<figure><img src="image_14.png" /></figure>

**Figure 4.** A possible asymptotic behavior of the unitary four-fermion problem ($N_↑ = 2$, $N_↓ = 2$) in harmonic potentials $U_σ(r)$: two pairons oscillate furiously (with large-amplitude motions); fermions $↑$ and $↓$ in each pairon have relative energy $O(1)$ (in $ħω$ units if dimension is to be respected) and remain strongly correlated; the two pairons have relative energy $→ +∞$ and are decoupled, so they can be seen as two identical bosons that have an internal structure (that of relative motion $↑↓$ within a pairon) and do not interact. The quantum statistical correction CorrStat$_{2,2}$ is then non-zero in Equation (33).

for any $(n_↑, n_↓) \in \mathbb{N}^{*2} \setminus \{1, 1\}^{20}$. Here, the prefactor relates the homogeneous case to the trapped case, and the first contribution in square brackets is modelled on the $N=3$ result of reference [45]; the second contribution in square brackets is an ideal-gas quantum statistical correction originating from the indistinguishable non-monoatomic subclusters into which the internal eigenstates of the trapped system $(n_↑, n_↓)$ decouple at high energy (the center of mass of the system remains in its ground state).

Let's explain this subcluster decoupling story a little better with some examples. If $(n_↑, n_↓) = (1, 1)$, the eigenstates asymptotically (for arbitrarily large energy values) take the form of two uncorrelated fermions $↑$ and $↓$ in large-amplitude oscillatory levels$^{21}$; the subclusters being monoatomic, we have CorrStat = 0. If $(n_↑, n_↓) = (2, 1)$, a new decoupling is possible, alongside that into three decorrelated fermions: the particles can separate into an atom $↑$ and a “pairon” $↑↓$ of strongly correlated fermions (the relative motion within the pairon remaining of low amplitude), with highly excited oscillatory levels for the atom $↑$ and for the center of mass of the pairon $↑↓$; the pairon is alone in its category, and we again have CorrStat = 0. The conclusion remains the same for $(n_↑, n_↓) = (3, 1)$, except that the triplon $↑↑↓$ appears as a new decoupled subcluster. On the other hand, for $(n_↑, n_↓) = (2, 2)$, there is possible decoupling into two $↑↓$ pairons of very high relative energy, see Figure 4; they no longer interact but are indistinguishable and lead, like identical bosons in an ideal gas, to a quantum statistical correction ignored by the integral over $S$ in Equation (33); the calculation gives CorrStat = $1/32$ [44].

Conjecture (33) is well established for $N=3$, by inverse application of the residue theorem, which converts the sum over the spectra (31), i.e. over the roots $s$ of $\Lambda_{n_↑, n_↓}$, into an integral [45]. For $N=4$, the analytical properties of function $\Lambda_{n_↑, n_↓}$ in the complex plane are not sufficiently known to apply Cauchy's theorem$^{22}$; for the special case $m_↑/m_↓ = 1$, however, the conjecture

<footnote>$^{20}$Case $(n_↑ = 1, n_↓ = 1)$ is different and must be excluded; it corresponds, contrary to what we have assumed, to a $N$-body scattering resonance with $N=2$ in $s$ wave: in the unitary limit, we must keep only the solution $R^{-s}$, $s = 1/2$, in Equation (22), as if the right root of $\Lambda$ to keep was $-s$; in fact, the solution $R^s$ corresponds to the regular part $\propto 1/a$ of the zero-energy scattering state (see note 11 for further details).</footnote>

<footnote>$^{21}$If the relative wavenumber $k_{rel} \to +\infty$, the scattering amplitude $f_{k_{rel}} \to 0$ in Equation (11) so even at the unitary limit, interactions become negligible.</footnote>

<footnote>$^{22}$It must be possible to unfold the integration path surrounding the roots and poles of $\Lambda_{n_↑, n_↓}$ on the real axis onto the imaginary axis without crossing any singularity—pole or branch cut—in the upper and lower half-planes.</footnote>
has been confirmed by a very accurate few-body quantum Monte Carlo calculation [46] (on the other hand, the experimental values [47,48] are not confirmed, the problem arising from the impossibility of obtaining the correct polynomial of degree 4 in $z = \exp(\beta\mu)$ by fitting the pressure $P$ or density $\rho$ measured on the experimentally accessible fugacity interval [49])$^{23}$. The proof of expression (33) in the general case therefore remains open. Another interesting question concerns the behavior of the cluster coefficients $b_{n_1,n_1}$ at large orders $n_\sigma \to +\infty$, which is required to perform an efficient summation of the series (32) after calculation of its first terms, in order to extend its applicability to the degenerate regime $T \lesssim T_F$ (for example, if the radius of convergence is zero, one could implement a conformal Borel-type resummation as in references [50,51]).

# 4. Open questions from a macroscopic point of view

In this section, the interacting Fermi gas, considered in the thermodynamic limit and at non-zero but arbitrarily low temperature, is described by an effective low-energy Hamiltonian theory, Landau and Khalatnikov quantum hydrodynamics [52]$^{24}$.

## 4.1. Overview of the considered superfluid regime

The three-dimensional system of fermions is here spatially homogeneous (in a quantization volume $[0, L]^3$ close to the thermodynamic limit, which is taken at the end of the calculations), with particles of equal mass $m_\uparrow = m_\downarrow = m$ in both internal states, spin-unpolarized (there are the same number of particles in both components, $N_\uparrow = N_\downarrow$, to allow complete pairing) and at canonical thermal equilibrium in a low-temperature limit ($T \neq 0$ but $T \to 0$).

Under these conditions, (i) fermions assemble into bound pairs $\uparrow\downarrow$ in the $s$ wave; in the presence of Fermi seas in both internal states, this is what the attractive interaction between $\uparrow$ and $\downarrow$ in Section 3 leads to, via the famous Cooper mechanism; this is true even for a negative scattering length $a$, where there is no $\uparrow\downarrow$ bound state in free space, although the size of a pair tends to $+\infty$ as $a \to 0^-$ (in the $a > 0$ case, a dimer state does exist and, not surprisingly, this is what the bound pair state reduces to in the low-density limit $\rho \to 0)^{25}$; (ii) these bound pairs, being composite bosons of sorts, form a condensate in the mode of wave vector $\mathbf{K}_{\text{pair}} = \mathbf{0}$ of their center of mass (of coherence length limited by the size of the box, infinite at the thermodynamic limit) and a superfluid.

As a result, we expect the system to have, in its ground state, a branch of acoustic excitation (by sound waves) that starts out linear in wavenumber $q$ with a cubic correction,

$$ \omega_{\mathbf{q}} = c q \left[ 1 + \frac{\gamma}{8} \left( \frac{\hbar q}{mc} \right)^2 + O(q^4 \ln q) \right] \quad (34) $$

<footnote>$^{23}$References [47,48] have access only to the fourth total cluster coefficient $b_4 = (b_{4,0} + b_{3,1} + b_{2,2} + b_{1,3} + b_{0,4})/2$, which prohibits a comparison with the conjecture (33) sector by sector.</footnote>

<footnote>$^{24}$The term “quantum hydrodynamics”, in non-linear physics, is understood in contrast to classical fluid hydrodynamics and refers to an Euler equation for a real-valued rather than operator-valued velocity field $v(r)$ with a $\propto \hbar^2$ quantum pressure term, which allows to describe the motion of quantum vortices—with quantized circulation—in the superfluid (like the Gross–Pitaevskii equation for the wave function of a Bose condensate written in terms of density and phase gradient). Here, the name is to be taken in the sense of the second quantization, the velocity field now being operator-valued $\hat{v}(r)$.</footnote>

<footnote>$^{25}$It is not entirely obvious that the Wigner–Bethe–Peierls contact interaction is attractive. To see this, we obtain it as the continuous limit $b \to 0$ of a model on a cubic lattice $bZ^3$ with a coupling $\propto \hbar^2/mb^2$ between neighboring sites (to represent the kinetic energy) and an on-site interaction $g_0/b^3$; at a fixed scattering length $a \neq 0$, we find that $g_0 \approx -\hbar^2 b/m < 0$ as $b \to 0$ (the bare coupling constant $g_0$ is therefore, in the resonant scattering regime $b \ll |a|$, quite different from the effective coupling constant $g = 4\pi\hbar^2 a/m$) [53].</footnote>
<figure><img src="image_15.png" /><figcaption>**Figure 5.** Energy of different types of excitations of an unpolarized homogeneous two-component Fermi gas at zero temperature, as a function of their wave number *q* or *k*. (a) Acoustic excitation branch $\hbar\omega_q$ of linear departure $\hbar cq$ (*c* is the speed of sound), bounded at the top by the broken-pair continuum (hatched area). Here, the branch tangentially reaches the lower edge of the continuum at the terminal point of wave number $q_{max}$. (b) Dispersion relation $\varepsilon_k$ of a fermionic quasiparticle $\chi$ (see text). Under the effect of a percussive excitation of wave vector **q**, a bound pair $\uparrow\downarrow$ of the pair condensate, initially at rest, dissociates into two fermionic quasiparticles $\chi$ of opposite spins, wave vectors **k** and $\mathbf{k}' = \mathbf{q} - \mathbf{k}$ and energies $\varepsilon_k$ and $\varepsilon_{\mathbf{k}'}$; as vector **k** is not constrained (no conservation of energy for a percussive excitation), a continuum of final energies {$\varepsilon_k + \varepsilon_{\mathbf{q}-\mathbf{k}}, \mathbf{k} \in \mathbb{R}^3$} appears. For the interaction strength chosen in the figure ($|\Delta|/\mu = 0.84$ or $1/k_F a \approx -0.16$ according to BCS theory, with $\mu$ the chemical potential of the gas and $\Delta$ the complex order parameter of the pair condensate), the acoustic branch is concave at low $q$ ($\gamma < 0$ in Equation (34)) and the dispersion relation $\varepsilon_k$ has a minimum $E_{pair}/2$ at $k = k_0 > 0$; the lower edge of the continuum is therefore exactly $E_{pair}$, at least as long as one can have $k = k' = k_0$, i.e. as long as $q = |\mathbf{k} + \mathbf{k}'| \le 2k_0$. The approximate dispersion relations shown are derived from BCS theory for $\varepsilon_k$ and Anderson's RPA for $\hbar\omega_q$. Depending on the interaction strength, the domain of existence in *q* of the acoustic branch may also be non-compact, connected $q \in ]0, +\infty[$ or not $q \in ]0, q_{max}[$, $q_{min}, +\infty[$ ($q_{min} > q_{max}$) [56]; the concavity of the branch is also variable [57]; finally, $k_0 = 0$ and $E_{pair} = 2(\mu^2 + |\Delta|^2)^{1/2}$ if $\mu < 0$, $k_0 = (2m\mu)^{1/2}/\hbar$ and $E_{pair} = 2|\Delta|$ otherwise.</figcaption></figure>

and this branch to be energy-limited at the top by the broken-pair continuum, with a lower edge of the continuum given by the pair binding energy $E_{pair}$, as in Figure 5a. Here $\omega_q$ is the angular eigenfrequency at wave vector **q**, $c$ is the speed of sound at zero temperature and the curvature parameter $\gamma$ is scaled so that it tends to one in the limit $k_F a \to 0^+$ of a weakly interacting condensed gas of dimers (in agreement with Bogolioubov theory)$^{26}$. The acoustic branch is often referred to as Goldstone [55], because it is associated with $U(1)$ symmetry breaking in pair condensation; its Higgs counterpart is discussed in Section 5.

