We present quantum phase diagrams for the antiferromagnetic long-range Ising model with a linear coupling to a single bosonic mode on the square and triangular lattice. For zero coupling, the ground-state magnetization forms a devil's staircase structure of magnetization plateaux as a function of a longitudinal field. Apart from a paramagnetic superradiant phase with a finite photon density at strong light-matter couplings, the long-range interactions lead to a plethora of intermediate phases that break the translational symmetry and have a finite photon density at the same time. We apply an adaption of the unit-cell-based mean-field calculations, which captures all possible magnetic unit cells up to a chosen extent. Further, we exploit an exact mapping of the non-superradiant phases to an effective Dicke model to calculate upper bounds for phase transitions towards superradiant phases. Finally, to treat quantum fluctuations in a quantitative fashion, we employ a generalized wormhole quantum Monte Carlo algorithm. We discuss how these three methods are used in a cooperative fashion. In the calculated phase diagrams we see several features arising from the long-range interactions: The devil's staircases of distinct magnetically ordered normal phases and non-trivial magnetically ordered superradiant phases beyond the findings for nearest-neighbor interactions. Examples are a superradiant phase with a three-sublattice magnetic order on the square lattice and the superradiant Wigner crystal with four sites per unit cell on the triangular lattice. We find the transition between normal and superradiant phases with the same (different) magnetic order to be of second order with Dicke universality (first order). Further, between superradiant phases we find first-order phase transitions, besides specially highlighted regimes for which we find indications for second-order behavior.
In the thermodynamic limit, the Dicke-Ising chain maps exactly onto an effective self-consistent matter Hamiltonian with the photon field acting solely as a self-consistent effective field. As a consequence, no quantum correlations between photons and spins are needed to understand the quantum phase diagram. This enables us to determine the quantum phase diagram in the thermodynamic limit using numerical linked-cluster expansions combined with density matrix renormalization group calculations (NLCE+DMRG) to solve the resulting self-consistent matter Hamiltonian. This includes magnetically ordered phases with significantly improved accuracy compared to previous estimates. For ferromagnetic Ising couplings, we refine the location of the multicritical point governing the change in the order of the superradiant phase transition, reaching a relative accuracy of
10−4. For antiferromagnetic Ising couplings, we confirm the existence of the narrow antiferromagnetic superradiant phase in the thermodynamic limit. The effective matter Hamiltonian framework identifies the antiferromagnetic superradiant phase as the many-body ground state of an antiferromagnetic transverse-field Ising model with longitudinal field. This phase emerges through continuous Dicke-type polariton condensation from the antiferromagnetic normal phase, followed by a first-order transition to the paramagnetic superradiant phase. Thus, NLCE+DMRG provides a precise determination of the Dicke-Ising phase diagram in one dimension by solving the self-consistent effective matter Hamiltonian.
We describe the quantum phase transitions in the ferromagnetic Dicke-Ising model using a Landau theory approach. The theory quantitatively captures the change from a second- to a first-order transition between the normal and superradiant phases through a tricritical point. We identify virtual nearest-neighbor double spin-flip processes as the crucial mechanism responsible for this behavior. The tricritical point constitutes a quantum phase transition above the upper critical dimension. We discuss the modifications to finite-size scaling required for the correct interpretation of numerical data at the tricritical point. Our results emphasize the need for adapted finite-size scaling forms in all-to-all interacting quantum systems and establish the ferromagnetic Dicke-Ising model as a paradigmatic platform for quantum phase transitions above the upper critical dimension, encompassing both standard
φ4 criticality and beyond.
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.
We investigate a generalized Dicke model by introducing two interacting spin ensembles coupled with a single-mode bosonic field. Apart from the normal to superradiant phase transition induced by the strong spin-boson coupling, interactions between the two spin ensembles enrich the phase diagram by introducing ferromagnetic, antiferromagnetic and paramagnetic phases. The mean-field approach reveals a phase diagram comprising three phases: paramagnetic-normal phase, ferromagnetic-superradiant phase, and antiferromagnetic-normal phase. Ferromagnetic spin-spin interaction can significantly reduce the required spin-boson coupling strength to observe the superradiant phase, where the macroscopic excitation of the bosonic field occurs. Conversely, antiferromagnetic spin-spin interaction can strongly suppress the superradiant phase. To examine higher-order quantum effects beyond the mean-field contribution, we utilize the Holstein-Primakoff transformation, which converts the generalized Dicke model into three coupled harmonic oscillators in the thermodynamic limit. Near the critical point, we observe the close of the energy gap between the ground and the first excited states, the divergence of entanglement entropy and quantum fluctuation in certain quadrature. These observations further confirm the quantum phase transition and offer additional insights into critical behaviors.
We give a thermodynamic-limit account of Dicke models with one cavity mode coupled collectively to interacting matter. Integrating out the cavity yields an exact self-consistent functional of the magnetisation
m,
e~(m)=λm2/2+emat(λm): a classical penalty on the bare-matter energy
emat in the self-consistent field
h=λm, with
λ=g2/(2ωc) the collective coupling. Supplying only that scalar field, the photon creates no phase the matter does not already possess. States holding a minimum form one connected curve,
λ(m)=μmat−1(m)/m, so superradiant first-order transitions are folds of one equation of state not crossings of disjoint sheets, and a fold can straighten into a continuous line. The remaining rules are local, each with a spectral counterpart: onset by the leading singularity of
emat (a softening polariton), order by one bare response -- the Landau quartic, or a divergent susceptibility forcing a Larkin-Pikin (LP) fold. For the Dicke-Ising model the Landau coefficients are exact, giving in closed form the second-order boundary and both zero-quartic fields, one tricritical; a
1/d expansion maps all four phases, with the AS-PS transition first order for
d≤duc=3=4−z (LP) and tricritical points in the
(d,ε) plane above. At the degenerate quadruple point the matter is a Rydberg-blockade chain, solved by strict-blockade iDMRG: the antiferromagnetic superradiant (AS) phase persists as a finite 1D wedge, first order into the corner. Other magnets: the triangular antiferromagnet keeps a continuous superradiant-superradiant line (3D-XY, no fold forced); the compass chain a BKT-functional onset; the Heisenberg and XX chains, via a conserved operator, a spectrally silent first-order onset; and the Dicke-Heisenberg diagram an exact tricritical point at the saturation corner.