Time-reversal symmetry breaking, collective modes, and Raman spectrum in pair-density-wave states

Andrey V. Chubukov; Steven A. Kivelson; Yi-Ming Wu; Yuxuan Wang

REVIEW 3 major objections 4 minor 105 references

Time-reversal symmetry breaking, collective modes, and Raman spectrum in pair-density-wave states

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A coexisting PDW and d-wave superconductor state spontaneously breaks time-reversal symmetry and hosts a sharp, Raman-active Higgs mode.

desk verdict A careful theory paper whose genuinely new claim — a sharp Raman-active Higgs mode in coexisting PDW+SC — is plausible but conditional on the unquantified decay into the theta- phason and on Padé continuation. read the letter →

arxiv 2501.14138 v1 pith:4IHUPEYN submitted 2025-01-23 cond-mat.str-el cond-mat.supr-con

Yi-Ming Wu , Andrey V. Chubukov , Yuxuan Wang , Steven A. Kivelson This is my paper · ORCID

classification cond-mat.str-elcond-mat.supr-con

keywords pair-densitywavetime-reversalsymmetrybreakingHiggsmodeRamanscatteringcollectivemodescupratesuperconductorsGinzburg-Landaufreeenergydensityorder

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper predicts two experimentally accessible signatures that together would identify a pair-density-wave superconductor (PDW+SC) state and distinguish it from a charge-density-wave superconductor (CDW+SC) state. First, coexisting PDW and uniform d-wave superconducting order spontaneously breaks time-reversal symmetry. Second, of the three amplitude (Higgs) modes in this state, one is a nearly undamped symmetric fluctuation of the two PDW components and should appear as a sharp peak in non-resonant Raman scattering. The authors argue that neither feature is present in the CDW+SC competitor, so a Raman measurement could settle whether a material hosts a true pair-density wave.

What carries the argument

The central object is the Ginzburg-Landau free energy for the three order parameters $\Delta_0$, $\Delta_Q$, and $\Delta_{-Q}$, with the quartic mixture term $\beta_5(\Delta_0^2\Delta_{-Q}^*\Delta_Q^* + \mathrm{c.c.})$ that, because $\beta_5>0$, fixes the phase relation and forces time-reversal symmetry breaking. Collective modes are obtained from the Gaussian fluctuation matrix $\hat{\Gamma}^{-1}(q)$ for amplitude and phase fields; the relevant mode is the symmetric Higgs combination $A_+=(A_Q+A_{-Q})/\sqrt{2}$, which in the presence of SC hybridizes with the SC amplitude mode $A_0$ and acquires a nearly delta-function spectral weight because SC gaps the quasiparticle decay channels.

What would settle it

A non-resonant Raman measurement on a candidate PDW+SC material (e.g., LBCO near x=1/8) that fails to show a sharp A-channel peak below $2|\Delta_0|$, or a Kerr/Nernst measurement showing no broken time-reversal symmetry, would contradict the prediction.

Watch

Extended reading notes

Core claim

Working from a mean-field Ginzburg-Landau treatment of a single Cu-O layer with period-8 unidirectional d-wave PDW order ($\mathbf{Q}=(\pi/4,0)$) and coexisting d-wave SC order, the authors find that the quartic coefficient $\beta_5$ is positive, locking the phases via $\varphi_0 - (\varphi_Q+\varphi_{-Q})/2 = \pi/2$ and thereby spontaneously breaking time-reversal symmetry. Fluctuations around this saddle point give three Higgs modes. In the regime $|\Delta_0|\gg|\Delta_Q|$, the mode $A_{+,-}$ is predominantly the symmetric PDW amplitude fluctuation $A_+=(A_Q+A_{-Q})/\sqrt{2}$, and its spectral function is a nearly delta-function peak at $\omega\approx 2|\Delta_Q|$. The uniform SC order strongly suppresses the damping that would otherwise over-damp this mode in a pure PDW state. The same mode is Raman active in the A-channel, whereas the antisymmetric mode $A_-$ is Raman inactive; a direct calculation of the dressed Raman susceptibility shows a sharp intensity peak in the PDW+SC case, compared with a broad shoulder for CDW+SC. The paper concludes that a sharp Raman peak, together with TRSB, is a unique marker of PDW+SC order.

