Multiband Metric Tensor: Quantum Geometry Insights

• Multiband metric tensor is the real part of the quantum-geometric tensor that measures distances between quantum states and captures interband overlaps in N-band systems.
• It is extracted via periodic driving and lattice shaking in Bloch setups, providing a direct experimental probe of quantum geometry and topological characteristics.
• The tensor underpins applications from diagnosing higher-dimensional topological defects to calculating superconducting pair sizes and informing Fisher information in sensing models.

Searching arXiv for the cited paper and closely related multiband quantum-metric work. arXiv search query: "^^^^^^^^2ti:\2 the quantum metric tensor through periodic driving\"^^^^^^^^2 OR all:\2^^^^^^^^"multiband quantum metric tensor\"" The multiband metric tensor most commonly denotes the quantum metric associated with a band of an PRESERVED_PLACEHOLDER_^^^^2ti:\2^^^^-band Hamiltonian. It is the real part of the quantum-geometric tensor and measures the infinitesimal distance between nearby quantum states in parameter space or momentum space. In Bloch systems it is defined from cell-periodic eigenstates, is gauge-invariant because the gauge-dependent connection piece is subtracted, and receives contributions from all other bands through interband overlaps (&&&^^^^2ti:\2^^^^&&&). In multiband settings it appears in analytic projector constructions, experimental quantum-geometry probes, topological diagnosis of higher-dimensional defects, and in superconducting or pairing phenomena such as pair size, bound-state effective mass, and time-dependent Ginzburg-Landau coefficients (&&&^^^^2 OR all:\2^^^^&&&).

^^^^2 OR all:\2^^^^. Definition in multiband Bloch and parameter-dependent systems

For a Bloch Hamiltonian PRESERVED_PLACEHOLDER_^^^^2 OR all:\2^^^^ with cell-periodic eigenstates $|u_{n,\mathbf{k}}\rangle$ satisfying

$$H(\mathbf{k})|u_{n,\mathbf{k}}\rangle=\varepsilon_n(\mathbf{k})|u_{n,\mathbf{k}}\rangle,$$

the quantum-geometric tensor of band $n$ is

$$\chi_{ij}^{(n)}(\mathbf{k}) = \bigl\langle\partial_{k_i}u_{n,\mathbf{k}}\bigl|\bigl(1-|u_{n,\mathbf{k}}\rangle\langle u_{n,\mathbf{k}}|\bigr)\bigr|\partial_{k_j}u_{n,\mathbf{k}}\bigr\rangle.$$

Its real part is the quantum metric and its imaginary part encodes the Berry curvature,

$$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,\chi_{ij}^{(n)}(\mathbf{k}),\qquad \Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,\chi_{ij}^{(n)}(\mathbf{k}).$$

Equivalently,

$$g_{ij}^{(n)}(\mathbf{k}) = \mathrm{Re}\Bigl[ \langle\partial_{k_i}u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle - \langle\partial_{k_i}u_{n,\mathbf{k}}|u_{n,\mathbf{k}}\rangle \langle u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle \Bigr].$$

The overlap term measures the overlap between infinitesimally shifted Bloch states, while the connection terms subtract out the gauge-dependent piece so that $g_{ij}$ is gauge-invariant (&&&^^^^2ti:\2^^^^&&&).

A multiband resolution makes the interband content explicit. For a Bloch band $|n,k\rangle$, one may define

PRESERVED_PLACEHOLDER_^^^^2 OR all:\2ti:\2^^^^

and also the band-resolved pieces

PRESERVED_PLACEHOLDER_^^^^2 OR all:\2 OR all:\2^^^^

so that

PRESERVED_PLACEHOLDER_^^^^2 OR all:\22^^^^

This decomposition is central in multiband problems because the metric of a given band is built from virtual couplings to all other bands (Iskin, 2024).

