Fates of Exact Finite-$N$ First-Order Transitions: Coherent Suppression versus Spectral-Collapse Termination

Abstract

Symmetry-protected ground-state crossings can produce exact first-order transitions (FOTs) even at finite $N$, in contrast to continuous transitions governed by soft-mode criticality. We establish exact Juddian ground-state crossings for arbitrary finite $N$ in the one- and two-photon anisotropic Dicke models and identify a common two-channel structure: the eigenvalues of the coupling matrix set the local normal-state instability thresholds, whereas its determinant fixes the exact crossing. Despite this common structure, the resulting FOTs have distinct fates determined by the bosonic response. In the one-photon model, the overlap between coherent Juddian branches vanishes in the thermodynamic or infinite frequency-ratio limits, asymptotically suppressing the FOT cusp. In the two-photon model, squeezed Juddian states lose normalizability at spectral collapse, thereby terminating the physical FOT line at finite $N$. Although the cusp strength tends to zero as the endpoint is approached, the terminating FOT retains a collective spectral signature: the different-parity gap is predicted to close with an $N$-dependent exponent $\mu_N=(2N+3)/4$, one greater than the cusp termination exponent. By contrast, the same-parity soft mode retains $N$-independent square-root scaling and governs the critical response. Thus spectral collapse can terminate an exact finite-$N$ FOT while preserving its collective imprint in the parity-resolved spectrum.