Folds of one curve: the superradiant phase diagram of Dicke modes with interacting matter

Technical Summary: Folds of one curve: the superradiant phase diagram of Dicke models with interacting matter

Problem Statement
The paper addresses the thermodynamic limit of Dicke models where a single cavity mode couples collectively to interacting matter. While the reduction of such systems to a self-consistent matter functional is established, a complete analytic account of the phase diagram—specifically the order of transitions, the existence of specific phases (such as the antiferromagnetic superradiant phase in one dimension), and the structural nature of first-order transitions in general dimensions—remained unresolved. Previous works relied heavily on numerical methods (Quantum Monte Carlo, DMRG) or mean-field approximations that failed to capture non-perturbative effects in low dimensions or the precise mechanism driving first-order transitions.

Methodology
The author adopts a rigorous thermodynamic reduction. By integrating out the cavity photon, the problem is mapped exactly to the minimization of a single classical functional of the magnetization $m$:
$$\tilde{e}(m) = \frac{\lambda}{2}m^2 + e_{\text{mat}}(\lambda m)$$
where $\lambda = g^2/2\omega_c$ is the collective coupling and $e_{\text{mat}}$ is the ground-state energy density of the bare matter in a self-consistent field $h = \lambda m$.

The analysis proceeds by studying the stationarity curve $\lambda(m) = \mu_{\text{mat}}^{-1}(m)/m$, where $\mu_{\text{mat}}$ is the bare magnetization response. Key methodological tools include:

1. Geometric Analysis of the Curve: The phase diagram is interpreted as the geometry of this single curve. Continuous transitions correspond to the curve leaving the axis or passing through critical points, while first-order transitions are identified as "folds" (saddle-node bifurcations) of the curve.
2. Landau Expansion and Response Coefficients: The order of the onset is determined by the sign of the Landau quartic coefficient $a_4$, derived from the bare matter's response coefficients (susceptibility $\chi_{\text{mat}}$ and higher derivatives).
3. Larkin–Pikin Mechanism: The paper applies the Larkin–Pikin criterion to cavity-QED: if the bare matter susceptibility $\chi_{\text{mat}}$ diverges at a critical point (e.g., at the matter's critical field), the cavity transition is forced to be first-order, regardless of the Landau quartic sign.
4. Analytic Expansions: Closed-form results are derived for the Dicke–Ising model using low-field linked-cluster expansions and a $1/d$ expansion (where $J \to J/d$) to access large dimensions.
5. Numerical Verification: Infinite-system Density Matrix Renormalization Group (iDMRG/VUMPS) is used to solve the bare matter and the self-consistent functional in one dimension, particularly near the quadruple point and for the antiferromagnetic phase.

Key Contributions and Results

• Unified Structural Picture: The paper establishes that for matter with continuous transitions, the entire phase diagram is a single connected curve in the $(\lambda, m)$ plane. Superradiant first-order transitions are not crossings of disjoint free-energy sheets but "folds" of this single curve. This allows the onset, transition order, and phase count to be read from one object.
• Classification of Onsets: Onsets are classified by the leading singularity of the bare susceptibility $\chi_{\text{mat}}(0)$:
  • Regular: Finite $\chi_{\text{mat}}(0)$ leads to a finite threshold $\lambda_c = 1/\chi_{\text{mat}}(0)$.
  • Cusp: Divergent $\chi_{\text{mat}}(0)$ (linear response) leads to an immediate onset ($\lambda_c = 0$).
  • Marginal: Log-divergent $\chi_{\text{mat}}(0)$ leads to a Berezinskii–Kosterlitz–Thouless (BKT) type essential singularity onset.
• Dicke–Ising Model in General Dimension:
  • Tricritical Points: Exact closed-form expressions are derived for the tricritical points in the ferromagnetic ($\varepsilon_{\text{tri}}^{\text{ferro}} = |J|/d$) and antiferromagnetic ($\varepsilon_{\text{tri}}^{\text{AF}} = 2|J|/\sqrt{8d-3}$) cases.
  • Antiferromagnetic Superradiant (AS) Phase: In one dimension, the AS phase exists only as a narrow wedge near the quadruple point. The paper proves that the $a_4=0$ point (where the transition might locally become first-order) is preempted by a direct first-order transition to the polarized superradiant (PS) phase in all dimensions.
  • AS–PS Transition Order: The transition between the antiferromagnetic superradiant (AS) and polarized superradiant (PS) phases is first-order in physical dimensions ($d=1, 2, 3$). This is driven by the Larkin–Pikin mechanism: the bare matter susceptibility diverges (belonging to the $(d+1)$-Ising class), forcing the stationarity curve to fold. This result holds even at $d=3$ where the divergence is logarithmic.
  • Quadruple Point: At the quadruple point (where normal/polarized and antiferromagnetic/superradiant phases meet), the matter reduces to a detuned Rydberg-blockade chain. iDMRG confirms the AS phase exists as a finite wedge and that the AS–PS transition remains first-order with a finite, scale-invariant jump ($\Delta m \approx 0.011$) even at the corner.
• Other Models:
  • Triangular Antiferromagnet: Due to a negative specific-heat exponent ($\alpha < 0$), the susceptibility remains finite at the critical point. Consequently, the AS–PS transition remains second-order (continuous), escaping the Larkin–Pikin fold.
  • Compass Chain: Exhibits a marginal onset (BKT type) at the symmetric point, transitioning to a regular onset with a finite threshold when bond asymmetry opens a gap.
  • Heisenberg and XX Chains: These conserve the coupled operator ($\sigma_z$). The onset is first-order (driven by the saturation edge singularity) but spectrally silent (no soft polariton mode), as the photon does not hybridize with the matter.
  • Dicke–Heisenberg Diagram: Adding a transverse field breaks conservation, creating an exactly solvable tricritical point at the saturation corner where the first-order jump vanishes continuously.

Significance and Claims
The paper claims to provide the first complete analytic account of the Dicke–Ising phase diagram in general dimensions, resolving long-standing questions about the order of the AS–PS transition and the existence of the AS phase in 1D.

• Mechanism of First-Order Transitions: It demonstrates that superradiant first-order transitions are generically "folds" of a single equation of state, driven either by the shape of the response (shape fold) or by a divergent susceptibility (Larkin–Pikin fold).
• Role of Divergence: It establishes that the divergence of the bare matter susceptibility is the sole condition required to force a superradiant transition to be first-order, overriding Landau quartic predictions.
• Spectral Signatures: The work links thermodynamic stability to spectral properties: stable branches carry a softening polariton mode at folds, except where the coupling is to a conserved quantity (spectrally silent) or where the response is divergent (broadening into a continuum).
• Completeness: A "single-hump" assumption on the susceptibility ensures that no hidden superradiant phases exist beyond the local verdict, making the local Larkin–Pikin criterion globally sufficient.

The paper concludes that the continuity of the bare matter transitions is the fundamental structural feature that allows the entire phase diagram to be described by a single connected curve, distinguishing these systems from those with intrinsic first-order matter transitions (which would result in disconnected sheets). The results are presented as exact within the thermodynamic limit, with numerical verification provided for the non-perturbative regimes.