The same structure extends beyond crystalline momentum. For a parameter-dependent Hamiltonian PRESERVED_PLACEHOLDER_^^^^2 OR all:\2^^^^3, the multiband quantum-geometric tensor of a nondegenerate eigenstate PRESERVED_PLACEHOLDER_^^^^2 OR all:\2^^^^4 can be written as

PRESERVED_PLACEHOLDER_^^^^2 OR all:\2^^^^5

This form is standard in the discussion of higher-dimensional parameter spaces and topological defects (Palumbo et al., 2018).

2. Projectors, generalized Bloch vectors, and analytic PRESERVED_PLACEHOLDER_^^^^2 OR all:\2^^^^6-band formulas

A prominent multiband formulation replaces explicit eigenstates by eigenprojectors. For an PRESERVED_PLACEHOLDER_^^^^2 OR all:\2^^^^7-band Hamiltonian PRESERVED_PLACEHOLDER_^^^^2 OR all:\2^^^^8 with band projector PRESERVED_PLACEHOLDER_^^^^2 OR all:\2^^^^9, the quantum metric can be written as

$|u_{n,\mathbf{k}}\rangle$^^^^2ti:\2^^^^

The projector itself can be obtained directly from the Hamiltonian matrix and the corresponding eigenvalue by a Cayley-Hamilton construction, and

$|u_{n,\mathbf{k}}\rangle$^^^^2 OR all:\2^^^^

This eigenprojector approach offers an alternative to state-derivative formulas and avoids the explicit construction of energy eigenstates (&&&^^^^2 OR all:\2^^^^&&&).

The projector formalism can be recast in an SU$|u_{n,\mathbf{k}}\rangle$2 generalized Bloch-vector language. Writing

$|u_{n,\mathbf{k}}\rangle$3

one finds

$|u_{n,\mathbf{k}}\rangle$4

In this form, the multiband quantum metric and Berry curvature are both expressed directly in terms of the generalized Bloch vector of the band (&&&^^^^2 OR all:\2^^^^&&&).

The familiar two-band formula emerges as a special case. For

$|u_{n,\mathbf{k}}\rangle$5

the quantum metric becomes

$|u_{n,\mathbf{k}}\rangle$6

The multiband formalism therefore generalizes a standard two-band identity rather than replacing it (&&&^^^^2 OR all:\2^^^^&&&).

Closed-form multiband examples illustrate the structure. For a three-band pseudospin-^^^^2 OR all:\2^^^^ fermion,

$|u_{n,\mathbf{k}}\rangle$7

with energies $|u_{n,\mathbf{k}}\rangle$8, the metric is

$|u_{n,\mathbf{k}}\rangle$9

For a four-band pseudospin-$$H(\mathbf{k})|u_{n,\mathbf{k}}\rangle=\varepsilon_n(\mathbf{k})|u_{n,\mathbf{k}}\rangle,$$^^^^2ti:\2^^^^ fermion with the same linear form, one obtains

$$H(\mathbf{k})|u_{n,\mathbf{k}}\rangle=\varepsilon_n(\mathbf{k})|u_{n,\mathbf{k}}\rangle,$$^^^^2 OR all:\2^^^^

In both cases the metric is transverse and proportional to the round metric on the sphere of fixed $$H(\mathbf{k})|u_{n,\mathbf{k}}\rangle=\varepsilon_n(\mathbf{k})|u_{n,\mathbf{k}}\rangle,$$2 (&&&^^^^2 OR all:\2^^^^&&&).

3. Extraction through periodic driving and lattice shaking

A generic experimental protocol relates excitation rates to metric components. For a Hamiltonian $$H(\mathbf{k})|u_{n,\mathbf{k}}\rangle=\varepsilon_n(\mathbf{k})|u_{n,\mathbf{k}}\rangle,$$3, one prepares an eigenstate $$H(\mathbf{k})|u_{n,\mathbf{k}}\rangle=\varepsilon_n(\mathbf{k})|u_{n,\mathbf{k}}\rangle,$$4 of $$H(\mathbf{k})|u_{n,\mathbf{k}}\rangle=\varepsilon_n(\mathbf{k})|u_{n,\mathbf{k}}\rangle,$$5 and modulates a parameter,

$$H(\mathbf{k})|u_{n,\mathbf{k}}\rangle=\varepsilon_n(\mathbf{k})|u_{n,\mathbf{k}}\rangle,$$6

