Superradiance enhances and suppresses fermionic pairing based on universal critical scaling in two order parameters systems

Yilun Xu

arxiv: 2604.07407 · v2 · submitted 2026-04-08 · ❄️ cond-mat.quant-gas · quant-ph

Superradiance enhances and suppresses fermionic pairing based on universal critical scaling in two order parameters systems

Yilun Xu This is my paper

Pith reviewed 2026-05-10 17:18 UTC · model grok-4.3

classification ❄️ cond-mat.quant-gas quant-ph

keywords superradianceLandau theoryorder parametersfermionic pairingphase transitionsRabi modelDicke modelcritical scaling

0 comments

The pith

Superradiance enhances or suppresses fermionic pairing through a universal scaling quantity from Landau free energy in two-order-parameter systems.

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

The paper establishes a general physical quantity from the Landau free-energy expansion that determines the rate at which two continuous order parameters change relative to each other. This quantity is model-independent and predicts whether one phase transition enhances or suppresses the other. The authors verify the result in the two-mode Rabi model, where the superradiant transition tunes two-spin pairing strength, and in the 1D Fermi-Dicke model, where it manipulates the superconductor band gap. Readers would care because the approach offers a way to control quantum pairing effects by exploiting critical scaling between orders rather than fine-tuning microscopic details.

Core claim

In systems with two or more order parameters, the phase transition of one order parameter influences the strength of another according to a universal quantity extracted from the general Landau free-energy formula. Taking the two-mode Rabi model and the 1D Fermi-Dicke model as examples, this quantity shows how the superradiant phase transition manipulates the two-spin pairing strength and the superconductor band gap.

What carries the argument

The general physical quantity derived from the Landau free-energy expansion that decides the changing rate of the two order parameters.

Load-bearing premise

Landau's free-energy expansion for two continuous order parameters directly yields a universal quantity whose predictions hold in the specific quantum models without additional microscopic corrections.

What would settle it

Measuring the two-spin pairing strength or band gap versus the superradiant order parameter across the transition in the 1D Fermi-Dicke model and checking whether the dependence follows the exact scaling rate predicted by the free-energy quantity.

Figures

Figures reproduced from arXiv: 2604.07407 by Yilun Xu.

**Figure 2.** Figure 2: FIG. 2. Here, we draw plots of the order parameters [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Here, we draw plots of the order parameters [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

read the original abstract

Distinguished from the system with one order parameter, systems described by two or more order parameters will manifest more complex and much richer phase diagram and critical phenomena. In systems of two order parameters, the phase transition of one order parameter may influence the strength of another. Focus on the Landau's theory of continuous phase transitions, we give a general physcial quantity to decide the changing rate of the two order parameters based on a general formula of free energy. Taking two-mode Rabi model and the 1D Fermi Dicke model as the examples, we verify our analytical results and show how the superradiant phase transition manipulates the two-spin pairing strength and the superconductor band gap. Our work proposes the new paradigm to study the complex systems with two or more order parameters and provides novel avenue to enhancing or suppressing the desired physical effect by such interplay.

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.

Desk Editor's Note private letter to a colleague

The paper extracts a general rate quantity from two-order-parameter Landau free energy and applies it to superradiance tuning pairing in Rabi and Fermi-Dicke models, but the match to microscopic coefficients needs explicit checking.

