We study one-dimensional quantum gases in continuous space with cavity-mediated infinite-range interactions using the Variational and the Diffusion Monte Carlo methods. Starting from the exact two-body solution, we construct a nontranslationally invariant Jastrow wave function that accurately captures the spatial structure induced by the cavity field and provides an efficient many-body ansatz for both bosonic and fermionic systems. We analyze properties of three characteristic quantum systems, subject to long-range interactions: (i) ideal Bose gas, (ii) interacting Bose gas, (iii) ideal Fermi gas. In the absence of short-range interactions, we identify a crossover from a stable, weakly modulated phase realized at finite particle number for repulsive interactions to a self-organized state for attractive interactions, marked by clustering, loss of superfluidity, and, at fixed interaction strength, the absence of a thermodynamic limit. Introducing short-range repulsion, either through contact interactions or fermionic statistics, leads to the formation of a mesoscopic gas-like regime that disappears in the thermodynamic limit. A qualitative phase diagram is proposed to illustrate the combined effects of short- and long-range interactions, highlighting the emergence of distinct regimes with characteristic structural properties. Applying the Kac prescription, we obtain a well-defined thermodynamic limit and show that the periodic modulation found for a finite number of particles in the repulsive regime is a finite-size artifact, while the self-organized state survives with its onset shifted to substantially stronger coupling.

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• Bose gases
• Fermi gases
• Long-range interactions