In the following, our low-temperature regime satisfies two conditions:

$$0 < k_B T \ll mc^2 \quad \text{and} \quad 0 < k_B T \ll E_{\text{pair}} \qquad (35)$$

The first ensures that only the linear part of the acoustic branch is thermally populated, the second that there is a negligible density of broken pairs (according to Boltzmann's law, this density has an activation factor $\exp(-E_{pair}/2k_B T)$, the fragments resulting from the dissociation of a bound pair—the $\chi$ fermionic quasiparticles in Figure 5b—individually having a minimum

<footnote>$^{26}$Our sign convention on $\gamma$ differs from that used in liquid helium-4, see reference [54].</footnote>
energy $E_{\text{pair}}/2$). Our system then reduces to a thermal gas of phonons, if we agree to call the quanta of the acoustic branch as such (in view of its linear departure)$^{27}$.

This raises three types of partially open questions:

(1) as we shall see, phonons (abbreviated to $\phi$) interact, the underlying Fermi superfluid constituting a nonlinear medium for sound. What are the effects of these interactions on phonons of wave vector $q$? In particular, they are expected to damp with a rate $\Gamma_q(T)$ and to undergo a thermal angular frequency shift $\Delta_q(T)$ (we don't count the shift at zero temperature, which by definition gives rise to the spectrum (34)—the neglected term $q^5 \ln q$ arises precisely from the cross-effect of interactions and quantum fluctuations in the phonon field [62,63])$^{28}$.

(2) what are the consequences of phonon collisional dynamics on the evolution of a particularly interesting macroscopic gas variable, the phase operator $\hat{\phi}_0(t)$ of the pair condensate?

(3) we enrich the problem by considering the partially polarized case $N_1 \neq N_\downarrow$. In the weakly polarized case, e.g. $N_1 - N_\downarrow = O(1)$, the unpaired supernumerary fermions of the majority spin component form, in the interacting gas, fermionic quasiparticles (abbreviated to $\chi$) with a dispersion relation $\epsilon_k$ different from those of the free fermions: it has a non-zero minimum, i.e. a gap, given by half the binding energy of a pair $\uparrow\downarrow$, where it varies quadratically with the wave number $k$ (see Figure 5b). The question then arises of the $\phi-\chi$ and of the $\chi-\bar{\chi}$ interactions; in particular, disagreement persists over the expression of the low-energy $\phi-\chi$ scattering amplitude (references [64] and [65] differ)$^{29}$. In the highly polarized case, where $N_1 - N_\downarrow$ is extensive like $N$, it is expected that $T=0$ pair condensation may take place in a plane-wave superposition of their center of mass (rather than in $K_{\text{pair}} = 0$ as assumed here), see references [66,67], giving rise to a spatially modulated superfluid (a supersolid following the fashionable terminology), with an imperfectly known domain of existence in parameter space $((N_1 - N_\downarrow)/N, 1/k_F a, T/T_F)$, the problem being complicated by its high sensitivity to thermal fluctuations [68] (there are as yet no experimental results in three-dimensional cold atom gases [69]).

The most interesting case for points 1 and 2 is that of a concave acoustic branch, $\gamma < 0$ in Equation (34), very different from the fairly well-known weakly interacting Bose gas (where $\gamma \approx 1 > 0$ as we said)$^{30}$. In particular, phonon damping for $\gamma < 0$ can only occur at $T \neq 0$, since

<footnote>$^{27}$ Equation (34) holds only for a short-range interaction $V(r_{ij})$, decreasing fast enough when $r_{ij} \to +\infty$. In the case of a dipolar interaction, as in gases of cold magnetic atoms, the speed of sound is anisotropic [58], see also Wilhelm Zwerger's contribution to this special issue [59] and Jean Dalibard's 2023-2024 lecture at the Collège de France [60]. In the case of a Coulomb interaction, as in superconducting electron gases, the acoustic branch gives way to a gapped plasmon branch ($\omega_q$ has a positive limit at $q=0$) [61]. Here, our atoms are neutral and of negligible dipole moment.</footnote>

<footnote>$^{28}$ In the convex case $\gamma > 0$, it is accompanied—still at $T=0$—by a non-zero imaginary part $\approx q^5$, corresponding to the Beliaev damping mechanism $q \to k \cdot k'$, see below.</footnote>

<footnote>$^{29}$ This problem is relevant for superfluid helium-4, whose excitation branch also has a quadratic minimum, the roton minimum; our $\phi-\chi$ scattering problem is therefore formally equivalent to the roton-phonon scattering already studied in reference [52], with the difference that rotons are bosons. The predictions of [52] are, however, incomplete and at odds with [64,65].</footnote>

<footnote>$^{30}$ Qualitatively, the case $\gamma < 0$ is obtained when the binding energy $E_{\text{pair}}$ is low enough: if we reduce $E_{\text{pair}}$, the broken pair continuum in Figure 5a lowers, pushes on the acoustic branch and eventually bends it downwards. This is what happens in the BCS limit $k_F a \to 0^-$ where $E_{\text{pair}}/mc^2 = O(\exp(-\pi/2k_F|a|))$ tends rapidly to zero; it no longer happens in the BEC limit $k_F a \to 0^+$ where $E_{\text{pair}} \sim E_{\text{dim}} = \hbar^2/ma^2 \gg mc^2$. It is not clear on which side of the unitary limit $1/k_F a = 0$ (i.e. for which sign of the scattering length $a$) the curvature parameter $\gamma$ changes sign, see Section 5. Given the shape of the lower continuum edge in Figure 5a—regardless of the sign of $\gamma$, the repulsion effect on the acoustic branch is strongest at large $q$ and weakest at small $q$ (where the energy difference between the continuum edge and the branch is greatest). We therefore expect to have an interval of values of $1/k_F a$ over which the branch is convex at small $q$ and concave at large $q$ [57].</footnote>
the decay of a phonon into any number $n > 1$ of phonons is forbidden by energy–momentum conservation for a concave acoustic branch; at the leading order in temperature, it results for the same reason from the $\phi\phi \to \phi\phi$ four-phonon processes of Landau and Khalatnikov [52] (for $\gamma > 0$ it results from the Beliaev $\phi \to \phi\phi$ or Landau $\phi\phi \to \phi$ three-phonon processes [70,71]). To our knowledge, this four-phonon damping has not yet been observed experimentally in any system. It could be observed in a Fermi gas of cold atoms in a box trap [72]. It could also be observed in superfluid helium-4 (a liquid of bosons) if the pressure is increased sufficiently to make $\gamma < 0$ (the roton minimum is lowered, which ends up making the acoustic branch concave at low $q$) and if the temperature is lowered sufficiently to reduce the density of rotons (through the activation factor $\exp(-E_{\text{roton}}/k_B T)$) and make the parasitic damping of phonons by rotons negligible [73]$^{31}$.

## 4.2. Which macroscopic theory to use?

An effective low-energy theory gives up describing the system below a certain length scale $\ell$; on the other hand, the theory is expected to be accurate at long wavelengths, in this case at leading order in temperature. Under these conditions, it is legitimate to cut the gas into portions of size $\ell$, for example into cubic boxes of side $\ell$ centered on the cubic lattice $\ell Z^3$, see Figure 6. Let's set out some constraints on the choice of $\ell$:

(1) we must have $\ell \gg \xi$ (here $\xi = \hbar/mc$ is the so-called healing or correlation length of the superfluid) and $\rho\ell^3 \gg 1$ (there are a large number of fermions per site) so that (i) each cubic portion can be considered mesoscopic with a well-defined equation of state linking pressure or chemical potential to density (as is the case in the thermodynamic limit), and (ii) the lattice spacing $\ell$ provides a wave number cutoff $\pi/\ell \ll mc/\hbar$ to the phononic excitations of the gas, restricting them to the quasi-linear part of the branch (34), which is universal because it is described by two parameters, $c$ and $\gamma$.

(2) we must have $k_B T \ll \hbar c\pi/\ell$ (this is the energy of the ground phonon mode in a portion) so that we can consider that (i) each portion is at zero temperature, and (ii) each portion is spatially homogeneous on the scale of the typical wavelength $q_{\text{th}}^{-1} = \hbar c/k_B T$ of thermal sound waves.

(3) it is also necessary that $\ell \ll \ell_{\text{coh}}$ where $\ell_{\text{coh}}$ is the coherence length of the fermion pairs, so that the notion of global phase $\hat{\phi}$ makes sense in each portion (as for a condensate). This constraint is inoperative here since $\ell_{\text{coh}} \approx L$ (bound pairs are condensed in 3D).

We can then represent the system by two field operators, the density field $\hat{\rho}(\mathbf{r})$ and the phase field $\hat{\phi}(\mathbf{r}')$, with $\mathbf{r}, \mathbf{r}' \in \ell Z^3$; these are canonically conjugate variables,

$$[\hat{\rho}(\mathbf{r})\ell^3, \hat{\phi}(\mathbf{r}')] = i\delta_{\mathbf{r},\mathbf{r}'} \qquad (36)$$

as if $\hat{\phi}$ were a momentum operator and $\hat{\rho}\ell^3$ a position operator in ordinary quantum mechanics [52,75]$^{32}$. The phase field gives access to the velocity field by simple differentiation (this is a

<footnote>$^{31}$In liquid helium-4, four-phonon scattering processes between intentionally produced (non-thermal) phonon beams have already been the subject of theoretical and experimental studies [74].</footnote>

<footnote>$^{32}$These historical references use a continuous space description for simplicity. The need to discretize the space to avoid infinities and make the theory renormalizable is emphasized in publication [76]. Here, we sweep these difficulties under the carpet; for example, we do not distinguish in (38) between the notion of bare equation of state $e_{0,0}(\rho)$—which enters the Hamiltonian—and true or effective equation of state $e_0(\rho)$—which is observed in an experiment.</footnote>
<figure><img src="image_16.png" /><figcaption>**Figure 6.** Cutting the gas into mesoscopic cubic portions of side $l$, in the quantum hydro-dynamics of Landau and Khalatnikov (this effective theory does not describe length scales $<l$). See text for choice of $l$.</figcaption></figure>

discrete gradient)<fnref n="33" />:

$$ \hat{\mathbf{v}}(\mathbf{r}) = \frac{\hbar}{m} \mathrm{grad} \hat{\phi}(\mathbf{r}) \qquad (37) $$

The Hamiltonian is obtained by summing the internal energy and the kinetic energy associated with the local fluid velocity in each portion:

$$ H = \sum_{\mathbf{r}} \frac{1}{2} m \hat{\mathbf{v}}(\mathbf{r}) \cdot \hat{\rho}(\mathbf{r}) \ell^3 \hat{\mathbf{v}}(\mathbf{r}) + \ell^3 e_0(\hat{\rho}(\mathbf{r})) \qquad (38) $$

Here, $e_0(\rho)$ is the zero-temperature energy density of the homogeneous Fermi gas with density $\rho$, and $m\hat{\rho}(\mathbf{r})\ell^3$ is the amount of matter (the mass) in the portion centered at $\mathbf{r}$. The equations of motion for $\hat{\rho}$ and $\hat{v}$ in the Heisenberg picture, derived from the Hamiltonian $H$, take the form of a continuity equation and an operator-valued Euler equation (without viscosity term)<fnref n="34" />, hence the name quantum hydrodynamics given to the theory (with the risk of confusion pointed out in note 24).

