--- title: Extended Dicke (g–J) Model url: https://www.emergentmind.com/topics/extended-dicke-g-j-model type: topic --- # Extended Dicke (g–J) Model Searching arXiv for recent and foundational papers on the extended Dicke \(g\)–\(J\) model, including critical phenomena, exact Dicke-model solution methods, and solid-state realizations. arXiv search: "extended Dicke model g J ErFeO3" The extended Dicke (\(g\)–\(J\)) model denotes a class of Dicke-type spin-boson Hamiltonians in which the standard collective coupling between an ensemble of two-level systems and a single bosonic mode is supplemented by a direct interaction within the atomic, qubit, or spin sector. In this setting, the usual Dicke coupling \(g\) competes with an additional interaction scale—denoted \(J\), \(\eta\), or \(\varepsilon\) depending on the formulation—and this competition modifies the normal and superradiant structures familiar from the standard Dicke model. Across semiclassical, finite-size quantum, and condensed-matter realizations, the extension produces additional fixed points, new spectral singularities, precursors of multiple excited-state quantum phase transitions (ESQPTs), and, in a magnetic-solid implementation, an atomically ordered phase absent from the unextended model [1808.03193] [2302.06028] [2209.11273]. ## 1. Standard Dicke structure and the meaning of the extension The standard finite-\(N\) Dicke model describes \(N\) identical two-level systems collectively coupled to one bosonic mode. In one exact finite-size formulation, its Hamiltonian is written as \[ H=-\Delta J_x+\omega d^\dagger d+2\lambda/\sqrt{N}\,(d^\dagger+d)J_z, \] with \(J_z=\frac12\sum_i \sigma_z^{(i)}\), \(J_x=\frac12\sum_i \sigma_x^{(i)}=\frac12(J_++J_-)\), total spin \(j=N/2\), and a \(Z_2\) parity symmetry [1404.7834]. This standard model provides the reference point for all extended versions. The extension consists of introducing a direct interaction among the two-level systems. In the critical-phenomena study of interacting qubits coupled to a single quantized mode, the additional term is a nonlinear qubit-qubit interaction with coupling \(\eta\), added to the usual Dicke collective coupling \(\gamma\); in that notation, \(\gamma\) plays the Dicke-like role analogous to \(g\), while \(\eta\) governs the internal qubit-qubit interaction [1808.03193]. In the semiclassical “bound luminosity” formulation, the extension appears as an all-to-all quadratic spin term, \[ \hat H = \frac{\hat p^2 + \omega^2 \hat q^2}{2} + g \hat p \hat S_y - \omega_0 \hat S_z + (1+\varepsilon)\frac{g^2}{2}\hat S_y^2, \] where \(\varepsilon\) parameterizes the direct dipole-dipole interaction between the two-level systems [2209.11273]. In the solid-state \(g\)–\(J\) realization in ErFeO\(_3\), the extension is phrased explicitly as competition between a Dicke-like coupling \(g\) and a short-range exchange \(J\) [2302.06028]. Accordingly, the expression “extended Dicke model” does not refer to a unique universal operator form. Across the literature considered here, it denotes a Dicke framework augmented by a direct intra-ensemble interaction, with the operator content depending on whether the platform is formulated in terms of collective spins \(J_\alpha\), collective spins \(S_\alpha\), or two-sublattice operators \(\hat\Sigma^\pm_\alpha\). ## 2. Hamiltonian formulations and competing interactions In the interacting-qubit formulation, the model extends the standard Dicke Hamiltonian by adding a qubit-qubit interaction with coupling \(\eta\). Its