What is it about?
Electrons in a crystal form energy bands, that, for some exotic materials, are also attached to topological labels: these are characteristics that are resistant to defects and disorder in the solid. Knowing these topological labels is useful for predicting materials properties. Here a formula is given for one such labels (the Berry phase).
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Why is it important?
To avoid lengthy calculations of the electronic band structure, formulas have been proposed in the past for the topological invariants, typically these are protected by symmetries of the crystal, for example the centre of inversion. Here it is shown that also when there is no inversion centre in the crystal, it is possible to define its action as an element of the modular group: the hidden symmetry is brought to light and it is related to the Berry phase, a topological invariant.
Perspectives
The reason why this symmetry has gone unnoticed for so long is that only by using a representation of the electron's wavefunction in terms of Riemann theta functions it is possible to define and appreciate the impact of such hidden symmetries. The systematic study of the modular group for different material classes can thus help establish new topological labels and discover new materials with interesting properties.
Emanuele Maggio
Scuola Superiore Meridionale
Read the Original
This page is a summary of: Berry phase of bloch states through modular symmetries, Scientific Reports, May 2026, Springer Science + Business Media,
DOI: 10.1038/s41598-026-53130-1.
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