Quantum Metric-Oscillator Correspondence

• Quantum Metric-Oscillator Strength Correspondence defines how the symmetric quantum geometry governs interband transition rates for linearly polarized light.
• It distinguishes the role of the quantum metric in optical selection rules from the Berry curvature associated with circular dichroism.
• The framework integrates optical spectral sum rules and time-domain responses to capture measurable geometric properties of electronic bands.

Searching arXiv for papers on quantum metric and oscillator strength correspondence to ground the article in current literature. Quantum metric–oscillator strength correspondence is the set of relations by which transition weights built from interband matrix elements are governed by the quantum metric, the symmetric real part of quantum geometry. In its sharpest optical form, the correspondence states that interband oscillator strength for linearly polarized light is controlled by the quantum metric in momentum space, in direct analogy with the established relation between circular-polarization contrast and Berry curvature. In this sense, linear and circular optical selection rules resolve the real and imaginary sectors of the quantum geometric tensor, respectively (Li et al., 12 Jul 2025). Complementary formulations express the quantum metric as an equal-time symmetric sum of interband dipole matrix elements, connect it to inverse-frequency-weighted optical spectral weight through the Souza-Wilkens-Martin sum rule, and show that time-domain step response can access the same geometric content (Verma et al., 2024).

1. Quantum-geometric foundation

The correspondence is formulated within the quantum geometric tensor of Bloch eigenstates. For a Bloch state $|u_{n\mathbf{k}}\rangle$, the tensor is

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

Its real part defines the quantum metric,

$$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,T_{ij}^{(n)}(\mathbf{k}),$$

and its imaginary part gives the Berry curvature,

$$\Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,T_{ij}^{(n)}(\mathbf{k}).$$

The correspondence problem is therefore a question of how products of interband matrix elements partition into a symmetric metric sector and an antisymmetric curvature sector (Li et al., 12 Jul 2025).

Several equivalent matrix-element formulations make this structure explicit. In a band and dipole representation, the time-dependent quantum geometric tensor can be written as

$$\mathcal{Q}_{\mu\nu}(t) = \int_{\mathbf k}\sum_{m\neq n} f_n(1-f_m)\, \hat r_\mu^{nm}\hat r_\nu^{mn} e^{i\hbar\omega_{mn}t},$$

with

$$\hat r_\mu^{nm}\hat r_\nu^{mn} = g_{\mu\nu}^{mn} + i\Omega_{\mu\nu}^{mn}/2 .$$

In a length-gauge multiband transport formulation, the position operator is

$$[r_{\mathbf k}]_{nm}= i\partial_{\mathbf k}\delta_{nm}+\mathcal A_{nm},$$

and the interband quantum metric is

$$g_{nm}^{ab}=\Re(\mathcal A_{nm}^a\mathcal A_{mn}^b).$$

These expressions show that the metric is not an auxiliary geometric ornament; it is the symmetric quadratic combination of interband dipole or Berry-connection matrix elements themselves (Verma et al., 2024, Zeng et al., 2024).

A recurrent conceptual point is that the same interband products generate distinct observables depending on how energies and polarizations enter. Berry-curvature observables extract the antisymmetric imaginary component. Quantum-metric observables extract the symmetric real component. The correspondence is therefore not a statement about one isolated response coefficient, but about how symmetry and weighting project quantum geometry into measurable transition strengths.

2. Linear-polarization oscillator strengths

The most direct correspondence is the optical one established for interband transitions in solids driven by linearly polarized light. For polarization $\hat{\boldsymbol\epsilon}$, the interband oscillator strength is defined as

$$f^{(\hat{\boldsymbol{\epsilon}})}_{nm} = \frac{2m_e}{\hbar\omega_{nm,\mathbf{k}}} \left| \langle n|\hat{\boldsymbol{\epsilon}}\cdot \mathbf{v}_{\mathbf{k}}|m\rangle \right|^2 .$$

For the two orthogonal linear polarizations associated with $$T_{ij}^{(n)}(\mathbf{k}) = \left\langle \partial_{k_i} u_{n\mathbf{k}} \middle| \bigl(1-|u_{n\mathbf{k}}\rangle\langle u_{n\mathbf{k}}|\bigr) \middle| \partial_{k_j} u_{n\mathbf{k}} \right\rangle .$$0 and $$T_{ij}^{(n)}(\mathbf{k}) = \left\langle \partial_{k_i} u_{n\mathbf{k}} \middle| \bigl(1-|u_{n\mathbf{k}}\rangle\langle u_{n\mathbf{k}}|\bigr) \middle| \partial_{k_j} u_{n\mathbf{k}} \right\rangle .$$1, one uses

$$T_{ij}^{(n)}(\mathbf{k}) = \left\langle \partial_{k_i} u_{n\mathbf{k}} \middle| \bigl(1-|u_{n\mathbf{k}}\rangle\langle u_{n\mathbf{k}}|\bigr) \middle| \partial_{k_j} u_{n\mathbf{k}} \right\rangle .$$2

