Part I — Fermi Liquid Theory

Chapter 5: Zero Sound & Collective Modes

One of the most striking predictions of Landau's Fermi liquid theory is the existence of zero sound — a collective oscillation of the Fermi surface that propagates even in the absence of collisions. Unlike ordinary (first) sound, which requires local thermal equilibrium maintained by frequent quasiparticle scattering, zero sound is a purely collisionless mode sustained by the coherent forward scattering of quasiparticles through the Landau interaction.

The distinction between the two regimes is controlled by $\omega\tau$, where $\tau$ is the quasiparticle collision time. In the hydrodynamic limit $\omega\tau \ll 1$, ordinary first sound propagates. In the collisionless limit $\omega\tau \gg 1$, zero sound takes over. This chapter derives both modes from the linearized Boltzmann equation and explores the crossover between them.

Linearized Boltzmann Equation for Quasiparticles

The quasiparticle distribution function $n_{\mathbf{p}}(\mathbf{r}, t)$ obeys the Boltzmann transport equation. For small deviations from equilibrium, we write:

$n_{\mathbf{p}}(\mathbf{r}, t) = n^0(\varepsilon_{\mathbf{p}}) + \delta n_{\mathbf{p}}(\mathbf{r}, t)$

where $n^0$ is the equilibrium Fermi function. The quasiparticle energy itself depends on the distribution through the Landau interaction:

$\varepsilon_{\mathbf{p}} = \varepsilon^0_{\mathbf{p}} + \sum_{\mathbf{p}'} f_{\mathbf{p}\mathbf{p}'}\, \delta n_{\mathbf{p}'}$

The linearized Boltzmann equation takes the form:

$\frac{\partial \delta n_{\mathbf{p}}}{\partial t} + \mathbf{v}_{\mathbf{p}} \cdot \nabla_{\mathbf{r}} \delta n_{\mathbf{p}} + \frac{\partial n^0}{\partial \varepsilon}\, \mathbf{v}_{\mathbf{p}} \cdot \nabla_{\mathbf{r}} \sum_{\mathbf{p}'} f_{\mathbf{p}\mathbf{p}'}\, \delta n_{\mathbf{p}'} = I[\delta n]$

The collision integral $I[\delta n]$ vanishes in the collisionless regime. For a plane wave disturbance $\delta n_{\mathbf{p}} \propto e^{i(\mathbf{q}\cdot\mathbf{r} - \omega t)}$, the deviation near the Fermi surface can be parametrized as:

$\delta n_{\mathbf{p}} = -\frac{\partial n^0}{\partial \varepsilon}\, \nu(\hat{\mathbf{p}})$

where $\nu(\hat{\mathbf{p}})$ describes the angular distortion of the Fermi surface.

Zero Sound: Collisionless Collective Oscillation

Setting the collision integral to zero ($\omega\tau \gg 1$), the linearized equation becomes an eigenvalue problem for the Fermi surface distortion. For an isotropic Fermi liquid with only the $\ell = 0$ Landau parameter $F_0^s$, the equation for $\nu(\theta)$ reduces to:

$\nu(\theta) = \frac{F_0^s \cos\theta}{s - \cos\theta} \int \frac{d\Omega'}{4\pi}\, \nu(\theta')$

where $s = \omega / (v_F q)$ is the dimensionless phase velocity and $\theta$ is the angle between $\hat{\mathbf{p}}$ and $\hat{\mathbf{q}}$. Self-consistency requires that $\int (d\Omega'/4\pi)\,\nu(\theta')$ equal a constant, yielding the fundamental equation for the zero sound velocity:

$1 = \frac{F_0^s}{2} \int_{-1}^{1} \frac{\cos\theta}{s - \cos\theta}\, \frac{d(\cos\theta)}{1}$

Evaluating the integral analytically:

$1 = \frac{F_0^s}{2}\left[s \ln\left|\frac{s+1}{s-1}\right| - 2\right]$

This equation has a real solution $s > 1$ for any $F_0^s > 0$. The condition $s > 1$ ensures the mode propagates faster than the Fermi velocity, avoiding Landau damping. In the weak-coupling limit $F_0^s \to 0^+$, the velocity approaches $s \to 1^+$ from above; for strong coupling $F_0^s \gg 1$, we find $s \approx \sqrt{F_0^s / 3}$.

