Chiral Dicke Model Overview

• The Chiral Dicke Model is a family of light–matter Hamiltonians that encode chirality via synthetic gauge flux or angular-momentum–selective coupling.
• It extends the standard Dicke model by introducing directional superradiant phases and rich symmetry properties including U(1) and Z2 invariances.
• Research explores tricritical behavior, Goldstone modes, and experimental realizations in optical cavities, circuit QED, and chiral quantum-optics architectures.

The chiral Dicke model denotes a family of Dicke-type light–matter Hamiltonians in which chirality is encoded either through synthetic gauge flux in coupled cavities or through angular-momentum–selective coupling to multiple cavity modes. In current arXiv usage, the term therefore does not identify a single canonical Hamiltonian but a class of constructions sharing collective light–matter coupling, superradiant ordering, and an additional directional or handed structure. Two representative realizations are the Dicke triangle, where artificial magnetic flux produces a chiral superradiant phase with circulating photon current and tricritical manifolds (Chen et al., 2022), and the two-mode chiral Dicke model, where an atomic ensemble couples chirally to two degenerate cavity modes and the Hamiltonian has an exact continuous $U(1)$ symmetry for arbitrary couplings (Yegovtsev et al., 23 Apr 2026). Earlier analyses of the full Dicke model with independent rotating and counter-rotating couplings provide the natural symmetry benchmark for these developments (Alcalde et al., 2010).

1. Hamiltonian architectures

One line of development starts from a tricritical Dicke cavity. In the Dicke triangle, each cavity contains an ensemble of $N$ identical three-level atoms in a $\Lambda$-like configuration with levels $\{|1\rangle,|0\rangle,|-1\rangle\}$, energies $\{+1,0,-1\}$ in units of $\Omega$, and cavity photons of frequency $\omega$. The single-cavity Hamiltonian is

$$H_D=\omega a^\dagger a+\frac{\sqrt{2}g}{\sqrt{N}}(a^\dagger+a)\sum_{k=1}^N d^{(k)}+\Omega\sum_{k=1}^N h^{(k)},$$

where the parameter $\gamma$ controls the relative strength of the two dipole-allowed transitions $|1\rangle\leftrightarrow|0\rangle$ and $N$0, and the effective dimensionless coupling is $N$1. Three such cavities are then coupled on an equilateral triangle by photon hopping with a Peierls phase,

$N$2

with cyclic boundary condition $N$3. The total artificial magnetic flux through the triangular plaquette is, up to gauge, $N$4, so $N$5 parameterizes a synthetic magnetic field (Chen et al., 2022).

A second construction is the two-mode chiral Dicke model. Its Hamiltonian is

$N$6

where mode $N$7 couples through co-rotating terms and mode $N$8 through counter-rotating terms. The two cavity modes are interpreted as circularly polarized or counter-propagating running-wave modes carrying opposite angular momentum along the cavity axis, and chirality is tied to the handedness of angular-momentum exchange between light and matter (Yegovtsev et al., 23 Apr 2026).

The single-mode full Dicke model furnishes the antecedent reference point. With a single bosonic mode $N$9, $\Lambda$0 two-level atoms, and independent rotating and counter-rotating couplings $\Lambda$1 and $\Lambda$2, its Hamiltonian is

$\Lambda$3

This model already distinguishes the rotating-wave, pure counter-rotating, and generic full regimes, and thereby clarifies which symmetry survives when both processes coexist (Alcalde et al., 2010).

2. Symmetry content and the meaning of chirality

In the Dicke triangle, chirality is flux-induced. For $\Lambda$4, the hopping is real and time-reversal symmetry is preserved. For general $\Lambda$5, the Hamiltonian is complex, time-reversal symmetry is broken, and the ground state can carry nontrivial circulating photon currents. The model retains a global parity symmetry,

$\Lambda$6

and at $\Lambda$7 the full symmetry includes $\Lambda$8 parity and discrete rotation $\Lambda$9 (Chen et al., 2022).

In the two-mode chiral Dicke model, chirality is angular-momentum selective rather than flux-driven. The defining feature is an exact continuous $\{|1\rangle,|0\rangle,|-1\rangle\}$0 symmetry for all $\{|1\rangle,|0\rangle,|-1\rangle\}$1, $\{|1\rangle,|0\rangle,|-1\rangle\}$2, and $\{|1\rangle,|0\rangle,|-1\rangle\}$3: $\{|1\rangle,|0\rangle,|-1\rangle\}$4 The symmetry is generated by the conserved quantity

$\{|1\rangle,|0\rangle,|-1\rangle\}$5

interpreted as the total angular momentum along the cavity axis. The model is therefore chiral in the sense that the two modes carry opposite chirality and angular momentum conservation enforces a robust continuous symmetry rather than a fine-tuned one (Yegovtsev et al., 23 Apr 2026).

