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arxiv: 2606.26081 · v1 · pith:2M4MKHHYnew · submitted 2026-06-24 · ❄️ cond-mat.str-el · cond-mat.stat-mech· quant-ph

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

Max H\"ormann This is my paper

Pith reviewed 2026-06-25 19:49 UTC · model grok-4.3

classification ❄️ cond-mat.str-el cond-mat.stat-mechquant-ph
keywords Dicke modelsuperradiancephase diagraminteracting spinsthermodynamic limitequation of statefirst-order transitions
0
0 comments X

The pith

The photon supplies only a scalar field that creates no phase the interacting matter does not already possess.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Integrating out the cavity mode in Dicke models with interacting matter produces an exact self-consistent energy functional of the magnetisation. This functional shows that the photon creates no new phases beyond those of the bare matter. Superradiant first-order transitions therefore appear as folds of one connected curve given by the inverse of the matter chemical potential over magnetisation. The onset and nature of transitions are set by local properties of the matter energy such as singularities or response functions. Specific models including Ising, triangular antiferromagnets, and Rydberg chains illustrate how the phase diagram is determined entirely by the matter equation of state.

Core claim

States holding a minimum form one connected curve λ(m) = μ_mat^{-1}(m)/m so that superradiant first-order transitions are folds of one equation of state not crossings of disjoint sheets, and a fold can straighten into a continuous line. The photon creates no phase the matter does not already possess.

What carries the argument

The self-consistent functional ilde e(m) = λ m^2/2 + e_mat(λ m) that reduces all superradiant physics to properties of the bare matter energy e_mat.

If this is right

  • The remaining rules for transitions are local, each with a spectral counterpart such as softening polariton for onset.
  • Order is determined by the Landau quartic or a divergent susceptibility forcing a Larkin-Pikin fold.
  • For the Dicke-Ising model the Landau coefficients are exact, giving closed form boundaries and tricritical points.
  • A 1/d expansion maps all four phases with first order transitions for d ≤ 3.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar reduction might hold for multi-mode cavities if the integration remains exact.
  • The approach could extend to finite-size systems where fluctuations modify the folds.
  • Experimental probes of the matter susceptibility could predict the entire superradiant diagram without cavity.

Load-bearing premise

The integration out of the cavity yields an exact self-consistent functional of the magnetisation in the thermodynamic limit with collective coupling.

What would settle it

A matter system exhibiting a superradiant phase transition that has no corresponding minimum in its bare energy functional would contradict the claim.

Figures

Figures reproduced from arXiv: 2606.26081 by Max H\"ormann.

Figure 1
Figure 1. Figure 1: FIG. 1. The three onset classes set by the leading singularity of [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The single-valued stationarity curve [PITH_FULL_IMAGE:figures/full_fig_p029_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The landscape is complete: no hidden superradiant branch (schematic, generic single [PITH_FULL_IMAGE:figures/full_fig_p031_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The guided example: the one-dimensional Dicke–Ising chain at [PITH_FULL_IMAGE:figures/full_fig_p039_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The Dicke–Ising stationarity curve [PITH_FULL_IMAGE:figures/full_fig_p042_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Ferromagnetic phase diagram in the rescaled [PITH_FULL_IMAGE:figures/full_fig_p045_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Ferromagnetic stationarity curves [PITH_FULL_IMAGE:figures/full_fig_p046_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Antiferromagnetic Dicke–Ising phase diagram in the rescaled units [PITH_FULL_IMAGE:figures/full_fig_p048_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Preemption of the antiferromagnetic phase in one dimension (iDMRG), and the role of the [PITH_FULL_IMAGE:figures/full_fig_p053_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. The Larkin–Pikin fold in one dimension, from infinite-system DMRG. Stationarity curves [PITH_FULL_IMAGE:figures/full_fig_p055_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. First-order AS–PS jumps near the quadruple point (QP), from strict-blockade iDMRG. [PITH_FULL_IMAGE:figures/full_fig_p060_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Triangular-lattice transverse-field Ising antiferromagnet—the [PITH_FULL_IMAGE:figures/full_fig_p065_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Self-consistent solution and second-order test for the triangular antiferromagnet ( [PITH_FULL_IMAGE:figures/full_fig_p066_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Cavity-coupled compass chain at the symmetric compass angle ( [PITH_FULL_IMAGE:figures/full_fig_p068_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Stationarity curves [PITH_FULL_IMAGE:figures/full_fig_p071_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. The Dicke–Heisenberg diagram [PITH_FULL_IMAGE:figures/full_fig_p073_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FIG. 17. The three connected clusters of the low-field expansion through sixth order, drawn for [PITH_FULL_IMAGE:figures/full_fig_p082_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. The bare independent-set transition at the quadruple point (strict-blockade iDMRG, [PITH_FULL_IMAGE:figures/full_fig_p095_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: FIG. 19. Convergence of the near-corner AS–PS jumps ( [PITH_FULL_IMAGE:figures/full_fig_p097_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: FIG. 20. Convergence of the scale-invariant QP-ray jump (strict blockade, [PITH_FULL_IMAGE:figures/full_fig_p098_20.png] view at source ↗
read the original abstract

