of vicinities onto vicinities in metric space (D'Amico et al., 2011). This usage is conceptually distinct from integrated band geometry, but it preserves the same emphasis on real-space distance and localization.

4. Disorder, topology, and phase transitions

Real-space integrated quantum metric methods have become particularly important in systems without translational symmetry. In the disordered Haldane model, the integrated quantum metric is nonzero in both trivial and topological phases and is bounded below by the Chern number. Inside the gap, the trivial phase has a smaller integrated quantum metric than the topological phases, consistent with stronger localization; outside the gap, the clean system is metallic and the integrated quantum metric diverges in the thermodynamic limit (Romeral et al., 2024). Under Anderson disorder, the Chern number and integrated quantum metric remain robust at midgap over a broad range of disorder strengths, begin to decay when disorder becomes comparable to or larger than the clean gap, roughly $\Omega_{\alpha\beta}$9, and can even show an initial disorder-induced increase for $$H=d_0(\mathbf{k})I+\mathbf d(\mathbf k)\cdot\boldsymbol\sigma$$0 (Romeral et al., 2024). Under vacancies, the topological phases display nonmonotonic behavior linked to overlapping impurity wave functions, which is visible directly in real-space metric maps (Romeral et al., 2024).

The Bott metric yields an allied view of the same problem class. In the clean Qi-Wu-Zhang model, $$H=d_0(\mathbf{k})I+\mathbf d(\mathbf k)\cdot\boldsymbol\sigma$$1 closely tracks $$H=d_0(\mathbf{k})I+\mathbf d(\mathbf k)\cdot\boldsymbol\sigma$$2 as a function of mass $$H=d_0(\mathbf{k})I+\mathbf d(\mathbf k)\cdot\boldsymbol\sigma$$3, with sharp peaks at the topological transition points $$H=d_0(\mathbf{k})I+\mathbf d(\mathbf k)\cdot\boldsymbol\sigma$$4 (Chatterjee et al., 6 Apr 2026). In the disordered QWZ model, the disorder-averaged Bott index remains quantized over a mobility-gap plateau, while the disorder-averaged Bott metric and trace of the integrated quantum metric show matching dependence on disorder and position: $$H=d_0(\mathbf{k})I+\mathbf d(\mathbf k)\cdot\boldsymbol\sigma$$5 is moderate deep inside the topological plateau and develops a bright ridge near the phase boundary where localization weakens and plaquette-induced leakage is enhanced (Chatterjee et al., 6 Apr 2026). In amorphous Chern insulators, the Bott index identifies a broad topological plateau but $$H=d_0(\mathbf{k})I+\mathbf d(\mathbf k)\cdot\boldsymbol\sigma$$6 varies substantially within that plateau and resolves asymmetry between the two plateau edges (Chatterjee et al., 6 Apr 2026).

The fidelity-marker program adds a transition-sensitive nonlocal diagnostic. The nonlocal fidelity marker,

$$H=d_0(\mathbf{k})I+\mathbf d(\mathbf k)\cdot\boldsymbol\sigma$$7

reduces in the thermodynamic limit to the Fourier transform of the momentum-space quantum metric and is postulated as a universal indicator of quantum phase transitions provided the crystalline momentum remains a good quantum number (Sousa et al., 2023). In SSH, Chern-insulator, and Bi$$H=d_0(\mathbf{k})I+\mathbf d(\mathbf k)\cdot\boldsymbol\sigma$$8Se$$H=d_0(\mathbf{k})I+\mathbf d(\mathbf k)\cdot\boldsymbol\sigma$$9-type models, it becomes longer-ranged near criticality (Sousa et al., 2023).

A more literal geometric interpretation of disorder appears in the real-space metric of solids. Once $$g_{\mu\nu}^{\pm}(\mathbf k)=\frac{1}{4}\,\partial_{k_\mu}\hat{\mathbf d}\cdot \partial_{k_\nu}\hat{\mathbf d},$$0 is defined from projected local states, disorder induces a curved real-space manifold with Christoffel symbols, Riemann tensor, Ricci tensor, Ricci scalar, and Einstein tensor defined from finite differences on the lattice (Oliveira et al., 2024). In both a 2D metal with a single impurity and a two-orbital Chern insulator with an impurity, the Ricci scalar oscillates around the defect, while the Euler characteristic remains zero for a single impurity (Oliveira et al., 2024). This suggests a differential-geometric reading of disorder in which impurities modify not only spectra and localization but also the effective geometry of real space itself.

