Magnonic Dicke Models: Theory & Applications

• Magnonic Dicke models are defined by collective spin excitations (magnons) interacting with bosonic modes, generalizing the classic Dicke framework.
• They capture cooperative phenomena such as quantum and thermal superradiant phase transitions, with phase diagrams enriched by long-range and short-range interactions.
• The models offer insights into universal dynamics, including mappings to Painlevé II and the necessity of gauge invariance in the ultrastrong coupling regime.

Magnonic Dicke models generalize the paradigmatic Dicke model from quantum optics to systems where the collective quantum degrees of freedom are spin excitations (magnons) in magnetic materials, and the role of the photonic field may be played by bosonic magnon modes, electromagnetic resonators, or hybrid magnon–photon fields. These models capture the cooperative interaction between an ensemble of magnetic two-level systems and a bosonic mode, supporting phenomena such as quantum and thermal equilibrium superradiant phase transitions, spin glass order, non-equilibrium pattern formation, and higher-order symmetry breaking. Experimental realizations span from solid-state magnetic lattices such as ErFeO₃ to driven-dissipative hybrid quantum platforms and opto-magnonic arrays. Below, key theoretical and experimental developments are organized to illuminate the structural principles, critical phenomena, and distinctive features of magnonic Dicke models.

1. Theoretical Structure of Magnonic Dicke Models

The canonical Dicke Hamiltonian models N two-level atoms collectively coupled to a single bosonic field mode:

$$H_{\text{Dicke}} = \omega a^\dagger a + \omega_0 S_z + \frac{2g}{\sqrt{N}} (a + a^\dagger) S_x.$$

Magnonic Dicke models reinterpret the collective “atomic” degrees of freedom as spin excitations, i.e., magnons, in a macroscopic magnetic system. The bosonic mode, instead of being solely photonic, may be a collective magnon, cavity photon, or a hybridized magnon–photon polariton.

A prototypical material realization appears in ErFeO₃, where the strongly coupled Fe³⁺ (iron) spins form an ordered antiferromagnetic lattice, quantized into magnon modes. The Er³⁺ (erbium) ions provide effective two-level systems, collectively interacting with the Fe³⁺ magnon via a coupling analogous to light–matter interaction (Bamba et al., 2020). A general magnonic Dicke Hamiltonian thus contains both the long-range (magnon-mediated) coupling $g$ and short-range exchange interactions $J$ between the effective spins:

$$\mathcal{H} = \omega_{\pi} a_{\pi}^{\dagger} a_{\pi} + \omega_{\text{Er}} \Sigma_x^+ + g \sqrt{\frac{2}{N_0}} i(a_{\pi}^{\dagger} - a_{\pi}) \Sigma_z^- + J\,\text{[short-range terms]}$$

where $a_{\pi}^{\dagger}$ creates a quasi-antiferromagnetic magnon, and $\Sigma_z^-$ encodes staggered Er³⁺ ordering across sublattices (Peraca et al., 2023).

Magnonic Dicke models may be further extended to encompass:

• Multiple interacting spin ensembles and corresponding bosonic channels (Liu et al., 2023).
• Site- or mode-dependent couplings introducing frustration and disorder (Strack et al., 2011).
• Higher-order and phase-imprinted couplings leading to $\mathbb{Z}_n$ symmetry and non-reciprocity (Ho et al., 5 Oct 2025).
• Coupling consistently derived within a gauge-invariant formalism, crucial for the ultrastrong regime (2002.04241).

2. Quantum and Thermal Superradiant Phase Transitions

Superradiant phase transitions (SRPTs) are signature collective phenomena in Dicke and magnonic Dicke models, characterized by the emergence of a macroscopic expectation value for the bosonic field (photon or magnon) and spontaneous symmetry breaking of parity or higher-order discrete symmetry.

In ErFeO₃, the equilibrium SRPT manifests at $T_c \approx 4$ K, where cooperative ultrastrong coupling between the Er³⁺ spins and the Fe³⁺ magnon mode induces simultaneous Er³⁺ sublattice ordering and Fe³⁺ magnon condensation. The phase boundary is governed by the effective transverse Dicke coupling,

$D_{g_z} = \frac{4g_z^2}{\omega_\pi} > 1,$

and is substantially elevated by the presence of magnon-mediated long-range interactions in addition to direct Er–Er exchange (Bamba et al., 2020, Peraca et al., 2023). The SRPT in this system occurs in genuine thermal equilibrium, contrasting most prior cavity QED experiments where driving and dissipation dominate.

