Two-dimensional (2D) magnetic van der Waals (vdW) materials, with thicknesses down to a monolayer, offer unique opportunities to study exotic phenomena such as topological phase transitions, magnetic skyrmions, and quantum spin liquids due to the interplay of strong anisotropy, fluctuations, and spin-orbit effects. Their properties are highly tunable via techniques like strain, nanostructuring, electric fields, and moiré twisting. Optical engineering, specifically using optical cavities, provides an alternative route to functionalize quantum materials, particularly advantageous as it avoids excessive heating associated with direct laser driving. Cavity quantum electrodynamics (c-QED) enhances light-matter coupling through mode volume compression, enabling equilibrium state modifications. While c-QED has been successfully applied to polaritonic control of electronic and phonon-mediated phase transitions, applications to magnetic systems have been limited, often requiring external drives or focusing on excited-state properties. This work extends c-QED engineering to the magnetic regime using α-RuCl₃ as a model system, demonstrating control of its magnetic ground state via cavity quantum fluctuations. The study explores how different cavity parameters can transition the equilibrium zigzag antiferromagnetic order into other magnetic phases within the extended Kitaev model, showcasing the possibility of achieving a ferromagnetic state through vacuum fluctuations and a Kitaev quantum spin liquid state through controlled cavity pumping.

The introduction extensively reviews the literature on 2D magnetism in vdW materials, highlighting the potential for exotic phenomena and the various techniques used to tune their properties. It emphasizes the recent progress in optical engineering and c-QED, contrasting the challenges and advantages of laser driving versus cavity-based methods. The authors discuss the limitations of previous studies focusing on electronic and phonon systems and the scarcity of experiments demonstrating polaritonic control of magnetic materials. This review sets the stage for the presented research by establishing the need for and potential of cavity-based control of magnetic ground states in suitable materials.

The study uses α-RuCl₃, a material known for its proximity to a quantum spin liquid, as the target system. A low-energy model, based on the extended Kitaev Hamiltonian, is employed to describe its magnetic behavior. The model incorporates interactions between the material's spins and a single effective cavity mode with circular polarization. First-principles calculations using the OCTOPUS and Wannier90 codes are performed to determine the electronic and spin parameters of α-RuCl₃, including Hubbard-Kanamori and hopping Hamiltonian parameters (Table 1) and equilibrium spin parameters (Table 2). The coupled spin-photon system is then analyzed via exact diagonalization on a 24-site spin cluster (Fig. 2a), allowing the calculation of the photo ground state (PGS) and its magnetic phase diagram as a function of light-matter coupling and cavity frequency. The authors consider both a dark cavity (no external pumping) and a driven cavity (with external pumping in the few-photon limit). The methodology accounts for the possibility of multiple cavity modes and their polarization states in a Fabry-Pérot cavity, as well as discussing the impact of cavity dissipation and its influence on the observed effects. The simulations use a perfect cavity approximation, but the authors consider the effects of realistic cavity decay rates (κ) and show that the reported results should be robust against such effects. Different scenarios (varying photon energy and light-matter coupling) are investigated to determine their influence on the system's magnetic state. The choice of using a number state representation for photons instead of a coherent state is justified.

The key findings center around the ability to control the magnetic ground state of α-RuCl₃ using cavity QED. The authors demonstrate that the system's magnetic phase can be manipulated by adjusting the cavity's properties. Specifically: 1. **Vacuum Fluctuation-Induced Transition:** In the THz regime, the interaction with cavity vacuum fluctuations alone is sufficient to induce a transition from the equilibrium zigzag antiferromagnetic order to a ferromagnetic state. This transition represents a true equilibrium state, termed the photo ground state (PGS), unlike metastable states achieved through classical light driving. 2. **Controlled Phase Transitions:** By varying the cavity frequency, photon occupation, and light-matter coupling, it's possible to stabilize any of the magnetic phases supported by the extended Kitaev model. The calculated phase diagram (Fig. 2b) illustrates the evolution of the system's magnetic state as a function of these parameters. The paths taken through this phase diagram highlight the tunability of the magnetic interactions (J, K, Γ) via the light-matter interaction. 3. **Quantum Spin Liquid State:** By pumping the cavity in the few-photon regime, the system can be driven into the Kitaev quantum spin liquid state, providing access to the non-equilibrium phase diagram of the semiclassical limit. 4. **First-Principles Parameterization:** The study utilizes first-principles calculations to determine the electronic and spin parameters of α-RuCl₃, providing a rigorous foundation for the theoretical model. The electronic band structure is shown in Figure 4. The detailed analysis of spin-spin correlation functions (Fig. 2d) and plaquette flux operator expectation values (Fig. 2e) further supports the observed magnetic phase transitions.

This research successfully demonstrates the potential of cavity QED as a tool for controlling the magnetic properties of quantum materials. The ability to manipulate the magnetic state of α-RuCl₃ by tuning cavity parameters opens up exciting possibilities for designing and controlling quantum states of matter. The findings highlight the significance of quantum vacuum fluctuations in driving phase transitions, demonstrating a new route to modify magnetic systems without resorting to high-intensity light sources or external magnetic fields. The ability to reach the Kitaev quantum spin liquid phase via cavity pumping paves the way for exploring this exotic phase in more detail. The use of first-principles calculations to parameterize the model adds to the rigor and reliability of the results. Future studies could extend this work to other quantum materials and explore the interplay between light-matter interactions and other external control parameters.

This study successfully demonstrates the control of magnetic phases in the quantum spin liquid candidate α-RuCl₃ using an optical cavity. The ability to achieve both ferromagnetic and Kitaev quantum spin liquid states through manipulation of cavity properties represents a significant advancement in cavity QED engineering of quantum materials. Future work could investigate the effects of cavity dissipation more comprehensively, explore the influence of different cavity geometries, and extend this approach to other magnetic materials.

The study employs a simplified model with a single effective cavity mode, potentially neglecting the complex interplay of multiple cavity modes and their polarization states. While the effect of cavity dissipation is discussed, a full quantitative analysis with realistic decay rates would require further computational effort. The 24-site spin cluster used for exact diagonalization calculations might not fully capture the macroscopic behavior of the system. The study focuses on equilibrium properties for the dark cavity and a quasi-equilibrium analysis for the driven cavity. A fully time-dependent description would provide a more complete picture.