semiclassical limit is \[ H_{\rm cl} = \frac{\omega}{2}(q^2+p^2) + \omega_0 j_z + 2\gamma q \left(\frac{j^2-j_z^2}{j}\right)^{1/2}\cos\phi + \frac{\eta}{2j}j_z^2, \] where \(q,p\) are field quadratures, \(j_z\) is the classical spin projection, \(\phi\) is the conjugate spin angle, and \(j=N_q/2\) [1808.03193]. This form makes explicit that the extension modifies the spin sector while retaining a single bosonic degree of freedom. In the quasiclassical Hamiltonian used to analyze the “bound luminosity” state, the direct interaction is encoded by \((1+\varepsilon)\frac{g^2}{2}\hat S_y^2\). The paper states that \(\varepsilon=0\) corresponds to the standard “extended” Hamiltonian arising from gauge transformation or circuit derivation, \(\varepsilon=-1\) cancels the quadratic \(\hat S_y^2\) term and recovers the ordinary Dicke-like structure, \(\varepsilon<0\) is ferroelectric-like and supports superradiance, and \(\varepsilon>0\) is anti-ferroelectric-like and stabilizes the subradiant or normal phase [2209.11273]. In the magnetic-solid realization, the effective Hamiltonian is \[ \begin{aligned} \hat{\mathcal{H}}/\hbar &= \omega_{\pi}\hat{a}_\pi^\dagger\hat{a}_\pi +\omega_\text{Er}\hat{\Sigma}_x^+ +\omega_z\hat{\Sigma}_z^+ \\ &\quad +g\sqrt{\frac{2}{N_0}}\, i(\hat{a}_\pi^\dagger-\hat{a}_\pi)\hat{\Sigma}_z^- \\ &\quad +J\frac{6}{N_0\hbar}\Big[(\hat{\Sigma}_x^+)^2+(\hat{\Sigma}_z^+)^2-(\hat{\Sigma}_x^-)^2-(\hat{\Sigma}_z^-)^2\Big], \end{aligned} \] with Er\(^{3+}\) spins playing the role of atoms and Fe\(^{3+}\) qAFM magnons the role of photons [2302.06028]. In that formulation, the \(g\)-term couples the Fe magnon quadrature to the Er staggered order and favors the superradiant phase, whereas the \(J\)-term encodes direct Er-Er antiferromagnetic exchange and enables an atomically ordered phase. A central consequence of all three formulations is the same: the extension creates a competition between boson-mediated collective ordering and direct spin-sector ordering. This competition is the source of the additional fixed points, phases, and nonanalytic spectral structures reported in the literature. ## 3. Semiclassical regimes, fixed points, and critical energies For the interacting-qubit model, the semiclassical equations of motion derived from \(H_{\rm cl}\) are \[ \dot q = \omega p, \] \[ \dot p = -\omega q - 2\gamma\left(\frac{j^2-j_z^2}{j}\right)^{1/2}\cos\phi, \] \[ \dot \phi = \omega_0 + \frac{j_z}{j}\left[\eta - 2\gamma q\left(\frac{j^2-j_z^2}{j}\right)^{-1/2}\cos\phi \right], \] \[ \dot j_z = 2\gamma q\left(\frac{j^2-j_z^2}{j}\right)^{1/2}\sin\phi. \] The fixed-point topology is organized by \[ f = \frac{4\gamma^2+\eta\omega}{\omega\omega_0}, \] which splits the model into three regimes [1808.03193]. For \(f<1\) (Region I), only the two fixed points \(\{q,p,j_z\}=\{0,0,\pm j\}\) exist. The point \(\{0,0,-j\}\) is stable and \(\{0,0,j\}\) is unstable. This reproduces the normal-phase structure of the standard Dicke model. For \(f\ge 1\) and \(\eta<\omega_0\) (Region II), two additional stable fixed points appear, \[ \left\{-q_{(s)},0,-jf^{-1},0\right\}, \qquad \left\{q_{(s)},0,-jf^{-1},\pi\right\}, \] with \[ q_{(s)} = \frac{2\gamma}{\omega}\sqrt{j-jf^{-2}}, \] and the original minimum becomes a saddle. This is the usual Dicke-model superradiant-type structure. For \(f\ge 1\) and \(\eta\ge \omega_0\) (Region III), two more fixed points appear, \[ \left\{0,0,-\frac{\eta j_z}{\omega_0}, \pm \frac{\pi}{2}\right\}, \] and the phase portrait acquires an additional saddle point absent from the