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Landau -roton theory revisited for superfluid helium 4 and Fermi Yvan Castin, Alice Sinatra, Hadrien Kurkjian

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Yvan Castin, Alice Sinatra, Hadrien Kurkjian. Landau phonon-roton theory revisited for superfluid helium 4 and Fermi gases. Physical Review Letters, American Physical Society, 2017, 119, pp.260402. ￿10.1103/PhysRevLett.119.260402￿. ￿hal-01570314v3￿

HAL Id: hal-01570314 https://hal.archives-ouvertes.fr/hal-01570314v3 Submitted on 25 Nov 2019

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Landau phonon-roton theory revisited for superfluid helium 4 and Fermi gases

Yvan Castin and Alice Sinatra Laboratoire Kastler Brossel, ENS-PSL, CNRS, Sorbonne Universit´es, Coll`ege de France, 75005 Paris, France

Hadrien Kurkjian TQC, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerp, Belgium

Liquid helium and -1/2 cold- Fermi gases both exhibit in their superfluid two dis- tinct types of excitations, gapless and gapped rotons or fermionic pair-breaking excitations. In the long wavelength limit, revising and extending Landau and Khalatnikov’s theory initially de- veloped for helium [ZhETF 19, 637 (1949)], we obtain universal expressions for three- and four-body couplings among these two types of excitations. We calculate the corresponding phonon damping rates at low and compare them to those of a pure phononic origin in high- helium and in strongly interacting Fermi gases, paving the way to experimental observations.