As you will have gathered, the great strength of this effective theory is that it does not depend on the nature of the bosonic or fermionic particles constituting the underlying superfluid, nor on their interactions (strong or weak, in liquid or gas phase) as long as they remain short-range, except through the equation of state $e_0(\rho)$ and the curvature parameter $\gamma$ at zero temperature. It therefore applies equally well to weakly interacting Bose gases, strongly interacting Fermi gases and liquid helium-4 (an extremely dense system that defies microscopic theory).

However, the formalism can only be trusted in a low-temperature limit, $T \to 0$, where spatial density fluctuations and phase gradients are small; Equation (38) must therefore be expanded to the relevant order (here, fourth order) in powers of

$$ \delta\hat{\rho}(\mathbf{r}) \equiv \hat{\rho}(\mathbf{r}) - \hat{\rho}_0 \quad \text{and} \quad \delta\hat{\phi}(\mathbf{r}) \equiv \hat{\phi}(\mathbf{r}) - \hat{\phi}_0 \qquad (39) $$

<footnote n="33">³³The fact that the velocity field operator is a gradient vector in no way implies that the flow is entirely superfluid (this would be physically false even at thermal equilibrium, at non-zero temperature). Let's explain this in two points. (i) Don't confuse the $\hat{v}(\mathbf{r})$ operator of quantum hydrodynamics (which contains all possible quantum and thermal fluctuations) with the $v(\mathbf{r})$ mean velocity field of ordinary hydrodynamics; in particular, whether or not the flow is superfluid depends on whether or not $v(\mathbf{r})$ (without hat) is irrotational. (ii) In general, we have $v(\mathbf{r}) \neq \langle\hat{v}(\mathbf{r})\rangle$ where the expectation value is taken in the quantum state of the system, since $v(\mathbf{r})$ is defined in terms of the mean matter current density, $v(\mathbf{r}) = \langle\hat{j}(\mathbf{r})\rangle/\langle\hat{\rho}(\mathbf{r})\rangle$ with here $\hat{j}(\mathbf{r}) = [\hat{\rho}(\mathbf{r})\hat{v}(\mathbf{r}) + \hat{v}(\mathbf{r})\hat{\rho}(\mathbf{r})]/2$ (by definition, the evolution equation of $\hat{\rho}$ in the Heisenberg picture is written $\partial_t\hat{\rho} + \text{div}\hat{j} = 0$ and that for the mean density $\rho(\mathbf{r}) = \langle\hat{\rho}(\mathbf{r})\rangle$ is written $\partial_t\rho + \text{div}(\rho v) = 0$); it would therefore be wrong to believe that $v(\mathbf{r}) = (\hbar/m)\mathrm{grad}(\langle\hat{\phi}(\mathbf{r}) - \hat{\phi}(\mathbf{r}_0)\rangle)$ (where $\mathbf{r}_0$ is an arbitrary reference position) and deduce that $v(\mathbf{r})$ is necessarily a gradient vector.</footnote>

<footnote n="34">³⁴See note 33 for the equation for $\hat{\rho}(\mathbf{r})$. The equation for $\hat{v}(\mathbf{r})$ is obtained by taking the gradient of that for $\hat{\phi}(\mathbf{r})$, $\hbar\partial_t\hat{\phi} = -\mu_0(\rho) - m\hat{v}^2/2$, where $\mu_0(\rho)$ is the zero-temperature chemical potential function, as in Equation (43).</footnote>
where $\hat{\rho}_0$ and $\hat{\phi}_0$ are the zero-wave-vector Fourier components of the fields $\hat{\rho}(\mathbf{r})$ and $\hat{\phi}(\mathbf{r})$ (physically, $\hat{\rho}_0 = \hat{N}/L^3$, where $\hat{N}$ is the total number of fermions operator, $L^3$ is the volume of the quantization box $[0, L]^3$ and $\hat{\phi}_0$ is the phase operator of the pair condensate [76]). The expanded Hamiltonian is formally written as

$$H = H_0 + H_2 + H_3 + H_4 + \dots \quad (40)$$

where $H_n$ is the contribution of total degree $n$ in $\delta\hat{\rho}$ and $\delta\hat{\phi}$.

The 0-order contribution $H_0$ is a constant of little interest; the 1-order contribution is exactly zero (because $\sum_{\mathbf{r}} \ell^3 \delta\hat{\rho}(\mathbf{r}) = 0$ by construction) and has been omitted directly from Equation (40). The quadratic contribution $H_2$ is diagonalized by a Bogolioubov transformation<fnref n="35" />:

$$H_2 = \text{const} + \sum_{\mathbf{k} \neq 0} \hbar \omega_{\mathbf{k}} \hat{b}_{\mathbf{k}}^{\dagger} \hat{b}_{\mathbf{k}} \quad (41)$$

where the creation $\hat{b}_{\mathbf{k}}^{\dagger}$ and annihilation $\hat{b}_{\mathbf{k}}$ operators of an elementary excitation (a phonon) of wave vector $\mathbf{k}$ obey the usual bosonic commutation relations

$$[\hat{b}_{\mathbf{k}}, \hat{b}_{\mathbf{k}'}] = 0 \quad \text{and} \quad [\hat{b}_{\mathbf{k}}, \hat{b}_{\mathbf{k}'}^{\dagger}] = \delta_{\mathbf{k},\mathbf{k}'} \quad (42)$$

The spectrum obtained here is exactly linear, $\omega_{\mathbf{k}} = ck$, with the speed of sound given by

$$mc^2 = \rho \frac{d^2}{d\rho^2} e_0(\rho) = \rho \frac{d}{d\rho} \mu_0(\rho) \quad (43)$$

where $\mu_0(\rho)$ is the zero-temperature chemical potential of the Fermi gas with density $\rho$. The relation (43) is exact (it's well known from superfluid hydrodynamics [75]), but the systematic absence of curvature in the spectrum is not physically realistic: this pathology stems from the fact that we have omitted so-called gradient corrections [77] from Hamiltonian $H$; to simplify, as illustrious predecessors [52] have done, we replace $\omega_{\mathbf{k}}$ in $H_2$ by hand with its cubic approximation (34), which is justified by reference [78].

The approximation $H_2$ corresponds to an ideal gas of phonons, and cannot describe sound attenuation. The interaction between phonons that causes their damping comes from the cubic contribution $H_3$ and the quartic contribution $H_4$. For the sake of simplicity, we give here only the expression of the most useful part of $H_3$, simplified to make the physics clear:

$$H_3|_{\text{simpl}} = \frac{\mathcal{A}}{2L^{3/2}} \sum_{\mathbf{k},\mathbf{k}',\mathbf{q}} (kk'q)^{1/2} \hat{b}_{\mathbf{k}}^{\dagger} \hat{b}_{\mathbf{k}'}^{\dagger} \hat{b}_{\mathbf{q}} \delta_{\mathbf{k}+\mathbf{k}',\mathbf{q}} + (k'kq)^{1/2} \hat{b}_{\mathbf{k}}^{\dagger} \hat{b}_{\mathbf{k}} \hat{b}_{\mathbf{q}} \delta_{\mathbf{k}',\mathbf{k}+\mathbf{q}} + \dots \quad (44)$$

with the constant amplitude (independent of the wave numbers) as a factor,

$$\mathcal{A} = (\xi/2)^{3/2} \rho^{-1/2} \left[ 3mc^2 + \rho^2 \frac{d^3}{d\rho^3} e_0(\rho) \right] \quad (45)$$

and $\xi = \hbar/mc$ as before<fnref n="36" />. The ellipse in Equation (44) contains terms $\hat{b}^{\dagger}\hat{b}^{\dagger}\hat{b}^{\dagger}$ and $\hat{b}\hat{b}\hat{b}$ of no great importance, as they do not conserve the energy $H_2$. The terms $\hat{b}^{\dagger}\hat{b}^{\dagger}\hat{b}$ and $\hat{b}^{\dagger}\hat{b}\hat{b}$, on the other hand, are central to damping: they correspond respectively to the Beliaev process (phonon $\mathbf{q}$ decays into two phonons $\mathbf{k}$ and $\mathbf{k}'$) and to the Landau process (phonon $\mathbf{q}$ merges with a phonon

<footnote n="35">³⁵This transformation corresponds to the modal expansions $\delta\hat{\rho}(\mathbf{r}) = L^{-3/2} \sum_{\mathbf{k} \neq 0} \rho_{\mathbf{k}}(\hat{b}_{\mathbf{k}} + \hat{b}_{-\mathbf{k}}^{\dagger}) \exp(i\mathbf{k} \cdot \mathbf{r})$ and $\delta\hat{\phi}(\mathbf{r}) = L^{-3/2} \sum_{\mathbf{k} \neq 0} \phi_{\mathbf{k}}(\hat{b}_{\mathbf{k}} - \hat{b}_{-\mathbf{k}}^{\dagger}) \exp(i\mathbf{k} \cdot \mathbf{r})$ where $\rho_{\mathbf{k}} = (\hbar\rho k/2mc)^{1/2}$ and $\phi_{\mathbf{k}} = (-i)(mc/2\hbar\rho k)^{1/2}$ are the amplitudes of the quantum fluctuations of density and phase in the phonon mode of wave vector $\mathbf{k}$. Note the relation $-i\omega_{\mathbf{k}}\delta\rho_{\mathbf{k}} - \rho(\hbar/m)k^2\phi_{\mathbf{k}} = 0$ imposed by the linearized continuity equation.</footnote>

<footnote n="36">³⁶The true coupling amplitude depends on the angles between the three wave vectors $\mathbf{k}_i$ involved; as damping is actually dominated at low temperatures by processes at small angles between wave vectors, due to the small denominator effect described below, we have written the amplitude directly at zero angles $\mathbf{k}_i \cdot \mathbf{k}_j / k_i k_j = 1$.</footnote>
k to form a single phonon $k'$), which can be represented diagrammatically as follows, where the phonon whose damping we are studying plays a privileged role:

<figure><img src="image_17.png" /></figure>

As $H_3$ is cubic, each vertex of the diagram representing its action is the meeting point of three phonon lines. We could proceed in the same way with $H_4$ (the vertices would be four lines), but we won't do so, as quartic processes generally play a sub-dominant role in damping compared with cubic processes (for example, in the four-phonon damping $\phi\phi \to \phi\phi$ for $\gamma < 0$, the amplitude of the direct process $k, q \to k', k''$ induced by $H_4$ in first-order perturbation theory is in practice negligible compared to that of the indirect process of the same initial and final state $k, q \to k+q \to k', k''$ induced by $H_3$ treated in second-order, due to the appearance in the latter of a very small energy denominator at small angles between $k$ and $q$)$^{37}$.

