Spontaneous Symmetry Breaking of Cavity Vacuum and Emergent Gyrotropic Effects in Embedded moiré Superlattices

In an electronic system, spontaneous symmetry breaking can arise from many-body interaction between electrons, leading to degenerate ground states distinguishable by emergent effects otherwise prohibited by the symmetry. Here we show that ultrastrong coupling of a mesoscopic electronic system to the vacuum of a cavity resonator can lead to another paradigm of spontaneous breaking of spatial symmetries in both systems. As a pertinent example, we consider the orbital gyrotropic effects in a moiré superlattice embedded in a THz split ring cavity resonator. Our mean-field and exact diagonalization calculations consistently demonstrate a spontaneous parity symmetry breaking in both the electronic ground state and the cavity vacuum, leading to two degenerate hybrid ground states distinguished by their opposite orbital gyrotropic Hall and magnetic effects. These sizable responses in the cavity-embedded moiré superlattice are highly tunable by both the cavity field polarization and interlayer bias on the moiré superlattice, providing an advanced platform for manipulating gyrotropic effects.

💡 Research Summary

This paper investigates a novel form of spontaneous symmetry breaking that occurs simultaneously in a mesoscopic electronic system and the vacuum field of a terahertz (THz) split‑ring cavity resonator. By embedding a twisted bilayer transition‑metal dichalcogenide (TMD) moiré superlattice within the sub‑wavelength gap of a split‑ring resonator, the authors achieve ultrastrong light‑matter coupling (coupling strength χ≈0.03) at a resonant frequency of 0.512 THz. The electronic degrees of freedom are modeled by a two‑band Haldane‑type tight‑binding Hamiltonian on a honeycomb moiré lattice, with nearest‑neighbor hopping t₁≈0.35 meV, next‑nearest‑neighbor hopping t₂≈0.1 meV, and complex phases φ_A=+2π/3, φ_B=−2π/3 that encode the intrinsic Berry curvature of each valley.

The total Hamiltonian H_tot = H_m + H_c includes the bare photon term H_c = ℏω a†a and the minimal coupling of the cavity vector potential to the electrons. By performing a Schrieffer‑Wolff transformation and projecting onto the zero‑photon subspace (n=0), an effective Hamiltonian H_eff is derived that contains a photon‑mediated four‑fermion interaction. Although this interaction respects spatial parity (inversion) symmetry, a self‑consistent mean‑field treatment yields non‑zero expectation values ⟨U_l⟩ and ⟨V_l⟩, indicating that the parity symmetry is spontaneously broken in the ground state.

Both mean‑field calculations and exact diagonalization (ED) on a 3 × 5 unit‑cell cluster (14 holes) reveal a two‑fold degenerate hybrid ground state. The two states differ markedly in their band dispersions, thereby breaking the C₃ rotational symmetry of the bare moiré lattice. More importantly, they possess opposite Berry‑curvature dipoles (BC‑dipole) and opposite orbital‑magnetic‑moment dipoles (OM‑dipole). Consequently, the orbital gyrotropic Hall effect (OGHE) and orbital gyrotropic magnetic effect (OGME) – described by tensors R_H^a and R_M^a – acquire finite, opposite values in the two states. The gyrotropic tensors are forbidden by parity in a symmetric system; their emergence here is a direct signature of the spontaneous parity breaking.

ED results corroborate the mean‑field picture: as χ increases, the energy splitting between the two lowest eigenstates vanishes for χ ≳ 0.24, confirming exact degeneracy. The orbital polarization σ_z = ⟨c_A†c_A – c_B†c_B⟩ becomes non‑zero and opposite in the two states, serving as an electronic order parameter for parity breaking. Simultaneously, the cavity field expectation ⟨a⟩ acquires opposite signs, demonstrating that the vacuum field itself undergoes spontaneous symmetry breaking—distinct from the conventional super‑radiant phase transition, which requires a macroscopic photon condensate in the thermodynamic limit.

The authors explore tunability of the gyrotropic response. The cavity polarization vector e_p = (cos θ, sin θ) controls the direction of the effective electric field. The gyrotropic components R_x and R_y exhibit sinusoidal dependence on θ; for θ = 0 (field along x) only R_x is non‑zero, while for θ = π/2 (field along y) only R_y survives. This selection follows from residual C₂^x or C₂^y symmetries that forbid the orthogonal component. Additionally, applying an interlayer bias Δ creates an on‑site energy difference between the two sublattices, explicitly breaking parity. Under finite Δ the two‑fold degeneracy disappears, new gyrotropic components appear, and the system can undergo a topological phase transition (change of Chern number). Remarkably, the Berry‑curvature dipole R_H^x changes sign continuously across the transition, offering a clear experimental hallmark.

In the discussion, the authors contrast their findings with the long‑standing debate on photon condensation in cavity QED. Their mesoscopic setting (∼10⁴ electrons) lies far from the thermodynamic limit, yet the exact diagonalization unambiguously shows ⟨a⟩ ≠ 0 as a manifestation of parity breaking rather than a macroscopic coherent state. This “vacuum‑induced symmetry breaking” provides a new route to generate otherwise forbidden linear responses in non‑magnetic crystals.

In summary, the work demonstrates that ultrastrong coupling to a THz cavity can transmit the cavity’s C₃ symmetry breaking to a moiré superlattice, and the collective electron‑photon interaction can spontaneously break parity, yielding two degenerate hybrid ground states with opposite orbital gyrotropic Hall and magnetic effects. The magnitude and orientation of these effects are highly tunable via cavity field polarization and interlayer bias, establishing cavity‑embedded moiré systems as a versatile platform for engineering exotic electromagnetic responses and suggesting broader applicability to other strongly correlated phases such as superconductivity or charge‑density waves.