Load-bearing premise

The sharp Raman peak assumes that higher-order damping channels, such as the decay of the Higgs mode into two $\theta_-$ phase-mode quasiparticles, do not broaden the mode—a process the paper leaves out as higher-order.

Editorial extensions

If this is right

Non-resonant Raman scattering becomes a bulk probe for PDW order, since the sharp $A_{+,-}$ peak appears in the computed Raman intensity for PDW+SC and not for CDW+SC.
The phase relation $\varphi_0 - (\varphi_Q+\varphi_{-Q})/2 = \pi/2$ makes PDW+SC thermodynamically distinct from CDW+SC, which does not break time-reversal symmetry.
The spontaneous TRSB may explain the anomalous Kerr and Nernst effects observed in underdoped LBCO.
In the $|\Delta_0|\ll|\Delta_Q|$ limit the sharp mode is still present, now as $A_{+,+}$, so the Raman fingerprint is robust across the ratio range studied numerically.
Weak disorder preserves the TRSB and generates a weak 1Q CDW with correlation length that diverges as disorder vanishes, keeping the PDW+SC state distinct from CDW+SC even when long-range order is absent.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same near-delta-function amplitude mode might also appear in optical or terahertz conductivity, not just Raman, because it carries charge-density fluctuations; extending the calculation to those probes is a natural next step the paper does not take.
The mechanism of damping suppression—the larger uniform SC gap cutting off decay channels that would otherwise over-damp the PDW amplitude fluctuation—is generic and could produce sharp sub-gap collective modes in other coexisting-order superconductors.
A doping- or pressure-dependent Raman study could map the ratio $|\Delta_0|/|\Delta_Q|$ in a real material, since the composition of the sharp mode switches between $A_{+,-}$ and $A_{+,+}$ as this ratio crosses unity.
The disorder-stabilized 1Q CDW correlation implies that scanning probes may see short-range charge order at the PDW wavevector even though bulk TRSB forbids true long-range 1Q order; this is an implicit, testable consequence.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. The genuinely new result is the sharp, Raman-active Higgs mode in the coexisting PDW+SC state. It is absent in pure PDW and in CDW+SC, so if real it gives a bulk probe that can distinguish these states. The paper also gives a clean microscopic derivation that PDW+SC spontaneously breaks TRS: the explicit beta5 computation, the positive sign over the temperature range, and the phase-mode stability check all hang together. That part is solid mean-field/GL work and worth having on its own.

What is good: the authors do real calculation rather than hand-waving. The beta5 expression is explicit and diagrammatically specified; the mode-mode coupling analysis for |Delta0| > |DeltaQ| is transparent and physically explains why SC suppresses damping. The supplement is detailed, and the CDW+SC comparison is a fair contrast, even though it omits phonons. There is no fitted target; self-citations are to actual prior results and not load-bearing.

Soft spots, in proportion. The sharp-peak claim has two real holes. Footnote 79 says the Higgs mode can in principle decay into two theta- phase-mode quasiparticles, but calls it higher-order and excludes it without a smallness estimate. That is not obviously safe: theta- is the composite CDW phason, its mass is set by the commensurability term which can be small, the two-body phase space is open, and the cubic vertex A+ theta-^2 is symmetry-allowed. A one-loop width could easily broaden the peak. Second, in the |Delta0| << |DeltaQ| regime the sharp peak rests only on Padé-continued numerical data; the authors state the analytic mechanism is left for future work, and there is no convergence or error analysis. No code or data are shipped, so I could not independently check the continuation. These gaps do not sink the TRSB conclusion, but they make the headline Raman prediction conditional. The Raman calculation also neglects vertex corrections and screening, which they acknowledge.

Bottom line: a careful, honest theory paper that will be useful to anyone working on PDW order, Higgs modes, or Raman probes of intertwined orders. It deserves a serious referee, not a desk reject. I would send it to specialists with the specific request that they quantify the A+ -> theta- theta- decay before final acceptance.

