Closed Dicke Lattice Dynamics

• Closed Dicke lattice is a family of non-dissipative, Hamiltonian light–matter models characterized by varying lattice constructions such as Peres, momentum-space, and spatial arrays.
• The framework employs spectral diagnostics including Peres lattices and energy–momentum maps to identify integrability, chaos, ESQPTs, and monodromy defects across different regimes.
• These models predict equilibrium superradiant phase transitions and critical scaling behaviors through coupled cavity–spin arrays and related constructions in both real and momentum space.

Searching arXiv for recent and foundational papers on "closed Dicke lattice" and related usages. “Closed Dicke lattice” is not a single standardized object but a family of closely related constructions built from Dicke or Tavis–Cummings light–matter models under Hamiltonian, non-dissipative dynamics. In one usage, “closed” means a Hermitian Dicke Hamiltonian and “lattice” means a Peres lattice, namely a plot of eigenstate expectation values versus energy used to diagnose integrability, chaos, and excited-state singularities. In another, it denotes ordered eigenstate lattices in the integrable Tavis–Cummings limit, where monodromy produces topological defects in energy–momentum or Peres plots. In a third, it refers to genuine spatial lattices or rings of coupled Dicke units, whose equilibrium superradiant phases are selected by photon hopping and lattice symmetry. Further uses occur in momentum-space “superradiance lattices” built from timed Dicke states and in ordered atomic arrays where superradiant avalanches effectively close competing decay channels (Bastarrachea-Magnani et al., 2013, Kloc et al., 2017, Wei et al., 1 Jul 2026, Wang et al., 2014, Masson et al., 2023).

1. Terminological scope and defining features

Across these usages, “closed” always denotes the absence of dissipation, baths, or Lindblad terms. In the single-mode Dicke studies, it means unitary dynamics of a Hermitian Hamiltonian. In coupled-cavity Dicke lattices it is implemented by setting the photon loss rate to zero, so that phase structure is determined by equilibrium energy minimization and Gaussian fluctuations rather than by steady-state balance. In the Tavis–Cummings monodromy setting, it similarly refers to an isolated Hamiltonian system whose regular lattices of eigenstate expectation values are meaningful precisely because one is working with Hamiltonian eigenstates rather than Liouvillian modes (Bastarrachea-Magnani et al., 2013, Kloc et al., 2017, Wei et al., 1 Jul 2026).

The word “lattice” is used in at least three technically distinct senses. First, it can mean a Peres lattice, i.e. a visual grid-like arrangement of points obtained by plotting $I_O(n)=\langle \psi_n|O|\psi_n\rangle$ against the eigenenergy $E_n$. Second, it can mean a joint spectral or expectation-value lattice in the integrable Tavis–Cummings model, typically organized by energy and the conserved excitation number $M$. Third, it can mean a genuine spatial lattice of coupled cavity–spin units, such as a one-dimensional ring with periodic boundary conditions, a trimer with flux, or a Dicke–Ising chain or square lattice. The superradiance-lattice literature introduces yet another usage: a tight-binding lattice in momentum space whose sites are timed Dicke states generated by EIT standing waves, while the ordered-array literature uses “closed” in the different sense that cooperative decay effectively closes weaker internal transitions (Wang et al., 2014, Masson et al., 2023).

This semantic multiplicity is not merely lexical. It separates three different research programs: spectral diagnostics of chaos and ESQPTs, topological defects in integrable spectra, and equilibrium many-body phase structure of spatially extended Dicke systems. The same phrase therefore names different objects depending on whether the “lattice” is a visualization, a quantum-number grid, a momentum-space construction, or a real-space array.

2. Peres lattices in the closed Dicke model

In the single-mode Dicke setting, the closed model is studied in the symmetric atomic subspace $j=N/2$ with

$$H=\omega a^\dagger a+\omega_0 J_z+\frac{2\gamma}{\sqrt{N}}(a+a^\dagger)J_x,$$

with $[a,a^\dagger]=1$ and $[J_i,J_j]=i\epsilon_{ijk}J_k$. The Hamiltonian commutes with the parity operator $\Pi=\exp(i\pi\Lambda)$, where $\Lambda=a^\dagger a+J_z+j$, so parity is used to block-diagonalize the spectrum. Because $J_x$ flips parity, $E_n$0 itself is not used as a Peres operator; $E_n$1 is used instead. Integrable limits occur for $E_n$2 and $E_n$3, and the rotating-wave Tavis–Cummings approximation restores conservation of $E_n$4. In the thermodynamic limit the normal-to-superradiant critical coupling is

$E_n$5

which becomes $E_n$6 at resonance $E_n$7 (Bastarrachea-Magnani et al., 2013).