standard Dicke model. The corresponding scaled energy is \[ \epsilon = \frac{E}{\omega_0 j}, \] with minimum energy \[ \epsilon_{\min}= \begin{cases} -1+\dfrac{\eta}{2\omega_0}, & f<1,\\[6pt] -\dfrac12\left(f+f^{-1}\right)+\dfrac{\eta}{2\omega_0}, & f\ge 1. \end{cases} \] The other critical energies are \[ \epsilon_+ = 1+\frac{\eta}{2\omega_0} \] in Region I, \[ \epsilon_- = -1+\frac{\eta}{2\omega_0}, \qquad \epsilon_+ = 1+\frac{\eta}{2\omega_0} \] in Region II, and \[ \epsilon_s = -\frac{\omega_0}{2\eta}, \qquad \epsilon_- = -1+\frac{\eta}{2\omega_0}, \qquad \epsilon_+ = 1+\frac{\eta}{2\omega_0} \] in Region III. Region I therefore has two critical energies, Region II three, and Region III four relevant stationary energies. The semiclassical density of states is computed through Weyl’s law, \[ \nu(E)=\frac{1}{(2\pi)^2}\int dq\,dp\,d\phi\,dj_z\; \delta\!\left(E-H_{\rm cl}(q,p,\phi,j_z)\right), \] and the singular structure of \(d\nu/d\epsilon\) distinguishes the three regimes. Region I exhibits only a jump-type discontinuity at \(\epsilon_+\). Region II exhibits a logarithmic singularity at \(\epsilon_-\) and a jump at \(\epsilon_+\). Region III exhibits a new logarithmic singularity at \(\epsilon_s\), a new jump discontinuity at \(\epsilon_-\), and the original jump at \(\epsilon_+\). This one-logarithmic-plus-two-jump pattern is the main semiclassical signature of the extended model’s third regime and is interpreted as the classical imprint of multiple ESQPTs [1808.03193]. ## 4. Finite-size quantum signatures and ESQPT precursors The finite quantum analysis of the interacting-qubit extension is performed by numerical diagonalization for \(N_q=200\) qubits, a bosonic basis with up to \(n_{\max}=600\) photons, the positive-parity sector only, and about \(5\times 10^4\) converged eigenstates [1808.03193]. The averaged quantum density of states, \[ \bar\nu(\bar\epsilon)=\frac{\Delta \bar n}{\Delta \bar E}, \] closely tracks the semiclassical density of states. In the finite spectrum, the nonanalytic semiclassical structures appear as level clustering near the critical energies. The clustering pattern follows the same regime count as the semiclassical analysis: one critical clustering in Region I, two in Region II, and three in Region III. These are interpreted as finite-size precursors of ESQPTs. The interpretation follows the standard semiclassical logic stated in the same work: logarithmic divergences or discontinuities are associated with saddle points of the classical energy surface, while jump-type discontinuities are associated with maxima, minima, and changes in accessible phase-space volume. Peres lattices provide the principal quantum diagnostic of order and chaos in this setting. The relevant plots use \(\langle J_z\rangle/(\omega_0 j)\) against energy for each eigenstate. In Regions I and II, the lattices behave much like those of the standard Dicke model. Around \(\epsilon_+\), the authors identify a precursor of the static ESQPT associated with a maximum in \(\langle J_z\rangle\); around \(\epsilon_-\), a precursor of the dynamic ESQPT is associated with a minimum in \(\langle J_z\rangle\) [1808.03193]. Region III is qualitatively distinct. At low energies, the Peres lattice becomes irregular, indicating chaos. Near the new critical structure linked to the nonlinear interaction, order reappears: a regular section of the lattice emerges again at larger values of \(\langle J_z\rangle\), near the second jump-type discontinuity in the semiclassical density of states. The paper