The symmetry distinction from the single-mode full Dicke model is sharp. In the path-integral analysis of the full Dicke model, the rotating-wave limit $\{|1\rangle,|0\rangle,|-1\rangle\}$6 conserves total excitation number $\{|1\rangle,|0\rangle,|-1\rangle\}$7, the pure counter-rotating limit $\{|1\rangle,|0\rangle,|-1\rangle\}$8 conserves the excitation difference $\{|1\rangle,|0\rangle,|-1\rangle\}$9, and both limits have a continuous $\{+1,0,-1\}$0 symmetry. By contrast, when $\{+1,0,-1\}$1 and $\{+1,0,-1\}$2, only parity $\{+1,0,-1\}$3 remains (Alcalde et al., 2010). This shows that chirality in the two-mode chiral Dicke model is not simply the asymmetry $\{+1,0,-1\}$4; the robust $\{+1,0,-1\}$5 structure requires the two-mode angular-momentum construction itself (Yegovtsev et al., 23 Apr 2026).

A recurrent misconception is that all chiral Dicke models must break time-reversal symmetry in the same way. The current literature suggests two distinct mechanisms: synthetic-flux chirality, which produces circulating currents and explicitly complex hopping amplitudes, and angular-momentum chirality, which produces exact $\{+1,0,-1\}$6 symmetry and a Goldstone sector without requiring a triangular flux geometry.

3. Ordered phases and order parameters

For the Dicke triangle, the normal phase is defined by $\{+1,0,-1\}$7 in all cavities. In the superradiant regime one writes

$\{+1,0,-1\}$8

so the six real variables $\{+1,0,-1\}$9 and $\Omega$0 serve as order parameters. The conventional superradiant phase has

$\Omega$1

so the coherent fields are uniform and real. The chiral superradiant phase instead has complex, nonuniform $\Omega$2, and can be represented in a convenient gauge by

$\Omega$3

More generally,

$\Omega$4

These configurations break both parity $\Omega$5 and the threefold rotation $\Omega$6, producing a six-fold degenerate ground state after accounting for the three choices of the distinguished cavity and the two parity-related signs (Chen et al., 2022).

The signature of chirality in the triangle is the photon current operator

$\Omega$7

At mean field,

$\Omega$8

In the conventional superradiant phase all phase differences vanish and $\Omega$9. In the chiral superradiant phase, $\omega$0, its sign changes under $\omega$1, and tuning $\omega$2 drives a jump from zero to finite positive or negative current when entering the CSR region (Chen et al., 2022).

The two-mode chiral Dicke model also has a normal and a superradiant phase, but the structure is different. Using a Holstein–Primakoff mapping and coherent-state shifts $\omega$3, the normal phase is

$\omega$4

The superradiant solution has

$\omega$5

with an arbitrary phase $\omega$6. The phase boundary is

$\omega$7

and for $\omega$8 this reduces to $\omega$9, or equivalently $$H_D=\omega a^\dagger a+\frac{\sqrt{2}g}{\sqrt{N}}(a^\dagger+a)\sum_{k=1}^N d^{(k)}+\Omega\sum_{k=1}^N h^{(k)},$$0 in the ordered phase. The order parameter therefore has a magnitude that separates normal and superradiant regimes and a phase that parameterizes the spontaneously broken $$H_D=\omega a^\dagger a+\frac{\sqrt{2}g}{\sqrt{N}}(a^\dagger+a)\sum_{k=1}^N d^{(k)}+\Omega\sum_{k=1}^N h^{(k)},$$1 manifold $$H_D=\omega a^\dagger a+\frac{\sqrt{2}g}{\sqrt{N}}(a^\dagger+a)\sum_{k=1}^N d^{(k)}+\Omega\sum_{k=1}^N h^{(k)},$$2 (Yegovtsev et al., 23 Apr 2026).

