Coexistence of long- and quasi-long range spatial order in 1D quantum quasicrystals

In standard one-dimensional systems, continuous symmetry breaking typically leads to gapless Goldstone modes (phonons or phasons) that destroy true long-range order through thermal or quantum fluctuations. This paper investigates whether it is possible to engineer a quantum quasicrystal (QC) where some Goldstone modes are gapped while others remain gapless, thereby allowing for the coexistence of different types of spatial order within the same system.

The researchers study a Bose-Einstein condensate (BEC) coupled to an optical cavity and a structured scanning laser. The cavity provides a periodic potential that breaks translational invariance, while the scanning laser introduces a second, incommensurate wave vector, creating the competition necessary for quasicrystalline order. Using a spectral variational method and a coherent-state path-integral formalism, the authors derive the hydrodynamic action of the system to analyze its low-energy excitations and the stability of its spatial order.

The study reveals that the cavity's standing-wave potential selectively gaps out the Goldstone mode associated with the cavity's wave vector, while the mode associated with the second, incommensurate wave vector remains gapless. This selective gapping results in a hybrid state: the Fourier component locked to the cavity displays true long-range positional order, whereas the component associated with the gapless mode exhibits only quasi-long-range (algebraic) order. This represents a novel phase of matter that combines the properties of pinned density waves with those of fluctuating one-dimensional quantum systems.

This work provides a theoretical framework for realizing and controlling distinct types of spatial order in quantum quasicrystals. By demonstrating that one can manipulate the infrared behavior of specific Fourier components through external potentials, the authors open new avenues for studying the interplay between symmetry breaking, fluctuations, and topology in synthetic quantum matter.