Controlling topological phases of matter with quantum light

The study asks how coupling quantum materials to cavity photons modifies and controls their topological properties. Prior work has shown control via classical light (Floquet driving), but this can suffer from heating. Here, the authors explore the impact of quantum light within a cavity on a paradigmatic 1D topological insulator, the Su-Schrieffer-Heeger (SSH) model. They aim to determine whether quantum fluctuations in a cavity can induce or suppress topological phases, how finite-size spectra and edge modes are affected, and how photonic observables can probe topological transitions. The work is important for non-dissipative control of topology and for leveraging hybrid light-matter systems in quantum technologies.

The paper situates itself within efforts to control quantum materials using light: Floquet engineering of topology has been demonstrated in solids and ultracold atoms, but heating is a concern. Cavity quantum electrodynamics offers new control avenues, including topological polaritons and anomalous Hall responses. Experiments have observed cavity-induced breakdown of topological protection in quantum Hall systems. The SSH model is a canonical 1D topological system realized across platforms (ultracold atoms, graphene nanoribbons, photonics, mechanical metamaterials). The authors reference gauge-invariant treatments of light-matter coupling via the Peierls substitution and prior results on the collective ultrastrong coupling regime, no-go theorems for bulk changes in the thermodynamic limit, and methods to read out cavity spectral functions.

- Model: A spinless SSH chain with L unit cells and sublattices A,B, intracell hopping v and intercell hopping w. Photons are a single cavity mode with frequency ωc. Light-matter coupling is introduced via the gauge-invariant Peierls substitution dressing hoppings by phase factors dependent on bond lengths: v→ve^{-ieA l_AB}, w→we^{-ieA l_BB}, with l_AB=b0, l_BB=a0−b0 and A∝(a+a†). - Unitary (Peierls) gauge: The Peierls Hamiltonian is obtained by a unitary Ω=exp(ieA∑jRj) acting on fermions, yielding bond-dependent phases and the full Hamiltonian H=Hph+Ω†HSSHΩ. The coupling strength is scaled as g=eA/√L (collective ultrastrong coupling). - Finite chain mean-field decoupling: Use a factorized ansatz |Ψ⟩=|ϕ⟩|ψ⟩, neglecting electron–photon correlations. This yields an effective SSH model with renormalized hoppings ṽ=v φ(g,b0), w̃=w φ(g,1−b0), where φ(g,b0)=⟨ϕ|e^{-ig(a+a†)}|ϕ⟩ is found self-consistently together with the photonic ground state. The renormalization is purely real, implying zero ground-state current. Finite-size spectra (edge modes) are computed as a function of w/v, g, and b0. - Bulk (periodic) mean-field: For periodic boundary conditions, write H=∑k ψk† Hk(a,a†) ψk + ωc(a†a+1/2), with d-vector components depending on K=k+g(a+a†): dkx=v cos(K b0) − w cos(K(1−b0)); dky=−v sin(K b0) − w sin(K(1−b0)); dkz=0, preserving chiral symmetry. Mean-field decoupling yields a bulk dispersion εk=√(ṽ² + w̃² − 2 ṽ w̃ cos k). The finite-L gap minimum at k=π/L defines the phase boundary. The phase boundary condition is w/v = φ(g,b0)/φ(g,1−b0). - Photonic state properties: Analyze the photonic ground state in the Fock basis; odd-photon coefficients vanish due to a discrete symmetry (no linear terms). Define quadratures X=(a+a†)/√2, P=i(a†−a)/√2, and obtain an effective photonic Hamiltonian Hph=ωc(a†a+1/2)+r(a+a†)² with squeezing parameter r controlled by the renormalized electronic kinetic energy. - Polariton spectrum and spectral function: Compute the photon spectral function A(ω)=Im ∫ dt e^{iωt}⟨−i[a(t),a†]⟩ including 1/L Gaussian fluctuations via a current-current response χ(ω). A(ω)=1/[π((ω²−ωc²−2ωc χ′(ω))²+(2ωc χ″(ω))²)], with paramagnetic and diamagnetic currents Jp and Ja specified. Additionally, derive an effective spin-wave Hamiltonian near k=0: Ĥ=ωc a†a + ωb b†b − i g ωs (a†+a)(b†−b) − g ωs (a†+a)² − const, with ωb=2|v−w|, yielding analytical polariton energies via Bogoliubov–Hopfield transformation.