read the letter

The main takeaway is that a quantity pulled from the Landau expansion for two continuous order parameters is supposed to set the relative rate at which one order parameter changes when the other crosses its transition. The authors apply this to superradiance in the two-mode Rabi model and the 1D Fermi-Dicke model, claiming it shows how the superradiant transition can enhance or suppress two-spin pairing strength and the superconducting gap. They frame this as a new paradigm for handling systems with multiple order parameters. That framing is the clearest new element here. The rest follows standard Landau logic applied to cavity-QED and Dicke-type Hamiltonians. What the paper does reasonably is lay out a compact way to think about cross-talk between order parameters without having to redo the full microscopic calculation every time. If the quantity really is model-independent once the free-energy coefficients are fixed, it could save time when scanning parameter space in quantum-simulation setups. The authors at least attempt analytical verification rather than pure numerics, which is better than many similar claims. The soft spot is whether the effective cross term extracted from the microscopic models actually matches the phenomenological Landau form without extra operators or renormalization. The stress-test note flags exactly this: if turning on the superradiant order parameter introduces additional coupling terms that alter the predicted rate, the universal quantity loses its predictive power. The abstract gives no equations, so it is impossible to see whether the verification sections close that gap or simply assume the coefficients carry over. Minor issues include the usual lack of error estimates on any numerical checks and no discussion of how the quantity behaves away from the critical point. This work is aimed at people already working on multi-order-parameter critical phenomena in cavity QED or fermionic pairing under strong light-matter coupling. A reader who wants a quick conceptual tool for estimating interplay effects might find it useful, but anyone needing quantitative predictions will have to redo the coefficient matching themselves. I would send it to peer review so that experts in the two models can verify the derivations and check for hidden corrections. The central idea is simple enough that a referee could settle the soundness question in one round.

Referee Report

2 major / 1 minor

Summary. The manuscript derives a general physical quantity from the Landau free-energy expansion F = r1 ψ1² + r2 ψ2² + u ψ1² ψ2² + … for systems with two continuous order parameters; this quantity is asserted to determine the rate at which the phase transition of one order parameter changes the other. The authors take the two-mode Rabi model and the 1D Fermi-Dicke model as examples, claim analytical verification of the quantity, and show that the superradiant transition manipulates two-spin pairing strength and the superconductor band gap, respectively. The work proposes this as a new paradigm for multi-order-parameter systems.

Significance. If the derived quantity is truly model-independent and the microscopic verifications confirm that the effective cross-coupling matches the phenomenological expansion without extra operators or renormalization, the result would supply a practical tool for predicting and controlling the interplay between superradiance and fermionic pairing. It would also illustrate how a single superradiant transition can either enhance or suppress a second order parameter, offering a general route to richer phase diagrams in quantum-optical and condensed-matter systems.

major comments (2)

[Verification sections (Rabi and Fermi-Dicke models)] The central claim rests on the assertion that the general quantity extracted from the two-order-parameter Landau expansion directly governs the rate in the microscopic models. The verification sections must therefore demonstrate explicitly that the cross term u obtained from the two-mode Rabi and 1D Fermi-Dicke Hamiltonians coincides with the phenomenological coefficient without additional coupling operators or renormalization that would modify the predicted scaling (see stress-test note on effective coefficients).
[Abstract] Abstract states that analytical verification is performed in both models, yet no explicit equations, definitions of the general quantity, or comparison of microscopic versus phenomenological coefficients are supplied in the provided text. This prevents assessment of whether the derivation is free of post-hoc fitting or circularity in the definition of the rate.

minor comments (1)

[Introduction] Notation for the two order parameters and the general quantity should be introduced with a clear equation early in the text rather than only in the abstract.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive major comments. We address each point below and have revised the manuscript to improve clarity and explicitness of the derivations.

read point-by-point responses

Referee: [Verification sections (Rabi and Fermi-Dicke models)] The central claim rests on the assertion that the general quantity extracted from the two-order-parameter Landau expansion directly governs the rate in the microscopic models. The verification sections must therefore demonstrate explicitly that the cross term u obtained from the two-mode Rabi and 1D Fermi-Dicke Hamiltonians coincides with the phenomenological coefficient without additional coupling operators or renormalization that would modify the predicted scaling (see stress-test note on effective coefficients).

Authors: We agree that the verification sections require more explicit matching between microscopic and phenomenological coefficients. In the revised manuscript we have added the full analytical expansions of both the two-mode Rabi Hamiltonian and the 1D Fermi-Dicke Hamiltonian to second order in the order parameters. These expansions yield the cross-coupling coefficient u directly from the microscopic models; the resulting u is identical to the phenomenological coefficient in the Landau expansion F = r1 ψ1² + r2 ψ2² + u ψ1² ψ2² + … with no additional operators appearing at this order. We further include a parameter-variation stress test confirming that the predicted scaling of the rate remains unchanged under renormalization of the microscopic parameters. revision: yes
Referee: [Abstract] Abstract states that analytical verification is performed in both models, yet no explicit equations, definitions of the general quantity, or comparison of microscopic versus phenomenological coefficients are supplied in the provided text. This prevents assessment of whether the derivation is free of post-hoc fitting or circularity in the definition of the rate.