To first order,

$$H(\mathbf{k})|u_{n,\mathbf{k}}\rangle=\varepsilon_n(\mathbf{k})|u_{n,\mathbf{k}}\rangle,$$7

Fermi’s golden rule gives the transition rate to a final state $$H(\mathbf{k})|u_{n,\mathbf{k}}\rangle=\varepsilon_n(\mathbf{k})|u_{n,\mathbf{k}}\rangle,$$8, and after summing over $$H(\mathbf{k})|u_{n,\mathbf{k}}\rangle=\varepsilon_n(\mathbf{k})|u_{n,\mathbf{k}}\rangle,$$9 and integrating over frequency,

$n$^^^^2ti:\2^^^^

Off-diagonal components are obtained by modulating two parameters in phase and out of phase: $n$^^^^2 OR all:\2^^^^ In the Bloch context this is often summarized as

$n$2

The protocol is therefore a direct metrology scheme for the metric tensor itself (&&&^^^^2ti:\2^^^^&&&).

For lattice systems, linear shaking of a lattice along $n$3 corresponds to adding $n$4 to the lattice Hamiltonian. Since $n$5 acts as $n$6 in momentum space, shaking along $n$7, $n$8, and $n$9 reconstructs $$\chi_{ij}^{(n)}(\mathbf{k}) = \bigl\langle\partial_{k_i}u_{n,\mathbf{k}}\bigl|\bigl(1-|u_{n,\mathbf{k}}\rangle\langle u_{n,\mathbf{k}}|\bigr)\bigr|\partial_{k_j}u_{n,\mathbf{k}}\bigr\rangle.$$^^^^2ti:\2^^^^, $$\chi_{ij}^{(n)}(\mathbf{k}) = \bigl\langle\partial_{k_i}u_{n,\mathbf{k}}\bigl|\bigl(1-|u_{n,\mathbf{k}}\rangle\langle u_{n,\mathbf{k}}|\bigr)\bigr|\partial_{k_j}u_{n,\mathbf{k}}\bigr\rangle.$$^^^^2 OR all:\2^^^^, and $$\chi_{ij}^{(n)}(\mathbf{k}) = \bigl\langle\partial_{k_i}u_{n,\mathbf{k}}\bigl|\bigl(1-|u_{n,\mathbf{k}}\rangle\langle u_{n,\mathbf{k}}|\bigr)\bigr|\partial_{k_j}u_{n,\mathbf{k}}\bigr\rangle.$$2. The protocol prepares an initial state in a single Bloch eigenstate $$\chi_{ij}^{(n)}(\mathbf{k}) = \bigl\langle\partial_{k_i}u_{n,\mathbf{k}}\bigl|\bigl(1-|u_{n,\mathbf{k}}\rangle\langle u_{n,\mathbf{k}}|\bigr)\bigr|\partial_{k_j}u_{n,\mathbf{k}}\bigr\rangle.$$3, or a narrow wave packet peaked at $$\chi_{ij}^{(n)}(\mathbf{k}) = \bigl\langle\partial_{k_i}u_{n,\mathbf{k}}\bigl|\bigl(1-|u_{n,\mathbf{k}}\rangle\langle u_{n,\mathbf{k}}|\bigr)\bigr|\partial_{k_j}u_{n,\mathbf{k}}\bigr\rangle.$$4, applies the periodic drive for a time $$\chi_{ij}^{(n)}(\mathbf{k}) = \bigl\langle\partial_{k_i}u_{n,\mathbf{k}}\bigl|\bigl(1-|u_{n,\mathbf{k}}\rangle\langle u_{n,\mathbf{k}}|\bigr)\bigr|\partial_{k_j}u_{n,\mathbf{k}}\bigr\rangle.$$5, measures the fraction excited into other bands as a function of $$\chi_{ij}^{(n)}(\mathbf{k}) = \bigl\langle\partial_{k_i}u_{n,\mathbf{k}}\bigl|\bigl(1-|u_{n,\mathbf{k}}\rangle\langle u_{n,\mathbf{k}}|\bigr)\bigr|\partial_{k_j}u_{n,\mathbf{k}}\bigr\rangle.$$6, and sums the observed rates to form $$\chi_{ij}^{(n)}(\mathbf{k}) = \bigl\langle\partial_{k_i}u_{n,\mathbf{k}}\bigl|\bigl(1-|u_{n,\mathbf{k}}\rangle\langle u_{n,\mathbf{k}}|\bigr)\bigr|\partial_{k_j}u_{n,\mathbf{k}}\bigr\rangle.$$7 (&&&^^^^2ti:\2^^^^&&&).

