Harmonic band theory: rigidity of non-zero degree harmonic maps from 2-torus to complex projective space

TL;DR

This work establishes a rigidity principle for isotropic harmonic maps from an elliptic curve to complex projective space, showing that such maps derived from holomorphic embeddings associated to complete linear systems are unique up to unitary equivalence. By leveraging Cukierman’s framework and Calabi rigidity, the authors prove that if two harmonic maps at a fixed level are isometric, they must arise from the same holomorphic data and are related by a unitary transformation, with the same extension across associated maps. The results generalize to embeddings without special hyperosculation points under a pullback condition on the Fubini–Study form, connecting rigidity of harmonic bands to stability of generalized Landau levels in condensed matter. Overall, the paper provides a rigorous, elliptic-curve–based analogue of Calabi rigidity for higher Landau levels and strengthens the theoretical foundation for the quantum-geometry description of harmonic bands. This has potential implications for classifying and stabilizing exotic phases in flat-band systems.

Abstract

We prove the rigidity of isotropic harmonic maps from a 2-torus to a complex projective space, when they are constructed from holomorphic embeddings associated to complete linear systems. We also prove that this rigidity holds for any holomorphic embeddings without special hyperosculation points, with an extra assumption on the pullbacks of Fubini--Study symplectic forms. These results ensure the rigidity of towers of harmonic bands in condensed matter physics.

Theorems & Definitions (12)

• Theorem 1
• Remark 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Remark 2
• proof : Proof of Theorem \ref{['thmrgd']}