PACS numbers: 03.75.Kk, 67.85.Lm, 47.37.+q

Introduction – Homogeneous superfluids with short- sion to the hydrodynamic regime ωqτγ . 1 or ωqτφ . 1 range interactions exhibit, at sufficiently low tempera- may be obtained from kinetic equations [2]. An experi- ture, phononic excitations φ as the only microscopic de- mental test of our results seems nowadays at hand, either grees of freedom. In this universal limit, all superfluids in liquid helium 4, extending the recent work of Ref.[3], of this type reduce to a weakly interacting phonon or in homogeneous cold Fermi gases, which the break- with a quasilinear dispersion relation, irrespective of the through of flat-bottom traps [4] allows one to prepare statistics of the underlying and of their inter- [5] and to acoustically excite by spatio-temporally mod- action strength. Phonon damping then only depends on ulated laser-induced optical potentials [6, 7]. the dispersion relation close to zero wavenumber (namely, Landau-Khalatnikov revisited – We recall the reasoning its slope and third derivative) and on the phonon non- of Ref.[1] to get the phonon-roton coupling in liquid he- linear coupling, deduced solely from the system equation lium 4, extending it to the phonon-fermionic quasipar- of state through Landau-Khalatnikov quantum hydrody- ticle coupling in unpolarised spin-1/2 Fermi gases. We namics [1]. first treat in first quantisation the case of a single roton In experiments, however, are not al- or fermionic excitation, considered as a γ- ways low enough to make the dynamics purely phononic. of position r, p and spin s = 0 or s = 1/2. Other elementary excitations can enrich the problem, In a homogeneous superfluid of density ρ, its Hamilto- such as spinless bosonic rotons in liquid helium 4 and nian is given by ǫ(p,ρ), an isotropic function of p such spinful fermionic BCS-type pair-breaking excitations in that p 7→ ǫ(p,ρ) is the γ-quasiparticle dispersion rela- spin-1/2 cold-atom Fermi gases. These excitations, de- tion. In presence of acoustic waves (phonons), the super- noted here as γ-, exhibit in both cases an fluid acquires position-dependent density ρ(r) and veloc- gap ∆ > 0. Remarkably, as shown by Landau and ity v(r). For a phonon wavelength large compared to Khalatnikov [1], the phonon-roton coupling, and more the γ-quasiparticle coherence length [8], here its thermal ~2 1/2 generally phonon coupling to all gapped excitations as wavelength (2π /m∗kBT ) [44], and for a phonon an- we shall see, depend to leading order in temperature gular frequency small compared to the γ-quasiparticle only on a few parameters of the dispersion relation of “internal” energy ∆, we can write the γ-quasiparticle the γ-quasiparticles, namely the value of the minimum Hamiltonian in the local density approximation [9, 10]: ∆ and its location k0 in wavenumber space, their deriva- H = ǫ(p,ρ(r)) + p · v(r) (1) tives with respect to density, and the effective mass m∗ close to k = k . We have discovered however that the 0 The last term is a Doppler effect reflecting the energy φ − γ coupling of Ref.[1] is not exact, a fact apparently difference in the lab frame and in the frame moving with unnoticed in the literature. Our goal here is to com- the superfluid. For a weak phononic perturbation of the plete the result of Ref.