A Peres lattice plots, for each eigenstate $E_n$8, the expectation value

$E_n$9

against $M$0, equivalently the long-time average of $M$1 for that eigenstate. The observables used are $M$2 with $M$3, and lattices are constructed separately in each parity sector. Ordered point patterns indicate regular dynamics, while scrambled point clouds indicate quantum chaos; local distortions inside otherwise ordered regions reveal mixed regular–chaotic structure (Bastarrachea-Magnani et al., 2013).

At resonance and for $M$4 atoms ($M$5), the Peres lattices are highly regular near $M$6, consistent with integrability. The eigenenergies follow $M$7, and degeneracies appear as vertical point stacks at identical energies but different expectation values. For nonzero couplings below the superradiant threshold, regular lattices persist at lower energies while irregular regions appear at higher energies, notably for $M$8. The key point is that chaos appears well before the equilibrium superradiant transition. For $M$9, global irregularity increases, but an ordered low-energy island develops below $j=N/2$0 in all three lattices and grows with increasing coupling; at $j=N/2$1 and $j=N/2$2 the plots show a sharp visual separation between this ordered region and chaotic higher-energy sectors (Bastarrachea-Magnani et al., 2013).

The same Peres geometry exposes two ESQPTs. In the $j=N/2$3 lattice, structural slope changes occur at $j=N/2$4 and $j=N/2$5. The static ESQPT at $j=N/2$6 is associated with saturation of the atomic excitation manifold and is independent of $j=N/2$7. The dynamic ESQPT at $j=N/2$8 depends on $j=N/2$9 and is tied to the fact that the configuration $$H=\omega a^\dagger a+\omega_0 J_z+\frac{2\gamma}{\sqrt{N}}(a+a^\dagger)J_x,$$0, which is the ground state in the normal phase, ceases to be the ground state in the superradiant phase; strong coupling populates many-photon, many-excitation states with $$H=\omega a^\dagger a+\omega_0 J_z+\frac{2\gamma}{\sqrt{N}}(a+a^\dagger)J_x,$$1. The emergence and growth of the regular island below $$H=\omega a^\dagger a+\omega_0 J_z+\frac{2\gamma}{\sqrt{N}}(a+a^\dagger)J_x,$$2 are visually linked to that dynamic ESQPT (Bastarrachea-Magnani et al., 2013).

The numerical diagonalization uses an extended displaced bosonic basis defined by

$$H=\omega a^\dagger a+\omega_0 J_z+\frac{2\gamma}{\sqrt{N}}(a+a^\dagger)J_x,$$3

with basis states $$H=\omega a^\dagger a+\omega_0 J_z+\frac{2\gamma}{\sqrt{N}}(a+a^\dagger)J_x,$$4 satisfying $$H=\omega a^\dagger a+\omega_0 J_z+\frac{2\gamma}{\sqrt{N}}(a+a^\dagger)J_x,$$5, and parity-adapted combinations

$$H=\omega a^\dagger a+\omega_0 J_z+\frac{2\gamma}{\sqrt{N}}(a+a^\dagger)J_x,$$6

Typical truncations are $$H=\omega a^\dagger a+\omega_0 J_z+\frac{2\gamma}{\sqrt{N}}(a+a^\dagger)J_x,$$7–$$H=\omega a^\dagger a+\omega_0 J_z+\frac{2\gamma}{\sqrt{N}}(a+a^\dagger)J_x,$$8, and the reported truncation error bound is $$H=\omega a^\dagger a+\omega_0 J_z+\frac{2\gamma}{\sqrt{N}}(a+a^\dagger)J_x,$$9 for the displayed spectral windows. The method yields large sets of converged excited states and makes the fine structure of regular, chaotic, and ESQPT-organized sectors directly visible (Bastarrachea-Magnani et al., 2013).