Referee Report

3 major / 4 minor

Summary. The paper studies collective modes of a unidirectional d-wave pair-density-wave (PDW) state with and without coexisting uniform d-wave superconductivity (SC), using a microscopic square-lattice model at 1/8 hole doping and a Gaussian (RPA-level) fluctuation action. The main claims are: (i) the PDW+SC state spontaneously breaks time-reversal symmetry because the quartic coefficient β5 is positive; (ii) the pure PDW state has two overdamped Higgs modes, whereas the PDW+SC state has three Higgs modes, one of which (the symmetric PDW amplitude combination A+) acquires a nearly delta-function-like spectral peak whose damping is strongly reduced by the uniform SC; and (iii) this sharp mode is Raman active and provides a spectroscopic distinction from the CDW+SC state, whose corresponding amplitude mode is broad. The computations are explicit: mean-field gap equations, GL coefficients (including a numerically evaluated β5), collective-mode spectral functions obtained by Pade analytic continuation, and a dressed Raman susceptibility. The paper is a Letter with a substantial supplement containing the derivations.

Significance. If the near-delta-function peak survives higher-order corrections, the paper delivers a falsifiable and experimentally accessible Raman signature that distinguishes PDW+SC from CDW+SC in candidate cuprates, together with a microscopic mechanism for TRSB that is relevant to Kerr and Nernst experiments. The paper is largely self-contained: the key coefficient β5 is computed from the fermionic model, and the collective-mode and Raman calculations are derived rather than fitted. The main caveats are that the sharp-peak prediction relies on numerical analytic continuation and on the neglect of a decay channel that the authors themselves flag in footnote 79; these issues need to be quantified before the central prediction can be considered robust.

major comments (3)

[Main text, footnote 79] The sharp A+ mode is claimed to be nearly undamped, but the decay A+ -> θ− + θ− is excluded as "a higher-order process, not included in our analysis" without any estimate of its rate. θ− is the antisymmetric PDW phase combination (Eqs. 6 and 7), i.e., the phason of the composite CDW ρ_{2Q}; its mass is set by the eighth-order commensurability term (footnote 66), which can be much smaller than the A+ frequency ~2|ΔQ|, so the two-body phase space is open. The cubic vertex A+ θ−^2 is even under Q ↔ -Q and is symmetry allowed. Because the near-delta peak is the central observable prediction, the authors should either compute the one-loop imaginary self-energy Im Σ(ω*) from this vertex or provide a parametric argument that the coupling is small; without this, the statement that the mode is "almost completely propagating" is not established.
[Supplement III.3 and Fig. 2(c)] In the |Δ0| << |ΔQ| regime, the sharp peak in the predominantly-A+ mode (A+,+) is a numerical observation from Pade-continued data; the paper explicitly states that the analytic verification "requires more work and we leave it for further study." Because the main text claims the delta-function-like peak for both regimes and bases the Raman prediction on it, this regime needs additional support, either through a Pade convergence/error analysis or through the analytic mechanism. As it stands, the claim in this regime is supported only by a single numerical continuation with no stated robustness check.
[Main text, paragraph after Eq. (7); Supplement II.2] The analytic continuation used throughout is by Pade approximants [78], but the manuscript reports no details on the number of Matsubara frequencies used, the averaging scheme of Ref. [78], or the sensitivity of the sharp peak to these choices. Near-delta-function peaks are exactly the features most vulnerable to spurious Pade poles. Please provide the numerical parameters and a robustness check (for example, varying the input Matsubara set or comparing with a maximum-entropy continuation) for the spectral functions in Fig. 2(b), Fig. 2(c), and Fig. 3(b).

minor comments (4)

[Footnote 66] The statement "We have computed the prefactor v and found it is positive" is not backed by any calculation shown in the main text or the Supplement; since the phase-locking φQ - φ−Q = mπ/2 is used in the collective-mode calculations, please include the computation or state explicitly that the phase-mode stability analysis in Supplement II.2 yields the same condition.
[Fig. 2 and Fig. 3] The spectral functions are given in arbitrary units and the peak widths are not quantified; a statement of the extracted Γ/ω* for the sharp mode in each regime would support the "nearly delta-function" characterization and would make the Raman visibility claim more concrete.
[Supplement I.1] There is a typo: "Bogoniubov" should be "Bogoliubov".
[Abstract and Concluding remarks] The wording "should be visible in Raman experiments" is stronger than what the calculation strictly supports, given the RPA-level treatment and the neglect of vertex corrections noted in Supplement V; please consider tempering to "is predicted to be visible" or add a sentence on expected robustness.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the collective-mode and Raman predictions come from the paper's own microscopic RPA calculation, with self-citations only supplying supporting pure-PDW coefficients.