Dispersion Relation

The zero sound dispersion relation is linear at long wavelengths:

$\omega = s\, v_F\, k$

where $s > 1$ is the solution of the zero sound equation above. The physical content is that the mode propagates faster than any individual quasiparticle ($v_F$ is the maximum quasiparticle speed). This is essential: if $s < 1$, the mode would lie in the particle-hole continuum and suffer Landau damping.

The Fermi surface distortion for zero sound is anisotropic, concentrated in the forward direction ($\theta \approx 0$ along $\hat{\mathbf{q}}$). This contrasts with first sound, where the distortion is purely $\ell = 1$ (a rigid shift).

First Sound: Hydrodynamic Regime

In the hydrodynamic regime $\omega\tau \ll 1$, frequent collisions establish local thermodynamic equilibrium. The distribution function is parametrized by local density, velocity, and temperature. Sound propagation is then described by the Euler equation and continuity equation, yielding ordinary (first) sound with velocity:

$c_1 = v_F \sqrt{\frac{1 + F_0^s}{3}}$

The factor of $1/\sqrt{3}$ comes from averaging over the Fermi surface, while$(1 + F_0^s)$ reflects the renormalization of the compressibility by interactions. In a non-interacting gas ($F_0^s = 0$), $c_1 = v_F / \sqrt{3}$.

First sound exists for both repulsive ($F_0^s > 0$) and attractive interactions as long as $F_0^s > -1$ (the Pomeranchuk stability condition for the compressibility). For $F_0^s < -1$, the compressibility becomes negative, signaling a thermodynamic instability toward phase separation.

Crossover from Zero to First Sound

The crossover between the two regimes is governed by $\omega\tau$. As temperature increases, $\tau \propto T^{-2}$ (from Pauli blocking of quasiparticle scattering), so low temperatures favor the collisionless zero-sound regime while higher temperatures push toward first sound.

Zero Sound ($\omega\tau \gg 1$)

• Collisionless propagation
• Anisotropic Fermi surface distortion
• Speed: $c_0 = s\, v_F$ with $s > 1$
• Dominant at low $T$

First Sound ($\omega\tau \ll 1$)

• Hydrodynamic (collision-dominated)
• Isotropic local equilibrium
• Speed: $c_1 = v_F\sqrt{(1+F_0^s)/3}$
• Dominant at higher $T$

An important relationship: zero sound is always faster than first sound. For large $F_0^s$, both velocities grow as $\sqrt{F_0^s/3}\,v_F$, and their ratio approaches $\sqrt{3}$. The crossover was first observed experimentally in liquid $^3\text{He}$ by Abel, Anderson, and Wheatley (1966).

Spin Zero Sound and Paramagnon Modes

In addition to the density (charge) channel, collective modes can propagate in the spin channel. Spin zero sound involves oscillations where spin-up and spin-down quasiparticles move out of phase, creating a propagating spin-density wave. The relevant Landau parameter is $F_0^a$ (the antisymmetric or exchange interaction).

The spin zero sound equation has the same form as the charge channel, with $F_0^s$ replaced by $F_0^a$:

$1 = \frac{F_0^a}{2}\left[s_a \ln\left|\frac{s_a+1}{s_a-1}\right| - 2\right]$

For $F_0^a > 0$, a propagating spin zero sound exists. However, in many systems (including $^3\text{He}$), $F_0^a < 0$, and there is no undamped spin collective mode. Instead, there is a strongly enhanced spin response near the Stoner instability ($F_0^a \to -1$) — these are the paramagnon modes.

Paramagnons are overdamped spin fluctuations characterized by a diffusive pole in the spin susceptibility:

$\chi(q, \omega) \sim \frac{\chi_0}{1 + F_0^a - i\omega / (D_s q^2)}$

Near the ferromagnetic instability, the paramagnon spectral weight is large at low frequencies, contributing an anomalous $T^3 \ln T$ correction to the specific heat and enhancing the spin-lattice relaxation rate in NMR experiments.

Attenuation of Zero Sound

Although zero sound is undamped in the strictly collisionless limit, residual quasiparticle collisions produce a finite attenuation. The attenuation coefficient $\alpha$ in the collisionless regime scales as:

$\alpha \propto \frac{\omega^2}{\omega^2 \tau^2} \sim \frac{1}{(\omega\tau)^2} \propto T^4$

Since $\tau \propto T^{-2}$, the attenuation at fixed frequency grows as $T^4$ with increasing temperature. At fixed temperature, higher-frequency zero sound propagates with less relative damping ($\alpha/k \propto 1/(\omega\tau)^2$).

