Quantum Metric-Based Optical Selection Rules

• Quantum metric-based optical selection rules are a framework that uses the real part of the quantum geometric tensor to determine allowed interband transitions in solids.
• They utilize linear polarization to map oscillator strengths, enabling differential valley responses even when Berry curvature effects vanish.
• This approach paves the way for valleytronic and spintronic applications by facilitating controllable, valley-specific optical excitations.

Quantum metric-based optical selection rules constitute a paradigm shift in the understanding and engineering of optical transitions in solids, elevating the real part of the quantum geometric tensor (the quantum metric) to a fundamental status equal to the well-known Berry curvature (its imaginary part). Whereas traditional selection rules have been tied exclusively to Berry curvature via circular polarization, quantum metric-based rules govern interband absorption, oscillator strength, and valley-contrasted transitions under linear polarization. This new framework enables direct mapping of quantum metric tensor elements, the establishment of valley-specific selection for orthogonal linear polarizations, and novel device functionalities in valleytronics and spintronics, particularly for materials where Berry curvature vanishes due to symmetry constraints (Li et al., 12 Jul 2025).

1. General Formalism for Interband Optical Absorption

Quantum metric-based optical selection rules are formulated starting from the interband absorption rate for linearly polarized light, as derived from Fermi’s golden rule: $$W(\omega,\hat\epsilon)=\frac{\pi e^2 E_0^2}{\hbar \Omega} \sum_{n\ne m}\int\frac{d^d k}{(2\pi)^d}|\langle u_{m\bm k}|\hat\epsilon\cdot v_{\bm k}|u_{n\bm k}\rangle|^2 \delta(\varepsilon_{m\bm k}-\varepsilon_{n\bm k}-\hbar\omega)$$ where $v^a_{\bm k}=(1/\hbar)\partial_{k_a}H(\bm k)$ is the velocity operator. The $k$-resolved oscillator strength for polarization $\hat\epsilon$ is defined as: $$f_{nm}^{(\hat\epsilon)}(\bm k)=\frac{2m_e}{\hbar\omega_{nm,\bm k}}|\hat\epsilon\cdot v_{nm}(\bm k)|^2 \quad \hbar\omega_{nm,\bm k}=\varepsilon_{m\bm k}-\varepsilon_{n\bm k}$$ This formalism organizes interband transitions according to symmetry and polarization, enabling precise determination of allowed and forbidden optical transitions (Li et al., 12 Jul 2025).

2. Quantum Geometric Tensor and Metric–Oscillator Strength Correspondence

The quantum geometric tensor in band space,

$$T^{ab}_{nm}(\bm k)=g^{ab}_{nm}(\bm k)-\frac{i}{2}\Omega^{ab}_{nm}(\bm k)$$

splits into two parts: the quantum metric $g^{ab}_{nm}$ (real, symmetric) and Berry curvature $\Omega^{ab}_{nm}$ (imaginary, antisymmetric). The key fundamental correspondences are:

• Circular polarization (Berry curvature): $$\Omega^{xy}_{nm}(\bm k)=-\frac{\hbar}{4m_e\omega_{nm,\bm k}}\bigl[f_{nm}^{(x+iy)}(\bm k)-f_{nm}^{(x-iy)}(\bm k)\bigr]$$
• Linear polarization (quantum metric): $$g^{xy}_{nm}(\bm k)=\frac{\hbar}{8m_e\omega_{nm,\bm k}}\bigl[f_{nm}^{(\hat x+\hat y)}(\bm k)-f_{nm}^{(\hat x-\hat y)}(\bm k)\bigr]$$ Isolating oscillator strengths for orthogonal linear polarizations yields a universal, analytic relationship directly linking the quantum metric to optical absorption differences. This establishes the quantum metric as an optically accessible geometric quantity, independent of Berry curvature-related effects.

