  We provide strong evidence that the effective spin-spin interaction in a
multimodal confocal optical cavity gives rise to a self-induced glassy phase,
which emerges exclusively from the peculiar euclidean correlations and is not
related to the presence of disorder as in standard spin glasses. As recently
shown, this spin-spin effective interaction is both non-local and
non-translational invariant, and randomness in the atoms positions produces a
spin glass phase. Here we consider the simplest feasible disorder-free setting
where atoms form a one-dimensional regular chain and we study the
thermodynamics of the resulting effective Ising model. We present extensive
results showing that the system has a low-temperature glassy phase. Notably,
for rational values of the only free adimensional parameter $\alpha=p/q$ of the
interaction, the number of metastable states at low temperature grows
exponentially with $q$ and the problem of finding the ground state rapidly
becomes computationally intractable, suggesting that the system develops high
energy barriers and ergodicity breaking occurs.
