Dissipative phase transitions of the Dicke-Ising model

The dissipative phase transitions in the open transverse and longitudinal Dicke-Ising model (DIM), which incorporates nearest-neighbor Ising-type spin interactions into the Dicke framework, are investigated within a mean-field approach and further validated by detailed stability analysis. While the dissipative phase diagram of the transverse DIM is only slightly shifted upward compared with its ground-state counterpart, dissipation in the longitudinal DIM stabilizes bistable nonequilibrium steady states and induces first-order phase transitions that are absent in the ground-state phase diagram. This bistable phase is characterized by the coexistence of superradiant and antiferromagnetic orders, and it converts a ground-state triple point into a tetracritical point, at which the boundaries of the first- and second-order transitions intersect. Our results reveal that the interplay among spin interactions, light-matter coupling, and dissipation supports a diverse set of nonequilibrium phase transitions and provides broad tunability of the phase diagram. These findings offer a theoretical foundation for exploring nonequilibrium physics in realistic open solid-state quantum systems.

💡 Research Summary

In this work the authors investigate nonequilibrium dissipative phase transitions in two extensions of the Dicke model that incorporate nearest‑neighbor Ising interactions, the so‑called Dicke‑Ising models (DIM). The two variants differ by the orientation of the Ising coupling relative to the light‑matter interaction: (i) the “transverse” DIM, where the Ising term is along σz while the Dicke coupling is along σx, and (ii) the “longitudinal” DIM, where both the Ising and Dicke couplings act on σx. Starting from the Hamiltonian H_DIM = H_Dicke + H_Ising, the authors add cavity photon loss at rate κ and describe the open dynamics with a Lindblad master equation.

First, the equilibrium (ground‑state) phase diagrams are obtained within a mean‑field treatment. The transverse DIM displays four possible ordered phases: a paramagnetic normal (PN) phase, an antiferromagnetic normal (AFN) phase, a paramagnetic superradiant (PS) phase, and an antiferromagnetic superradiant (AFS) phase where superradiance coexists with staggered magnetization. By contrast, the longitudinal DIM lacks the AFS phase because the short‑range antiferromagnetic interaction (favoring staggered σx) competes directly with the long‑range photon‑mediated interaction (favoring uniform σx), leaving only PN, AFN and PS phases in the ground‑state diagram.

The core of the paper is the analysis of the steady‑state phase diagrams when photon loss is present. By factorizing operator expectations (mean‑field) and neglecting quantum fluctuations, a set of semiclassical equations of motion for the two sublattice spin components and the cavity field amplitude is derived. Stationary solutions are classified into the same four (transverse) or three (longitudinal) families. Linear stability is then examined by perturbing each fixed point and constructing the Jacobian matrix M; the eigenvalue spectrum of M determines whether a solution is dynamically stable.

For the transverse DIM the stability analysis yields explicit critical lines in the (J,g) plane: a second‑order line separating PN from AFN (J = Ω/4), a second‑order line separating AFN from PS (g = g_zc2), and a second‑order line separating PS from AFS (g = g_zc1). All transitions respect the underlying Z₂ spin‑flip and Dicke parity symmetries, so the order of the transition (first vs second) follows the usual Landau classification.

The longitudinal DIM behaves qualitatively differently under dissipation. Although the equilibrium diagram contains only three phases, the open system develops a bistable region where both a superradiant (PS) steady state and an antiferromagnetic normal (AFN) steady state are simultaneously stable. The boundary between these two stable solutions is a first‑order (discontinuous) transition. Importantly, the line where the second‑order AFN–PN transition meets the first‑order PS–AFN line defines a tetracritical point: a point at which four phase‑boundary segments intersect (PN, AFN, PS, and the bistable coexistence region). This tetracritical point does not exist in the closed (ground‑state) model and is a genuine nonequilibrium feature induced by photon loss.

The authors discuss the physical origin of these findings. In the transverse case the Ising interaction is orthogonal to the Dicke coupling, allowing the staggered σz order and the uniform σx superradiant order to coexist without direct competition, which explains the presence of the AFS phase. In the longitudinal case the two interactions act on the same spin component, leading to a direct competition between short‑range antiferromagnetism and long‑range photon‑mediated ferromagnetism; this competition suppresses the AFS phase but, when dissipation is present, stabilizes a bistable coexistence of the two competing orders.

Finally, the paper emphasizes experimental relevance. The parameters J (nearest‑neighbor exchange), g (collective light‑matter coupling), Ω (atomic transition frequency), ω (cavity frequency), and κ (cavity loss) are all tunable in platforms such as quantum‑dot arrays, superconducting qubit lattices, and Rydberg‑atom ensembles coupled to microwave or optical cavities. The predicted nonequilibrium phase diagram, especially the bistable region and the tetracritical point, provides a concrete target for future experiments probing dissipative quantum many‑body physics beyond equilibrium. The authors also note that while mean‑field theory captures the qualitative structure, quantitative shifts of the phase boundaries are expected from quantum fluctuations, suggesting that more sophisticated numerical methods or experimental verification will be valuable next steps.