### 4.3. How to calculate phonon damping?

Let's imagine that we apply a short pulse of Bragg excitation to the gas, initially at thermal equilibrium, inducing a small coherent Glauber displacement of amplitude $\alpha \in C^*$ in the phonon mode of wave vector $q$ without affecting the other modes, which corresponds to the unitary evolution operator $U_{exc} = \exp(\alpha \hat{b}_q^\dagger - \alpha^* \hat{b}_q)$ $^{38}$. Immediately after excitation, we have a non-zero mean for the corresponding annihilation operator:

$$ \langle \hat{b}_q(0^+) \rangle = \alpha \neq 0 \qquad (47) $$

This leads to an observable modulation of the mean gas density at wave vectors $\pm q$ since $\langle \delta\hat{\rho}(\mathbf{r}, 0^+) \rangle$, a linear combination of $\langle \hat{b}_k(0^+) \rangle$ and $\langle \hat{b}_k^\dagger(0^+) \rangle$ as in note 35, is then $\neq 0$.

In the limit $\alpha \to 0$, i.e. in the linear response regime, the many-body Green's function formalism applied to the effective low-energy theory, hence to the phonon Hamiltonian (40) (rather than to a microscopic description of the Fermi gas with interaction potential $V(\mathbf{r}_{ij})$ as references [75,80] for example do), leads to the exact expression

$$ \langle \hat{b}_q(t) \rangle \stackrel{t \to 0}{=} \alpha e^{-i\omega_q t} \int_{C_+} \frac{d\zeta}{2i\pi} \frac{e^{-i\zeta t/\hbar}}{\zeta - \Sigma_q(\zeta)} \qquad (48) $$

In this expression, $C_+$ is the integration path parallel to the real axis in the upper complex half-plane, followed from right to left (from $\text{Re}\zeta = +\infty$ to $\text{Re}\zeta = -\infty$), see Figure 7, and $\Sigma_q(\zeta)$ is the self-energy at wave vector $q$ and complex energy $\zeta$ $^{39}$, which is not known explicitly but is defined by its perturbative expansion to all orders in the phonon-phonon interaction. Here, we restrict

<footnote>$^{37}$This energy denominator $\hbar\omega_q + \hbar\omega_{k-} - \hbar\omega_{k+q}$ would be exactly zero at zero angle without the curvature terms in the dispersion relation (34), which incidentally shows the singularity of a $\gamma = 0$ theory (to which we'll return in Section 4.3).</footnote>

<footnote>$^{38}$In a cold atom experiment, Bragg excitation is induced by the superposition of two far off-resonance laser beams of wave vectors $k_1$ and $k_2$ with $k_1 - k_2 = q$; even if the $\pm q$ acoustic modes are initially empty ($\hbar\omega_q \gg k_B T$) and can only receive phonons, Raman (two-photon) processes—absorption of a photon in one laser beam, stimulated emission in the other—induce $\pm \hbar(k_1 - k_2) = \pm \hbar q$ momentum changes in the Fermi gas and generally excite the two modes; however, the duration of the Bragg excitation can be adjusted so that the $-q$ mode emerges intact from the excitation procedure [79].</footnote>

<footnote>$^{39}$In comparison with the usual energy variable $z$ (of reference [80], for example), the energy variable used here is shifted by the unperturbed energy of mode $q$, $\zeta = z - \hbar\omega_q$. This explains why we were able to take out the unperturbed phase factor in (48) and why $\Sigma_q(\zeta)$ is taken at $\zeta = i0^+$ in the approximation (50) to come (it actually corresponds to $z = \hbar\omega_q + i0^+$).</footnote>
<figure><img src="image_18.png" /><figcaption>Figure 7. Integration path in the complex plane followed by Equation (48).</figcaption></figure>

ourselves to the cubic interaction $H_3$ (see Section 4.2) and the expansion takes the following diagrammatic form

$$ \Sigma_q(\zeta) = \underset{n=2}{\stackrel{q}{\bigcirc}} + \underset{n=4}{\stackrel{q}{\bigcirc}} + \underset{n=4}{\stackrel{q}{\bigcirc}} + \underset{n \ge 6}{\stackrel{q}{\bigcirc}} + \dots \qquad (49) $$

where the integer $n$ gives the order in $H_3$ [63]. We only indicate the topology; it remains to sum over all possible non-redundant orientations of the internal lines, see example of diagram section (46)$^{40}$; a precise value can then be assigned to each diagram, involving a sum over the wave vectors and Matsubara frequencies of the internal lines [80].

In order to obtain explicit results for the damping, we traditionally perform the following two approximations$^{41}$:

(1) the Markov approximation (the phonon gas seen by mode $q$ constitutes a reservoir with no memory, i.e. with negligible correlation time): we ignore the energy dependence of the self-energy as follows (see note 39),

$$ \Sigma_q(\zeta) \approx \Sigma_q(i0^+) \qquad (50) $$

The integral in Equation (48) is then calculated by the residue theorem (one closes the contour with an infinite semicircle in the lower complex half-plane),

$$ \langle \hat{b}_q(t) \rangle |_{\text{Markov}} \stackrel{t \ge 0}{\approx} \alpha e^{-i\omega_q t} e^{-i\Sigma_q(i0^+)t/\hbar} \qquad (51) $$

The signal decay in this case is exponential, with a rate corresponding to the imaginary part of $\Sigma_q(i0^+)$ (the real part gives the change in the mode angular frequency).

<footnote>$^{40}$In the first diagram of (49) (with one loop), (i) orienting the top line to the right and the bottom line to the left and (ii) orienting the top line to the left and the bottom line to the right correspond to the same contribution, by invariance of the diagram by angle $\pi$ rotation around its horizontal axis. The same applies to the inner loop of the second diagram in (49), which is locally rotationally symmetric. For the same reason of rotational symmetry (this time global) of the diagrams of order $n=4$, we decide, in order to avoid double counting, to put the inner loop in the upper branch and, in the third diagram of (49), to orient the bridge downwards.</footnote>

<footnote>$^{41}$Our effective theory being exact to the leading order in temperature, we use it only in the $T \to 0$ limit, fixing the ratio $\bar{q} = \hbar c q / k_B T$ so that the mode $q$ is also described exactly. We then have $\omega_q / \Gamma_{th} \to +\infty$ where $\Gamma_{th} = \Gamma_q = k_B T / \hbar c$ is the thermalization rate of the phonon gas and $\Gamma_q$ is the function (52), since the exponent $\nu$ introduced in Figure 8 on page 450 is always $> 1$, see Table 1 on page 451: the thermalization rate $\Gamma_{th}$ tends towards zero faster than the angular eigenfrequency $\omega_q$ and the mode by definition enters the collisionless regime. In the opposite, hydrodynamic regime $\omega_q \ll \Gamma_{th}$, the phonon gas has time to reach local thermal equilibrium at each oscillation point of the sound wave $q$, and damping is described by viscosity-type coefficients in the classical hydrodynamic equations of a two-fluid model [81].</footnote>
(2) Born approximation: Im $\Sigma_q(i0^+)$ is calculated perturbatively to leading order $n$ in $H_3$. The damping rate of phonon $q$ is then$^{42,43}$

$$ \Gamma_q|_{\text{Born-Markov}} = -\frac{2}{\hbar} \text{Im} \Sigma_q^{(n)}(i0^+) \quad (52) $$

where the exponent gives the order in $H_3$. In the case of a convex acoustic branch ($\gamma > 0$) in 3D, it's sufficient to go to order $n = 2$: this is the three-phonon Beliaev-Landau damping, much studied theoretically and observed in liquid helium-4 [83,84] and, to a lesser extent, in bosonic cold-atom gases, only Beliaev damping having been seen there [85]. In the concave case ($\gamma < 0$), we need to go to order $n = 4$ (the first contribution in Equation (49) is purely real for $\zeta = i0^+$ because its energy denominators, of the form $\hbar\omega_q + \hbar\omega_k - \hbar\omega_{k+q}$ and $\hbar\omega_q - (\hbar\omega_k + \hbar\omega_{q-k})$, cannot vanish, but the next two are not in $\phi\phi \to \phi\phi$ processes); this case has been little studied theoretically (reference [78] noted and corrected an error in the original calculation [52], and reference [86] obtained a much more explicit expression of the result, even generalizing it to a non-zero phonon chemical potential $\mu_\phi$)$^{44}$ and it has, to our knowledge, never been observed experimentally (no precise measurement of $\Gamma_q^{\gamma<0}$ has been made in any system).

Let's determine the validity of the Born approximation by means of the estimate of order $n \in 2\mathbb{N}^*$ given in reference [63]:

$$ \Sigma_q^{(n)}(i0^+) \approx \int \left( \prod_{i=1}^{n/2} d^d k_i \right) \frac{\langle |L^{d/2} H_3| \rangle^n}{(\Delta E)^{n-1}} \approx |\gamma| T^3 \left( \epsilon_{dD} = \frac{T^{2d-4}}{|\gamma|^{(5-d)/2}} \right)^{n/2} \quad (53) $$

where $d \ge 2$ is the dimension of space. The writing in the second expression symbolically represents the product of $n$ matrix elements of the cubic phonon interaction in the numerator and the product of $n-1$ energy denominators (associated with $n-1$ intermediate states) in the denominator, and the integral is taken over the independent phonon wave vectors $k_i$. The order of magnitude in the third expression is obtained as in reference [63] by restricting to the small $O(|\gamma|^{1/2}T)$ angles between $k_i$ and $q$, which is legitimate when $T \to 0$ [52]; we omit here the dependence in the density $\rho$, in $\xi = \hbar/mc$ and in the coupling constant $\mathcal{A}$ of Equation (45), unlike reference [63], but we keep that in the curvature parameter $\gamma$, as we will soon link $\gamma$ and $T$.

<footnote>$^{42}$To shed further light on the approximation (52), let's point out for $n=2$ that it can be obtained by the quantum master equation method well known in quantum optics (we obtain a closed evolution equation for the density operator $\hat{\rho}_S(t)$ of a small system $S$—here the phonon mode $q$—coupled to a large reservoir $R$—here the other phonon modes $k \ne q$, by resorting to the Born-Markov approximation) [36,82] or, more simply, using Fermi's golden rule (we calculate $d\langle\hat{n}_q\rangle/dt$, where $\langle\hat{n}_q\rangle = \langle\hat{b}_q^\dagger\hat{b}_q\rangle$ is the mean number of phonons in the non-equilibrium mode $q$, by summing the incoming fluxes—population processes $k, k' \to q$ and $k' \to k, q$—and outgoing fluxes—inverse depopulation processes $q \to k, k'$ and $q, k \to k', k'$—and linearizing just after the Bragg excitation, as $(d/dt)\delta n_q(t=0^+) = -\Gamma_q\delta n_q(t=0^+)$, where $\delta n_q(t) = \langle n_q \rangle(t) - \bar{n}_q$ is the deviation from thermal equilibrium). For $n=4$, we find the same result (52) by extending Fermi's golden rule to higher orders [38].</footnote>