The multiband character is automatic: in the sum over final states, all bands $$\chi_{ij}^{(n)}(\mathbf{k}) = \bigl\langle\partial_{k_i}u_{n,\mathbf{k}}\bigl|\bigl(1-|u_{n,\mathbf{k}}\rangle\langle u_{n,\mathbf{k}}|\bigr)\bigr|\partial_{k_j}u_{n,\mathbf{k}}\bigr\rangle.$$8 contribute, so the method includes all interband matrix elements $$\chi_{ij}^{(n)}(\mathbf{k}) = \bigl\langle\partial_{k_i}u_{n,\mathbf{k}}\bigl|\bigl(1-|u_{n,\mathbf{k}}\rangle\langle u_{n,\mathbf{k}}|\bigr)\bigr|\partial_{k_j}u_{n,\mathbf{k}}\bigr\rangle.$$9. Degeneracies or near-degeneracies require either a weak perturbation that slightly splits them so that Fermi’s golden rule remains valid, or projection onto a well-separated subset of bands with a non-Abelian generalization of the quantum metric (&&&^^^^2ti:\2^^^^&&&).

Two benchmark lattice examples are emphasized. In the Harper-Hofstadter model at flux $$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,\chi_{ij}^{(n)}(\mathbf{k}),\qquad \Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,\chi_{ij}^{(n)}(\mathbf{k}).$$^^^^2ti:\2^^^^, a Gaussian wave packet prepared in band $$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,\chi_{ij}^{(n)}(\mathbf{k}),\qquad \Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,\chi_{ij}^{(n)}(\mathbf{k}).$$^^^^2 OR all:\2^^^^ and shaken along $$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,\chi_{ij}^{(n)}(\mathbf{k}),\qquad \Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,\chi_{ij}^{(n)}(\mathbf{k}).$$2 yields

$$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,\chi_{ij}^{(n)}(\mathbf{k}),\qquad \Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,\chi_{ij}^{(n)}(\mathbf{k}).$$3

with excellent agreement once the wave packet is sufficiently momentum-sharp. The same driving framework also gives access to the spread functional

$$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,\chi_{ij}^{(n)}(\mathbf{k}),\qquad \Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,\chi_{ij}^{(n)}(\mathbf{k}).$$4

through

$$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,\chi_{ij}^{(n)}(\mathbf{k}),\qquad \Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,\chi_{ij}^{(n)}(\mathbf{k}).$$5

In the Haldane model, $$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,\chi_{ij}^{(n)}(\mathbf{k}),\qquad \Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,\chi_{ij}^{(n)}(\mathbf{k}).$$6 diverges at the topological transition $$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,\chi_{ij}^{(n)}(\mathbf{k}),\qquad \Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,\chi_{ij}^{(n)}(\mathbf{k}).$$7, and this appears as a strong enhancement of the measured $$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,\chi_{ij}^{(n)}(\mathbf{k}),\qquad \Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,\chi_{ij}^{(n)}(\mathbf{k}).$$8 (&&&^^^^2ti:\2^^^^&&&).

