Resonances create energy gaps in quasicrystals: a long-sought analytical breakthrough

New analytical work by Cavendish Laboratory researchers shows how true energy gaps can arise in quasicrystals, answering a longstanding open question about these extraordinary materials.  

Quasicrystals sit between ordinary crystals and fully disordered materials: their atoms form ordered patterns that never exactly repeat. Once thought impossible, they have since been found in meteorites and even in material created during the first atomic bomb test. Yet making analytical predictions for quasicrystals has long been a major challenge, because the standard band theory used for ordinary crystals no longer applies. It is therefore not obvious that quasicrystals should even support true energy gaps: such gaps are a basic feature of periodic crystals but are generally absent in fully disordered materials.

In a new theoretical study, recently published with an Editors’ suggestion in Physical Review B, the researchers have shown that an eightfold optical quasicrystal, of the kind that is currently studied experimentally by trapping ultracold atoms at the centre of four crossed laser beams, does support true energy gaps. 

“These gaps matter because they help determine how particles move through a material and whether it behaves as a conductor, an insulator, or a more exotic quantum phase,” said Dr Emmanuel Gottlob, who led this research as a PhD Student at the Cavendish Laboratory.  

The team used a configuration-space approach that works directly in the infinite-size limit. They traced these gaps to resonances between neighbouring sites and predicted how many states lie below the main gaps from simple geometric regions in configuration space, including irrational values linked to the silver ratio.  

“Our large-scale numerical simulations agree closely with these predictions,” said Gottlob. “It confirms that we can now use our new analytical tools to make exact closed-form predictions about the energy spectrum of quasicrystals, that are valid in the infinite-size limit.” 

“These tools finally allow us to make definitive statements that are true beyond finite-size numerical calculations,” said Prof Ulrich Schneider, head of the Many-body Quantum Dynamics group at the Cavendish Laboratory. “This is an essential step in bringing our understanding of quasicrystals on par with their periodic counterparts, and provides a stronger foundation for future experiments on optical quasicrystals.” 

Reference: 

Emmanuel Gottlob, David Gröters and Ulrich Schneider, ‘Origin of energy gaps in quasicrystalline potentials’, Phys. Rev. B 113, 16 April 2026. DOI: 10.1103/tpqy-pdmz

Contributors to this research:

• PhD Students (former and current): Chelsea Ou, Michael Wu,  Jr-Chiun Yu, Shaurya Bhave, Georgia Nixon
• Postdoctoral Researchers (former and current): Dr Bo Song, Dr Leanne Reeve
• Group Lead: Prof Ulrich Schneider

Image:

Repeated 8-fold optical quasicrystal. Credit: Emmanuel Gottlob