full rationale

The paper's central claim is that in the PDW+SC state one amplitude (Higgs) mode is nearly undamped and Raman active, with a sharp peak in the non-resonant Raman intensity. The derivation chain is self-contained: a microscopic t-t' model with d-wave PDW order at Q=(pi/4,0) is specified, mean-field gap magnitudes are chosen, the corresponding couplings g1 and g2 are solved from the gap equations, and the fluctuating action is computed at Gaussian level with polarization matrices built from fermion loops. The spectral functions and Raman susceptibilities are then evaluated by numerical analytic continuation. No experimental data, fitted target, or externally imposed line shape enters these calculations, so the sharp peak is not a fitted parameter renamed as a prediction, and no quantity is defined in terms of the result it is used to derive. The TRSB claim is supported by an explicit computation of beta5 in the Supplemental Material, not by an imported uniqueness theorem or by a self-citation. Self-citations appear in one supporting place: the pure-PDW Ginzburg-Landau coefficients gamma, kappa0, and u are quoted from Ref. [49] (also Ref. [2] of the Supplement), a prior work by one of the present authors and collaborators. This citation explains the pure-PDW ordering of the A+ and A- mode energies but is not load-bearing for the central PDW+SC prediction, which is obtained from the paper's own fermion-loop calculation and numerical spectral functions. The cited pure-PDW coefficients are parameter-free results of a separate model calculation with stated assumptions that do not include the target sharp Raman mode, so they count as independent support rather than circularity. The main caveats in the paper are explicitly acknowledged rather than hidden: footnote 79 states that the Higgs decay into two theta- phase-mode quasiparticles is a "higher-order process, not included in our analysis," and Supplement III.3 states that the analytic mechanism for the sharp peak in the |Delta0| << |DeltaQ| regime "requires more work and we leave it for further study." These are unquantified approximations and gaps in rigor, not cases where a conclusion reduces to an input by construction. No quoted equation or argument equates a prediction with a fitted input, and no load-bearing step rests solely on a self-citation chain. The correct verdict is therefore no significant circularity.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The paper is a model calculation: all collective-mode observables are computed from the BdG/RPA machinery, with the mean-field gap magnitudes and band parameters supplied as inputs. No new fundamental entities are introduced.

free parameters (5)

|Delta0| (uniform SC gap) = 0.01t to 0.1t
Mean-field gap magnitude chosen by hand in Figs. 2 and S6-S7; all collective-mode frequencies scale with this input.
|DeltaQ| (PDW gap) = 0.05t to 0.1t
PDW gap magnitude chosen by hand; central for setting the peak position.
t' (next-nearest hopping) = -t/4
Band parameter fixed to a standard cuprate value, not derived.
mu (chemical potential) = set to 1/8 hole doping
Chosen to match the La-based cuprate filling of interest.
T (temperature) = 0.005t
Temperature fixed well below the gap scale, used in all numerics.

assumptions (6)

standard math Bogoliubov-de Gennes mean-field theory and Gaussian (RPA) fluctuations capture the collective modes.
Used in Eq. (5) and Supplement II.
standard math Pade approximants provide a valid analytic continuation from Matsubara to real frequencies.
Used in the 'Collective modes' section; no convergence checks reported.
domain assumption The single-band t-t' model with d-wave pairing and period-8 unidirectional PDW represents cuprate PDW materials.
Motivated by LBCO/LNSCO/LESCO at 1/8 doping; other microscopic mechanisms are possible.
domain assumption PDW and SC order coexist uniformly in space.
The authors themselves caution that mesoscopic phase separation could invalidate the comparison (Concluding remarks).
ad hoc to paper The relative phase phi_Q - phi_-Q is locked by an eighth-order commensurability term -upsilon[(Delta_Q Delta*_-Q)^4 + c.c.] with upsilon > 0.
Needed to specify the TRSB state completely; the coefficient is claimed positive but the term is specific to the period-8 choice.
domain assumption Vertex corrections, Coulomb screening, and optical phonons are negligible for the Raman response.
Neglected in the Raman calculation (Supplement V), but the authors note vertex corrections matter for constant Raman vertices.