3. Energy–momentum lattices, monodromy, and the integrable Tavis–Cummings sector

A different closed Dicke-lattice construction appears in the integrable limit of the extended Dicke model with counter-rotating strength $[a,a^\dagger]=1$0. Setting $[a,a^\dagger]=1$1 gives the Tavis–Cummings Hamiltonian, for which the $[a,a^\dagger]=1$2 excitation number

$[a,a^\dagger]=1$3

is conserved. In this regime, eigenstates can be organized simultaneously by energy and $[a,a^\dagger]=1$4, and both energy–momentum maps and Peres lattices form ordered arrays with topological defects related to monodromy rather than to chaos. The corresponding classical large-$[a,a^\dagger]=1$5 limit has two degrees of freedom, and after a canonical transformation the dynamics at fixed $[a,a^\dagger]=1$6 reduces to an effective one-degree-of-freedom Hamiltonian (Kloc et al., 2017).

At $[a,a^\dagger]=1$7 and above the coupling threshold

$[a,a^\dagger]=1$8

the point $[a,a^\dagger]=1$9 becomes an unstable stationary point at energy $[J_i,J_j]=i\epsilon_{ijk}J_k$0. The critical classical fiber $[J_i,J_j]=i\epsilon_{ijk}J_k$1 is a singular pinched torus of focus–focus type. Near the north pole of the Bloch sphere the motion spirals, takes infinite time to pass the focus, and realizes the closed-system analogue of dynamic superradiance: pulse-like exchange of excitations between field and atoms, slow near the poles and fastest near the equator. In the tuned case $[J_i,J_j]=i\epsilon_{ijk}J_k$2, the photon emission or absorption rate scales linearly with $[J_i,J_j]=i\epsilon_{ijk}J_k$3 in this closed setting (Kloc et al., 2017).

Quantum mechanically, the joint spectrum $[J_i,J_j]=i\epsilon_{ijk}J_k$4 forms a lattice with a point defect at $[J_i,J_j]=i\epsilon_{ijk}J_k$5. Encircling this defect transforms the lattice cell by

$[J_i,J_j]=i\epsilon_{ijk}J_k$6

which is the transpose of the classical monodromy matrix for one focus–focus singularity. Peres lattices built from $[J_i,J_j]=i\epsilon_{ijk}J_k$7 or $[J_i,J_j]=i\epsilon_{ijk}J_k$8 versus $[J_i,J_j]=i\epsilon_{ijk}J_k$9 show corresponding dislocations, although the local topology of the defect depends on the chosen observable. In the integrable case, chains at fixed principal quantum number across varying $\Pi=\exp(i\pi\Lambda)$0 make the kinked structure near the defect especially clear (Kloc et al., 2017).

The same energy marks an ESQPT. Within the critical $\Pi=\exp(i\pi\Lambda)$1 subspace, the unstable stationary point produces an ESQPT at

$\Pi=\exp(i\pi\Lambda)$2

The semiclassical density of states in that subspace has a logarithmic divergence at $\Pi=\exp(i\pi\Lambda)$3, while in the full two-degree-of-freedom spectrum the first derivative of the level density has a downward jump. The alignment of the ESQPT energy with the monodromy defect increases the local spectral density and amplifies the visibility of the defect in both energy–momentum and Peres lattices (Kloc et al., 2017).

Once counter-rotating terms are restored, $\Pi=\exp(i\pi\Lambda)$4 symmetry is broken and only parity remains. Classically, the pinched-torus neighborhood becomes chaotic already for very small $\Pi=\exp(i\pi\Lambda)$5, and quantum lattices rapidly lose their point-defect structure. The monodromy point lies on a line of minimal level spacing, so generic perturbations induce strong level repulsion, tear the lattice along a vertical break rooted at the monodromy energy, and disorder the Peres plots (Kloc et al., 2017).

