Controlling topological phases of matter is a significant challenge in condensed matter physics. While classical light has been employed as a spectroscopic probe and to induce novel phases through irradiation or cavity coupling, the potential of quantum light remains largely unexplored. Topological phases, known for their robustness and applications in quantum technologies, are particularly interesting in this context. Previous research has explored Floquet topological insulators using classical, time-modulated light, demonstrating Floquet-engineered topological band structures experimentally. However, these approaches often suffer from heating problems. Quantum fluctuations of light fields in cavities offer a different avenue, potentially leading to polaritons with non-trivial topological properties or anomalous Hall responses. This paper investigates the impact of coupling to cavity photons on a topological material, building on recent observations of topological protection breakdown in a quantum Hall system coupled to a cavity. The Su-Schrieffer-Heeger (SSH) model, a prototypical one-dimensional model exhibiting a topological phase transition associated with a non-trivial Zak phase, serves as the basis for this study. The SSH model has been experimentally realized in various platforms, including ultracold atoms, graphene nanoribbons, and photonic and mechanical metamaterials. This work explores the interplay between topological phases and quantum light by coupling the SSH model to a single-mode photonic resonator via the gauge-invariant Peierls phase, aiming to understand how quantum light fluctuations affect the system's topological properties and edge modes.

The paper extensively reviews the literature on light control of quantum materials, highlighting the use of classical light to induce novel phases of matter. It cites examples of Floquet engineering of topological band structures and the use of time-modulated optical lattices with ultracold atoms. The authors also discuss previous work on topological polaritons, cavity QED effects on materials, and the observation of topological protection breakdown in quantum Hall systems coupled to cavities. The literature review positions the current work within the context of these previous efforts, emphasizing the novelty of using quantum light fluctuations to control topological properties and the advantages over classical Floquet driving schemes.

The study employs a one-dimensional SSH model coupled to a single-mode cavity. The SSH Hamiltonian describes spinless electrons hopping on a dimerized chain with alternating hopping amplitudes (v and w). A single-mode cavity Hamiltonian is included, and the coupling between the SSH chain and the cavity is modeled through the gauge-invariant Peierls substitution. This substitution introduces a Peierls phase that dresses the hopping amplitudes, leading to a dependence on the lattice geometry. The authors employ a factorized mean-field ansatz for the ground state, neglecting correlations between cavity modes and electrons. This approach is justified in the limit of a large number of unit cells (L). The resulting effective Hamiltonian yields renormalized hopping amplitudes (ṽ and w̃) dependent on the cavity photon field. The photonic ground state and electronic ground state are determined self-consistently. For the topological phase diagram analysis, periodic boundary conditions are considered, enabling a momentum-space representation. The chiral symmetry of the system, crucial for topological protection, is shown to be preserved even with light-matter coupling. A mean-field approach is used to calculate the bulk energy spectrum, revealing how light-matter coupling shifts the topological phase transition point. The photonic properties are investigated by analyzing the expectation values of position and momentum operators, revealing squeezing in the photonic state. Finally, the polariton spectrum, providing insights into the topological phase transition, is obtained by calculating the photon spectral function, incorporating Gaussian fluctuations. An analytical argument using a spin-wave analysis is presented to support the numerical results regarding the polariton spectrum behavior near the topological transition.

The key findings are: 1. Quantum light fluctuations can significantly affect the topological properties of the SSH model, shifting the topological phase transition point. 2. Depending on the lattice geometry (parameter b₀), coupling to quantum light can either stabilize or destabilize the topological phase. 3. A finite-length SSH chain coupled to a cavity exhibits edge modes whose behavior is modified by light-matter coupling, with the transition point shifted depending on b₀. 4. The topological phase diagram reveals that a trivial SSH model can be converted into a topological one by increasing the light-matter interaction (for certain b₀ values). 5. In the thermodynamic limit (large L), the topological transition point remains unaffected by the cavity, although the polariton spectrum reflects the underlying topology. 6. The lower polariton branch disappears at the topological transition point (v = w), providing a spectroscopic signature for the transition. 7. The photonic state exhibits squeezing due to the light-matter interaction. This squeezing is related to a discrete symmetry protected by the absence of electronic current in the ground state. The analytical spin-wave analysis supports the numerical findings regarding the disappearance of the lower polariton branch at the transition point.

The results demonstrate a novel approach to controlling topological phases using quantum light. This method avoids the heating problems associated with classical Floquet driving schemes. The dependence of the topological phase transition on both the light-matter coupling and the lattice geometry offers additional control parameters. The identification of the disappearing lower polariton branch as a signature of the topological transition suggests a promising experimental probe for these phenomena. This work contributes significantly to the field of cavity quantum electrodynamics (QED) and topological materials, opening new possibilities for manipulating topological phases in condensed matter systems.

This paper shows that coupling a quantum material to a cavity's quantum light field can control topological properties and drive phase transitions. The SSH model, coupled to a single-mode cavity via the Peierls phase, serves as a paradigmatic example. The study demonstrates control over the topological phase transition through light-matter coupling and lattice geometry, overcoming limitations of classical Floquet engineering. The disappearance of the lower polariton branch at the transition point offers a practical way to experimentally detect this transition. Future work could explore the use of inhomogeneous electromagnetic fields and their effects on topological properties.

The study employs a mean-field approximation, neglecting electron-photon correlations. While justified in the large-L limit, this approximation may not be entirely accurate for smaller system sizes. The analytical model for the polariton spectrum is based on a spin-wave approximation for k=0, which may not capture the full complexity of the system's dynamics for all momenta. The focus on a single-mode cavity limits the generalizability of the results to more complex cavity systems.