4. Critical manifolds, tricriticality, and multiversality

The Dicke triangle supports a richer phase diagram than the ordinary Dicke model. In a single cavity, the tricritical Dicke model already exhibits normal and superradiant phases separated by both first- and second-order lines meeting at a tricritical point. In the triangle, both the conventional superradiant sector and the chiral superradiant sector inherit this structure. The quartic Landau coefficient in both SR and CSR expansions is

$$H_D=\omega a^\dagger a+\frac{\sqrt{2}g}{\sqrt{N}}(a^\dagger+a)\sum_{k=1}^N d^{(k)}+\Omega\sum_{k=1}^N h^{(k)},$$3

so the normal-to-superradiant transition is second order for $$H_D=\omega a^\dagger a+\frac{\sqrt{2}g}{\sqrt{N}}(a^\dagger+a)\sum_{k=1}^N d^{(k)}+\Omega\sum_{k=1}^N h^{(k)},$$4 and first order for $$H_D=\omega a^\dagger a+\frac{\sqrt{2}g}{\sqrt{N}}(a^\dagger+a)\sum_{k=1}^N d^{(k)}+\Omega\sum_{k=1}^N h^{(k)},$$5. This yields a line of tricritical points for the SR sector and a line of chiral tricritical points for the CSR sector, both at $$H_D=\omega a^\dagger a+\frac{\sqrt{2}g}{\sqrt{N}}(a^\dagger+a)\sum_{k=1}^N d^{(k)}+\Omega\sum_{k=1}^N h^{(k)},$$6. The full three-dimensional phase diagram in $$H_D=\omega a^\dagger a+\frac{\sqrt{2}g}{\sqrt{N}}(a^\dagger+a)\sum_{k=1}^N d^{(k)}+\Omega\sum_{k=1}^N h^{(k)},$$7 contains two superradiant regions, SR and CSR, separated by a first-order surface, and at $$H_D=\omega a^\dagger a+\frac{\sqrt{2}g}{\sqrt{N}}(a^\dagger+a)\sum_{k=1}^N d^{(k)}+\Omega\sum_{k=1}^N h^{(k)},$$8 the normal, SR, and CSR phases meet at triple points (Chen et al., 2022).

The critical exponents separate ordinary criticality from tricriticality and also distinguish the uniform and chiral condensation channels. Defining the distance $$H_D=\omega a^\dagger a+\frac{\sqrt{2}g}{\sqrt{N}}(a^\dagger+a)\sum_{k=1}^N d^{(k)}+\Omega\sum_{k=1}^N h^{(k)},$$9 from a point to the nearest critical point along a line perpendicular to the relevant critical boundary, the total photon number satisfies

$\gamma$0

Along ordinary second-order NP–SR or NP–CSR lines,

$\gamma$1

whereas at both TCP and CTCP,

$\gamma$2

For the lowest excitation gap,

$\gamma$3

with

$\gamma$4

At the triple points, two excitation energies vanish simultaneously with exponents $\gamma$5 and $\gamma$6, reflecting the coexistence of distinct critical modes (Chen et al., 2022).

The two-mode chiral Dicke model exhibits a different phenomenon, termed multiversality. The normal and $\gamma$7-broken superradiant phases are always separated by the same critical circle $\gamma$8 in the $\gamma$9 plane, but the gap-closing exponent depends on the approach direction. Writing

$|1\rangle\leftrightarrow|0\rangle$0

the lowest normal-phase gap scales generically as

$|1\rangle\leftrightarrow|0\rangle$1

Along the special line

$|1\rangle\leftrightarrow|0\rangle$2

where two excitation branches become degenerate at criticality, the scaling changes to

$|1\rangle\leftrightarrow|0\rangle$3

The same pair of phases and the same phase boundary therefore support two distinct universality classes, depending only on the path taken in coupling space (Yegovtsev et al., 23 Apr 2026).

5. Excitations, soft modes, and Goldstone structure

In the Dicke triangle, the normal phase can be reduced, after a Schrieffer–Wolff transformation and projection onto the lowest atomic state, to an effective three-mode bosonic Hamiltonian. Fourier transforming to momenta $|1\rangle\leftrightarrow|0\rangle$4 gives mode energies

$|1\rangle\leftrightarrow|0\rangle$5

and a Bogoliubov transformation yields quasiparticle energies

$|1\rangle\leftrightarrow|0\rangle$6

The normal phase is stable as long as all $|1\rangle\leftrightarrow|0\rangle$7. Instability of the $|1\rangle\leftrightarrow|0\rangle$8 mode signals a continuous transition to the uniform SR phase, while instability of the $|1\rangle\leftrightarrow|0\rangle$9 modes signals condensation into the CSR phase (Chen et al., 2022).