5. Dynamical, transport, and collective manifestations

The integrated quantum metric is not only a static localization measure. In step-response theory for insulators, the relevant object is the time-dependent quantum geometric tensor

$$g_{\mu\nu}^{\pm}(\mathbf k)=\frac{1}{4}\,\partial_{k_\mu}\hat{\mathbf d}\cdot \partial_{k_\nu}\hat{\mathbf d},$$1

whose symmetric part at $$g_{\mu\nu}^{\pm}(\mathbf k)=\frac{1}{4}\,\partial_{k_\mu}\hat{\mathbf d}\cdot \partial_{k_\nu}\hat{\mathbf d},$$2 equals the quantum metric,

$$g_{\mu\nu}^{\pm}(\mathbf k)=\frac{1}{4}\,\partial_{k_\mu}\hat{\mathbf d}\cdot \partial_{k_\nu}\hat{\mathbf d},$$3

(Verma et al., 2024). After a step-like electric field $$g_{\mu\nu}^{\pm}(\mathbf k)=\frac{1}{4}\,\partial_{k_\mu}\hat{\mathbf d}\cdot \partial_{k_\nu}\hat{\mathbf d},$$4 is switched off, the relaxation of the bulk dipole moment obeys

$$g_{\mu\nu}^{\pm}(\mathbf k)=\frac{1}{4}\,\partial_{k_\mu}\hat{\mathbf d}\cdot \partial_{k_\nu}\hat{\mathbf d},$$5

with $$g_{\mu\nu}^{\pm}(\mathbf k)=\frac{1}{4}\,\partial_{k_\mu}\hat{\mathbf d}\cdot \partial_{k_\nu}\hat{\mathbf d},$$6 determined by the symmetric time-dependent quantum geometric tensor (Verma et al., 2024). In the high-temperature limit, $$g_{\mu\nu}^{\pm}(\mathbf k)=\frac{1}{4}\,\partial_{k_\mu}\hat{\mathbf d}\cdot \partial_{k_\nu}\hat{\mathbf d},$$7, providing a direct route to the real-space integrated quantum metric that differs from the Souza-Wilkens-Martin sum rule (Verma et al., 2024).

Disorder can also make the real-space quantum metric directly transport active. In a disordered multi-flatband stub-pyrochlore lattice, the real-space metric operator

$$g_{\mu\nu}^{\pm}(\mathbf k)=\frac{1}{4}\,\partial_{k_\mu}\hat{\mathbf d}\cdot \partial_{k_\nu}\hat{\mathbf d},$$8

enters a geometric conductivity

$$g_{\mu\nu}^{\pm}(\mathbf k)=\frac{1}{4}\,\partial_{k_\mu}\hat{\mathbf d}\cdot \partial_{k_\nu}\hat{\mathbf d},$$9

where $\hat{\mathbf d}=\mathbf d/|\mathbf d|$0 is a Lorentzian broadening matrix set by disorder (Yin et al., 19 Apr 2026). The resulting phase diagram contains a critical delocalized regime between flat-band localization and Anderson localization; its transport is described as percolation of quantum-metric puddles, with finite-size scaling exponents $\hat{\mathbf d}=\mathbf d/|\mathbf d|$1 and $\hat{\mathbf d}=\mathbf d/|\mathbf d|$2, consistent with a classical percolation universality class (Yin et al., 19 Apr 2026). With Rashba spin-orbit coupling, this regime becomes a diffusive metallic phase and the localization-length exponent is reported as approximately $\hat{\mathbf d}=\mathbf d/|\mathbf d|$3 (Yin et al., 19 Apr 2026).

In spin-orbit-coupled Fermi superfluids, the role of integrated quantum metric is indirect but explicit. Near the critical temperature, the coefficient of the time-dependent Ginzburg–Landau action splits as

$\hat{\mathbf d}=\mathbf d/|\mathbf d|$4

and the inter-helicity contribution is

$\hat{\mathbf d}=\mathbf d/|\mathbf d|$5

so the effective pair mass tensor $\hat{\mathbf d}=\mathbf d/|\mathbf d|$6 acquires a geometric contribution controlled by the quantum metric (Iskin, 2018). The paper explicitly states that the metric is not integrated over real space itself; rather, integrated momentum-space geometry enters the real-space kinetic term of the composite bosons through the effective mass tensor (Iskin, 2018). This is a concrete example in which integrated quantum geometry governs real-space propagation, superfluid density, pair dispersion, sound velocity, compressibility, spin susceptibility, and $\hat{\mathbf d}=\mathbf d/|\mathbf d|$7 (Iskin, 2018).