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Cite this review

Pith. "Pith review of Time-reversal symmetry breaking, collective modes, and Raman spectrum in pair-density-wave states." pith.science (2026). https://pith.science/paper/4IHUPEYN

@misc{pith2026250114138,
  author       = {Pith},
  title        = {Pith review of: Time-reversal symmetry breaking, collective modes, and Raman spectrum in pair-density-wave states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4IHUPEYN}},
  note         = {Machine review of arXiv:2501.14138}
}

abstract

Inspired by empirical evidence of the existence of pair-density-wave (PDW) order in certain underdoped cuprates, we investigate the collective modes in systems with unidirectional PDW order with momenta $\pm {Q}$ and a $d$-wave form-factor with special focus on the amplitude (Higgs) modes. In the pure PDW state, there are two overdamped Higgs modes. We show that a phase with co-existing PDW and uniform ($d$-wave) superconducting (SC) order, PDW/SC, spontaneously breaks time-reversal symmetry - and thus is distinct from a simpler phase, SC/CDW, with coexisting SC and charge-density-wave (CDW) order. The PDW/SC phase exhibits three Higgs modes, one of which is sharply peaked and is predominantly a PDW fluctuation, symmetric between ${Q}$ and -${Q}$, whose damping rate is strongly reduced by SC. This sharp mode should be visible in Raman experiments.

Figures

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Energy hierarchy of Higgs modes: PDW state vs PDW+SC state 14

PDW+SC 10 III. Energy hierarchy of Higgs modes: PDW state vs PDW+SC state 14
[106]

SC coexists with CDW 18 V

Mode-mode coupling in the PDW+SC state: composition of the longitudinal modes and the locations of the peaks in the spectral function 17 IV. SC coexists with CDW 18 V. Derivation of the Raman susceptibility 22 VI. The role of disorder 24 References 26 I. MEAN FIELD ANALYSIS
[107]

(S2) arXiv:2501.14138v1 [cond-mat.str-el] 23 Jan 2025 2 Throughout this paper we choose t′ =−t/4 and measure all energies scales in units of t

Bogoliubov quasiparticle band in the extended Brillouin zone We begin by introducing the non-interacting part of the fermions, which is a square lattice t−t′ model described by the following Hamiltonian H0 = ∑ k,σ ξ(k)ψ† σ(k)ψσ(k) (S1) where ψ† σ(k),ψσ(k) are fermion operators...

2025 arXiv
[108]

Mean field gap equations To obtain the mean field gap equations for the order parameters, we integrate out fermions, and restrict to the non-fluctuating part of the resulting effective action. The total action for our departure, which includes both the 3 - π 0 π - π 0 π - π 0 ...
[109]

(S23) Here we have used the abbreviation that fn(k) = fk+n 2 Q

Evidence of TRSB: explicit calculation of β5 By explicitly expanding SMF to the quartic order, we found that the coefficient β5 introduced in Eq.(S21) is given by β5 = 4T ∑ n ∫d2k (2π)2 [ f0(k)f2 1 (k)f2(k)Gp,0(k)Gp,1(k)Gh,7(k)(k)Gh,0(k) + f2(k)f2 3 (k)f4(k)Gp,1(k)Gp,2(k)Gh,6(...
[110]

Let’s first consider the case when only the unidirec- tional PDW ∆±Q is present

Pure PDW We now discuss collective modes from the fluctuating partSFL. Let’s first consider the case when only the unidirec- tional PDW ∆±Q is present. Their mean field gap values are solved from Eq.(S19) by setting ∆0 = 0. Assuming the 8 fluctuating fields h±Q(q) are small, w...
[111]

Π 1,3), it obeys Πi,j(q) =−Πj,i(q) and Πi,j(q) =−Πi,j(−q)

If Πi,j couples amplitude and phase modes (e.g. Π 1,3), it obeys Πi,j(q) =−Πj,i(q) and Πi,j(q) =−Πi,j(−q)
[112]

Π 1,2 or Π3,4), it obeys Πi,j(q) = Πj,i(q) and Πi,j(q) = Πi,j(−q)

If Πi,j is the coupling between amplitude modes, or between phase modes (e.g. Π 1,2 or Π3,4), it obeys Πi,j(q) = Πj,i(q) and Πi,j(q) = Πi,j(−q). Moreover, all Π i,j(q) are real. Thus, due to amplitude-phase coupling, the matrix ˆΓ−1(q) is in general a mixture of symmetric and ...
[113]