In the generalized Dicke model with competing spin–spin interactions, the SRPT can be tuned, enhanced, or suppressed by the nature of those interactions: ferromagnetic exchange (negative $J$) lowers the threshold for superradiance, whereas antiferromagnetic exchange (positive $J$) suppresses it, potentially favoring atomically ordered but non-superradiant phases (Liu et al., 2023, Peraca et al., 2023). Critical points exhibit closing of the excitation gap, divergence of fluctuations, and enhancement of entanglement entropy, in agreement with standard signatures of second-order quantum phase transitions.

3. Role of Short-Range Exchange and Disorder: Phase Structure and Glassiness

When short-range interactions (e.g., direct Er–Er exchange) are present alongside the long-range (magnon-mediated) Dicke coupling, the phase diagram is enriched. For instance, in the extended Dicke (g–J) model realized in ErFeO₃, three phases are found (Peraca et al., 2023):

• Normal (N): No long-range order.
• Superradiant (S): Both magnon condensation and Er³⁺ sublattice order.
• Atomic (A): Atomically ordered phase with no accompanying magnon condensation, selected by the dominance of $J$ over $g$.

Transitions between these phases can be first or second order, evidenced experimentally by discontinuities or kinks in magnon frequency (THz spectroscopy) and signatures in the magnetocaloric effect.

Spatial disorder or multimode couplings, as in generalizations where the coupling $g_{i\ell}$ depends on emitter position and photon mode, can induce frustrated, glassy phases. The multimode Dicke model supports a quantum spin glass (QSG) phase, characterized by (Strack et al., 2011):

• A random superradiant order parameter, as a linear combination of photonic eigenmodes.
• A gapless continuum in the atomic spectral response, contrasting the gapped FM superradiant phase.
• Nonzero Edwards–Anderson order parameter $q_{\text{QSG}}$, denoting frozen quantum disorder.

These features are accessible experimentally via rf spectroscopy on the spin ensemble and input–output measurements of the cavity spectrum, and are directly relevant to disordered magnonic Dicke models.

4. Universal Dynamical Features and Mapping to Painlevé II

When system parameters are sweept quasi-adiabatically through the critical point of the phase transition, the breakdown of adiabaticity is governed by classical bifurcation phenomena. Magnonic Dicke models exhibit a pitchfork bifurcation in the classical dynamics at the superradiant threshold. Universality in the redistribution of excitation is revealed by a phase-dependent “jump” in the action variable, with the quantum action change exhibiting a universal “log-sine” distribution,

$\Delta I \sim -\frac{2}{\pi} \ln(2 \sin \pi \xi)$

with $\xi \in (0,1)$ a uniformly distributed pseudophase (0901.4778). The semiclassical dynamics near criticality may be mapped to the second Painlevé equation (PII), and the resulting particle number statistics after the sweep are asymptotically independent of microscopic parameter details. This mapping applies even in the presence of counter-rotating (non-RWA) terms and in reductions of the Dicke model to the Lipkin–Meshkov–Glick limit.

These universal results are experimentally accessible in passage experiments, such as Feshbach sweeps in cold atoms or collective mode ramping in magnonic insulators.

5. Gauge Invariance and Theoretical Consistency in the Ultrastrong Coupling Regime

Theoretical consistency of the Dicke model, including its magnonic extensions, must carefully consider gauge invariance, especially in the ultrastrong coupling regime ($g \sim \omega$). Analysis demonstrates that naive truncation of the Hilbert space to two levels renders the matter potential nonlocal, invalidating the minimal coupling prescription unless a dressing via a unitary operator (implementing the full physical gauge transformation) is applied (2002.04241). The correct gauge-invariant Hamiltonian is required to predict polariton spectra and phase boundaries accurately. In the dilute and thermodynamic limit, the gauge-invariant Dicke model reduces to a bilinear form equivalent to the Hopfield model for bosonic polarization, ensuring equivalence of polariton branches and critical behavior. Application of these principles to magnonic cavity systems ensures theoretical predictions retain physical meaning in realistic experimental contexts, including those accessing the ultrastrong regime.