We give a thermodynamic-limit account of Dicke models with one cavity mode coupled collectively to interacting matter. Integrating out the cavity yields an exact self-consistent functional of the magnetisation $m$, $\tilde e(m) = \lambda m^2/2 + e_{\rm mat}(\lambda m)$: a classical penalty on the bare-matter energy $e_{\rm mat}$ in the self-consistent field $h = \lambda m$, with $\lambda = g^2/(2\omega_c)$ the collective coupling. Supplying only that scalar field, the photon creates no phase the matter does not already possess. States holding a minimum form one connected curve, $\lambda(m) = \mu_{\rm mat}^{-1}(m)/m$, so superradiant first-order transitions are folds of one equation of state not crossings of disjoint sheets, and a fold can straighten into a continuous line. The remaining rules are local, each with a spectral counterpart: onset by the leading singularity of $e_{\rm mat}$ (a softening polariton), order by one bare response -- the Landau quartic, or a divergent susceptibility forcing a Larkin-Pikin (LP) fold. For the Dicke-Ising model the Landau coefficients are exact, giving in closed form the second-order boundary and both zero-quartic fields, one tricritical; a $1/d$ expansion maps all four phases, with the AS-PS transition first order for $d\le d_{uc}=3=4-z$ (LP) and tricritical points in the $(d,\epsilon)$ plane above. At the degenerate quadruple point the matter is a Rydberg-blockade chain, solved by strict-blockade iDMRG: the antiferromagnetic superradiant (AS) phase persists as a finite 1D wedge, first order into the corner. Other magnets: the triangular antiferromagnet keeps a continuous superradiant-superradiant line (3D-XY, no fold forced); the compass chain a BKT-functional onset; the Heisenberg and XX chains, via a conserved operator, a spectrally silent first-order onset; and the Dicke-Heisenberg diagram an exact tricritical point at the saturation corner.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript claims that integrating out the cavity mode in the thermodynamic limit for Dicke models with collective coupling to interacting matter yields an exact self-consistent functional ilde e(m) = \lambda m^2/2 + e_mat(\lambda m), with \lambda = g^2/(2 \omega_c). Consequently, the photon introduces no new phases beyond those already present in the bare matter; superradiant transitions appear as folds of the single connected curve \lambda(m) = \mu_mat^{-1}(m)/m rather than crossings between disjoint sheets. The work supplies closed-form Landau coefficients for the Dicke-Ising model, a 1/d expansion that maps all four phases (with LP first-order behavior for d \le 3), an iDMRG solution of the Rydberg-blockade chain showing a finite AS wedge, and explicit results for triangular antiferromagnets, compass/Heisenberg/XX chains, and the Dicke-Heisenberg tricritical point.