4. Closed Dicke lattices as coupled cavity–spin arrays

In the spatial sense, a closed Dicke lattice is a ring or array of local Dicke units coupled by photon hopping. A representative Hamiltonian is

$\Pi=\exp(i\pi\Lambda)$6

with

$\Pi=\exp(i\pi\Lambda)$7

periodic boundary conditions $\Pi=\exp(i\pi\Lambda)$8, and large-$\Pi=\exp(i\pi\Lambda)$9 equilibrium analysis based on mean field plus Holstein–Primakoff theory. In the superradiant regime each site develops a real coherent field $\Lambda=a^\dagger a+J_z+j$0, and the sign pattern of the $\Lambda=a^\dagger a+J_z+j$1 defines the “configuration” class. For $\Lambda=a^\dagger a+J_z+j$2 the symmetry-inequivalent classes are $\Lambda=a^\dagger a+J_z+j$3, $\Lambda=a^\dagger a+J_z+j$4, $\Lambda=a^\dagger a+J_z+j$5, and $\Lambda=a^\dagger a+J_z+j$6, with degeneracies $\Lambda=a^\dagger a+J_z+j$7, $\Lambda=a^\dagger a+J_z+j$8, $\Lambda=a^\dagger a+J_z+j$9, and $J_x$0, respectively (Wei et al., 1 Jul 2026).

The closed-lattice ground state is determined by minimizing

$J_x$1

The on-site term favors nonzero $J_x$2 above threshold, while the hopping term enforces sign alignment or alternation. For $J_x$3, the energy is minimized by the uniform ferromagnetic-like configuration $J_x$4; for $J_x$5, it is minimized by the staggered antiferromagnetic-like configuration $J_x$6. The Hessian eigenvalues around the normal phase are

$J_x$7

which gives the mode-resolved threshold

$J_x$8

For $J_x$9, this reduces to $E_n$00. The normal-to-superradiant transition is continuous, whereas scanning $E_n$01 through zero at fixed $E_n$02 produces a first-order transition between the ferromagnetic and antiferromagnetic superradiant configurations. The lowest excitation gap and the ground-state photon fluctuations scale with exponent $E_n$03, matching the equilibrium mean-field Dicke universality class (Wei et al., 1 Jul 2026).

A closely related closed Dicke lattice model arises in hybrid arrays of superconducting microwave cavities coupled to ensembles of NV centers. The effective Hamiltonian is

$E_n$04

with cavity dispersion $E_n$05. In the closed limit $E_n$06, the momentum-resolved instability threshold is

$E_n$07

and the global threshold is the minimum over $E_n$08. For a 1D nearest-neighbor chain with $E_n$09, the minimum lies at $E_n$10, so the equilibrium superradiant phase is uniform. The open-system finite-$E_n$11 instabilities emphasized in the same work are explicitly described as non-equilibrium effects absent in the closed Hamiltonian limit (Zou et al., 2014).

The trimer provides the minimal odd ring. Its closed Hamiltonian,

$E_n$12

contains unbalanced rotating and counter-rotating couplings through $E_n$13. At $E_n$14, the model restores the $E_n$15 symmetry of the Tavis–Cummings limit and the superradiant phase contains a Goldstone mode. The classical ground-state functional supports a translationally invariant non-frustrated superradiant phase and a translation-breaking frustrated superradiant phase. The normal-phase Hessian has six exact eigenvalues, yielding non-frustrated and frustrated critical branches $E_n$16 and $E_n$17; the actual threshold is their minimum. At time-reversal-symmetric $E_n$18, two soft modes emerge at the normal-phase boundary with exponent $E_n$19, while on the frustrated side the low-energy mode scales as $E_n$20. For $E_n$21, the normal-phase soft mode instead scales as $E_n$22 and exhibits anomalous finite critical fluctuations because the conjugate quadratic stiffnesses vanish with the same exponent (Zhang et al., 2023).

5. Closed equilibrium Dicke lattices with additional matter interactions

The equilibrium Dicke–Ising model extends the closed Dicke lattice by adding short-range Ising couplings among the spins while retaining a single quantized cavity mode. The Hamiltonian is

$E_n$23

It is studied on 1D chains and 2D square lattices with periodic boundary conditions at zero temperature using sign-problem-free QMC. For $E_n$24 the model has $E_n$25 symmetry: Ising spin-flip in the $E_n$26 sector and Dicke parity in the light–matter sector. The natural order parameters are the superradiant amplitude $E_n$27, the photon density $E_n$28, the uniform magnetization $E_n$29, and the staggered magnetization $E_n$30 (Langheld et al., 2024).