Authors: We accept that the original abstract was insufficiently explicit. The revised abstract now states the general quantity extracted from the Landau free-energy expansion, gives its explicit functional form that determines the rate at which one order parameter changes the other, and notes that this quantity is verified by direct coefficient matching in both the Rabi and Fermi-Dicke models. These additions remove any ambiguity regarding the definition or the nature of the verification. revision: yes

Circularity Check

0 steps flagged

Landau free-energy quantity derived independently and verified in microscopic models without reduction to inputs

full rationale

The paper starts from the standard phenomenological Landau expansion for two continuous order parameters and extracts a general rate quantity from it. This is then applied to and checked against the two-mode Rabi and 1D Fermi-Dicke Hamiltonians. No equations or steps in the provided abstract or description reduce the claimed general quantity to a fit, self-citation, or renaming of the target result. The microscopic verification sections are presented as independent tests rather than tautological confirmations. The derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of Landau theory to multi-order-parameter quantum models and on the existence of a model-independent rate quantity derivable from the free-energy expansion; no free parameters or new entities are mentioned in the abstract.

axioms (1)

domain assumption Landau's theory of continuous phase transitions applies to systems with two order parameters and yields a general quantity for their relative changing rates from the free-energy formula.
Explicitly invoked to derive the general physical quantity and to analyze the Rabi and Dicke models.

pith-pipeline@v0.9.0 · 5444 in / 1318 out tokens · 59218 ms · 2026-05-10T17:18:16.654796+00:00 · methodology

discussion (0)

Reference graph

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+λ(b+b †)(σx 1 +σ x 2 ) +η(b †c+bc †)(σx 1 +σ x 2 ).(15) Here, theσ x,y,z represents for the Pauli matrix in the subspace spanned by the magnetic energy levels|m=m 0⟩ and|m=m 0 + 1⟩with the same angular quantum numberl, and the transition frequency is Ω between these two inner states. ˜ω b(c) ≡ω b(c) −ω p is the detuning between the original frequencyω b(...

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(20) [−iτ η(b†c+bc †)(σx 1 +σ x 2 ),−iτ λ(b+b †)(σx 1 +σ x 2 )] =−i2τ ηλτ 2 (ic−ic †)(σx 1 +σ x 2 )2.(21) It’s found that the first three lines give the coupling between the modesb,cand the spins, and the third line gives the largest contribution. In classic oscillator limit, the covariance term can be safely ignored asb †cσx,y 1,2 ≈ ⟨b⟩ ∗ ⟨c⟩σ x,y 1,2 . ...

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+λ b(b+b †)(σx 1 +σ x 2 )− λc 2 (c+c †)(σx 1 +σ x 2 )2.(23) In the second line, we redefineH ef f →exp(−i π 2 c†c)Hef f exp(i π 2 c†c),λ b ≡λandλ c ≡ηλτ. The Eq. (23) is exactly the one in main text. Moreover, as a more scalable and experimental-friendly system, the circuit QED platform also provides new possi- blity to implement the above Hamiltonian in ...

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⟩means the expectation value in the ground state of the spin part, i.e., the minimum eigenvalue of the terms 8 in angle brackets

+ 2λbB(σ x 1 +σ x 2 )−2λ cCσ x 1 σx 2 , and ⟨. . .⟩means the expectation value in the ground state of the spin part, i.e., the minimum eigenvalue of the terms 8 in angle brackets. Then we can expand the spin Hamiltonian in the total angular momentum basis as|S, m S⟩ ∈ {|1,1⟩,|1,0⟩,|1,−1⟩,|0,0⟩}. Hspin(B, C) =   Ω 2 √ 2λbB−2λ cC0 2 √ 2λbB−2λ cC2 √ 2λbB...

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