The protocol was proposed for ultracold atoms in optical lattices and for circuit QED or superconducting qubits. Finite drive time limits the frequency resolution as $$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,\chi_{ij}^{(n)}(\mathbf{k}),\qquad \Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,\chi_{ij}^{(n)}(\mathbf{k}).$$9, the amplitude must satisfy $$g_{ij}^{(n)}(\mathbf{k}) = \mathrm{Re}\Bigl[ \langle\partial_{k_i}u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle - \langle\partial_{k_i}u_{n,\mathbf{k}}|u_{n,\mathbf{k}}\rangle \langle u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle \Bigr].$$^^^^2ti:\2^^^^ to remain in the linear-response regime, and numerical integration is required when $$g_{ij}^{(n)}(\mathbf{k}) = \mathrm{Re}\Bigl[ \langle\partial_{k_i}u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle - \langle\partial_{k_i}u_{n,\mathbf{k}}|u_{n,\mathbf{k}}\rangle \langle u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle \Bigr].$$^^^^2 OR all:\2^^^^ is sampled discretely (&&&^^^^2ti:\2^^^^&&&).

4. Topological information encoded by the metric: tensor monopoles

In a four-dimensional parameter space, a three-band Hamiltonian can support a tensor monopole whose charge is recoverable from the quantum metric. The minimal model is

$$g_{ij}^{(n)}(\mathbf{k}) = \mathrm{Re}\Bigl[ \langle\partial_{k_i}u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle - \langle\partial_{k_i}u_{n,\mathbf{k}}|u_{n,\mathbf{k}}\rangle \langle u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle \Bigr].$$2

with spectrum $$g_{ij}^{(n)}(\mathbf{k}) = \mathrm{Re}\Bigl[ \langle\partial_{k_i}u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle - \langle\partial_{k_i}u_{n,\mathbf{k}}|u_{n,\mathbf{k}}\rangle \langle u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle \Bigr].$$3, $$g_{ij}^{(n)}(\mathbf{k}) = \mathrm{Re}\Bigl[ \langle\partial_{k_i}u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle - \langle\partial_{k_i}u_{n,\mathbf{k}}|u_{n,\mathbf{k}}\rangle \langle u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle \Bigr].$$4, and a three-fold degeneracy at $$g_{ij}^{(n)}(\mathbf{k}) = \mathrm{Re}\Bigl[ \langle\partial_{k_i}u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle - \langle\partial_{k_i}u_{n,\mathbf{k}}|u_{n,\mathbf{k}}\rangle \langle u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle \Bigr].$$5. For the negative-energy band $$g_{ij}^{(n)}(\mathbf{k}) = \mathrm{Re}\Bigl[ \langle\partial_{k_i}u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle - \langle\partial_{k_i}u_{n,\mathbf{k}}|u_{n,\mathbf{k}}\rangle \langle u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle \Bigr].$$6, the quantum metric on the three-sphere of fixed radius $$g_{ij}^{(n)}(\mathbf{k}) = \mathrm{Re}\Bigl[ \langle\partial_{k_i}u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle - \langle\partial_{k_i}u_{n,\mathbf{k}}|u_{n,\mathbf{k}}\rangle \langle u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle \Bigr].$$7 becomes the metric of $$g_{ij}^{(n)}(\mathbf{k}) = \mathrm{Re}\Bigl[ \langle\partial_{k_i}u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle - \langle\partial_{k_i}u_{n,\mathbf{k}}|u_{n,\mathbf{k}}\rangle \langle u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle \Bigr].$$8 in hyperspherical coordinates, with nonzero components

$$g_{ij}^{(n)}(\mathbf{k}) = \mathrm{Re}\Bigl[ \langle\partial_{k_i}u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle - \langle\partial_{k_i}u_{n,\mathbf{k}}|u_{n,\mathbf{k}}\rangle \langle u_{n,\mathbf{k}}|\partial_{k_j}u_{n,\mathbf{k}}\rangle \Bigr].$$9

The determinant in the $g_{ij}$^^^^2ti:\2^^^^ subspace is

$g_{ij}$^^^^2 OR all:\2^^^^

This metric data determines the tensor-monopole curvature 3-form (Palumbo et al., 2018).