For ferromagnetic $E_n$31, any $E_n$32 explicitly breaks the spin $E_n$33, so only one phase boundary remains, separating a normal phase from a superradiant phase. At small longitudinal field $E_n$34 the transition is first order, whereas at large $E_n$35 it is continuous and follows Dicke criticality. The line separating these regimes ends at a multicritical point at $E_n$36. In the continuous regime, QMC agrees with the infinite-dimensional variational mean-field condition

$E_n$37

while the first-order segment at small $E_n$38 is a genuine beyond-mean-field effect (Langheld et al., 2024).

For antiferromagnetic $E_n$39, both $E_n$40 symmetries remain intact even at finite $E_n$41, giving four phases: PN, AN, PS, and AS. The AS phase carries simultaneous superradiant and AFM order and is identified as the light–matter analogue of a lattice supersolid. At $E_n$42, the AN–PS transition is first order and persists to finite $E_n$43 until it intersects the mean-field AN–AS line, where it splits and opens an intermediate AS region. On the square lattice, the AS–PS boundary is first order for $E_n$44 and continuous with 3D Ising universality for $E_n$45; on the chain it remains first order throughout the explored parameter range (Langheld et al., 2024).

All continuous superradiant transitions in this closed model are in the Dicke universality class,

$E_n$46

with the modified finite-size scaling expected above the upper critical dimension. The effective exponent entering drift and rounding is $E_n$47, leading to

$E_n$48

The QMC formulation integrates out the photon exactly and yields a retarded all-to-all transverse interaction among spins, which is sampled by a wormhole generalization of the directed-loop algorithm. This provides unbiased equilibrium benchmarks for closed Dicke lattices with competing local and global interactions (Langheld et al., 2024).

6. Alternative but related constructions

The phrase also appears in settings that are not real-space Hamiltonian Dicke lattices. In the superradiance-lattice construction, $E_n$49 fixed $E_n$50-type atoms support timed Dicke states

$E_n$51

which act as momentum-space lattice sites. A standing-wave EIT coupling generates nearest-neighbor hopping in the timed-Dicke basis,

$E_n$52

Detuning asymmetry produces a uniform force in momentum space with Bloch frequency $E_n$53, leading to Bloch oscillations and Wannier–Stark ladders, while periodic modulation renormalizes the hopping through Bessel functions and yields band collapse and dynamic localization. The system is operationally “closed” because EIT confines dynamics to a dark-state manifold, suppressing spontaneous emission from $E_n$54 and leaving ground-state dephasing as the principal residual decoherence (Wang et al., 2014).

A different usage appears in ordered two-dimensional arrays of multilevel alkaline-earth(-like) atoms. There, the array is not a Hamiltonian Dicke lattice but a free-space cooperative emitter governed by a channel-resolved Lindblad master equation. “Closed” refers to transition closing: a superradiant avalanche can funnel emission overwhelmingly into one dominant branch, effectively suppressing weaker fine-structure or Zeeman channels. For a channel $E_n$55, the effective branching ratio is

$E_n$56

with $E_n$57 under strong cooperative enhancement. In square arrays at $E_n$58, the integrated bright-branch share increases from approximately $E_n$59 for a single Sr atom in the quoted multilevel example to approximately $E_n$60 for a $E_n$61 array, while predicted burst intensities scale superlinearly, with exponents such as $E_n$62–$E_n$63 depending on species, spacing, and polarization. Directional superradiance can persist even when global superradiance is weak, and shows revivals near $E_n$64 and $E_n$65 due to geometric resonances (Masson et al., 2023).

These constructions remain adjacent to the main closed-Dicke-lattice program because they retain collective Dicke-state organization while relocating the notion of “lattice” from real space to momentum space, or the notion of “closed” from Hamiltonian isolation to effective suppression of alternative decay channels. They therefore extend, rather than unify, the semantics of the term.