[1], and to quantitatively obtain superfluid, we expand the Hamiltonian to second order phonon damping rates due to the φ − γ coupling as func- in density fluctuations δρ(r)= ρ(r) − ρ: tions of temperature, a nontrivial task in the considered strongly interacting systems. We restrict to the colli- 1 H≃ ǫ(p,ρ)+ ∂ ǫ(p,ρ)δρ(r)+ p · v(r)+ ∂2ǫ(p,ρ)δρ2(r) sionless regime ωqτγ ≫ 1 and ωqτφ ≫ 1, where ωq is the ρ 2 ρ angular eigenfrequency of the considered phonon mode (2) of wavevector q, and τγ (τφ) is a typical collision time of not paying attention yet to the noncommutation of r thermal γ-quasiparticles (thermal phonons). An exten- and p. Phonons are bosonic quasiparticles connected to 2 the expansion of δρ(r) and v(r) on eigenmodes of the γ scattering is obtained by adding the contributions of quantum-hydrodynamic equations linearised around the the direct process (terms of Hˆ quadratic in ˆb), and of homogeneous solution at rest in the quantisation volume the absorption-emission or emission-absorption process V: (terms linear in ˆb) treated to second order in perturbation theory [1]: δρ(r) 1 ρq ˆ ρq ˆ† iq·r = 1/2 bq + b−q e (3) v(r) V vq −vq   q=6 0      X q ′ q ′ q ′ eff ′ ′ k k k A (k, q; k , q )= ′ + ′ + ′ 1/2 2 k q k q k q with modal amplitudes ρq = [~ρq/(2mc)] and vq = 1/2 [~c/(2mρq)] q, m being the mass of a superfluid parti- A (k, q; k + q)A (k′, q′; k′ + q′) = A (k, q; k′, q′)+ 1 1 cle and c the sound velocity. The annihilation and cre- 2 ~ω + ǫ − ǫ ˆ ˆ† q k k+q ation operators bq and bq of a phonon of wavevector q ′ ′ ′ ′ A1(k − q , q ; k)A1(k − q , q; k ) and energy ~ωq = ~cq obey usual commutation relations + (7) † ǫk − ~ωq′ − ǫk q′ ˆ ˆ ′ − [bq, bq′ ]= δq,q . For an arbitrary number of γ-quasiparticles, we switch where in the second (third) term the γ-quasiparticle first to second quantisation and rewrite Eq.(2) as absorbs phonon q (emits phonon q′) then emits phonon q′ q ′ (absorbs phonon ). Up to this point this agrees with † A1(k, q; k ) † Hˆ = ǫkγˆ γˆk + (ˆγ ′ γˆk ˆbq+h.c.) Ref.[1], except that the first derivative ∂ρ∆ in Eq.(5), kσ σ V1/2 k σ σ k k k′ q thought to be anomalously small in low-pressure helium, X,σ ,X, ,σ ′ ′ was neglected in Ref.[1]. A2(k, q; k , q ) † × δk+q,k′ + γˆ ′ γˆkσδk+q,k′+q′ Eq. (7), issued from a local density approximation, V k σ k k′ q q′ , X, , ,σ holds to leading order in a low-energy limit. We then † 1 take the T → 0 limit with scaling laws × [ˆb ′ˆbq + (ˆb q′ˆbq + h.c.)] (4) q 2 − 1/2 q ≈ T, k − k0 ≈ T (8) † whereγ ˆkσ andγ ˆkσ are bosonic (rotons, s = 0, σ = 0) or fermionic (s =1/2, σ =↑, ↓) annihilation and creation reflecting the fact that the thermal energy of a phonon operators of a γ-quasiparticle of wavevector k = p/~ in is ~cq ≈ kBT and the effective kinetic energy of a γ- spin component σ, obeying usual commutation or anti- quasiparticle, that admits the expansion commutation relations. The first sum in the right-hand ~2 2 side of Eq.(4) gives the γ-quasiparticle energy in the un- (k − k0) 3 ǫk − ∆ = + O(k − k0) (9) k→k0 perturbed superfluid, with ǫk ≡ ǫ(~k,ρ). The second 2m∗ sum, originating from the Doppler term and the term is also ≈ k T . The coupling amplitudes A and energy linear in δρ in Eq.(2), describes absorption or emission B 1 denominators in Eq.(7) must be expanded up to relative of a phonon by a γ-quasiparticle, characterised by the corrections of order T [45]. On the contrary, it suffices to amplitude expand A2 to leading order T in temperature. We hence