describes this as a revival of order, and uses it to confirm Peres’ conjecture relating spectral regularity in the quantum model to regularity in the corresponding semiclassical dynamics. In this extended model, the additional interaction does not merely increase irregularity; it creates a new energy window in which regularity re-emerges after a chaotic sector. ## 5. Dynamical solutions, bound luminosity, and chaotic onset A different semiclassical line of work studies the extended Dicke model through quasiclassical equations of motion derived from \[ \hat H = \frac{\hat p^2 + \omega^2 \hat q^2}{2} + g \hat p \hat S_y - \omega_0 \hat S_z + (1+\varepsilon)\frac{g^2}{2}\hat S_y^2. \] Replacing operators by commuting real variables yields \[ \begin{aligned} \dot S_z &= -g p S_x - (1+\varepsilon)g^2 S_x S_y, \\ \dot S_x &= g p S_z + \omega_0 S_y + (1+\varepsilon)g^2 S_y S_z, \\ \dot S_y &= -\omega_0 S_x, \\ \dot p &= -\omega^2 q, \\ \dot q &= p + g S_y, \end{aligned} \] together with the conserved spin-length relation \[ S_x^2+S_y^2+S_z^2 = S^2+S \approx S^2,\qquad S\gg 1. \] Under the approximation \(q \approx 0\), \(\dot q \approx 0\), and \(\dot p \approx 0\), one has \[ p \approx -g S_y, \] and the dynamics reduces to \[ \begin{aligned} \dot S_z &= -\varepsilon g^2 S_x S_y, \\ \dot S_x &= \omega_0 S_y + \varepsilon g^2 S_y S_z, \\ \dot S_y &= -\omega_0 S_x. \end{aligned} \] This reduced system is analytically solvable in the superradiant regime \[ g > g_c,\qquad g_c=\sqrt{\frac{\omega_0}{|\varepsilon|S}}, \] specifically for \(\varepsilon<0\) [2209.11273]. The solution is expressed in Jacobi elliptic functions and describes the “bound luminosity” state. After eliminating \(S_z\), the equation for \(S_y\) is \[ \ddot S_y = -\omega_0^2 S_y - \frac{\varepsilon^2 g^4}{2}S_y \left(S_y^2 + 2C\frac{\omega_0}{\varepsilon g^2}\right), \] with \[ S_z = \frac{\varepsilon g^2}{2\omega_0}S_y^2 + C. \] The normalized solution takes the form \[ x(t)=\frac{S_y(t)}{S} = k\frac{2\Omega}{\omega_0}\left(\frac{g_c}{g}\right)^2 \operatorname{cn}(\Omega t,k), \] and the period is \[ T = \frac{4K(k)}{\Omega}. \] Physically, the state is a periodic exchange of energy between cavity field and collective matter excitation: the energy is initially stored in the field, then absorbed by the two-level-system ensemble, and then released back to the cavity. The paper identifies \[ \mathbf E \sim \langle \hat p\rangle,\qquad \mathbf d \sim -\langle \hat S_y\rangle,\qquad p(t)=-gS_y(t), \] and interprets the evolution through the oscillatory dipole and Zeeman energies. The same work investigates chaos numerically via Poincaré sections defined by \(q=0\), \(p=p(E)\), with intersections plotted in the spin-angle variables \(\theta=\arccos(S_z/S)\) and \(\varphi=\arctan(S_y/S_x)\). For \(g\ll g_c\), trajectories are regular; near \(g_c\), chaotic regions appear; for \(g>g_c\), phase space becomes largely chaotic. A notable numerical conclusion is that the onset of chaos is suppressed for \(\varepsilon>0\): a larger ratio \(g/g_c\) is required to generate comparable chaos than in the \(\varepsilon<0\) case [2209.11273]. ## 6. Condensed-matter realization and phase structure in ErFeO\(_3\) The extended Dicke \(g\)–\(J\) model has been realized as an effective low-energy description of the magnetic solid ErFeO\(_3\), where Er\(^{3+}\) spins act as the atomic ensemble and Fe\(^{3+}\) qAFM magnons act as the bosonic mode [2302.06028]. The work uses a two-sublattice approximation, with \[ \hat{\Sigma}_p=\sum_{i=1}^{2N_0}\hat{\sigma}_{i,p}/2,\qquad