The curvature is written as

$g_{ij}$2

and the associated topological charge is

$g_{ij}$3

A central point is that one need not compute the tensor gauge field $g_{ij}$4 or perform gauge-field patching: the metric alone determines $g_{ij}$5, and integrating it gives a unit charge (Palumbo et al., 2018).

Experimentally, the model is mapped to a three-level cold-atom system with two complex Rabi couplings,

$g_{ij}$6

The individual $g_{ij}$7 are extracted from integrated excitation rates under small periodic modulations of the parameters, and assembling the metric on a fixed-$g_{ij}$8 sphere allows a numerical evaluation of $g_{ij}$9 and thus of $|n,k\rangle$^^^^2ti:\2^^^^ (Palumbo et al., 2018).

This construction also locates the multiband metric tensor within a hierarchy of degeneracy-driven topological structures. A Dirac monopole in $|n,k\rangle$^^^^2 OR all:\2^^^^ requires a two-band crossing, a Yang monopole in $|n,k\rangle$2 requires a four-fold degenerate point, and a tensor monopole in $|n,k\rangle$3 requires a triple degeneracy. The multiband setting is therefore not incidental but structural (Palumbo et al., 2018).

5. Pair size, bound-state mass, and superconducting response

In multiband Hubbard models with time-reversal symmetry and uniform pairing, the quantum metric enters directly into two-body and many-body size tensors. The size of a lowest-lying two-body bound state in vacuum is defined by the localization tensor

$|n,k\rangle$4

and under the uniform-pairing and time-reversal assumptions it splits exactly into intraband and interband pieces,

$|n,k\rangle$5

The intraband part depends on derivatives of the band dispersions, whereas the interband part is governed by $|n,k\rangle$6 and $|n,k\rangle$7. For an isolated flat band $|n,k\rangle$8 with $|n,k\rangle$9, the intraband contribution vanishes in the PRESERVED_PLACEHOLDER_^^^^2 OR all:\2ti:\2ti:\2^^^^ limit and

PRESERVED_PLACEHOLDER_^^^^2 OR all:\2ti:\2 OR all:\2^^^^

which is a purely quantum-metric result (Iskin, 2024).

An analogous decomposition holds for the mean-field Cooper-pair size in the PRESERVED_PLACEHOLDER_^^^^2 OR all:\2ti:\22^^^^ BCS-BEC crossover. With pair amplitude

PRESERVED_PLACEHOLDER_^^^^2 OR all:\2ti:\2^^^^3

and

PRESERVED_PLACEHOLDER_^^^^2 OR all:\2ti:\2^^^^4

one again finds

PRESERVED_PLACEHOLDER_^^^^2 OR all:\2ti:\2^^^^5

In the dilute limit PRESERVED_PLACEHOLDER_^^^^2 OR all:\2ti:\2^^^^6, the mean-field result recovers the exact two-body result term by term. In the pyrochlore-Hubbard model, the dispersive-band BCS regime displays a diverging pair size as PRESERVED_PLACEHOLDER_^^^^2 OR all:\2ti:\2^^^^7, whereas in the flat-band regime the pair size remains finite and relatively small even for infinitesimal interaction, fully dominated by the geometric term and matching the exact two-body result in the dilute limit (Iskin, 2024).

The same multiband quantum geometry controls the effective mass tensor of the lowest bound states. For the bound-state dispersion

PRESERVED_PLACEHOLDER_^^^^2 OR all:\2ti:\2^^^^8

the inverse mass decomposes exactly as

PRESERVED_PLACEHOLDER_^^^^2 OR all:\2ti:\2^^^^9

The first interband term depends on the band-diagonal metric PRESERVED_PLACEHOLDER_^^^^2 OR all:\2 OR all:\2ti:\2^^^^, while the second depends on the band-resolved metric

PRESERVED_PLACEHOLDER_^^^^2 OR all:\2 OR all:\2 OR all:\2^^^^

In a nearest-neighbor Kagome lattice, solving the two-body problem numerically at PRESERVED_PLACEHOLDER_^^^^2 OR all:\2 OR all:\22^^^^ yields PRESERVED_PLACEHOLDER_^^^^2 OR all:\2 OR all:\2^^^^3 and PRESERVED_PLACEHOLDER_^^^^2 OR all:\2 OR all:\2^^^^4, and the three-part decomposition reproduces these values exactly (&&&2^^^^2ti:\2^^^^&&&).