′ ~ ~ ′ get our main result, the effective coupling amplitude of ′ ∂ρǫk + ∂ρǫk k + k A (k, q; k )= ρ + vq · (5) 1 q 2 2 the φ − γ scattering to leading order in temperature: ′ where q, k and k are the wavevectors of the incoming ~ ~ ′ 2 ~2 2 eff ′ ′ q 1 2 ′′ ( ρk0) k0 phonon and the incoming and outgoing γ-quasiparticles. A2 (k, q; k , q ) ∼ ρ ∆ + + T →0 mcρ 2 2m 2m Eq.(5) is invariant under exchange of k and k′. This re- ( ∗ ∗ sults from symmetrisation of the various terms, in the ′ 2 ′ ′ ρ∆ ′ ρ∆ ′ ′ ρk0 2m∗c form [f(p)eiq·r +eiq·rf(p)]/2 with r and p canonically × uu + (u + u ) uu − + w ~ck0 ~ck0 k0 ~k0 conjugated operators, ensuring that the correct form of (      ′ Eq.(2) is hermitian. The third sum in Eq.(4), originat- m∗c ′ 2 ′2 ρk0 2 ′2 ing from the terms quadratic in δρ in Eq.(2), describes + ~ (u + u )w + u u − (u + u ) (10) k0 k0 )) direct scattering of a phonon on a γ-quasiparticle, with the symmetrised amplitude Here ∆′, k′ , ∆′′ are first and second derivatives of ∆ and 0 ′ ′ q·k ′ q ·k q·q 2 2 k0 with respect to ρ; u = , u = ′ , w = ′ are ∂ ǫk + ∂ ǫk′ qk q k qq ′ ′ ρ ρ ′ A2(k, q; k , q )= ρqρq′ (6) cosines of the angles between k, q and q ; our results hold 2 for k0 = 0 provided the limit k0 → 0 is taken in Eq.(10). where the primed wavevectors are the ones of emerging In Eq.(3.17) of Ref.[1], the ∆′ terms were neglected as ′ quasiparticles. It also describes negligible two-phonon said, but the last term in Eq.(10), with the factor ρk0/k0, absorption and emission. The effective amplitude for φ− was simply forgotten. 3