6. Extensions: Lattices, Higher-Order Symmetry, Non-Reciprocity, and Driven-Dissipative Physics

Hybrid quantum architectures now realize spatially structured or multi-ensemble extensions of the Dicke model. Examples include:

• Dicke lattice models: Ensembles of NV centers in diamond coupled to arrays of superconducting cavities, supporting collective phases, pattern formation (finite-momentum superradiance), and non-equilibrium phase transitions sensitive to inhomogeneous broadening and dissipation (Zou et al., 2014).
• Spin-1 and multi-level Dicke models: Implemented using atomic F = 1 hyperfine manifolds, with imbalanced drives creating steady-state and oscillatory (limit-cycle) superradiant phases, mapped by cavity output spectroscopy (Zhiqiang et al., 2016).
• n-phase Dicke models with higher-order discrete symmetry: Engineered in driven atom tweezer arrays, where phase-imprinted couplings result in $\mathbb{Z}_n$ or $\mathbb{Z}_{2n}$ symmetry-breaking superradiant states and non-reciprocal, non-Hermitian dynamics. The transition between symmetric and broken-symmetry phases may become first order at finite detuning and cavity loss (Ho et al., 5 Oct 2025). Effective Hamiltonian couplings take the form $$H_{\mathrm{eff},jl} \sim \Delta_{\mathrm{pc}} \cos \phi(j − l) − \kappa \sin \phi(j − l)$$, creating tunable non-reciprocity between subgroups—implementable in both optomechanical and opto-magnonic systems.

These model extensions provide testbeds for exploring dynamical instabilities, pattern formation, directionality and symmetry of collective states, and non-Hermitian quantum phenomena relevant to quantum information and simulation platforms.

7. Universality and Mapping of Strongly Correlated Light–Matter Systems

Recent theoretical work has established that a broad class of light–matter models with complex matter–matter interactions, including Ising chains and strongly correlated magnetic systems, are reducible in their non-superradiant regimes to an effective Dicke model that captures the low-energy physics (Schellenberger et al., 23 Feb 2024). This mapping is achieved by expressing spins in hard-core boson language, transforming to momentum space, and showing that only the $k=0$ magnon couples to the light field for large system size. Correlated matter–matter terms decouple, and the Dicke model captures the criticality and collective excitation spectrum. This framework enables analytic tractability in the analysis of magnonic and hybrid cavity systems across a wide domain, with the phase transition signalled by the closing of the $k=0$ excitation gap even in the presence of strong correlations.

Table: Key Experimental and Theoretical Parameters in Magnonic Dicke Model Realizations

Parameter/Feature	Example Value/Setting	Reference System/Paper
Coupling strength $g_z$	$2\pi \times 0.116$ THz	ErFeO₃ (Bamba et al., 2020)
Magnon frequency $\omega_\pi$	$2\pi \times 0.896$ THz	ErFeO₃ (Bamba et al., 2020)
Critical temp. $T_c$	$\sim 4$ K	ErFeO₃ (Bamba et al., 2020)
Short-range exchange $J$	$0.60$ meV	ErFeO₃ (Bamba et al., 2020, Peraca et al., 2023)
Collective enhancement	$g_{\rm eff} \propto \sqrt{N}$	All Dicke-type models
Spin glass order param. $q$	Nonzero in QSG phase	Multimode Dicke (Strack et al., 2011)

References and Outlook

• The superradiant phase transition in solids (ErFeO₃) is an intrinsic thermal equilibrium effect, in contrast to the driven-dissipative transitions in cold atom or cavity QED systems (Bamba et al., 2020, Peraca et al., 2023).
• The phase diagram of extended models reveals atomic-ordered, superradiant, and normal phases, with phase boundaries mapped experimentally using THz spectroscopy and the magnetocaloric effect (Peraca et al., 2023).
• Disorder and random couplings, as well as spatially modulated or complex-valued couplings in lattice and multi-mode settings, can yield spin-glass, pattern-forming, or non-reciprocal phases (Strack et al., 2011, Zou et al., 2014, Ho et al., 5 Oct 2025).
• Universality of dynamical statistics and phase transition signatures (e.g., logarithmic corrections to action, log-sine fluctuation distribution) provides robust experimental markers (0901.4778).

Magnonic Dicke models thus constitute a versatile class of collective quantum models, unifying theoretical advances in light–matter criticality, quantum simulation, and symmetry-breaking phenomena with experimentally accessible platforms in both atomic and solid-state settings. Their continued paper enables the controlled exploration of collective quantum phase transitions, quantum criticality, and the interplay of disorder, symmetry, and non-equilibrium physics in many-body systems.