3. Valley-Contrasted Selection Rules under Linear Polarization

Valley-contrasted selection arises when valleys $K$, $K'$ (points of high-symmetry in momentum space) are related by mirror-like symmetries. The quantum metric exhibits the transformation: $g^{xy}_{nm}(K)=-g^{xy}_{nm}(K')$ while

$$g^{xx}_{nm}(K)=g^{xx}_{nm}(K'),\quad g^{yy}_{nm}(K)=g^{yy}_{nm}(K')$$

Defining the degree of linear polarization for a transition $v\to c$ at valley $K$: $$\eta(K)=\frac{f^{(\hat x+\hat y)}-f^{(\hat x-\hat y)}}{f^{(\hat x+\hat y)}+f^{(\hat x-\hat y)}}=\frac{2g^{xy}_{vc}}{g^{xx}_{vc}+g^{yy}_{vc}}$$ By symmetry, $\eta(K')=-\eta(K)$. For valleys on mirror-invariant lines, $g^{xy}=\pm g^{xx}=\pm g^{yy}$, which produces $\eta(K)=\pm 1$, $\eta(K')=\mp 1$. In practice, only one linear polarization couples to each valley, with

$$\hat\epsilon_{45^\circ} = \frac{\hat x+\hat y}{\sqrt2} \longleftrightarrow K, \qquad \hat\epsilon_{-45^\circ} = \frac{\hat x-\hat y}{\sqrt2} \longleftrightarrow K'$$

enabling deterministic valley selection via linear polarization—a property unattainable from Berry curvature alone.

4. Demonstrations in Tight-Binding and First-Principles Models

Three classes of quantum metric-based selection rules are confirmed:

Model	Valley Points	Selection Rule Realization
Altermagnet (AM)	$X$, $Y$	$\eta(X)=+1$, $\eta(Y)=-1$
Kane–Mele (KM)	$M$, $M'$	$\eta(M)=-\eta(M')$
monolayer V₂SeSO	$X$, $Y$	$\eta(X)=+1$, $\eta(Y)=-1$

• Altermagnet (AM) model: Tight-binding calculations with mirror symmetries $M_{x}$, $M_{y}$ show analytical and numerical valley-selective absorption: perfect $\pm 1$ degree of polarization at invariant lines.
• Kane–Mele model: Both analytic $k\cdot p$ and ab initio approaches demonstrate metric-based valley selection at $M$, $M'$, with $\eta\to\pm 1$ when spin-orbit coupling vanishes.
• Monolayer V₂SeSO: DFT reveals two spin-split valleys ($X$, $Y$), each favoring a distinct linear polarization, producing spin-polarized photocurrents observable in absorption microscopy and photoemission spectroscopy.

Figures in (Li et al., 12 Jul 2025) illustrate the polarization locking and valley selectivity in each material, confirming theoretical predictions.

5. Implications for Valleytronic and Spintronic Device Engineering

Quantum metric-based optical selection rules unlock several functionalities:

• Direct mapping of $g^{xy}(\bm k)$: Polarization-differential absorption yields experimental access to quantum metric tensor components, complementing nonlinear optics and shift current protocols.
• Valley control with linear beams: In materials with weak or null Berry curvature (e.g., time-reversal-broken, inversion-symmetric magnets), linear polarization at $\pm45^\circ$ can select electronic valleys, expanding device design space.
• Exciton and condensate engineering: Quantum metric enhancement of exciton binding and superfluid stiffness enables selective valley pumping to stabilize anisotropic condensates.
• Material platforms: Principally 2D transition-metal Janus altermagnets, twisted magnets, conventional 2D magnets under symmetry breaking, and engineered heterostructures exploiting valley index and mirror symmetry.
• Experimental signatures: Valley-split peaks in polarization-resolved absorption/reflectivity, valley-selective populations in time- and angle-resolved photoemission under linear pump, and nonlinear optical harmonics modulated by quantum metric-based selection.

A plausible implication is that these metric-driven rules provide a robust pathway for valleytronic and spintronic applications, with polarization as a direct control knob for valley degrees of freedom, even in systems where Berry curvature-based selection fails.

6. Contextual Significance and Theoretical Advancement

By assigning the quantum metric an equal role to Berry curvature in optical selection rules, a complete geometric theory of interband transitions emerges. This framework clarifies polarization-dependent valley selection, allows explicit control over valley quantum states by external linear fields, and bridges the gap between quantum geometry and device-level optoelectronic engineering. The approach also suggests fundamentally new optical measurement strategies and experimental designs for exploring quantum geometry in crystals and engineered materials.