Addition in the corrected-augmented version: here to a scaling on k different from Eq.(8) as it forces Expression (7) of the coupling amplitude is incomplete k to be at a nonzero distance from k0, even in the low- because it does not take into account the interactions phonon-energy limit: When q → 0 at fixed k, the Dirac among phonons. This omission affects expression (10) delta in Eq.(13) becomes of the effective amplitude, the angular integral in note [47] and the dashed lines of Figs. 1 and 2. The erratum dǫk ~ ′ ~ −1 dk δ( ωq + ǫk − ǫk ) ∼ ( cq) δ 1 − u ~ (14) [42] that corrects this omission is reproduced here in q→0 c ! appendix, see in particular Eqs. (19) and (20), and it is supplemented by a verification of the final result using 1 dǫk and imposes that the group velocity ~ dk of the incom- a microscopic approach based on Bogoliubov theory with ing γ-quasiparticle is larger in absolute value than that, an arbitrary short-range interaction potential. – End of c, of the phonons. This condition, reminiscent of Lan- addition. dau’s criterion, restricts wavenumber k to a domain D not containing k0. In the low-q limit, that is for q much Damping rates – A straightforward application of smaller than the k significantly contributing to Eq.(13), Eq.(10) is a Fermi-golden-rule calculation of the damp- but with no constraint on the ratio ~cq/kBT , we write scat q 1/2 ing rate Γq of phonons due to scattering on γ- A1 in Eq.(5) to leading order q in q, and integrate over quasiparticles. The γ-quasiparticles are in thermal equi- the direction of k, to obtain librium with Bose or Fermi mean occupation numbers 2s −1 2 n¯γ,k = [exp(ǫk/kBT ) − (−1) ] . So are phonons in 2 −ǫk/kB T ~2 2 a-e (2s + 1)ρ dkk e c k ′ ′ modes q 6= q, with Bose occupation numbersn ¯b,q = Γq ≃ dǫ ∂ρǫk + dǫ (15) 4πmc | k | 1+¯nb,q ρ k ~ ′ −1 D dk dk [exp( ωq /kBT ) − 1] ; mode q is initially excited (e.g. Z 2 2 ~ −ǫk∗ /kB T by a sound wave) with an arbitrary number nb,q of (2s + 1)ρk∗ ck∗ k B T e ′ ′ q k q k ∼ 2 3 ∂ρǫk∗ + (16) phonons. By including both loss + → + and T →0 4π~ mc ρη∗ 1+¯nb,q ′ ′ gain q + k → q + k processes [46] and summing over σ, d scat ~ one finds that dt nb,q = −Γq (nb,q − n¯b,q) with Eq.(16) is an equivalent when T → 0 at fixed cq/kBT ; dǫk ~ k∗ is the element of the border of D ( dk |k=k∗ = η∗ c, 3 3 ′ 2π d kd q 2 η∗ = ±) with minimal energy ǫk (when more than one of Γscat = (2s + 1) Aeff (k, q; k′, q′) q ~ (2π)6 2 such k∗ exists, one has to sum their contributions). As Z 2s ǫk∗ > ∆, the damping rate due to scattering dominates n¯b,q′n¯γ,k′ [1 + (−1) n¯γ,k] × δ(ǫk + ~ωq − ǫk′ − ~ωq′ ) the one due to absorption-emission in the mathematical n¯b,q limit T → 0 ; we shall see however that this is not always (11) so for typical temperatures in current experiments. and k′ = k + q − q′. As our low-energy theory only To be complete, we give a low-temperature equivalent of the damping rate of the γ-quasiparticle k due to in- holds for kBT ≪ ∆, the gas of γ-quasiparticles is non- 1/2 teraction with thermal phonons. With k − k0 = O(T ) degenerate, andn ¯γ,k ≃ exp(−ǫk/kBT ) ≪ 1 in Eq.(11). as in Eq.(8), we find ~Γγφ ∼ (πI/42)(k T )7/(~cρ1/3)6, By taking the T → 0 limit at fixed ~cq/kBT and setting k B ~ωq where the factor 2s + 1 is gone (no summation over σ is Aeff = f, where the dimensionless quantity f only 2 ρ needed) but I is the same angular integral as in Eq.(12). depends on angle cosines, we obtain the equivalent γφ Here scattering dominates [48]. Using τγ ≃ 1/Γk , we −∆/kB T 2 4 checked that the figures 1 and 2 below are in the col- ~ scat e k0q c 1/2 Γq ∼ (2s + 1) (m∗kBT ) I (12) T →0 (2π)9/2 ρ2 lisionless regime ωqτγ ≫ 1. Similarly, we checked that ωqτφ ≫ 1 on the figures. 2 2 2 ′ with I = d Ωk d Ωq′ f (u,u , w) an integral over Application to helium – Precise measurements of the angles of direction k and q′ [47]. (relating ρ to pressure) and of the roton R R One proceeds similarly for the calculation of the damp- dispersion relation for various were performed a-e ′ 4 2 ing rate Γq of phonons q due to absorption q + k → k in liquid He at low temperature (kBT ≪ mc , ∆). They ′ or emission k → q + k processes by thermal equilibrium give access to the parameters k0, ∆, their derivatives γ-quasiparticles. We obtain and m∗. The measured sound velocities agree with the 2 dµ thermodynamic relation mc = ρ dρ , where µ is the zero- 2π d3k temperature of the liquid. We plot Γa-e = (2s + 1) [A (k, q; k′)]2 q ~ (2π)3 1 in Fig. 1 the phonon damping rates as functions of tem- Z ′ ′ perature, for a fixed angular frequency ω . At the cho- × δ(~ωq + ǫk − ǫk )(¯nγ,k − n¯γ,k ) (13) q sen high pressure, the phonon dispersion relation is con- with k′ = k + q. Low degeneracy of the γ-quasiparticles cave at low q, therefore the Beliaev-Landau [11–16] three- and energy conservation allow us to writen ¯γ,k − n¯γ,k′ ≃ phonon process φ ↔ φφ is energetically forbidden at low exp(−ǫk/kBT )/(1 +n ¯b,q). Energy conservation leads temperature and the Landau-Khalatnikov [1, 6, 16] pro- 4