Expanding the squared matrix elements produces a sum-and-difference decomposition whose cross term is precisely the symmetric interband velocity product that defines the off-diagonal metric component. The resulting quantum metric–oscillator strength correspondence is

$$T_{ij}^{(n)}(\mathbf{k}) = \left\langle \partial_{k_i} u_{n\mathbf{k}} \middle| \bigl(1-|u_{n\mathbf{k}}\rangle\langle u_{n\mathbf{k}}|\bigr) \middle| \partial_{k_j} u_{n\mathbf{k}} \right\rangle .$$3

Thus the absorption difference between two orthogonal linear polarizations directly measures the off-diagonal quantum metric (Li et al., 12 Jul 2025).

The same algebra yields a more general projection rule. For a linear polarization along a unit vector $$T_{ij}^{(n)}(\mathbf{k}) = \left\langle \partial_{k_i} u_{n\mathbf{k}} \middle| \bigl(1-|u_{n\mathbf{k}}\rangle\langle u_{n\mathbf{k}}|\bigr) \middle| \partial_{k_j} u_{n\mathbf{k}} \right\rangle .$$4,

$$T_{ij}^{(n)}(\mathbf{k}) = \left\langle \partial_{k_i} u_{n\mathbf{k}} \middle| \bigl(1-|u_{n\mathbf{k}}\rangle\langle u_{n\mathbf{k}}|\bigr) \middle| \partial_{k_j} u_{n\mathbf{k}} \right\rangle .$$5

up to the prefactors fixed by the oscillator-strength definition. The oscillator strength for linear polarization therefore probes the quadratic form of the interband quantum metric projected onto the polarization direction. This is the precise sense in which the quantum metric acts as the geometric weight of linearly polarized interband absorption (Li et al., 12 Jul 2025).

The same framework clarifies the contrast with conventional circular-polarization selection rules. The Berry-curvature relation quoted in the same work is

$$T_{ij}^{(n)}(\mathbf{k}) = \left\langle \partial_{k_i} u_{n\mathbf{k}} \middle| \bigl(1-|u_{n\mathbf{k}}\rangle\langle u_{n\mathbf{k}}|\bigr) \middle| \partial_{k_j} u_{n\mathbf{k}} \right\rangle .$$6

Circular dichroism therefore isolates the antisymmetric geometric sector, whereas linear-polarization anisotropy isolates the symmetric one. The correspondence is not merely analogous to Berry-curvature optics; it completes it.

3. Quantum metric-based optical selection rules

From the foregoing relations follows a new class of optical selection rules. The central claim is that symmetry-allowed and symmetry-forbidden interband transitions for linearly polarized light can be dictated by the quantum metric, not only by Berry curvature. In systems with suitable mirror or mirror-like symmetries, this yields valley-contrasted optical selection rules that lock orthogonal linear polarizations to distinct valleys (Li et al., 12 Jul 2025).

The decisive special case arises in two-band, nondegenerate settings at symmetry-invariant valleys. There the polarization contrast can become complete, with degree of linear polarization satisfying

$$T_{ij}^{(n)}(\mathbf{k}) = \left\langle \partial_{k_i} u_{n\mathbf{k}} \middle| \bigl(1-|u_{n\mathbf{k}}\rangle\langle u_{n\mathbf{k}}|\bigr) \middle| \partial_{k_j} u_{n\mathbf{k}} \right\rangle .$$7

Under this condition one linear polarization is fully allowed while the orthogonal one is forbidden. This is a strict linear-polarization analogue of familiar valley-selective circular dichroism, but its governing geometry is the metric rather than the curvature (Li et al., 12 Jul 2025).

