---
title: Quantum Metric-Based Optical Selection Rules
url: https://www.emergentmind.com/topics/quantum-metric-based-optical-selection-rules
type: topic
---

# Quantum Metric-Based Optical Selection Rules

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 [2507.09260].

## 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 [2507.09260].

## 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 [2507.09260] 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.

Source: https://www.emergentmind.com/topics/quantum-metric-based-optical-selection-rules