6 -2 10 10 (a) 5 -3 10 10 -4 4 10

10 q -5

ω 10

3 / -6 10 q 10 [1/s] Γ

q -7 2 Γ 10 10 -8 10 1 -9 10 10 -10 0 10 10 -2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 10 (b) -3 T [K] 10 -4 10

FIG. 1: Phonon damping rates at angular frequency ωq = q -5 − 1 4 ω 10 2π × 165 GHz (q = 0.3A˚ ) in liquid He at pressure P = 20 / -6 q 10 bar as functions of temperature. Solid line: purely phononic Γ -7 damping Γ due to Landau-Khalatnikov four-phonon pro- 10 φφ -8 cesses [1, 6, 16]; it depends on the curvature parameter γ 10 γ ~q 2 4 -9 defined as ωq = cq[1 + 8 ( mc ) + O(q )]. Interpolating mea- 10 surements of P 7→ γ(P ) in Refs.[20, 21] gives γ = −6.9. -10 10 0.02 0.04 0.06 0.08 0.1 0.15 Dashed black line/dash-dotted black line: damping due to µ scattering/absorption-emission by rotons, see Eq.(12)/(15). kBT/ Red dashed line: original formula of Ref.[1] for the damping rate due to phonon-roton scattering. The roton parameters FIG. 2: Phonon damping rates at wavenumber q = mc/2~ in are extracted from their dispersion relation k 7→ ǫk measured −1 unpolarized homogeneous cold-atom Fermi gases in thermo- at various pressures [22]: ∆/kB = 7.44K, k0 = 2.05A˚ , dynamic limit as functions of temperature. (a) At unitarity ′ ′ 2 ′′ − m∗/m = 0.11, ρk0/k0 = 0.39, ρ∆ /∆ = −1.64, ρ ∆ /∆ = 1 ′ a = 0, where most parameters of the phonons and fermionic −8.03, ρm∗/m∗ = −4.7. In Eq.(15), parabolic approxima- quasiparticles are measured (see text). (b) On the BCS side tion (9) is used (hence ǫk∗ /∆ ≃ 1.43). The speed of sound 1/kFa = −0.389, these parameters are estimated in BCS the- c = 346.6 m/s, and the Gr¨uneisen parameter dln c = 2.274 m∗ ∆ ′ dln ρ ory (µ/ǫF ≃ 0.809, ∆/µ ≃ 0.566, m = 2µ , ρµ /µ ≃ 0.602, ′ ′′ entering in Γφφ, are taken from equation of state (A1) of ρ∆ /∆ ≃ 0.815, ρ2∆ /∆ ≃ −0.209, dln c ≃ 0.303). In both ~q kB T −2 dln ρ Ref.[23]. The low values mc = 0.13 and mc2 < 10 jus- cases the curvature parameter γ defined in the caption of tify our use of quantum hydrodynamics. Fig. 1 is estimated in the RPA [37]. Solid line: phonon- phonon (a) Beliaev-Landau damping φ ↔ φφ (for γ > 0) as in Eqs.(121,122) of Ref.[16] (independent of |γ|) and (b) Landau- Khalatnikov damping φφ ↔ φφ (for γ ≃ −0.30 < 0) [6, 16]. cess φφ ↔ φφ is dominant. Our high yet experimen- Dashed line/dash-dotted line: scattering/absorption-emission tally accessible [17, 18] value of ωq leads to attenuation phonon-fermionic quasiparticle processes, as in Eq.(12)/(15). lengths 2c/Γq short enough to be measured in centimet- In Eq.(15), we took for ǫk (a) the form proposed in Ref.[34] ric cells. As visible on Fig. 1, the damping of sound is (hence ǫk∗ /∆ ≃ 1.12) and (b) the BCS form (hence ǫk∗ /∆ ≃ in fact dominated by four-phonon Landau-Khalatnikov 1.14). µ is the T = 0 gas chemical potential, and the plotted quantities are in fact inverse quality factors. Here processes up to a temperature T ≃ 0.6 K. In this regime 2 2 kBT/mc > 0.03 in contrast to Fig.1 where kB T/mc < 0.01: one would directly observe this phonon-phonon damping cold are effectively farther from the T → 0 limit than scat a-e mechanism, which would be a premiere. The sound at- liquid helium, hence the inversion of the Γq -Γq hierarchy. tenuation measurements of Ref.