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Quantum Geometric Tensor Example

Demonstrates the Quantum Geometric Tensor (QGT) and its connection to observable transport properties in condensed matter systems like Twisted Bilayer Graphene (TBG).

How to Run

cargo run -p physics_examples --example quantum_geometric_tensor

Physics Overview

The Quantum Geometric Tensor (QGT) is a fundamental object in quantum physics that encapsulates both the geometry and topology of quantum states in parameter space.

Mathematical Definition

$$Q_{ij}^n(\mathbf{k}) = \sum_{m \neq n} \frac{\langle n | v_i | m \rangle \langle m | v_j | n \rangle}{(E_n - E_m)^2}$$

where $v_i = \partial_{k_i} H$ is the velocity operator.

QGT Decomposition

The QGT naturally splits into two physically distinct parts:

Component Definition Physical Meaning
Quantum Metric $g_{ij} = \text{Re}(Q_{ij})$ Distance between quantum states in k-space
Berry Curvature $\Omega_{ij} = -2 \cdot \text{Im}(Q_{ij})$ "Magnetic field" in momentum space

Key Concepts

1. Quantum Metric (Real Part)

The quantum metric measures how much quantum states change as you move through the Brillouin zone. It sets a geometric lower bound on transport:

$$D \geq g_{ii} \cdot E_{gap}$$

Even in perfectly flat bands where conventional transport vanishes, the quantum metric ensures non-zero conductivity.

2. Berry Curvature (Imaginary Part)

The Berry curvature acts like a magnetic field in momentum space, causing:

  • Anomalous Hall Effect: Transverse current without external B-field
  • Orbital Magnetization: Intrinsic magnetic moment of Bloch electrons
  • Topological Invariants: Chern number = $\frac{1}{2\pi} \int \Omega , d^2k$

3. Effective Band Drude Weight

For flat-band systems like magic-angle TBG:

$$D = D_{conv} + D_{geom} = \text{(curvature)} + g \cdot E_{gap}$$

When $D_{conv} \approx 0$ (flat band), the geometric term provides metallic behavior!

Application: Twisted Bilayer Graphene

At the magic angle (~1.1°), TBG exhibits:

  • Nearly flat electronic bands
  • Strong electron correlations
  • Unconventional superconductivity
  • QGT-dominated transport

The Quasi-QGT connects directly to experimental observables:

  • Real part → Band Drude Weight (optical conductivity)
  • Imaginary part → Orbital Angular Momentum (ARPES, dichroism)

APIs Demonstrated

API Purpose
quantum_geometric_tensor Computes QGT component $Q_{ij}^n$
effective_band_drude_weight Transport weight including geometric contribution
QuantumEigenvector Bloch state wavefunctions
QuantumVelocity Velocity operator matrix elements
QuantumMetric Real part of QGT

References

  • Kang et al., arXiv:2412.17809 - Experimental probe of Quasi-QGT
  • Provost & Vallee (1980) - Original QGT formulation
  • Xie et al., Nature (2021) - Spectroscopic signatures of QGT in TBG