Magnonic Superradiant Phase Transition

The paper addresses whether a true superradiant phase transition (SRPT) can occur in thermal equilibrium, circumventing no-go theorems that forbid equilibrium SRPTs for charge-based light-matter coupling. The authors focus on ErFeO3 and propose that its known low-temperature phase transition (LTPT) at about 4 K is a magnonic analogue of the SRPT, arising from ultrastrong coupling between Fe3+ magnons (spin-wave field) and Er3+ spins. They motivate the significance by linking SRPTs to intrinsically stable quantum squeezing and decoherence-robust quantum technologies, contrasting with previously observed driven, dissipative nonequilibrium SRPTs in cold-atom systems. The study aims to show, using realistic parameters and experimental inputs, that ErFeO3 undergoes an equilibrium magnonic SRPT, establishing a long-sought equilibrium SRPT in a real material system.

The authors situate their work within the Dicke model framework, originally predicting an SRPT in 1973. They note extensive discussion of no-go theorems for equilibrium SRPTs in minimal-coupling (charge-based) systems and subsequent debates and proposed counterexamples, including circuit QED scenarios and the role of spin degrees of freedom or spatially varying multimode fields in circumventing no-go constraints. Nonequilibrium SRPTs have been demonstrated in laser-driven cold atoms. In magnetism, ultrastrong photon-magnon, spin-magnon, and magnon-magnon couplings have been observed, but no SRPT evidence had been reported. Prior discussions linked ErFeO3 LTPT to cooperative Jahn-Teller physics, which has analogies to SRPT, yet a direct SRPT interpretation with quantitative evidence was lacking. Earlier experiments (Li et al., 2018) showed Dicke cooperativity (N-scaling) in ErxY1−xFeO3, supporting cooperative coupling between Er3+ spins and Fe3+ magnons.

• Spin model: The system ErxY1−xFeO3 (0 ≤ x ≤ 1) is modeled with three Hamiltonian parts: H = H_Fe + H_Er + H_Er-Fe. Fe3+ spins are treated in a two-sublattice (A/B) canted antiferromagnet model (S = 5/2) with isotropic exchange J_Fe, Dzyaloshinskii–Moriya (DM) interactions D_z^Fe, and magnetic anisotropy terms A_x, A_z, A_xz, plus anisotropic g-tensor and external DC magnetic field B_DC. Er3+ spins are represented as two sublattices with anisotropic g-tensor and Er–Er exchange J_Er; Y3+ sites are nonmagnetic (R=0). The Er–Fe interactions include isotropic exchange J_Er-Fe and antisymmetric (DM-like) terms D_Er-Fe with symmetry-constrained components (D_x, D_y) within each unit cell.
• Assumptions: Long-wavelength (k≈0) limit; two-sublattice description for both Fe and Er sublattices; homogeneous B_DC; Er y-components effectively unaffected by Er–Fe interactions (due to higher energy cost along b-axis); Er–Fe coupling only within a unit cell; neglect of qFM magnons for LTPT description.
• Parameter determination: Parameters are constrained/fitted using terahertz magnetospectroscopy (spin resonance frequencies, anti-crossings) and magnetization phase diagrams from literature. Reported parameter examples include Fe3+ subsystem (J_Fe ≈ 4.96 meV; D ≈ −0.107 meV; anisotropy A1 ≈ 0.0073 meV, A2 ≈ 0.0150 meV), anisotropic g-factors (Er: g_h ≈ 6, g_⊥ ≈ 3.4, g_∥ ≈ 9.6; Fe: g_h ≈ 2, g_⊥ ≈ 2, g_∥ ≈ 0.6), and Er–Er/Er–Fe exchange components (J_⊥ ≈ 0.037 meV, J_∥ ≈ 0.60 meV, D_⊥ ≈ 0.034 meV, D_∥ ≈ 0.003 meV).
• Mean-field analysis: Thermal equilibrium spin expectation values for Er3+ (two sublattices) and Fe3+ (two sublattices) are computed versus temperature and DC magnetic field along a, b, or c, yielding phase diagrams and order parameters (e.g., Er AFM vector along c; Fe AFM vector rotation in bc-plane). Order parameters include the Er z-component difference and Fe AFM rotation angle φ = arctan(S_x/S_z).
• Extended Dicke Hamiltonian derivation: The spin model is transformed into an extended Dicke Hamiltonian by quantizing Fe magnons (qAFM mode, ω ≈ 2π × 0.896 THz) and expressing Er ensemble in collective spin operators. The Er resonance scale is E_Er = ħ × 0.023 THz. Two key Er–magnon couplings appear: transverse (g_⊥ ≈ 2π × 0.116 THz) and longitudinal (g_∥, smaller), alongside direct Er–Er exchange. The minimal Hamiltonian relevant to LTPT (with B_DC//a maintaining Γ2 symmetry) contains: magnon energy, Er spin energy, Er–Er exchange terms (Σ^zΣ^z, Σ^xΣ^x), and transverse/longitudinal Er–magnon couplings ((a+a†)Σ^x, (a−a†)Σ^z/√N forms).
• Semiclassical method: Following Wang–Hioe/Hepp–Lieb, in the thermodynamic limit, the partition function is evaluated by replacing magnon operators with coherent-state c-numbers; minimizing an effective action yields equilibrium values of magnon and spin order parameters as functions of T and B_DC. This is applied to the extended Dicke Hamiltonian to compute LTPT boundaries and order parameters.
• SRPT condition via Holstein–Primakoff: Linearizing around the normal phase provides an analytic SRPT condition in the extended Dicke model: 4g^2/(ħω_Er ħω_p) + 4g_x^2/(ħω_Er ħω_p) + 4 z_Er J_Er/(ħ ω_p) > 1, where the three terms quantify transverse and longitudinal Er–magnon couplings and direct Er–Er exchange contributions (termed coupling depths). Numerical values are extracted from parameters to assess each contribution quantitatively.

