Unilateral Criticality and Phase Transition in the Cavity-Ising Model

Superradiant phase transitions from cavity light-matter coupling have been widely explored across platforms. Here, we report a unilateral critical endpoint (UCEP) and a tricritical point (TCP) in the phase diagram of the cavity-coupled transverse Ising model with $\mathbb{Z}_2$ symmetry. At zero temperature, we demonstrate that this model hosts three phases separated by two second-order and one first-order transitions. These lines intersect at a TCP and a UCEP, the latter not captured by existing phase-transition paradigms. The UCEP displays one-sided criticality: approaching the point from one side, the system behaves as a second-order transition, while from the other side it is first-order. Correspondingly, two order parameters, respectively, undergo the first- and the second-order phase transitions at the same point. We construct a minimal description of UCEP with the density of the free energy $f = c_{1}(\tilde{\alpha}^{2}+c_{2})+(\tilde{\alpha}^{2}+c_{2})^{2}\ln{\vert\tilde{\alpha}^{2}+c_{2}\vert}$, with the UCEP at $(c_{1},c_{2})=(1/e,0)$ and $\tilde{\alpha}$ being the order parameter. We further map the finite-temperature phase diagram and perform a symmetry analysis. By unifying first- and second-order signatures in a single, direction-dependent endpoint, the UCEP introduces a qualitatively new class of phase transition and may have applications in fields such as quantum measurement and quantum sensing. This work also provides an intriguing platform for exploring novel critical phenomena in cavity-coupled many-body systems with or without dissipation.