The theory was checked in both model and materials settings. Tight-binding and first-principles calculations were reported for an altermagnet model, a Kane-Mele model, and monolayer $$T_{ij}^{(n)}(\mathbf{k}) = \left\langle \partial_{k_i} u_{n\mathbf{k}} \middle| \bigl(1-|u_{n\mathbf{k}}\rangle\langle u_{n\mathbf{k}}|\bigr) \middle| \partial_{k_j} u_{n\mathbf{k}} \right\rangle .$$8-wave altermagnet $$T_{ij}^{(n)}(\mathbf{k}) = \left\langle \partial_{k_i} u_{n\mathbf{k}} \middle| \bigl(1-|u_{n\mathbf{k}}\rangle\langle u_{n\mathbf{k}}|\bigr) \middle| \partial_{k_j} u_{n\mathbf{k}} \right\rangle .$$9. The significance of these examples is not only illustrative. They show that the correspondence survives beyond abstract two-level derivations and can organize valley-selective excitation protocols relevant to valley-based spintronic and optoelectronic applications (Li et al., 12 Jul 2025).

A common misconception is to treat linear-polarization selectivity as a trivial rotation of circular dichroism. The metric-based framework indicates otherwise. Circular and linear polarization resolve different tensor structures, and complete selectivity for orthogonal linear polarizations requires the specific symmetry and band-structure conditions stated above. The selection rule is therefore genuinely geometric and not a mere basis change in polarization space.

4. Optical spectral weight, sum rules, and step response

A broader version of the correspondence emerges when optical response is formulated through the time-dependent quantum geometric tensor. In that formulation, the metric is the equal-time symmetric sum of interband dipole matrix elements, but ordinary conductivity does not expose it directly because the same matrix elements are dressed by transition-energy factors. The conductivity is related to the antisymmetric part of the tQGT by

$$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,T_{ij}^{(n)}(\mathbf{k}),$$0

and its frequency expansion takes the form

$$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,T_{ij}^{(n)}(\mathbf{k}),$$1

Because the metric enters through $$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,T_{ij}^{(n)}(\mathbf{k}),$$2, it appears in conductivity with transition-energy prefactors. The paper’s formulation is explicit that metric matrix elements are thereby “convoluted with energy prefactors” in conventional linear response (Verma et al., 2024).

This obstruction is removed by generalized sum rules. The family

$$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,T_{ij}^{(n)}(\mathbf{k}),$$3

contains the Souza-Wilkens-Martin case $$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,T_{ij}^{(n)}(\mathbf{k}),$$4,

$$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,T_{ij}^{(n)}(\mathbf{k}),$$5

Taking the symmetric part yields the total quantum metric. In this formulation, inverse-frequency-weighted optical spectral weight is the global optical realization of the metric–oscillator strength correspondence (Verma et al., 2024).

The time-domain analogue is a step-response protocol. After switching off a static electric field, the relaxation function satisfies

$$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,T_{ij}^{(n)}(\mathbf{k}),$$6

In the classical or high-temperature limit,

$$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,T_{ij}^{(n)}(\mathbf{k}),$$7

so that

$$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,T_{ij}^{(n)}(\mathbf{k}),$$8

This shifts the correspondence from frequency-domain spectroscopy to time-domain relaxation. The same interband dipole weights that generate optical spectral moments become directly visible through constrained-equilibrium decay (Verma et al., 2024).

5. Strong-field, pump–probe, and transition-weight analogues

Outside weak-field optical absorption, several works identify closely related correspondences in which the metric governs dynamical or spectroscopic transition weights. In strong dc fields, a fully quantum multiband density-matrix theory finds quantum metric induced oscillations generated by interband coherence. The basic building block is again the symmetric product of interband Berry connections,

$$g_{ij}^{(n)}(\mathbf{k})=\mathrm{Re}\,T_{ij}^{(n)}(\mathbf{k}),$$9

which enters both steady and oscillatory currents. The Bloch-frequency component is weighted by $$\Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,T_{ij}^{(n)}(\mathbf{k}).$$0, and a higher-frequency component is weighted by $$\Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,T_{ij}^{(n)}(\mathbf{k}).$$1. Since the paper does not use the term oscillator strength explicitly, the most careful formulation is that the spectral or dynamical weight of the oscillatory current is carried by interband dipole matrix elements whose quadratic combination is the quantum metric. This suggests a metric-controlled oscillator-strength density for field-driven interband-coherence oscillations (Zeng et al., 2024).