The following discussion is parallel to what we have outlined above

PDW+SC When the uniform SC component is also present in addition to the PDW state, more collective modes from the SC fluctuations need to be included. The following discussion is parallel to what we have outlined above. We define a new basis that incorporate the A0 and θ0 from...
[114]

As a result, it is easy to obtain δGk δ∆ ∗ Q =−iGk δ ˆE4 δ∆ ∗ Q Gk−iGk δ ˆE5 δ∆ ∗ Q Gk−iGk δ ˆE6 δ∆ ∗ Q Gk

is an implicit function of ∆Q (∆ ∗ Q) such thatδ∆ ∗ 0/δ∆ ∗ Q is nonzero. As a result, it is easy to obtain δGk δ∆ ∗ Q =−iGk δ ˆE4 δ∆ ∗ Q Gk−iGk δ ˆE5 δ∆ ∗ Q Gk−iGk δ ˆE6 δ∆ ∗ Q Gk. (S52) Substituting this relation back to the gap equation one gets, |∆Q|2 g1 + 1 8 ∑ k Tr [ Gk∆ ...
[115]

scales linearly with ∆Q (∆ ∗ Q) we will have ∆ ∗ Q δ ˆE6 δ∆ ∗ Q = 1 2 ( ˆE6−i ˆE3 ) . (S55) 12 Using the fact that Tr [ Gk ˆE1Gk ˆE4 ] = Tr [ Gk ˆE2Gk ˆE4 ] = Tr [ Gk ˆE3Gk ˆE4 ] = 0 (S56) it is then easy to see from Eq.(S53) that |∆Q|2 g1 + 1 16 ∑ k Tr [ Gk ˆE4Gk ˆE4 ] + 1 16...
[116]

From the gap equations we have ∆0 ∆Q g1 g2 = ∑ k Tr { [G−1 0 (k) + Σ]−1δ ˆE6 δ∆ ∗ 0 } ∑ k Tr { [G−1 0 (k) + Σ]−1δ ˆE4 δ∆ ∗ 0 }=S ( ∆0 ∆Q )

scales linearly with ∆Q (∆ ∗ Q). From the gap equations we have ∆0 ∆Q g1 g2 = ∑ k Tr { [G−1 0 (k) + Σ]−1δ ˆE6 δ∆ ∗ 0 } ∑ k Tr { [G−1 0 (k) + Σ]−1δ ˆE4 δ∆ ∗ 0 }=S ( ∆0 ∆Q ) . (S59) withS(...) being some function deduced from the ratio. Therefore, ∆0 scales linearly with ∆Q, and...
[117]

(S65) By minimizing this action, one can obtain the mean-field values ∆±Q and frequencies of the longitudinal (Higgs) modes δ±Q(τ) = ∆±Q(τ)− ∆±Q

PDW State For a pure PDW state, the GL expansion is justified for |∆±Q|≪ T , and we have, neglecting spatial fluctuations and nonanalytic ∂τ dependence that accounts for damping, L(∆±Q) =κ0(|∂τ∆Q|2 +|∂τ∆−Q|2) +r(|∆Q|2 +|∆−Q|2) +u(|∆Q|4 +|∆−Q|4) +γ(|∆Q|2|∆−Q|2). (S65) By minimi...
[118]

By symmetry, A− is not affected, while A+ and the Higgs mode of the uniform order A0 hybridize, leading to two eigenmodes A++ andA+−

PDW+SC State In the presence of a uniform SC order ∆ 0, there are three Higgs modes. By symmetry, A− is not affected, while A+ and the Higgs mode of the uniform order A0 hybridize, leading to two eigenmodes A++ andA+−. For our consideration, we assume that ∆0 is larger than ∆Q...
[119]

interaction

Due to the translation symmetry, all such choices are equivalent to (S72). Using (S72) we find that for 2 r + ˜r< 0, the mean-field values of ∆±Q are ∆Q = ∆−Q = √ −(2r + ˜r) 4u + 2γ + 2¯γ + 4˜γ. (S73) 17 Expanding around these mean-field values as ∆±Q = ∆±Q +δ±Q, we find the L...
[120]

(9) in the main text, q = (0,ω )

Mode-mode coupling in the PDW+SC state: composition of the longitudinal modes and the locations of the peaks in the spectral function In this subsection we apply a simple mode-mode coupling theory to get analytical understanding of the composition of the coupled symmetric fluc...
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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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