Dimensionality crossover for moir\'e excitons in twisted bilayers of anisotropic two-dimensional semiconductors

We study the energies and optical spectra of excitons in twisted bilayers of anisotropic van der Waals semiconductors exhibiting moir\'e patterns, taking phosphorene as a case study. Following the electronic Hamiltonian introduced by us in [Phys. Rev. B 105, 235421 (2022)], and leveraging the scale separation between the moir\'e lengthscale and the exciton Bohr radii, we introduce a continuous model for excitons that incorporates the spatial variation of their binding energies. Our zone-folding calculations reveal a dimensionality crossover for the exciton states, driven by the combined dispersion- and moir\'e potential anisotropies, from quantum-dot-like (0D) lattices at twist angles $\theta<\theta_*$, to quantum-wire-like (1D) arrays at $\theta>\theta_*$, with crossover angle $\theta_*=4^\circ$. We identify clear signatures of this dimensionality crossover in the twist angle dependence of the excitonic absorption spectra, which allows experimental verification of our theoretical results through standard optical measurements. Our results establish two-dimensional anisotropic moir\'e semiconductors as versatile solid-state platforms for exploring bosonic correlations across different dimensionalities.