Quantum Criticality Beyond Thermodynamic Stability

Rigorous study of correlation functions and criticality in the ground state (GS) of quadratic bosonic Hamiltonians (QBH) has been facilitated by the fact that such a state is zero-mean Gaussian, thus fully described by its covariance matrix. While the standard notion of criticality is given for thermodynamically stable Hamiltonians, physically relevant (mean-field) QBHs often lack a GS – notably, in the context of photonic, antimagnonic, or cavity-QED systems. We show that a generalized notion of criticality can be meaningfully given for such QBHs, by computing correlation functions in the quasiparticle vacuum, which is Gaussian and coincides with the GS in the thermodynamically stable regime. Crucially, we show this state is unique when a quantity we term the Krein gap, which replaces the usual many-body gap, is positive. In particular, we prove that, for finite-range couplings in one dimension, correlations are exponentially bounded except when the Krein gap closes, at exceptional points and Krein collisions, where long-range correlations can ensue. This points to deep connections between the dynamical stability boundaries and signatures of criticality (or multi-criticality) for this class of systems. Implications for entanglement entropy scaling are discussed.