[19] in helium at 23 bars and ωq =2π × 1.1 GHz are indeed limited to T > 0.8 K where damping is still dominated by the rotons. Application to – In cold-atom Fermi gases, in- µ = ξǫF, where ξ ≃ 0.376 [29], and the critical tem- teractions occur in s-wave between opposite-spin atoms. perature is Tc ≃ 0.167ǫF/kB [29]. For the effective mass Of negligible range, they are characterized by the scat- of the fermionic excitations and their dispersion relation tering length a tunable by Feshbach resonance [24–29]. at non vanishing k − k0, we must rely on results of a dimensional ǫ = 4 − d expansion, m∗/m ≃ 0.56 and Precise measurements of the fermionic excitation pa- 2 2 2 2 −1 ~ (k −k0 ) rameters k0 and ∆ were performed at unitarity a =0 ǫk ≃ ∆+ 2 [34]. We also trust Anderson’s RPA 8m∗k0 [30]. Due to the unitary-gas scale invariance [31–33], k0 is prediction [35, 36] that the q = 0 third derivative of the 2 1/3 proportional to the Fermi wavenumber kF = (3π ρ) , phononic dispersion relation is positive [37]. The damp- k0 ≃ 0.92kF [30], and ∆ is proportional to the Fermi ~ 2 2 ing rates of phonons with wavenumber q = mc/2 are ~ kF energy ǫF = 2m , ∆ ≃ 0.44ǫF [30]. This also de- plotted in Fig. 2a. The contribution of the three-phonon termines their derivatives with respect to ρ. Similarly, Landau-Beliaev processes φ ↔ φφ, here energetically the equation of state measured at T = 0 is simply allowed, is dominant; it is computed in the quantum- 5 hydrodynamic approximation where it is independent of Khalatnikov sound damping in a superfluid, unobserved the aforementioned third derivative. to this day, is dominant. The phononic excitation branch becomes concave in This project received funding from the FWO and the − the BCS limit kFa → 0 [38]. As visible on Fig. 2b, the EU H2020 program under the MSC Grant Agreement phonon-phonon damping (now governed by the Landau- No. 665501. Khalatnikov processes mentioned earlier) is much weaker, and dominates the φ − γ damping only at very low tem- Addition in the corrected-augmented version peratures. At commonly reached temperatures T > 0.05ǫF/kB [39], the damping is in fact dominated by absorption-emission φ − γ processes which, unlike in liq- Erratum : corrections due to three-body uid helium, prevail over scattering ones because of the interactions among phonons. smaller value of ǫk∗ /∆. Although the associated quality −1 factors ωq/Γq may seem impressive, the lifetimes Γq of the modes do not exceed one second in a gas of 6Li with The results on the effective φ − γ scattering ampli- tude and associated damping rate, based on the roton a typical Fermi temperature TF =1µK, which is shorter than what was observed in a Bose-Einstein condensate Hamiltonian (Eqs. (1) and (4)), do not account for the [40]. Our predictions, less quantitative than on Fig. 2a, phonon-phonon interaction, which is described, to lowest are based on the BCS approximation for the equation order in q, by the Hamiltonian [1]: of state and the fermionic excitation dispersion relation ~2 2 2↔1 BCS k 2 2 1/2 (3) Aφ−φ(q1, q2; q3) ǫk ≃ ǫ = [( − µ) + ∆BCS] and on the RPA ˆ k 2m H = δq1+q2,q3 for the q = 0 third derivative of ω (whose precise value φ−φ V1/2 q q1,q2,q3 here). A cutting remark on Ref.[41]: even in the X ˆ† ˆ† ˆ BCS approximation to which it is restricted, we disagree × bq1 bq2 bq3 + h.c. (17) a-e with its expression of Γq .   Conclusion – By complementing the local density ap- with the φ + φ ↔ φ amplitude [16]: proximation in Ref.[1] with a systematic low-temperature expansion, we derived the definitive leading order ex- 2 ~3 2↔1 mc q1q2q3 pression of the phonon-roton coupling in liquid helium A (q1, q2; q3)= φ−φ ρ1/2 32m3c3 and we generalized it to the phonon-pair-breaking exci- r d ln c q · q q · q q · q tation coupling in Fermi gases. The ever-improving ex- × 2 − 1+ 1 2 + 1 3 + 2 3 (18) d ln ρ q q q q q q perimental technics in these systems give access to the  1 2 1 3 2 3  microscopic parameters determining the coupling and al- dln c low for a verification in the near future. Our result We have used here the Gr¨uneisen parameter dln ρ = ′′ ′ also clarifies the regime of temperature and interaction (ρµ /µ +1)/2. To leading order, there are two diagrams ˆ (3) strength in which the purely phononic φφ ↔ φφ Landau- mediated by Hφ−φ missing in Eq. (7):

