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Brillouin Platycosms and Topological Phases
Chen Zhang1, Peiyuan Wang1, Junkun Lyu1
1The University of Hong Kong, Department of Physics and HK Institute of Quantum Science and Technology, Pokfulam Road, Hong Kong, China.
Physical Review Letters
|October 12, 2025
Summary
Ten unique 3D manifolds called platycosms are shown to model universes for Bloch particles. This research classifies topological insulators and identifies invariants, diversifying models for quantum mechanics.
Area of Science:
- * Condensed matter physics
- * Mathematical physics
- * Topology
Background:
- * Ten distinct closed flat 3D manifolds, known as "platycosms," are significant in mathematics.
- * These platycosms have been proposed as geometric models for the Universe.
- * Brillouin torus is a key concept in understanding momentum-space units.
Purpose of the Study:
- * To demonstrate the manifestation of platycosms as universes for Bloch particles.
- * To provide K-theoretical classifications of topological insulators over platycosms.
- * To formulate a complete set of topological invariants for identifying these systems.
Main Methods:
- * Utilizing the Atiyah-Hirzebruch spectral sequence for K-theoretical classifications.
- * Extending the concept of the Brillouin torus to ten Brillouin platycosms.
- * Applying projective crystallographic symmetries within the broader framework.
Main Results:
- * Platycosms are identified as momentum-space units (Brillouin platycosms) for Bloch particles.
- * Exact K-theoretical classifications and topological invariants for insulators on platycosms are established.
- * Topological phase transitions are characterized by Weyl semimetals, with chirality numbers depending on platycosm orientability.
Conclusions:
- * The study generalizes the Brillouin torus to ten Brillouin platycosms.
- * This work diversifies the theoretical stages for Bloch wave function behavior.
- * Findings offer new geometric models for the Universe and topological insulator classification.
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