Freestanding van der Waals crystals made of single-atom carbon chains (carbynes) have been recently realized technologically. Here we investigate their electronic and optical properties theoretically and compare the results with previously reported continuous-wave and time-resolved photoluminescence data obtained by our group. Employing a fully three-dimensional tight-binding formalism benchmarked against density functional theory calculations we predict the semimetal-semiconductor transition to occur in van der Waals carbyne crystals composed by the chains of about 42 atoms long. The semiconductor phase is characterized by a hyperbolic Van Hove singularity which gives rise to unconventional hyperbolic exciton states. The previously reported experimental data probe the semiconductor phase, where resonant features associated with hyperbolic excitons are clearly visible. The exciton oscillator strength is found to be strongly sensitive to the length of carbon chains: Our previously reported experimental data show that it decreases with increasing chain length. This tendency, confirmed by the theoretical modeling, manifests the evolution of the hyperbolic exciton state on the way to the semiconductor-semimetal crossover. Our approach accounts for the actual crystalline geometry, including alternating intrachain hoppings and interchain couplings. By fitting tight-binding dispersions to density functional theory data we extract consistent parameters and establish a comprehensive framework for the physics of carbyne crystals. This study paves the way towards efficient band-gap engineering in ultimate one-dimensional carbon crystals.

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• Dynamical phase transitions
• Electronic structure of atoms & molecules
• Excitons
• Phase separation
• Quantum phase transitions
• Quantum fluids & solids
• Superconductors
• van der Waals systems
• Density functional calculations
• Tight-binding model