Does an equilibrium superradiant phase transition exist in cavity QED?
In plain words
Simple models predict that enough atoms in a cavity will spontaneously fill it with light in their lowest energy state. General theorems forbid this, yet loopholes have been proposed.
Precise statement
$N$ atoms or electrons coupled to a cavity field with full minimal coupling, including the $A^{2}$ term, in thermal equilibrium. Determine whether the ground state can have $\langle a\rangle/\sqrt{N} \ne 0$ in the thermodynamic limit, given no-go theorems for uniform fields (Rzazewski et al. 1975; Andolina et al. 2019). Answer: yes for a specific realizable setting, or a general no-go proof covering spatially varying and magnetic couplings.
What would settle it
A proof covering all gauge-invariant multimode couplings, or a model and experiment showing an equilibrium photon condensate.
Status in the literature
Unverified note: it names recent work and was partly written without a live literature search; it awaits a check by an editor.
No-go theorems hold for uniform fields; photon condensation is permitted for spatially varying fields as a magnetostatic instability (Andolina et al., PRB 2020); an equilibrium magnonic analog was reported in $\mathrm{ErFeO_3}$ (Kim et al., Science Advances 2025), while a photonic equilibrium superradiant phase is unobserved.
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