<footnote>$^{43}$There's a little mathematical subtlety here: if the leading order $n$ is $\ge 4$, the Markov approximation must no longer simply replace $\Sigma_q(\zeta)$ with $\Sigma_q(i0^+)$ but must approximate it by a Taylor expansion around $\zeta = i0^+$. For example, for $n=4$, we take $\Sigma_q(\zeta) = \Sigma_q(i0^+) + \zeta(d/d\zeta)\Sigma_q(i0^+)$ so that at order 4 in $H_3$, $\zeta - \Sigma_q(\zeta) = [1 - (d/d\zeta)\Sigma_q(i0^+)][\zeta - \Sigma_q^{(2)}(i0^+) - \Sigma_q^{(4)}(i0^+)]$ with $\Sigma_q^{(4)}(i0^+) = \Sigma_q^{(4)}(i0^+) + \Sigma_q^{(2)}(i0^+) (d/d\zeta) \Sigma_q^{(2)}(i0^+)$. Then replace $\Sigma_q^{(4)}(i0^+)$ with $\Sigma_q^{(4)}(i0^+)$ in the damping rate expression (52). In the three-dimensional concave case, this makes no difference to $\Gamma_q|_{\text{Born-Markov}}$, since $(d/d\zeta)\Sigma_q^{(2)}(i0^+) = \text{a real quantity, as is } \Sigma_q^{(2)}(i0^+)$; in the two-dimensional concave case, the conclusion is less obvious but remains the same, see note 48. On the other hand, this substitution must be made in the calculation of the thermal angular frequency shift $\Delta_q$ of the mode, $\hbar\Delta_q|_{\text{Born-Markov}} = \text{Re}[\Sigma_q^{(2)}(i0^+) + \Sigma_q^{(4)}(i0^+)] - \text{idem at } T=0$.</footnote>

<footnote>$^{44}$If we restrict ourselves at low temperature to the leading collisional processes $\phi\phi \to \phi\phi$, the phonon number becomes a conserved quantity, allowing us to take $\mu_\phi < 0$.</footnote>
In short, the Born expansion is legitimate in dimension $d=3$ if its small parameter tends to zero at low temperature:

$$ \epsilon_{3D} = \frac{T^2}{|\gamma|} \underset{T \to 0}{\to} 0. \qquad (54) $$

To discuss the validity of the Markov approximation, we assume that the behavior of the self-energy in the vicinity of $\zeta = i0^+$ is characterized by two exponents, $v$ and $\sigma$, the one giving the typical values $\propto T^v$ of its imaginary part and the one giving its typical scale of variation $\propto T^\sigma$, as in Figure 8$^{45,46}$. From Equation (52), we therefore have

$$ \Gamma_q \underset{T \to 0}{\approx} T^v \qquad (55) $$

The function $\Sigma_q(\zeta)$ then has a slow (negligible) energy variation on the scale of the damping rate (which is indeed the inverse of the characteristic time in Equation (48)) if it is wider than high, which imposes

$$ v > \sigma \qquad (56) $$

The exponent $v$ is obtained by an explicit calculation of the right-hand side of Equation (52) in the limit $T \to 0$, as was done in reference [78] for the three-dimensional case$^{47}$ and in reference [63] for the two-dimensional convex case; more simply, we can use the estimate (53) with $n=2$ if $\gamma > 0$ and $d \in \{2,3\}$, $n=4$ if $\gamma < 0$ and $d=3$$^{48}$. The exponent $\sigma$ is obtained by generalizing the previous

<footnote>$^{45}$The exponents introduced here differ by one unit from those in reference [63] due to a different choice of convention.</footnote>

<footnote>$^{46}$For the sake of clarity, we have assumed in Figure 8 that the function shown has a maximum at the origin. This is not necessarily true (the two-dimensional convex case of reference [63] provides a counter-example, see its Equation (114)). The true definition of exponents $v$ and $\sigma$ is that the scaled function $\mathrm{Im}\,\Sigma_q(\zeta = \bar{\zeta}mc^2(k_B T/mc^2)^\sigma)/[mc^2(k_B T/mc^2)^v]$ has a finite, non-zero limit as $T \to 0$ with fixed reduced complex energy $\bar{\zeta}$ ($\mathrm{Im}\,\bar{\zeta} > 0$).</footnote>

<footnote>$^{47}$The historical reference [52] for fixed $\gamma < 0$ also finds $v=7$ but the $q$ dependence of $\Gamma_q$ is different, e.g. $\Gamma_q \approx qT^6$ in [52] instead of $q^3T^4$ in [78] as $q \to 0$.</footnote>

<footnote>$^{48}$For fixed $\gamma < 0$ and $d=2$, Alice Sinatra obtained in 2021, in the formulation of references [72,78], the unpublished result that $\mathrm{Im}\,\Sigma_q^{(n=4)}(i0^+) = 0$ at order $T^3$ (the expected leading order in temperature). To see this, it is actually simpler to use expressions (84) and (85) in reference [63]: (i) in (84), we can ignore processes $\phi \leftrightarrow \phi\phi\phi$ and restrict ourselves to process $\phi\phi \to \phi\phi$ (second contribution), the only one conserving energy–momentum; (ii) in the integrand of (85), we are entitled to replace $\zeta$ by 0 in the numerator of the large fraction for the similar reason that processes $\phi \leftrightarrow \phi\phi$ do not conserve energy–momentum—this makes the numerator real; (iii) we then check that, if the scaled energy difference $\Delta E/[k_B T(k_B T/mc^2)^2]$ next to $i0^+$ vanishes in the denominator of the large fraction, as required by the Dirac distribution $\delta(\Delta E)$ of the generalized Fermi golden rule [38], the numerator also vanishes (we show this by formally replacing $\gamma$ in the numerator by its $\Delta E$-cancelling expression, a rational function of the moduli and angles of the phonon wavevectors). In other words, the limit at $\zeta = 0$ (but also at $\zeta = -\Delta E$) of the transition amplitude in the numerator of the large fraction, considered as a rational fraction of the angles, can be written $\Delta E \times P/Q$, where the polynomials $\Delta E$, $P$ and $Q$ are two by two coprime.</footnote>

<footnote>However, this reasoning neglects possible edge effects in the integral on phonon wave numbers, in the sense of reference [87], where one of the wave numbers tends to zero, which makes one of the energy denominators of the $\phi \leftrightarrow \phi\phi$ processes tend to zero. Including these edge effects, we find that the processes $\phi \to \phi\phi\phi$, $\phi\phi \to \phi\phi$ and $\phi\phi\phi \to \phi$, abbreviated as $1 \to 3$, $2 \to 2$ and $3 \to 1$ in reference [63], each make a non-zero contribution to $\mathrm{Im}\,\Sigma_q^{(n=4)}(i0^+)$ at order $T^3$, but the sum of these contributions is exactly zero (the contribution of the edge $q'_1 + q'_2 = q$ in $2 \to 2$, i.e. $C \int_0^q d\bar{k} (\bar{n}_k^{\mathrm{lin}} + \bar{n}_{q-k}^{\mathrm{lin}} + 1) \bar{k}(\bar{q}-\bar{k})/\bar{q}$, is exactly offset by $1 \to 3$, and that of the edges $q'_1 = 0$ and $q'_2 = 0$ as a whole, i.e. $2C \int_{\bar{q}}^{+\infty} d\bar{k} (\bar{n}_{k-q}^{\mathrm{lin}} - \bar{n}_k^{\mathrm{lin}}) \bar{k}(\bar{k}-\bar{q})/\bar{q}$, is exactly offset by $3 \to 1$; here, $\bar{k} = \hbar ck/k_B T$, $\bar{n}_k^{\mathrm{lin}} = 1/(\exp \bar{k} - 1)$, $\Lambda = \rho^2((d^3/d\rho^3)e_0(\rho))/(3mc^2)$ and $C = k_B T(k_B T/mc^2)^2[9(1+\Lambda)^2/8\rho\zeta^2]/[\bar{q}(3|\gamma|)^{3/2}]$ for Landau at temperature leading order) but these contributions are exactly opposite, in particular because the energy denominators $\epsilon_q-(\epsilon_k+\epsilon_{q-k})$ and $\epsilon_q+\epsilon_k-\epsilon_{q+k}$ are opposite $\sim \mp\hbar ck[1-(v_q/c)\cos\theta]$ at leading order in $k$ ($v_q = d\epsilon_q/\hbar dq$ is the group velocity and $\theta$</footnote>

<footnote>The conclusion is not changed by the correction $\Sigma_q^{(2)}(i0^+)(d/d\zeta)\Sigma_q^{(2)}(i0^+)$ of our note 43, as we find that factors $\Sigma_q^{(2)}(i0^+)$ and $(d/d\zeta)\Sigma_q^{(2)}(i0^+)$ are both real. This was obvious for the first factor (the edge effects it presents in 1D [87] are suppressed in 2D by a lowering of the phonon density of states at low wavenumber). This was not the case for the second factor: because of edge effects in the integration over $k$ (see Equation (39) in reference [63]), the Beliaev and Landau processes each give a non-zero contribution to $\mathrm{Im}\,(d/d\zeta)\Sigma_q^{(2)}(i0^+)$ (it is $2[9(1+\Lambda)^2/8\rho\zeta^2]/[\bar{q}(3|\gamma|)^{3/2}]$ for Landau at temperature leading order) but these contributions are exactly opposite, in particular because the energy denominators $\epsilon_q-(\epsilon_k+\epsilon_{q-k})$ and $\epsilon_q+\epsilon_k-\epsilon_{q+k}$ are opposite $\sim \mp\hbar ck[1-(v_q/c)\cos\theta]$ at leading order in $k$ ($v_q = d\epsilon_q/\hbar dq$ is the group velocity and $\theta$</footnote>
<figure><img src="image_19.png" /><figcaption>**Figure 8.** In the limit $T \to 0$, the order of magnitude and typical width of function $\text{Im} \Sigma_q(\zeta)$ near $\zeta = i0^+$ are assumed to be characterized by two power laws in temperature, with exponents $\nu$ and $\sigma$ (taking into account a possible temperature dependence of the curvature parameter $\gamma$).</figcaption></figure>

estimate to the case $\zeta \neq 0$, i.e. by adding $\zeta$ to $\Delta E$ in (53); however, in an expansion at small angles between $\mathbf{k}_i$ and $\mathbf{q}$, the part of $\Delta E$ linear in the wave numbers vanishes, leaving only the cubic contributions $\approx \gamma T^3$, so that, independently of the dimension of space $d$,

$$ \frac{1}{\zeta + \Delta E} \approx \frac{1}{\zeta + \gamma T^3} \quad \text{and therefore} \quad T^{\sigma} \approx |\gamma| T^3 \qquad (57) $$

taking into account the dependence on the parameter $\gamma$, which—as we said—may vary with temperature. The resulting validity condition $T^\nu = o(\gamma T^3)$ in (56) can be given a simple interpretation: in the limit $T \to 0$ taken with the scaling law $q \approx T$, the damping rate $\Gamma_q$ must tend to zero faster than the $q^3$ term in $\omega_q$,

$$ \Gamma_q \stackrel{\hbar c q / k_B T \text{ fixed}}{=} o(\omega_q^{(3)}) \quad \text{with} \quad \omega_q^{(3)} = \gamma m c^2 (q \xi)^3 / 8 \hbar \qquad (58) $$

which is the true mark of the Markovian nature of damping (rather than the naive perturbative condition $\Gamma_q = o(\omega_q)$).

The situation is summarized in Table 1 below$^{49}$. The Born–Markov approximation is therefore usable in dimension 3, except over a narrow interval of values of $\gamma$, of width $\approx (k_B T / mc^2)^2$ around $\gamma = 0$; for $\gamma = 0$, the phonon dispersion relation (34) deviates quintically from the linear law $cq$, which is obviously a special case.

is the angle between $\mathbf{k}$ and $\mathbf{q}$) and enter the derivative of the Dirac distribution $\delta'(\epsilon)$, which is an odd function of its argument.