q ′ q ′ q ′ q ′ ′ q eff ′ ′ k k k k k A2 (k, q; k , q ) = ′ + ′ + ′ + ′ + ′ k q k q k q q k k q

′ ′ ′ ′ ′ ′ ′ ′ ′ ′ A1(k, q; k + q)A1(k , q ; k + q ) A1(k − q , q ; k)A1(k − q , q; k ) = A2(k, q; k , q )+ + ~ωq + ǫk − ǫk+q ǫk − ~ωq′ − ǫk−q′ 2↔1 ′ ′ ′ ′ 2↔1 ′ ′ ′ ′ 2A (q , q − q ; q)A1(k, q − q ; k ) 2A (q, q − q; q )A1(k , q − q; k) + φ−φ + φ−φ (19) ~(ωq − ωq−q′ − ωq′ ) ǫk − ǫk′ − ~ωq−q′

The energy denominators of the two new diagrams are equal to −~ωq−q′ to leading order (see note [45]). One should 6 thus add −(~q/mcρ)(ρ∆′/2)(2d ln c/d ln ρ − 1+ w) to Eq. (10):

~ ′ ~ ′ 2 ~2 2 eff ′ ′ q 1 2 ′′ ρ∆ d ln c ( ρk0) k0 A2 (k, q; k , q ) ∼ ρ ∆ − 2 − 1 + + T →0 mcρ 2 2 d ln ρ 2m 2m (   ∗ ∗ ρ∆′ 2 ρ∆′ ρk′ m c × uu′ + (u + u′) uu′ − 0 + ∗ w ~ck ~ck k ~k ( 0  0   0  0  ′ m∗c ′ 2 ′2 ρk0 2 ′2 + ~ (u + u )w + u u − (u + u ) (20) k0 k0 ))

This also changes the angular integral given in note [47]: where Vk is the Fourier transform of the interaction po- ~2 2 2 2 k0 2 1 4α 28 2 2β 2 4α 2 tential. This spectrum describes hydrodynamic phonons I/(4π) = ( 2m∗mc ) [ 25 − 15 + 45 α + 9 +A( 9 − 3 )+ 2 2 ~ 3 2 1 α 2 2 4α 2α β 2β 3 (with ǫq = cq + O(q )) provided V0 > 0 and ∂kVk|k=0 = A +4βB( 15 − 9 )+ B ( 15 − 9 + 3 + 3 )+ 9 B + 4 0. It has a roton minimum ∆ = ǫk0 in k = k0 pro- B m∗ 2 ′′ ′ ], and the definition of A = 2 [ρ ∆ + ρ∆ (1 − 2 ~2 9 (~k0) vided ∂kǫk|k=k0 = 0 and ∂kǫk|k=k0 = /m∗ > 0, which 2d ln c/d ln ρ)]+α2. Accordingly, the black dashed curves is always possible with a careful choice of the function in Figs. 1, 2a and 2b are multiplied respectively by 0.46, k 7→ Vk. 0.78 and 0.85. Note that the Gr¨uneisen parameter also appears in Ref. [43]; our coefficients of the uu′ and w monomials of Aeff are however still in disagreement with 2 Next, we use the 3- and 4-quasiparticle coupling am- that reference. plitudes derived from Bogoliubov theory (see Eqs. (E18), (E19) and (E20) of Ref. [16]), we compute the effective Microscopic verification of the effective coupling scattering amplitude in second order perturbation theory amplitude as prescribed by Eq. (105) of [16] (in which we set q1 = q, ′ ′ q2 = k, q3 = q and q4 = k ) and we take the limit of ′ To check the expression of the effective coupling am- q and q tending to 0. Since the expected result (20) plitude that we obtained through phonon-roton hydrody- does not depend on k to leading order, we choose k = k0, namics (Eq. (20)), we recompute it within a microscopic which greatly simplifies the microscopic calculation. As in the hydrodynamic approach, we then eliminate k′ us- approach in the particular case of a weakly-interacting ′ . In this well-known system, the excitation spec- ing momentum conservation, and the norm of q using trum takes the Bogoliubov form energy conservation (with the difference that one should go up to order q3 in the calculation of q − q′ because the scattering amplitude of the microscopic model diverges ~2k2 ~2k2 ǫ = +2ρV (21) as 1/q off-shell). We get finally: k 2m 2m k s  

~ 5 ~ ~ eff ′ ′ q 4 2∆ ′ 3 k0 ′ k0 ′ Amicro(k, q; k , q ) ∼ − 2Ek0 + uu + ∆Ek0 4(u + u )+ uu T →0 16∆3mcρ m c2 mc m c ( ∗  ∗  ~k 2~k − 4E2 ∆2 4(u2 + u′2)+ w − 1+ 0 uu′(u + u′) + 2∆4 2w − 1+ 0 uu′(u + u′) k0 m c m c  ∗   ∗  ~k ~k ~k ~k − 2∆3E 0 2(u + u′)+ 0 uu′ + 8∆3mc2 0 w(u + u′)+ 0 u2u′2 (22) k0 mc m c mc m c  ∗   ∗  )

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