- Quantum light modifies finite-size topological properties of the SSH chain: the presence and onset of edge modes and the apparent phase boundary shift depend on the sublattice spacing parameter b0 and coupling g. - For b0=0.8, increasing g shifts the topological transition to smaller w/v, favoring the topological phase; for b0=0.2, the transition shifts to larger w/v, suppressing topology. This asymmetry arises from different Peierls renormalizations of v and w tied to geometry. - The finite-size phase boundary is given implicitly by w/v = φ(g,b0)/φ(g,1−b0), where φ is the self-consistent photonic renormalization factor. As L→∞, the critical point approaches w/v→1, recovering the isolated SSH transition and indicating no bulk change by a single-mode cavity in the thermodynamic limit. - The photonic ground state is squeezed under coupling: ⟨X²⟩ decreases and ⟨P²⟩ increases with g, with odd-photon Fock components suppressed by symmetry. - Polariton spectroscopy reveals topology: in the trivial/topological phases (v≠w), three polariton branches appear; at the topological transition (v=w), the lowest polariton branch is pushed to zero frequency and loses spectral weight, leaving only two peaks in A(ω). Analytical spin-wave theory reproduces this via polariton energies and predicts the disappearance of the lower branch at v=w. - Chiral (sublattice) symmetry is preserved by the Peierls coupling (dkz=0), ensuring particle-hole symmetric spectra.

The study demonstrates that coupling to a single-mode quantum cavity can control finite-size topological characteristics of a 1D insulator by renormalizing hopping amplitudes in a geometry-dependent way, without breaking chiral symmetry. This addresses the core question of whether quantum light can induce or suppress topology: depending on b0 and g, a trivial SSH chain can be driven into a topological regime with emergent edge modes, or a topological chain can be pushed toward trivial behavior. Although bulk criticality in the thermodynamic limit remains unchanged (w=v), cavity photons encode the topological transition in the polariton spectrum: the vanishing and loss of spectral weight of the lowest polariton branch at v=w provides a direct, experimentally accessible probe via transmission/reflection. Compared to Floquet engineering with classical light, the quantum-cavity approach avoids heating issues and leverages ground-state properties and fluctuations. The observed squeezing reflects back-action of electronic topology on photonic states, highlighting bidirectional light-matter interplay.

The paper establishes a gauge-invariant theory of an SSH chain coupled to a single-mode cavity and shows that quantum light can control and induce topological phases in finite systems by geometry-dependent Peierls renormalization, while preserving chiral symmetry. It identifies a finite-size, geometry- and coupling-dependent phase boundary and demonstrates that in the thermodynamic limit the transition point remains w=v. The polariton spectrum serves as a probe: the lowest branch disappears at the topological transition. The approach avoids Floquet heating and suggests cavity photons as tools to both control and read out topology. Future directions include considering inhomogeneous or momentum-dependent electromagnetic fields (e.g., multimode or spatially varying cavities) that could further tailor topological properties and potentially enable superradiant/topological phases beyond the uniform single-mode scenario.

- Mean-field factorization between electrons and photons neglects correlations; results are justified for large L but may miss correlation-driven effects at finite size or strong coupling. - Single-mode, uniform cavity field with collective g∝1/√L scaling ensures no bulk changes in the thermodynamic limit; thus control is primarily a finite-size effect. - Numerical implementations use truncated photon Hilbert space (e.g., Nmax=10) and specific parameter choices; convergence and quantitative details may vary with truncation. - The analytical spin-wave treatment focuses on k=0 excitations and is approximate; full momentum-dependent effects are incorporated numerically but the simplified model may not capture all features. - No dissipation or disorder is included; realistic cavity losses and material imperfections could modify spectra and edge states in experiments.