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Clifford Algebras and Liquid Crystalline Fermions

N. Johnson1, L. C. Head2, O. D. Lavrentovich3,4, A. N. Morozov1, G. Negro1, E. Orlandini5, C. A. Smith1, G. M. Vasil6, and D. Marenduzzo1

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Phys. Rev. X 15, 041052 – Published 18 December, 2025

DOI: https://doi.org/10.1103/fnx3-htyx

Abstract

We show that Clifford algebras provide a natural language to describe the physics of liquid crystal defects in 3D. This framework shows that most of these defects have fermionic nature, as the director field profile on a 2D cross section can algebraically be represented by a spinor. Defects in uniaxial, biaxial nematics, and cholesterics are represented by elements belonging to different Clifford algebras, suggesting that there are fundamental distinctions between topological defects in each of these phases. Our theory allows nematic defects to be interpreted as Majorana-like spinors, as defects and antidefects are topologically equivalent, while some cholesteric defects, such as screw dislocations, are better viewed as Weyl-like spinors of well-defined chirality. Defects can be described by a “defect bivector,” an algebraic element that quantifies the rototranslation associated with them. In cholesterics, fermionic defects of different types can combine to yield composite quasiparticles with either fermionic or bosonic nature. Under cylindrical confinement, these quasiparticles provide the way to understand the structure of screw dislocations. In the bulk, they may condensate to form topological phases, such as blue phases or skyrmion lattices. Our results provide a surprising link between liquid crystals, particle physics, and topological quantum matter.

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Popular Summary

Liquid crystals—found in everyday display technologies—are remarkable materials made of elongated molecules that flow like a liquid yet tend to align like a crystal. Within them lie tiny defects, only nanometers across, where this alignment breaks down, forming complex swirling patterns similar to enlarged versions of the whorls in fingerprints. In this study, we uncover an unexpected connection: The same mathematical ideas that describe electrons and neutrinos in particle physics can also explain the behavior of these liquid crystal defects.

To reveal this link, we develop a theoretical framework based on Clifford algebras to describe defects in a type of liquid crystal called a cholesteric, composed of chiral molecules. This framework resolves a long-standing problem dating back to the 1970s about how to represent these defects consistently. Our analysis shows that cholesteric defects share key features with exotic quantum particles known as Majorana and Weyl fermions. Even more surprisingly, we find that different kinds of defects can merge to form composite structures that behave like either fermions or bosons—offering a classical analog to particle interactions seen in quantum systems.

This discovery reveals a profound link between the physics of soft materials and the mathematics of quantum matter. It shows that patterns and motions in everyday materials can mimic phenomena once thought exclusive to the quantum world. Beyond explaining puzzling patterns in liquid crystals, our approach suggests new directions for materials research and may even inspire ideas for topological computing, where information is processed in ways protected from errors by underlying geometric principles.

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Phys. Rev. X 15, 041052– Published 18 December, 2025
  • Received 24 March 2025
  • Accepted 24 September 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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