p\in\{x,y,z\}, \] and \(+\) and \(-\) indicating sublattice sum and difference. The model contains three phases. The normal phase (N) corresponds to paramagnetic or unpolarized Er\(^{3+}\) behavior and Fe\(^{3+}\) qAFM order without the extra magnon condensate associated with the SRPT-like ordering; in Bertaut notation this is the \(\Gamma_2\) phase. The superradiant phase (S) has cooperative ordering in both sectors, with \[ \langle \hat{\Sigma}_z^- \rangle \neq 0,\qquad \langle \hat{\Sigma}_x^- \rangle \approx 0, \] and a finite Fe magnon condensate, \[ \langle S_y \rangle \neq 0. \] This is the magnonic analog of the Dicke superradiant phase and corresponds to the \(\Gamma_{12}\) phase. The atomically ordered phase (A) is absent from the standard Dicke model; it is characterized by \[ \langle \hat{\Sigma}_x^- \rangle \neq 0,\qquad \langle \hat{\Sigma}_z^- \rangle \approx 0, \] together with canting along \(z\), \[ \langle \hat{\Sigma}_z^+ \rangle \neq 0, \] and no superradiant Fe order. The mean-field \(T\)–\(H\) phase diagram contains the S phase at low \(T\) and low/intermediate \(H\), the A phase at low \(T\) and intermediate \(H\), and the N phase at higher \(T\) or larger \(H\). Below about \(2.8\) K, the sequence with increasing field is S \(\to\) A \(\to\) N. The triple point is near \(T=2.8\) K and \(H=0.5\) T. Above \(\sim 4\) K, only the S \(\to\) N transition is observed, and fields above \(\sim 1\) T drive the system into the N phase. The S \(\to\) A transition is first-order, with an abrupt switch of Er order-parameter components and coexistence of spectral lines consistent with phase coexistence; the A \(\to\) N and S \(\to\) N transitions are second-order, with continuous vanishing of the order parameter [2302.06028]. The experimental support comes from magnetocaloric-effect and terahertz magnetospectroscopy measurements. In the magnetocaloric analysis, the Grüneisen ratio \[ \Gamma_H ~=~ -\frac{(\partial S/\partial H)_T}{C_H} ~=~ \frac{1}{T}\frac{\partial T}{\partial H}\bigg|_S \] is used, and peaks in \(dT/d(\mu_0 H)\) trace the phase boundaries. For \(T<2.8\) K, two peaks identify the S \(\to\) A and A \(\to\) N boundaries; for \(2.8\text{ K}0, \] with an additional pole at \(E=n\) for even \(N\) in the appropriate parity sector [1404.7834]. That exact solution is relevant to the extended Dicke \(g\)–\(J\) literature primarily as a baseline. The same work emphasizes that the finite-\(N\) Dicke model is non-integrable for \(N>1\) at any finite coupling in Braak’s sense, even though it is exactly solvable. The argument is based on the insufficiency of energy and parity to label all eigenstates uniquely, the pole-controlled level structure of the \(G\)-functions, and the dominance of avoided crossings over true crossings within parity sectors [1404.7834]. Because that analysis does not include an explicit \(g\)–\(J\) extension, it does not provide an exact solution of the extended model itself. A plausible implication is that direct intra-ensemble interactions should be viewed as enrichments of an already non-integrable finite-\(N\) Dicke background rather than as perturbations that restore integrability. In the literature summarized here, this expectation is borne out phenomenologically by additional fixed points, multiple ESQPT precursors, first-order as well as second-order phase transitions, and nontrivial order-chaos-order structures in semiclassical and quantum diagnostics. Source: https://www.emergentmind.com/topics/extended-dicke-g-j-model