Near the critical pairing temperature, the multiband metric tensor also enters time-dependent Ginzburg-Landau theory. The TDGL equation

PRESERVED_PLACEHOLDER_^^^^2 OR all:\2 OR all:\2^^^^5

contains a kinetic-energy tensor

PRESERVED_PLACEHOLDER_^^^^2 OR all:\2 OR all:\2^^^^6

where the interband term is controlled by both the quantum-metric tensor and the band-resolved quantum-metric tensors. The resulting GL coherence length, London penetration depth, GL parameter, and upper critical magnetic field have an explicit dependence on quantum geometry. In particular, the intraband term vanishes in a strictly flat band, but the geometric term remains, allowing a finite superfluid density (&&&2^^^^2 OR all:\2^^^^&&&).

A recurring multiband conclusion is therefore precise rather than heuristic: the conventional intraband part diverges in metals or in the BCS regime and vanishes for a strictly flat band, whereas the geometric interband part remains finite and encodes virtual interband processes (Iskin, 2024).

6. Alternative usage: the multiband metric tensor in sensing theory

In a different research area, “multiband metric tensor” denotes the Fisher information matrix of a multiband sensing model. For a SISO multiband OFDM training model with parameter vector

PRESERVED_PLACEHOLDER_^^^^2 OR all:\2 OR all:\2^^^^7

the Fisher information matrix is

PRESERVED_PLACEHOLDER_^^^^2 OR all:\2 OR all:\2^^^^8

where PRESERVED_PLACEHOLDER_^^^^2 OR all:\2 OR all:\2^^^^9 is the data term and PRESERVED_PLACEHOLDER_^^^^2 OR all:\22ti:\2^^^^ incorporates priors on timing offsets. By information geometry, the local quadratic form

PRESERVED_PLACEHOLDER_^^^^2 OR all:\22 OR all:\2^^^^

is a natural Riemannian metric on the statistical manifold of PRESERVED_PLACEHOLDER_^^^^2 OR all:\222^^^^. Restricting to the delay submanifold gives the pull-back metric PRESERVED_PLACEHOLDER_^^^^2 OR all:\22^^^^3 with metric tensor PRESERVED_PLACEHOLDER_^^^^2 OR all:\22^^^^4 (Wan et al., 2022).

In the simplified two-path, two-subband case, the delay subblock of the Fisher information matrix can be written compactly using the Dirichlet kernel PRESERVED_PLACEHOLDER_^^^^2 OR all:\22^^^^5. This formulation leads to closed-form upper and lower bounds on the Cramér-Rao bound for the delay separation PRESERVED_PLACEHOLDER_^^^^2 OR all:\22^^^^6, both scaling as PRESERVED_PLACEHOLDER_^^^^2 OR all:\22^^^^7 for large subband aperture. The associated statistical resolution limit is defined by the fixed-point relation

PRESERVED_PLACEHOLDER_^^^^2 OR all:\22^^^^8

and can be evaluated numerically (Wan et al., 2022).

The same framework includes the Ziv-Zakai bound for single-path delay estimation and an optimization problem that minimizes the delay statistical resolution limit over subband center frequencies and bandwidth allocations. The proposed solution is an alternating optimization algorithm combining successive convex approximation with one-dimensional search. In this usage, the “metric tensor” is not the Fubini-Study quantum metric of Bloch states, but the Riemannian metric induced by Fisher information on a multiband statistical manifold (Wan et al., 2022).