The two-dimensional concave case is therefore special: the quantity $\text{Im} \Sigma_q^{(4)\text{eff}}(i0^+)$ of note 43—considered at all temperature orders—does not give the correct temperature scaling law $\propto T^3$ of the self-energy at fourth order in $H_3$ on a $O(T^3)$ neighborhood of $\zeta = i0^+$; nor does it give the damping rate $\Gamma_q$, since the Born and Markov approximations fail, as in the two-dimensional convex case (see the last line of our Table 1). Note, however, still for $\gamma < 0$, that the limiting case $\rho\xi^2 \to +\infty$ of a very weakly interacting underlying superfluid must be set apart as there we have the additional small parameter $1/\rho\xi^2$ helping validity of Markov (as in Section 3.2 of reference [63]) and Born (as in Equation (17) of the same reference); this limiting case is inaccessible in a gas of spin-1/2 fermions with contact interaction—we have $\rho\xi^2 = O(1)$ when $\gamma < 0$ [88]—but it is in a Bose gas with an interaction range $\gtrsim \xi$ as considered in reference [78].

<footnote>$^{49}$In the third row of the table, possible logarithmic factors $\ln(1/T)$ are omitted for simplicity. These factors arise from the fact that, for scaling laws $\gamma \propto T^2$ and $q \propto T$, the terms $q^3$ and $q^5 \ln q$ are of the same order of magnitude in the dispersion relation (34): in this low curvature regime, the logarithmico-quintic contribution to $\omega_q$ is no longer a small correction and must be kept.</footnote>
<table><caption>**Table 1.** In the study of phonon damping in a superfluid, validity of the Born–Markov approximation in the low-temperature limit $T \to 0$ depending on the dimension of space $d$ and the curvature parameter $\gamma$ of the acoustic branch (more precisely, its sign and its temperature dependence, the third line holding regardless of the sign of $\gamma$).</caption><thead><tr><th></th><th>v</th><th>σ</th><th>Markov</th><th>ε<sub>dD</sub></th><th>Born</th></tr></thead><tbody><tr><td>d = 3, γ > 0 fixed</td><td>5</td><td>3</td><td>yes</td><td>≈ T<sup>2</sup> → 0</td><td>yes</td></tr><tr><td>d = 3, γ < 0 fixed</td><td>7</td><td>3</td><td>yes</td><td>≈ T<sup>2</sup> → 0</td><td>yes</td></tr><tr><td>d = 3, γ = O(T<sup>2</sup>)</td><td>5</td><td>5</td><td>no</td><td>≈ T<sup>0</sup> ≠ 0</td><td>no</td></tr><tr><td>d = 2, γ > 0 fixed</td><td>3</td><td>3</td><td>no</td><td>≈ T<sup>0</sup> ≠ 0</td><td>no</td></tr></tbody></table>

The exponents $v$ and $\sigma$ used in the definition (56) of the Markovian regime are shown in Figure 8, and the small parameter of the Born expansion $\epsilon_{dD}$ is given in Equation (53).

The precise calculation of the damping rate $\Gamma_q$ (or what takes its place for non-exponential decay, such as the inverse of the width of $|\langle\hat{b}_q(t)\rangle|^2$ at relative height $1/e$) for these small values of curvature is, to our knowledge, an open question; it is of great experimental relevance, as the interaction strength leading to $\gamma = 0$ seems to be close to the unitary limit (see reference [57] and our Section 5.1), the preferred point for cold atom experiments in a regime of fairly high values of $T_c/T_F$ and collisional properties conducive to evaporative cooling [18].

For good measure, we have also considered the two-dimensional convex case in Table 1: the Born–Markov approximation fails here, and reference [63] had to resort to a non-perturbative heuristic approximation on the self-energy $\Sigma_q(\zeta)$ to reach good agreement with classical field simulations (quantum hydrodynamics operators $\hat{b}_q, \hat{b}_q^\dagger$ replaced by complex numbers $b_q, b_q^*$) in the weakly interacting regime $\rho\xi^2 \gg 1$ of the underlying bosonic superfluid, where a small parameter was thought to be available and to ensure the success of Fermi’s golden rule even in the $k_B T/mc^2 \to 0$ limit (this reasonable expectation, confirmed in Section 3.2 of reference [63] to order two in $H_3$, is invalidated in Section 4.3 of the same reference by a calculation to order four)$^{50}$.

## 4.4. Phase diffusion of the pair condensate

A fundamental and practical question concerns the coherence time of the pair condensate at thermal equilibrium in a Fermi gas perfectly isolated from its environment.

For an infinite unpolarized system of $\uparrow$ and $\downarrow$ fermions, the coherence time is infinite, as asserted by the $U(1)$ symmetry-breaking phenomenon: in the grand canonical ensemble (term $-\mu\hat{N}$ added to the Hamiltonian of the Fermi gas where $\mu$ is the chemical potential and $\hat{N}$ the total

<footnote>$^{50}$We haven't even mentioned the very special case of dimension $d=1$, where two wave vectors make a very small angle (zero!) as soon as they are in the same direction. Let's just say that the Born small parameter is still given by Equation (53), even though this equation was obtained under the assumption $d \ge 2$. Restoring the density dependence as in [63], we find more precisely $\epsilon_{1D} = 1/[\gamma^2\rho\xi(k_B T/mc^2)^2]$, the prefactor in (53) being written $\gamma(k_B T)^3/(mc^2)^2$. In the low-temperature limit $k_B T/mc^2 \to 0$ at $\rho\xi$ fixed considered here, $\epsilon_{1D} \to +\infty$ and we must immediately resort to non-perturbative approximations on $\Sigma_q(\zeta)$ and $\Gamma_q$, such as the self-consistent calculation of references [89,90]. In the opposite weakly-interacting limit $\rho\xi \to +\infty$ at $k_B T/mc^2$ fixed, $\epsilon_{1D} \to 0$ and we can use Fermi's golden rule as in reference [87]; more precisely, we expect the validity condition of the golden rule to be written as $\rho\xi(k_B T/mc^2)^2 \gg \phi_{1D}(\bar{q})$ where $\phi_{1D}$ is some function of $\bar{q} = \hbar c q/k_B T$, forgetting the $\gamma$ dependence for simplicity (at a fixed $\bar{q}$, the Born approximation imposes this condition, but the Markov approximation is then also satisfied because we have $\hbar\Gamma_q^{\text{golden rule}} \approx \gamma[(k_B T)^3/(mc^2)^2]\epsilon_{1D} \approx k_B T/(\gamma\rho\xi) \ll \Delta E \approx \gamma(k_B T)^3/(mc^2)^2$ where $\Delta E$, the typical Beliaev–Landau energy denominator, gives the width in $\zeta$ of the self-energy as in Equation (57)). In 2D, as shown in reference [63], the perturbative $H_3$-expansion of the self-energy is subject to a similar validity condition, $\rho\xi^2(k_B T/mc^2)^2 \gg \phi_{2D}(\bar{q})$, see its Equation (96), which, unlike the 1D case, is not obtained by simple power counting.</footnote>
number of particles operator), the complex order parameter $\Delta(\mathbf{r}, t)$ is uniform and constant; in the canonical ensemble, it therefore evolves with the undamped phase factor $\exp(-2i\mu t/\hbar)^{51}$, advancing at the immutable angular frequency $2\mu/\hbar$; in any case, the coherence time is infinite.

What about a finite size system (quantization box $[0, L]^3$, fixed total number of fermions $N$)? To find out, let's follow reference [76] and write the evolution equation for the condensate phase operator, which we denote $\hat{\phi}_0$ as in Equation (39), in the quantum hydrodynamics regime$^{52}$:

$$ -\hbar \frac{d}{dt} \hat{\phi}_0 = \mu_0(\rho) + \sum_{\mathbf{q} \neq 0} \hat{b}_{\mathbf{q}}^{\dagger} \hat{b}_{\mathbf{q}} \frac{d}{dN} (\hbar \omega_{\mathbf{q}}) \equiv \hat{\mu} \quad (59) $$

In the second expression, $\mu_0(\rho) = dE_0/dN$ is the chemical potential of the $N$ fermions in the ground state of energy $E_0$ at density $\rho$, and the sum over $\mathbf{q}$ can be interpreted as the adiabatic derivative (meaning at fixed number operators $\hat{b}_{\mathbf{q}}^{\dagger}\hat{b}_{\mathbf{q}}$ of the phonon modes) with respect to $N$ of the corresponding sum in the phonon Hamiltonian $H_2$ (41). The second expression as a whole is therefore the isentropic derivative of the Hamiltonian with respect to the total number of particles. In this sense, it is a chemical potential operator for the fermions, hence the notation $\hat{\mu}$ in the third expression, and Equation (59) is a quantum version of the famous second Josephson relation, linking the time derivative of the (classical) phase of the order parameter to the equilibrium chemical potential $\mu$.

In a given realization of the experiment, which we assume to correspond to a $N$-body eigen-state $|\psi_\lambda\rangle$ of energy $E_\lambda$ sampling the canonical ensemble, the occupation numbers $\hat{b}_\mathbf{q}^\dagger\hat{b}_\mathbf{q}$ fluctuate and decorrelate under the effect of incessant collisions between phonons due in particular to $H_3$, see Equation (44). At times long enough for a large number of collisions to have taken place, we therefore expect a diffusive spreading of the condensate phase, the variance of the random phase shift increasing linearly with time:

$$ \mathrm{Var}_{\lambda}[\hat{\phi}_0(t) - \hat{\phi}_0(0)] \underset{\Gamma_{\mathrm{coll}}^{\phi} t \gg 1}{\sim} 2D_{\lambda}t \quad (60) $$

with a subintensive diffusion coefficient $D_\lambda$, i.e. $\approx 1/N$ in the thermodynamic limit. Here $\Gamma_{\mathrm{coll}}^\phi = \Gamma_{q=k_B T/\hbar c}$ is the typical collision rate between thermal phonons (the function $\Gamma_q$ is that of Equation (52)). The spreading (60) induces an exponential loss of temporal coherence of rate $D_\lambda$, by virtue of Wick's relation (the phase shift statistic in $|\psi_\lambda\rangle$ is expected to be approximately Gaussian [91]),

$$ \begin{aligned} \langle \exp\{-i[\hat{\phi}_0(t) - \hat{\phi}_0(0)]\} \rangle_{\lambda} &\approx \exp\{-i\langle\hat{\phi}_0(t) - \hat{\phi}_0(0)\rangle_{\lambda}\} \exp\{-\frac{1}{2}\mathrm{Var}_{\lambda}[\hat{\phi}_0(t) - \hat{\phi}_0(0)]\} \\ &\approx \exp\{-i\langle\hat{\mu}\rangle_{\lambda} t/\hbar\} \exp(-D_{\lambda}t) \end{aligned} \quad (61) $$

which is confirmed by the resolvent analysis of reference [76]$^{53}$.