6. Non-Hermitian and phase-space extensions

The real-space integrated quantum metric has been extended beyond Hermitian Bloch band theory. For non-Hermitian systems under open boundary conditions, the real-space integrated quantum metric of spectral sector $\hat{\mathbf d}=\mathbf d/|\mathbf d|$8 is defined as

$\hat{\mathbf d}=\mathbf d/|\mathbf d|$9

with $$\mathcal{Q}_{\alpha\beta}=\int_{\mathrm{BZ}} q_{\alpha\beta}(\mathbf{k})\, d\mathbf{k}, \qquad \mathcal{G}_{\alpha\beta}=\frac{1}{2}\Re\{\mathcal{Q}_{\alpha\beta}\},$$0 built biorthogonally from right and left eigenstates (Sun et al., 19 May 2026). The central result is an exact equivalence

$$\mathcal{Q}_{\alpha\beta}=\int_{\mathrm{BZ}} q_{\alpha\beta}(\mathbf{k})\, d\mathbf{k}, \qquad \mathcal{G}_{\alpha\beta}=\frac{1}{2}\Re\{\mathcal{Q}_{\alpha\beta}\},$$1

where $$\mathcal{Q}_{\alpha\beta}=\int_{\mathrm{BZ}} q_{\alpha\beta}(\mathbf{k})\, d\mathbf{k}, \qquad \mathcal{G}_{\alpha\beta}=\frac{1}{2}\Re\{\mathcal{Q}_{\alpha\beta}\},$$2 is the non-Bloch left-right quantum metric on the generalized Brillouin zone (Sun et al., 19 May 2026). The same quantity equals the gauge-invariant part of the spread of localized non-Bloch Wannier functions, showing that the real-space integrated quantum metric remains the correct localization functional even in the presence of the non-Hermitian skin effect (Sun et al., 19 May 2026).

A broader generalization treats real and momentum space on equal footing in phase space. In that framework, the full phase-space quantum geometric tensor is decomposed as

$$\mathcal{Q}_{\alpha\beta}=\int_{\mathrm{BZ}} q_{\alpha\beta}(\mathbf{k})\, d\mathbf{k}, \qquad \mathcal{G}_{\alpha\beta}=\frac{1}{2}\Re\{\mathcal{Q}_{\alpha\beta}\},$$3

with $$\mathcal{Q}_{\alpha\beta}=\int_{\mathrm{BZ}} q_{\alpha\beta}(\mathbf{k})\, d\mathbf{k}, \qquad \mathcal{G}_{\alpha\beta}=\frac{1}{2}\Re\{\mathcal{Q}_{\alpha\beta}\},$$4, and quantum-metric corrections enter wave-packet energy, Berry connection, equations of motion, and the density of states at order $$\mathcal{Q}_{\alpha\beta}=\int_{\mathrm{BZ}} q_{\alpha\beta}(\mathbf{k})\, d\mathbf{k}, \qquad \mathcal{G}_{\alpha\beta}=\frac{1}{2}\Re\{\mathcal{Q}_{\alpha\beta}\},$$5 (Maranzana et al., 22 Mar 2026). Spatial gradients of the momentum-space metric produce polarization,

$$\mathcal{Q}_{\alpha\beta}=\int_{\mathrm{BZ}} q_{\alpha\beta}(\mathbf{k})\, d\mathbf{k}, \qquad \mathcal{G}_{\alpha\beta}=\frac{1}{2}\Re\{\mathcal{Q}_{\alpha\beta}\},$$6

while mixed real/momentum metric components generate a Hall-like conductivity

$$\mathcal{Q}_{\alpha\beta}=\int_{\mathrm{BZ}} q_{\alpha\beta}(\mathbf{k})\, d\mathbf{k}, \qquad \mathcal{G}_{\alpha\beta}=\frac{1}{2}\Re\{\mathcal{Q}_{\alpha\beta}\},$$7

(Maranzana et al., 22 Mar 2026). This suggests a further enlargement of the subject: the integrated quantum metric need not be confined to either momentum space or real space alone, but can appear as part of a phase-space geometry whose spatial dependence drives measurable response.

Across these developments, a common pattern emerges. The integrated quantum metric may be introduced by Brillouin-zone integration, by projector traces in real space, by plaquette contraction, by local fidelity or localization markers, by projected local states, or by biorthogonal open-boundary projectors. In each case, it quantifies a gauge-invariant notion of spread or distinguishability, and it becomes experimentally or computationally consequential through localization, topology, transport, collective dynamics, or disorder-induced geometry (Matsuura et al., 2010, Romeral et al., 2024, Chatterjee et al., 6 Apr 2026, Sun et al., 19 May 2026).