• Identification of LTPT as equilibrium magnonic SRPT: The LTPT at Tc ≈ 4 K in ErFeO3 corresponds to a superradiant phase where Er3+ spins order antiferromagnetically along c (nonzero ⟨Σ^z⟩) and Fe3+ AFM vector rotates in the bc-plane due to spontaneous qAFM magnon population (nonzero magnon amplitude), all in thermal equilibrium.
• Phase diagrams and order parameters: Mean-field phase diagrams versus T and B_DC along a, b, c reproduce experimental trends, including different critical fields for B_DC parallel vs antiparallel to weak Fe magnetization along a. In zero field, the Fe AFM rotation angle is φ ≈ 46° at T = 0 K, close to the experimental estimate 49°.
• Coupling regime: The transverse Er–magnon coupling g_⊥ ≈ 2π × 0.116 THz is a sizable fraction of both Er resonance (E_Er/ħ ≈ 0.023 THz) and magnon frequency (ω ≈ 2π × 0.896 THz), placing the system in the ultrastrong coupling regime.
• LTPT can be driven solely by Er–magnon coupling: With Er–Er exchange set to zero (J_Er = 0), an SRPT still occurs with Tc ≈ 1.2 K at B_DC = 0, demonstrating that cooperative Er–magnon coupling alone can cause the transition (magnonic SRPT).
• Er–Er exchange alone yields lower Tc: Without Er–magnon coupling, Tc ≈ 2.6 K at B_DC = 0. The full model yields Tc ≈ 4 K, showing that Er–magnon coupling enhances Tc and critical fields beyond what Er–Er exchange alone provides.
• Quantified SRPT condition and contributions: From the extended Dicke model SRPT condition, coupling depths are D1 = 4g^2/(ω_Er ω_p) = 2.65 (transverse, positive), D3 = −4g_x^2/(ω_Er ω_p) = −0.51 (longitudinal, negative), D4 = 4 z_Er J_Er/(ħ ω_p) = 9.29 (Er–Er exchange). The sum D1 + D3 + D4 > 1 confirms SRPT. Relative contributions indicate the longitudinal term reduces the total, while the transverse Er–magnon coupling alone satisfies D1 > 1 and (D1 + D3) > 1, implying it is strong enough to cause SRPT by itself.
• Consistency across methods: Phase boundaries from the mean-field spin model and the semiclassical extended Dicke Hamiltonian are in close agreement (minor differences discussed in supplementary material).

The results establish that the ErFeO3 LTPT is a thermal-equilibrium superradiant phase transition mediated by cooperative Er–magnon coupling. The observed Er antiferromagnetic ordering (nonzero Σ^z) and concomitant spontaneous rotation of the Fe AFM vector (linked to nonzero qAFM magnon amplitude) are the magnetic analogues of spontaneous polarization and field emergence in the Dicke SRPT. Quantitative modeling with experimentally constrained parameters reproduces phase diagrams and critical behavior, and the derived extended Dicke Hamiltonian connects the microscopic spin model to SRPT theory. The analysis clarifies how antisymmetric Er–Fe exchange (DM-like) provides the dominant transverse coupling driving SRPT, while Er–Er exchange further elevates Tc and critical fields. This equilibrium realization bypasses no-go theorems applicable to minimal charge-based coupling by leveraging spin degrees of freedom and magnonic modes in a solid. The identification of an equilibrium SRPT in a real material has implications for robust quantum squeezing at criticality and decoherence-resistant quantum sensing and information processing.

The study develops a spin-based microscopic model for ErFeO3 that, validated against terahertz magnetospectroscopy and magnetization data, maps onto an extended Dicke Hamiltonian. It demonstrates that the LTPT at ≈4 K is a magnonic SRPT: cooperative, ultrastrong Er–magnon coupling can by itself induce the transition (Tc ≈ 1.2 K without Er–Er exchange), and together with Er–Er exchange reproduces the observed Tc and phase boundaries. This provides the first quantitative confirmation of a thermal-equilibrium SRPT in a material system since the original Dicke proposals. Future directions include experimental probes of quantum fluctuations and two-mode squeezing near the SRPT critical point in ErxY1−xFeO3, and searches for photonic SRPTs in materials with explicit spin degrees of freedom.

• Model assumptions: Two-sublattice descriptions for both Fe3+ and Er3+; long-wavelength (k≈0) limit; homogeneous external field; Er y-components assumed insensitive to Er–Fe interactions; Er–Fe coupling restricted within unit cells; neglect of qFM magnons for LTPT.
• Parameter determination: Several parameters (Er–Er and Er–Fe exchange components, anisotropic g-factors, anisotropies) are fitted within ranges consistent with phase diagrams and terahertz spectra; phase boundaries alone are insufficient to uniquely determine all parameters.
• Approximations in theory: Mean-field treatment of spins and semiclassical/coherent-state approximation in the thermodynamic limit; Holstein–Primakoff linearization for SRPT condition; some small coupling terms (e.g., g_z) neglected due to estimated weakness.
• Scope: The work provides theoretical confirmation using experimentally informed parameters; direct experimental verification of predicted equilibrium squeezing at criticality remains for future studies. Photonic (charge-based) equilibrium SRPT remains unconfirmed and subject to no-go debates.