In the case $\gamma > 0$ of a convex acoustic branch, the situation resembles that of weakly interacting Bose condensates well studied in reference [95]: the dominant collisions are the three-phonon Beliaev-Landau ones $\phi \leftrightarrow \phi\phi$, and $D_\lambda$ has been calculated for the low-temperature Fermi pair

<footnote>$^{51}$There is a factor of 2 under the exponential because $\Delta$ is a pair-order parameter whereas $\mu$ is the chemical potential of the fermions. There is no factor of 2 in Equation (59) because the phase operator $\hat{\phi}_0$ is conjugate to the fermion density.</footnote>

<footnote>$^{52}$To obtain this equation, we had to eliminate $\hat{b}_{\mathbf{q}}\hat{b}_{-\mathbf{q}}$ and $\hat{b}_{\mathbf{q}}^{\dagger}\hat{b}_{-\mathbf{q}}^{\dagger}$ terms by temporal smoothing; this is of no consequence, as they oscillate with a period $\approx \hbar/k_B T$ much shorter than the collisional timescales of interest here (see below) and automatically average to zero.</footnote>

<footnote>$^{53}$If we assume that the interacting phonon gas is an ergodic quantum system [92,93], the mean $\langle\hat{\mu}\rangle_\lambda$ in the steady state $|\psi_\lambda\rangle$ depends only on the two conserved quantities, the energy $E$ and the number of particles $N$, and coincides for a large system with the microcanonical chemical potential $\mu_{\mathrm{mc}}(E=E_\lambda, N)$. If the energy $E$ fluctuates from one realization of the experiment to the next around the mean value $\bar{E}$, as in the canonical ensemble, the phase factor in the third expression of Equation (61) fluctuates and leads to a Gaussian-in-time loss of coherence: linearization of $\mu_{\mathrm{mc}}(E, N)$ around $\bar{E}$ gives $\mathrm{Var}[\hat{\phi}_0(t) - \hat{\phi}_0(0)] \sim [\partial_E \mu_{\mathrm{mc}}(\bar{E}, N)]^2 (\mathrm{Var} E) t^2/\hbar^2$, a parasitic effect $\approx t^2/N$ rapidly masking the phase diffusion (60) $\approx t/N$ [94].</footnote>
condensate in reference [86]; we give here a simplified expression, keeping only the scaling laws in N, T and $\gamma$ (under the assumption $\gamma = O(1)$):

$$D_{\lambda}^{\gamma>0} \approx N^{-1} T^4 \gamma^0. \qquad (62)$$

There has been no experimental verification yet in cold atomic gases (the only quantum fluids sufficiently well isolated for the condensate loss of coherence to be intrinsic), even for bosons.

In the concave case $\gamma < 0$, on the other hand, the question remains largely open. An attempt to calculate $D_\lambda$ in reference [86], taking into account only the Landau-Khalatnikov four-phonon collision processes $\phi\phi \to \phi\phi$ at small angles, of typical rate $\Gamma_{\text{coll}}^\phi \propto (k_B T / mc^2)^7 mc^2 / \hbar |\gamma| \approx T^7$, led to an infinite diffusion coefficient,

$$D_{\lambda}^{\gamma<0} = +\infty \qquad (63)$$

more precisely to a superdiffusive spreading law (simplified as in (62))$^{54}$

$$\text{Var}_{\lambda}^{\gamma<0}[\hat{\phi}_0(t) - \hat{\phi}_0(0)] \approx N^{-1} T^{20/3} |\gamma|^{1/3} t^{5/3} \qquad (64)$$

in particular because collisions $\phi\phi \to \phi\phi$ preserve the total number $N_\phi$ of phonons (unlike $\phi \leftrightarrow \phi\phi$)$^{55,56}$. To go beyond this and obtain the true (a priori finite) value of $D_\lambda$ remains an open question: one would have to take into account the subdominant five-phonon processes $\phi\phi \leftrightarrow \phi\phi\phi$ which change $N_\phi$ and occur at a rate $\approx T^9$ [96], of the same order of magnitude as that of large-angle $\phi\phi \to \phi\phi$ processes [52,78], which is not easy$^{57}$.

# 5. Open questions requiring a microscopic theory of the many-body problem

The quantum hydrodynamics of Section 4 is only a low-energy effective theory. It therefore has limitations of two kinds, which we briefly review here, and which make it impossible to dispense with a many-body microscopic calculation.

## 5.1. Determining the ingredients of quantum hydrodynamics

Quantum hydrodynamics involves two quantities that are external to it, the equation of state of the unpolarized Fermi gas at zero temperature (through the energy density $e_0(\rho)$ or the chemical potential $\mu_0(\rho)$—its derivative—at density $\rho$) and the curvature parameter $\gamma$ of the acoustic branch (34).

In the present case of equal mass $m_1 = m_2 = m$, the equation of state has been measured experimentally [48,97] and various approximate calculation methods give satisfactory results, such as fixed-node diffusive quantum Monte Carlo [98,99] or the Gaussian fluctuations approximation in a path integral formulation [100,101].

The situation is much more open for the curvature parameter $\gamma$. The Anderson random phase approximation (RPA) [61], equivalent for this problem to the eigenfrequency calculation of linearized time-dependent BCS equations or even to the more powerful Gaussian fluctuations

<footnote>$^{54}$In all cases, see Equations (62) and (64), we find that there is no phase spreading at the thermodynamic limit $N \to +\infty$: in an isolated gas, the limited coherence time of the condensate is a finite-size effect.</footnote>

<footnote>$^{55}$$N_\phi$ should then be added to the list of constants of motion, alongside $E$ and $N$, in note 53.</footnote>

<footnote>$^{56}$The fact that, for $\gamma < 0$, the phonon damping rate $\Gamma_q$ tends towards zero as $q^3$ (instead of $q$ for $\gamma > 0$) also plays a role; however, without the conservation of $N_\phi$, it would lead to a marginally superdiffusive $t \ln t$ spreading law (see Equation (C.20) of reference [86] and the morphology stated below its Equation (72)).</footnote>

<footnote>$^{57}$Publication [86], misunderstanding reference [96], had seen there a five-phonon damping rate scaling as $T^{11}$. The error has been corrected here. Indeed, reference [96], considering a quasi-thermal equilibrium with a small non-zero phonon chemical potential $\mu_\phi \to 0^-$, obtains the evolution equation $L^{-3}dN_\phi/dt = -\Gamma_\phi \mu_\phi$ for the average phonon number, where $\Gamma_\phi \approx T^{11}$ is not the rate sought despite appearances; as $L^{-3}dN_\phi/dt \approx T^2 d\mu_\phi/dt$ for Bose's law $\bar{n}_q = 1 / \{\exp[(\hbar c q - \mu_\phi) / k_B T] - 1\}$, we actually have $-d\mu_\phi/dt \propto (\Gamma_\phi / T^2)\mu_\phi$, of rate $\approx T^9$.</footnote>
approximation [100,102]$^{58}$, leads to a fairly simple analytical expression of $\gamma$ in terms of $\mu/|\Delta|$ and $(\partial\mu/\partial|\Delta|)_a$, exact in the limit $k_F a \to 0^+$ of a condensate of dimers ($\gamma \to 1$), reasonable in the BCS limit $k_F a \to 0^-$ ($\gamma \to -\infty$ exponentially with $1/k_F|a|$ since the acoustic branch is crushed by the broken-pair continuum) and changing sign for $|\Delta|/\mu \approx 0.87$, i.e. $1/k_F a \approx -0.14$ for the rather approximate BCS equation of state, not far from the unitary limit in any case [57].

In particular, $\gamma$ has the same positive value at the unitary limit in all three approaches (convex acoustic branch at low $q$):

$$ \gamma_{a^{-1}=0}^{\text{RPA}} \approx 0.084 \qquad (65) $$

The error, however, is uncontrolled, and we're not even sure of the sign.

A completely different method proceeds by extending the problem to an arbitrary spatial dimension $d$ and expanding around dimension four, in powers of the small parameter $\epsilon = 4 - d$. At the unitary limit, it too predicts a convex branch at low $q$ [104]$^{59}$:

$$ \gamma_{a^{-1}=0}^{\text{dimension}} = \frac{1}{3} \left[ 1 - \frac{1}{4}\epsilon + O(\epsilon^2) \right] \stackrel{\epsilon=1}{\approx} \frac{1}{4} > 0 \qquad (66) $$

Experimentally, a recent measurement of the acoustic branch by Bragg excitation in a cold-atom Fermi gas leads, on the contrary, to a concave branch at the unitary limit [105]:

$$ \gamma_{a^{-1}=0}^{\text{exp}} = \frac{8\mu}{3E_F} \zeta \approx \zeta \quad \text{with} \quad \zeta = -0.085(8) < 0 \qquad (67) $$

where the ratio $\mu/E_F$ is $\approx 3/8$ for the ground-state unitary gas [48] and $\zeta$ is the acoustic-branch curvature parameter for a rescaling of $q$ by $k_F$, $\omega_q = cq(1 + \zeta q^2/k_F^2 + \dots)$. The result (67) suffers from two limitations [106]: (i) a cubic fit of the branch over an interval of rather high $q$ values, $q/k_F \in [0.29, 1.63]$, rather than over a narrow neighborhood of $q = 0$ (for the RPA dispersion relation, for example, which has an inflection point at $q \approx 0.5k_F$, such a fit, blindly mixing convex and concave parts, would not give the right sign of $\gamma_{a^{-1}=0}^{\text{RPA}}$), and (ii) a relatively high temperature, $T = 0.128(8)T_F \approx 0.8T_c$: even if we start from the RPA branch of parameter $\gamma > 0$ in the ground state, quantum hydrodynamics predicts a thermal change $\delta\gamma_{\text{th}}$ in curvature (by interaction of the mode $q$ with thermal phonons) negative enough to change its sign:

$$ \delta\gamma_{\text{th}} \sim -\frac{8\pi^2}{9(3\mu/E_F)^{1/2}} \left(\frac{T}{T_F}\right)^2 \approx -0.14 < -\gamma_{a^{-1}=0}^{\text{RPA}} \qquad (68) $$

The question of the sign of $\gamma$ at the unitary limit, which crucially determines the three-phonon ($\gamma > 0$) or four-phonon ($\gamma < 0$) nature of sound damping in the low-temperature collisionless regime, therefore remains largely open$^{60}$.