A more direct pump–probe correspondence appears in time-resolved ARPES for degenerate filled bands. For a linearly polarized pump along direction $$\Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,T_{ij}^{(n)}(\mathbf{k}).$$2, the momentum-resolved transition probability summed over filled-to-empty channels is

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

By combining two linear-polarization geometries, the off-diagonal metric component is obtained from

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

The same quantity is observable as the pump-induced loss of occupied spectral weight in trARPES. This is an oscillator-strength correspondence in all but name: the measurable spectral depletion is a sum over allowed interband transitions weighted by squared dipole-like derivative matrix elements, and it is directly proportional to a component of the many-body valence-state quantum metric (Gersdorff et al., 2021).

These extensions broaden the subject beyond equilibrium optical absorption. They show that whenever an observable is organized by quadratic interband couplings—dipole matrix elements, Berry connections, or derivative matrix elements—the symmetric component can inherit a quantum-metric interpretation. The exact optical meaning, however, depends on the response protocol and weighting factors.

6. Scope, limitations, and adjacent correspondences

The correspondence exists in both direct and indirect forms, and distinguishing them is essential. Direct forms include linearly polarized interband oscillator strengths in solids, inverse-frequency-weighted optical spectral weight, step-response relaxation, and pump-induced transition probabilities in trARPES (Li et al., 12 Jul 2025, Verma et al., 2024, Gersdorff et al., 2021). Indirect forms arise when the metric controls overlap or form-factor weights that are structurally analogous to transition strengths but are not themselves presented as optical oscillator strengths.

A prominent indirect example occurs in disorder and localization theory for isolated bands. There the metric is defined as

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

and the small-$$\Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,T_{ij}^{(n)}(\mathbf{k}).$$6 overlap form factor obeys

$$\Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,T_{ij}^{(n)}(\mathbf{k}).$$7

In a multiband formulation, the interband form factors satisfy

$$\Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,T_{ij}^{(n)}(\mathbf{k}).$$8

This does not amount to an optical sum rule, and the paper explicitly does not discuss oscillator strength. A plausible implication is that the same geometry governing optical transition weights also governs overlap-based transition amplitudes and disorder vertices in projected-band theories (Dai et al., 5 May 2026).

An adjacent conceptual generalization appears in operator formulations of the quantum metric tensor. For adiabatic parameter changes generated by operators $$\Omega_{ij}^{(n)}(\mathbf{k})=-2\,\mathrm{Im}\,T_{ij}^{(n)}(\mathbf{k}).$$9, one has

$$\mathcal{Q}_{\mu\nu}(t) = \int_{\mathbf k}\sum_{m\neq n} f_n(1-f_m)\, \hat r_\mu^{nm}\hat r_\nu^{mn} e^{i\hbar\omega_{mn}t},$$0

with Berry connection

$$\mathcal{Q}_{\mu\nu}(t) = \int_{\mathbf k}\sum_{m\neq n} f_n(1-f_m)\, \hat r_\mu^{nm}\hat r_\nu^{mn} e^{i\hbar\omega_{mn}t},$$1

That work does not derive optical oscillator strengths. Nevertheless, the covariance structure suggests sum-over-states representations in terms of transition matrix elements of the generators, and hence a route from parameter-space geometry to transition-weight formulas (Gonzalez et al., 2018).

Several misconceptions recur in discussions of the subject. The first is the identification of quantum metric with Berry curvature; the former is the symmetric real part of quantum geometry, the latter the antisymmetric imaginary part. The second is the assumption that ordinary conductivity directly measures the metric; in the cited tQGT formulation, conductivity carries extra transition-energy prefactors and therefore does not isolate the total metric without special weighting (Verma et al., 2024). The third is the assumption that all correspondences are $$\mathcal{Q}_{\mu\nu}(t) = \int_{\mathbf k}\sum_{m\neq n} f_n(1-f_m)\, \hat r_\mu^{nm}\hat r_\nu^{mn} e^{i\hbar\omega_{mn}t},$$2-resolved. Some are local in momentum space, as in the linear-polarization selection rules and trARPES transition probabilities, but the SWM sum rule and step-response protocol principally access global integrated metric tensors. Finally, the direct optical correspondence is not fully universal across all materials and response regimes: the strongest selection-rule statements require the two-band, nondegenerate, symmetry-invariant valley setting described above (Li et al., 12 Jul 2025).

In this sense, quantum metric–oscillator strength correspondence is best understood not as one isolated identity but as a hierarchy of geometric response principles. At its core lies a simple statement: whenever a response is built from quadratic interband matrix elements, its symmetric channel can encode the quantum metric. The clearest realization is the metric control of linearly polarized interband oscillator strength; the broader literature shows that the same structure extends to optical spectral moments, time-domain relaxation, pump–probe transition probabilities, and several closely related overlap and coherence weights.