The Dicke model, describing the cooperative interaction between two-level atoms and a single-mode photonic field, exhibits a quantum phase transition (QPT) dependent on light-matter coupling strength. Introducing short-range atom-atom interactions complicates the model but is expected to generate new phenomena. This paper addresses the challenge of solving this extended Dicke model, or the 'g-J' model, which incorporates both atom-field (g) and atom-atom (J) interactions. Analytical solutions are unavailable, making experimental simulation crucial. Previous computational studies under various approximations have revealed first-order QPTs, shifts in the superradiant phase transition (SRPT) boundary, and modifications of entanglement. Several experimental platforms have been proposed, but successful simulations have been lacking. This research presents a new approach using a crystal of Erbium orthoferrite (ErFeO3) as a solid-state quantum simulator for the g-J model. ErFeO3's magnetic properties are governed by the Er3+ and Fe3+ spin subsystems and their interplay. Previous work demonstrated Dicke cooperativity in the Er-Fe3+ interaction, showcasing the similarity between ErFeO3's magnetic Hamiltonian and the Dicke Hamiltonian. The paramagnetic Fe3+ ions (magnons of ordered Fe3+ spins) represent the atomic ensemble (light field), and the spin-magnon interaction mirrors the g-term in the Dicke model. A magnetic phase transition in ErFeO3, resembling a Dicke SRPT, further strengthens this analogy. Below 4 K, the Er lattice develops C-type AFM order, and a zone-boundary Fe3+ magnon mode condenses, mimicking the emergence of atomic and photonic polarizations in the standard SRPT. This transition (Γ2 → Γ12) is identified as a magnonic SRPT.

The Dicke model, introduced by Dicke in 1954, describes the cooperative coherent coupling of an ensemble of two-level atoms with a single-mode light field. Subsequent work has explored the model's rich phenomenology, including the superradiant phase transition (SRPT) at a critical coupling strength. Research has extended this model to incorporate atom-atom interactions, acknowledging their importance in explaining phenomena like dephasing and intensity correlations in fluorescent spectra. This extended Dicke model, or g-J model, considers the interplay between photonic-field-mediated long-range and direct short-range interactions. Computational studies under various approximations have revealed a range of new phenomena, including first-order quantum phase transitions, shifts in the SRPT boundary, and modifications of entanglement. Experimental platforms such as atomic Bose-Einstein condensates, superconducting qubits, and quantum dots have been suggested, but successful simulations remained elusive before this study.

This study employed ErFeO3, an antiferromagnetic insulator, as a solid-state quantum simulator for the g-J model. The methodology involved several key experimental techniques: 1. **Terahertz (THz) Time-Domain Spectroscopy:** This technique was used to monitor the quasi-antiferromagnetic (qAFM) magnon mode of Fe3+ spins. THz transmission measurements on a z-cut ErFeO3 crystal in the Faraday geometry yielded absorption coefficient spectra as a function of temperature and magnetic field. The evolution of the qAFM mode in different phases provided insights into the system's behavior. The analysis involved calculating the complex index of refraction from the transmitted THz electric field data, accounting for Fresnel transmission coefficients and propagators through the sample. 2. **Magnetocaloric Effect (MCE) Measurements:** MCE experiments, sensitive to the magnetic entropy landscape, were used to identify phase boundaries by measuring the temperature change (dT/dH) as a function of magnetic field. The observed peaks in dT/dH indicated phase transitions, providing another method to map the phase diagram. 3. **Magnetization Measurements:** Iso-field and isothermal magnetization measurements (M) were performed to complement the THz and MCE data, confirming the phase transitions observed through other methods. However, sample shape non-uniformities complicated the precise identification of some transition boundaries. 4. **Mean-Field Calculations:** A theoretical model based on a microscopic Hamiltonian was developed to describe the interactions within the ErFeO3 system. Mean-field calculations, assuming uniform mean fields experienced by each spin, predicted the phase diagram and the nature of the order parameters in each phase. The theoretical framework used a second-quantized form of the spin Hamiltonian and focused on the dominant interaction terms to simplify the complex system. The self-consistent equations were solved numerically to obtain the equilibrium spin configurations and free energies, thus predicting the phase diagram.

The study's key findings include: 1. **Identification of a Novel Atomic Phase:** In addition to the superradiant (S) and normal (N) phases expected from the standard Dicke model, a new atomically ordered (A) phase was identified. This phase is characterized by Er3+ ordering but without Fe3+ order parameter onset. The existence of the A phase directly demonstrates the impact of the J-term (atom-atom interaction) in the Hamiltonian. 2. **Detailed Phase Diagram Mapping:** A comprehensive temperature-magnetic field (T-H) phase diagram was constructed through a combined analysis of THz spectroscopy, MCE, and magnetization measurements. The diagram showed the boundaries between the S, A, and N phases. 3. **Order Parameter Identification:** The order parameters for each phase were identified and their evolution was monitored across phase transitions. The Fe3+ order parameter was found to be finite in the S phase but negligible in the A phase, providing further support for the distinct nature of the A phase. The Er3+ order parameter exhibited different configurations in the S and A phases. 4. **Transition Order Determination:** The study differentiated between first-order and second-order phase transitions. The S → A transition was identified as a first-order, abrupt transition, while the A → N and N → S transitions were characterized as continuous, second-order transitions. These transitions were identified through distinct features in the experimental data, specifically the abrupt changes observed in the first-order transition compared to the continuous evolution in the second-order transitions. 5. **Experimental Confirmation of Theoretical Predictions:** The experimental phase diagram and order parameter behavior demonstrated good agreement with predictions from mean-field calculations, which further validates the theoretical framework used in this study.

This work successfully demonstrates the feasibility of using ErFeO3 to simulate the extended Dicke model, a complex problem previously intractable analytically. The identification of the atomically ordered phase highlights the significant impact of short-range atom-atom interactions on the system's behavior, providing experimental confirmation of theoretical predictions. The detailed mapping of the phase diagram, including the distinction between first- and second-order transitions, expands our understanding of the rich phenomenology of this model. The use of multiple experimental techniques (THz spectroscopy, MCE, magnetization measurements) and theoretical modeling (mean-field calculations) ensured robustness and completeness of the results. The close agreement between theoretical predictions and experimental observations strongly supports the validity of the theoretical model employed. The tunability provided by an external magnetic field offers exciting prospects for future studies of quantum criticality and thermalization.

This study successfully simulated the extended Dicke model using ErFeO3, revealing a novel atomically ordered phase alongside the expected superradiant and normal phases. The precise characterization of phase boundaries and order parameters provides valuable insight into the interplay between light-matter and atom-atom interactions. This platform offers a unique opportunity to explore unconventional quantum criticality and chaos-assisted thermalization, advancing both quantum optics and many-body physics. Future research could explore other rare-earth orthoferrites or orthochromites to expand the range of models that can be simulated.

The irregular shape of the sample used for magnetization measurements introduced a non-uniform demagnetizing effect, potentially affecting the precise identification of some phase boundaries, particularly the A → N transition. The mean-field approximation, while simplifying the complex theoretical model, may not capture all the subtle details of the system's behavior. Future work might benefit from employing more sophisticated theoretical techniques beyond the mean-field approach to address these limitations.