<footnote>$^{58}$These different approaches lead to exactly the same implicit equation linking angular eigenfrequency $\omega_q$, chemical potential $\mu$ and order parameter $\Delta$, and exactly the same equation linking $\mu, \Delta$ and s-wave scattering length $a$ [57]; they differ only in the equation of state $\mu = \mu_0(\rho)$ linking $\mu$ to $\rho$ in the ground state, the one of Gaussian fluctuations being the most accurate. For example, at the unitary limit $a^{-1} = 0$, the approaches all give $mc^2/\mu = 2/3$ (this is exact by scale invariance, $\mu_0(\rho) \propto \rho^{2/3}$ in Equation (43)), $|\Delta|/\mu \approx 1.16$ (close to the experimental value $0.44E_F/0.376E_F \approx 1.17$ knowing that $|\Delta| = E_{\text{pair}}/2$ in these theories and that $E_{\text{pair}}/2E_F \approx 0.44$ in the experiment [103]) but the ratio $\mu/E_F \approx 0.376$ in the experiment [48], very poorly reproduced $\approx 0.59$ by RPA and BCS, is much better $\approx 0.40$ in Gaussian fluctuations.</footnote>

<footnote>$^{59}$We obtained expression (66) by directly inserting Equation (50) of reference [104] into the dispersion relation (48) of the same reference and using the exact property $mc^2 = 2\mu/3$ due to scale invariance. Proceeding differently, i.e. via its Equation (52) and its result $c_2/c_1 = O(\epsilon^2) \approx 0$ with $d = 3$ in its Equation (48), we find the fairly close value $\gamma_{a^{-1}=0}^{\text{dimension}} = 8/45 \approx 0.18$.</footnote>

<footnote>$^{60}$The damping studied experimentally in reference [107] is in the hydrodynamic regime, in the sense of note 41. This reference therefore does not allow us to resolve the problem.</footnote>
## 5.2. Describing high-frequency modes

Quantum hydrodynamics, with its almost linear acoustic branch, cannot reliably describe the sound waves of angular frequency $\omega_q > mc^2/\hbar$ in the Fermi superfluid. Ignoring the composite nature of bound pairs $\uparrow\downarrow$, it is totally inapplicable to angular frequencies $\omega \approx E_{\text{pair}}/\hbar$, where $E_{\text{pair}}$ is the binding energy of a pair: at these frequencies, pairs can break into two fermionic excitations $\chi$ (conservation of energy no longer prohibits this), see Figure 5a.

A microscopic description of the Fermi gas is then required. At zero temperature, the main method available is that of the time-dependent BCS variational theory [108]. Its specialization to the linear response regime yields the following eigenvalue equation for the energy $z$ of the modes of wave vector $\mathbf{q}$:

$$ \det M(\mathbf{q}, z) = 0 \quad \text{with} \quad M(\mathbf{q}, z) = \begin{pmatrix} M_{|\Delta||} (\mathbf{q}, z) & M_{|\Delta|\theta} (\mathbf{q}, z) \\ M_{\theta|\Delta|} (\mathbf{q}, z) & M_{\theta\theta} (\mathbf{q}, z) \end{pmatrix} \qquad (69) $$

where the coefficients of the $2 \times 2$ matrix correspond to a response in the modulus $|\Delta|$ or in the phase $\theta$ of the complex order parameter $\Delta(\mathbf{r}, t)$. In the weakly-interacting BCS limit $k_F a \to 0^-$, off-diagonal elements are usually (rightly) neglected and the dynamics decouples into modulus and phase modes; in the general case, this distinction no longer applies.

The exploration of solutions to Equation (69) has begun. At a fixed wavenumber $q$, we find at most one root under the edge $\epsilon_{\mathbf{q}}^{\text{bord}}$ of the broken-pair continuum, the root $\hbar\omega_{\mathbf{q}}$ of the acoustic branch. On the interval $z \in ]\epsilon_{\mathbf{q}}^{\text{bord}}, +\infty[$, the function $\det M(\mathbf{q}, z)$ has a branch cut$^{61}$, it is necessary to add an infinitesimal shift $i0^+$ to $z$ to make sense of it; the function then acquires an imaginary part, which cannot vanish simultaneously with the real part, and Equation (69) has no solution. On the other hand, a complex $z_{\mathbf{q}}$ with a non-infinitesimal $< 0$ imaginary part can be found, by performing an analytic continuation of function $z \to \det M(\mathbf{q}, z)$ from the upper half-plane to the lower half-plane through its branch cut (indicated by arrow $\downarrow$ in the subscript):

$$ \det M_{\perp}(\mathbf{q}, z_{\mathbf{q}}) = 0 \quad \text{with} \quad \text{Im} \, z_{\mathbf{q}} < 0 \qquad (70) $$

There is therefore a collective mode in the continuum, which decays exponentially in time through the emission of broken pairs. The calculation was first performed in the BCS limit $k_F a \to 0^-$, both for neutral fermions and for electrons in a superconductor, in reference [109]. It was then generalized to fermionic cold-atom gases for arbitrary values of $k_F a$, no longer neglecting the off-diagonal elements $M_{|\Delta|\theta}$ and $M_{\theta|\Delta|}$ [110,111]. The Andrianov-Popov branch persists up to $1/k_F a = 0.55$ (point of zero chemical potential $\mu = 0$ in BCS theory) and always starts at $2|\Delta|$ quadratically in $q$ with a complex coefficient:

$$ z_{\mathbf{q}}^{\mu>0} \underset{q \to 0}{\stackrel{\mu>0}{\equiv}} 2|\Delta| + \zeta \frac{\hbar^2 q^2}{4m_*} + O(q^3) \quad (\text{Im}\,\zeta < 0) \qquad (71) $$

where $m_*$ is the effective mass of a fermionic quasiparticle $\chi$ at the location $k = k_0$ of its energy minimum$^{62}$. We have written $2|\Delta|$ here rather than $E_{\text{pair}}$, where $\Delta$ is the equilibrium order parameter, even though BCS theory is unable to distinguish (we have exactly $E_{\text{pair}} = 2|\Delta|$ for all $\mu > 0$ in this theory), in order to evoke the Higgs mechanism [113] which we think

<footnote>$^{61}$The matrix elements of $M(\mathbf{q}, z)$ involve an integral over the wave vector $\mathbf{k}$ of one of the dissociation fragments of a bound pair of the condensate, and the integrand contains the corresponding energy denominator $z - (\epsilon_{\mathbf{k}} + \epsilon_{\mathbf{q}-\mathbf{k}})$; by definition, the denominator vanishes when $z$ belongs to the broken-pair continuum, see the legend to Figure 5.</footnote>

<footnote>$^{62}$The effective mass is such that $\epsilon_{\mathbf{k}} - E_{\text{pair}}/2 \sim \hbar^2(k-k_0)^2/2m_*$ as $k \to k_0$. Scaling by $m_*$ in Equation (71) ensures that $\zeta$ has a finite, non-zero limit when $k_F a \to 0^-$ [109]. In this regime, the well-known reference [112] predicts an incorrect behavior for $z_{\mathbf{q}}$ at low $q$, with an imaginary part tending linearly to zero $\propto q$, see its Equation (2.38). The quantity $\zeta$ here has nothing to do with that in Equation (67), there is an unfortunate coincidence of notations.</footnote>
the collective mode of the continuum comes under [114]$^{63}$; moreover, in the opposite limit $k_F a \to 0^+$ of a bosonic dimer condensate, where $2|\Delta| \ll E_{\text{pair}} \sim 2|\mu| \sim \hbar^2/ma^2$ (this time we have $\mu < 0$), we do indeed find a collective excitation branch starting quadratically at $2|\Delta|$ and not at $E_{\text{pair}}$ [111]. The extension of Equation (69) to non-zero temperature (beyond a simple BCS-type mean-field generalization, perhaps insufficient$^{64}$) remains to our knowledge an open question.

From an experimental point of view, in cold atoms or superconductors, excitation at angular frequencies $\omega > E_{\text{pair}}/\hbar$ has only been carried out at zero wavenumber, where there is, according to zero temperature theories, no collective mode in the continuum, the spectral weight of the mode tending to 0 as $q \to 0$ [110]; at long times, we simply observe oscillations of the order parameter at angular frequency $E_{\text{pair}}/\hbar$ (this is the effect of the non-zero edge of the continuum), which attenuate with a power law $t^{-\alpha}$ [117,118] by the same mechanism as the spreading of the Gaussian wave packet of a free particle in ordinary quantum mechanics (percussive excitation creates a “wave packet” of broken pairs $(\mathbf{k}, -\mathbf{k})$ in the continuum, whose evolution governed by the dispersion relation $2\epsilon_{\mathbf{k}}$ is effectively one-dimensional for $k_0 > 0$ ($\mu > 0$), in which case $\alpha = 1/2$ [119], and three-dimensional for $k_0 = 0$ ($\mu < 0$), in which case $\alpha = 3/2$ [120]$^{65}$. The observation of the continuum mode (at $q > 0$) and the precise measurement of its dispersion relation $z_{\mathbf{q}}$ therefore remain to be done (hints are given in references [110,121]).

## Declaration of interests

The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations.

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<footnote>$^{63}$As this mechanism results from $U(1)$ symmetry breaking, here by condensation of bound pairs, it must be characterized by the energy scale associated with the order parameter, i.e. $|\Delta|$ up to a factor; this is indeed what reference [113] finds, see its Equation (2b). On the other hand, the energy scale $E_{\text{pair}}$ is related to pair breaking, not pair condensation, and therefore has no a priori connection with the Higgs branch. Having $E_{\text{pair}} = 2|\Delta|$ is a source of confusion and prevents the two phenomena from being decoupled. It would also be interesting to see whether property $E_{\text{pair}} = 2|\Delta|$ remains rigorously true at zero temperature in a theory more elaborate than BCS or in experiments.</footnote>

<footnote>$^{64}$This generalization is not a panacea, as can be seen on the imaginary part of the acoustic branch. For $\gamma > 0$ (but not for $\gamma < 0$), this criticism can be leveled at RPA already at zero temperature, since it wrongly predicts a purely real angular eigenfrequency $\omega_{\mathbf{q}}$. However, this doesn't seem too serious, since the imaginary part $(-1/2)\Gamma_{\mathbf{q}}(T=0) \approx q^5$ obtained by quantum hydrodynamics (Belaev damping) gets lost in the neglected sub-subdominant terms in Equation (34). This problem is most visible at low $q$ at non-zero temperature, where $\Gamma_{\mathbf{q}}(T>0)$ starts linearly in $q$ with a coefficient $\propto T^4$ (exponent $\nu$ is 5 in Table 1 for the scaling law $q \propto T$ of note 41), which the BCS-type mean-field theory cannot account for (it predicts a coefficient $O[-\exp(E_{\text{pair}}/2k_B T)]$ [115,116] since the only thermal occupation numbers it brings out are those $\bar{n}_{\mathbf{k}} = 1/(\exp(\epsilon_{\mathbf{k}}/k_B T) + 1)$ of fermionic quasiparticles $\chi$). In other words, the linearization of the time-dependent BCS equations or, what amounts to the same thing, the Gaussian fluctuations approximation takes into account the $\phi-\chi$ coupling but not the $\phi-\phi$ coupling.</footnote>

<footnote>$^{65}$The observations of reference [118] in a unitary Fermi gas at $T \neq 0$ raise several questions: (i) the measured exponent $\alpha \approx 1 \pm 0.15$ is very different from the theoretically predicted value ($\alpha = 1/2$ at the unitary limit), (ii) it cannot be ruled out that the decay of the oscillation amplitude is in fact exponential, (iii) in contrast to the amplitude, the angular frequency of the oscillations shows no observable reduction as $T$ approaches the transition temperature $T_c$ (where there is no more $U(1)$ symmetry breaking and $|\Delta|$ tends to zero), which seems incompatible with the qualification of Higgs oscillations used in this reference (the measured angular frequency is not proportional to $|\Delta|/\hbar$), but also suggests a rather troubling lack of dependence of $E_{\text{pair}}$ with temperature (the measured angular frequency should be given by $E_{\text{pair}}/\hbar$ since the excitation is done at $q=0$). Incidentally, the temperature is never very low in the experiment, $T \gtrsim 0.1 T_F$, see our Section 2, which makes the theory at $T=0$ stricto sensu inapplicable.</footnote>
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