In the two-mode chiral Dicke model, quadratic expansion around the mean-field solution yields a Bogoliubov Hamiltonian for the two photon modes and the Holstein–Primakoff boson. In the superradiant phase there are two gapped polaritonic modes with explicitly tunable gaps and, in addition, a Goldstone mode

$N$00

throughout the superradiant phase in the thermodynamic limit. The spectrum is tunable by the chiral imbalance between $N$01 and $N$02, the detuning $N$03, and the dispersive coupling $N$04. The paper emphasizes that the structure of one Goldstone mode plus two gapped modes persists for $N$05 (Yegovtsev et al., 23 Apr 2026).

The contrast with the single-mode full Dicke model again clarifies the symmetry content. In the path-integral treatment of the full Dicke model, the rotating-wave and pure counter-rotating limits have a continuous $N$06 symmetry and the superradiant phase contains a zero-energy Goldstone mode. In the generic full model with both $N$07 and $N$08, only discrete $N$09 symmetry survives and the superradiant phase has no zero mode; all excitations remain gapped (Alcalde et al., 2010). This benchmark makes clear why the robust Goldstone mode of the two-mode chiral Dicke model is structurally nontrivial.

The two-mode chiral Dicke model also admits a dissipative mean-field extension. In the supplemental analysis summarized in the paper, a nontrivial steady-state solution exists for $N$10 and

$N$11

and the corresponding superradiant $N$12-broken state is linearly stable. The equilibrium Goldstone interpretation therefore remains relevant, at mean-field level, even in the presence of moderate cavity loss (Yegovtsev et al., 23 Apr 2026).

6. Experimental realizations, related models, and broader scope

The Dicke triangle was proposed as implementable both with atoms in optical cavities and in circuit QED. In the optical-cavity setting, three cavities or three sites of a cavity array form a triangle, each cavity hosts a large ensemble of atoms with a suitable three-level structure, photon hopping is engineered through evanescent coupling or fiber links, and time-periodic modulation of hopping generates the synthetic phase $N$13. In circuit QED, three superconducting resonators are coupled in a loop, each resonator is coupled to many effective three-level systems, and parametrically modulated couplers implement synthetic gauge fields. Superradiance can be detected through cavity transmission spectra, homodyne detection, or quadrature measurements; chirality can be accessed through direction-dependent output or phase correlations, with $N$14 expected to jump discontinuously across the SR–CSR boundary and hysteresis expected for first-order transitions (Chen et al., 2022).

The two-mode chiral Dicke model is motivated by platforms with two counter-propagating or two circularly polarized modes. The paper identifies high-finesse ring cavities supporting two running-wave modes, circuit-QED realizations with two resonator modes and a superconducting qubit ensemble, and chiral quantum-optics architectures such as waveguide QED or photonic-crystal waveguides with spin–orbit coupling of light as candidate settings. Observation of the ordered phase would require detecting macroscopic occupations $N$15, $N$16, a freely rotating relative phase constrained by the $N$17 symmetry, and a low-energy near-zero-frequency spectral line associated with the Goldstone mode. Observation of multiversality would require controllable variation of the ratio $N$18 and spectroscopic resolution of $N$19 versus $N$20 gap closing (Yegovtsev et al., 23 Apr 2026).

The broader literature contains adjacent chiral Dicke-like constructions. In chiral waveguide optomechanics, atoms trapped near a chiral ring waveguide map to a generalized quantum Rabi model, and the three-atom case realizes a $N$21-symmetric Dicke-like problem with a first-order quantum phase transition and multicomponent Schrödinger-cat ground states (Sedov et al., 2020). In quantum Hall systems, the coupling of nuclear spins to a linearly dispersing Nambu–Goldstone mode in the canted antiferromagnetic phase yields an effective Dicke model with a continuous-mode bosonic field, where the long-wavelength approximation produces collective coupling of the total nuclear spin to the NG mode (Hama et al., 2015). These constructions are not identical to the flux-threaded Dicke triangle or the two-mode chiral Dicke model, but they show that chiral Dicke physics extends beyond conventional cavity-QED geometries.

Several open directions are explicitly identified. For the Dicke triangle these include larger rings and $N$22D arrays under synthetic flux, quantum fluctuations beyond the $N$23 mean-field limit, real-time dynamics across TCP and CTCP, and the inclusion of drive and dissipation (Chen et al., 2022). For the two-mode chiral Dicke model they include a full driven-dissipative phase diagram, periodic and quasiperiodic driving, finite temperature, disorder, multimode or multiband generalizations, and the effect of chirality and the robust Goldstone mode on quantum chaos and scrambling (Yegovtsev et al., 23 Apr 2026). Taken together, these proposals suggest that the chiral Dicke model is best viewed as a developing family of long-range light–matter theories in which symmetry, chirality, and superradiance are intertwined in ways that are not available in the standard single-mode Dicke model.