Significance. If the central reduction holds, the result supplies a unifying and simplifying account of superradiant phase structure that reduces the problem to properties of the matter equation of state alone. Explicit constructions—exact Landau coefficients, the 1/d expansion, and strict-blockade iDMRG—are concrete strengths that render the framework directly testable and applicable across multiple magnets.

minor comments (2)
  1. [§2] §2 (or wherever the thermodynamic-limit integration is performed): the step from the collective-coupling Hamiltonian to the exact functional ilde e(m) would benefit from an explicit intermediate line showing how the photon quadratic term is completed to the square and the self-consistency h = \lambda m is imposed.
  2. [1/d expansion section] The 1/d expansion paragraph: the upper-critical dimension d_uc = 3 = 4 - z is stated without a short derivation of the dynamical exponent z that enters the hyperscaling relation; a one-sentence reminder would aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and accurate summary of the manuscript, which correctly captures the central claim that superradiant transitions arise as folds of a single connected equation of state rather than crossings between disjoint sheets. The recommendation for minor revision is noted.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives the effective functional ilde e(m) = λ m^2/2 + e_mat(λ m) directly from integrating out the cavity mode in the collective-coupling Hamiltonian in the thermodynamic limit, then obtains the self-consistency condition λ(m) = μ_mat^{-1}(m)/m from extremizing that functional. This reduction is a standard exact rewriting for infinite-range collective coupling and does not presuppose the target phase diagram; the subsequent statements that all superradiant transitions are folds of the bare matter equation of state follow immediately from the geometry of that single curve. Explicit constructions for the Dicke-Ising model (closed-form Landau coefficients), 1/d expansion, and iDMRG solution of the Rydberg-blockade chain are performed independently on specific microscopic Hamiltonians and confirm the absence of extra phases without relying on fitted parameters or self-citations. No step matches any enumerated circularity pattern.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The analysis relies on the exactness of the cavity integration and the independent knowledge of the matter energy functional e_mat from the bare matter system.

axioms (1)
  • domain assumption The cavity can be integrated out exactly in the thermodynamic limit to yield the self-consistent functional ilde e(m) = λ m^2/2 + e_mat(λ m)
    This is the foundational step stated in the abstract for the entire analysis.

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discussion (0)

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Reference graph

Works this paper leans on

130 extracted references · 1 canonical work pages · 1 internal anchor

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    Vacuum, broken symmetry, and the background field The Landau coefficients of eq. (13) are low-field Taylor coefficients of the bare-matter energy: writing the per-site ground-state energy in the fieldhconjugate toσz as emat(h) =e 0 − 1 2 χmat(0)h 2 + a4 λ4 h4 + a6 λ6 h6 +O(h 8),(A1) the substitutionh=λmin eq. (5) returns exactly the Landau series eq. (13)...

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    Clusters, denominators, and the quartic coefficient At orderh 2k each insertion ofV=−h P i σz i flips one spin and the2kinsertions must return to the vacuum, so every site is flipped an even number of times and a connected cluster spans at mostksites: at fourth order a single site or an adjacent pair; at sixth order additionally the three-site path (the b...

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    The coefficients through sixth order All coefficients and gaps below are written inbareunits (|J|literal). The rescaling J→J/dof section B—under which matter mean field is exact atd=∞—is applied only when converting the barea4 = 0roots and onsets to the tricritical loci quoted in the main text; it is the positive substitution|J| → |J|/d, which moves the l...

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    Embedding ind: bonds per site, not coordination The per-site assembly on the hypercubic latticeZd is a2k λ2k =W 1 +d W 2 +d(2d−1)W 3 +. . . ,(A11) withW b the reduced contribution of theb-site cluster and the prefactors the embedding numbers ofZ d:dbonds per site and 2d 2 =d(2d−1)three-site paths per site (for the Néel 84 vacuum the paths split evenly int...

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    Heisenberg chain in a field: the Bethe working equations Thecurvesµ B(h)ande B(h)ofsectionVD3comefromtheBethesolutionofthechain[76] in its zero-temperature dressed-energy form, the route by which Griffiths first computed the magnetisation curve [87]. One convention flag: our field couples toσz = 2Sz, so the field of the spin-1 2 literature is2hand ourm=⟨σ...

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