We propose physically reasonable systems capable of avoiding ergodicity at infinite time in the thermodynamic limit, even with generic perturbations and when coupled to a heat bath. In two dimensions, the rainbow loop soup has (stretched) exponentially numerous absolutely stable nonergodic states with diverging energy but vanishing energy density. In three dimensions the rainbow membrane soup has (stretched) exponentially numerous nonergodic states with diverging energy barriers, leading to infinite-time robust ergodicity breaking that even survives coupling to a nonzero temperature heat bath. We describe our results in the language of exact emergent symmetries and demonstrate how the systems avoid common instabilities. Our construction naturally connects to quantum dimer models, topologically ordered systems, the group word construction, and Hamiltonians whose low-energy eigenstates exhibit anomalous entanglement entropy.

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• Exotic phases of matter
• Ergodic theory
• Kinetically constrained models
• Thermal & statistical models