There is also a distinct contribution from Landau damping whenever the mode velocity enters the particle-hole continuum ($s < 1$). For the density channel with $F_0^s > 0$, the zero sound velocity satisfies $s > 1$, so Landau damping is absent. However, if higher-order Landau parameters are included, mode velocities can dip into the continuum at certain wave vectors, producing strong damping at those momenta.

Experimentally, the attenuation of zero sound in $^3\text{He}$ was measured via ultrasonic techniques and agrees well with the $T^4$ dependence predicted by theory.

Detailed Derivations

Derivation of the zero sound dispersion relation from the linearized Boltzmann equation

Step 1: Start with the collisionless ($I[\delta n] = 0$) linearized Boltzmann equation for a plane wave disturbance $\delta n_{\mathbf{p}} \propto e^{i(\mathbf{q}\cdot\mathbf{r} - \omega t)}$:

$$(-\omega + \mathbf{v}_\mathbf{p}\cdot\mathbf{q})\,\delta n_\mathbf{p} + \left(-\frac{\partial n^0}{\partial\varepsilon}\right)\mathbf{v}_\mathbf{p}\cdot\mathbf{q}\sum_{\mathbf{p}'} f_{\mathbf{p}\mathbf{p}'}\,\delta n_{\mathbf{p}'} = 0$$

Step 2: Parametrize the Fermi surface distortion as $\delta n_\mathbf{p} = (-\partial n^0/\partial\varepsilon)\,\nu(\hat{\mathbf{p}})$. Since $-\partial n^0/\partial\varepsilon \approx \delta(\varepsilon - E_F)$ at $T = 0$, the equation becomes:

$$(-\omega + v_F q\cos\theta)\,\nu(\hat{\mathbf{p}}) + v_F q\cos\theta\,N^*(0)\int\frac{d\Omega'}{4\pi}\,f(\hat{\mathbf{p}}, \hat{\mathbf{p}}')\,\nu(\hat{\mathbf{p}}') = 0$$

where $\theta$ is the angle between $\hat{\mathbf{p}}$ and $\hat{\mathbf{q}}$, and the sum over $\mathbf{p}'$ has been converted to an angular integral using the density of states at the Fermi surface.

Step 3: Define $s = \omega/(v_F q)$ and keep only the $l = 0$ Landau parameter $F_0^s$ (so that $N^*(0)\,f(\hat{\mathbf{p}}, \hat{\mathbf{p}}') \to F_0^s$ is isotropic). Solving for $\nu$:

$$\nu(\theta) = \frac{F_0^s\cos\theta}{s - \cos\theta}\,\langle\nu\rangle$$

where $\langle\nu\rangle = \int (d\Omega'/4\pi)\,\nu(\hat{\mathbf{p}}')$ is the angular average.

Step 4: Self-consistency requires integrating both sides over angles. Using $d\Omega = 2\pi\sin\theta\,d\theta$, with $u = \cos\theta$:

$$1 = F_0^s\int_{-1}^{1}\frac{du}{2}\,\frac{u}{s - u} = \frac{F_0^s}{2}\left[-1 + s\int_{-1}^{1}\frac{du}{s - u}\right]$$

Step 5: Evaluate the integral. Using $\int_{-1}^{1} du/(s - u) = \ln|(s+1)/(s-1)|$ (valid for $s > 1$):

$$1 = \frac{F_0^s}{2}\left[s\ln\left(\frac{s+1}{s-1}\right) - 2\right]$$

Step 6: Analyze limiting cases. For weak coupling $F_0^s \ll 1$, expand $s = 1 + \delta$ with $\delta \ll 1$. Then $\ln((s+1)/(s-1)) \approx \ln(2/\delta)$, and:

$$1 \approx \frac{F_0^s}{2}[\ln(2/\delta) - 2] \implies \delta \approx 2\,e^{-2/F_0^s - 2}$$

For strong coupling $F_0^s \gg 1$, expand $s \gg 1$: $\ln((s+1)/(s-1)) \approx 2/s + 2/(3s^3) + \cdots$, giving:

$$1 \approx \frac{F_0^s}{2}\cdot\frac{2}{3s^2} \implies \boxed{s \approx \sqrt{F_0^s/3} \quad (F_0^s \gg 1)}$$

This matches the first sound result $s_1 = \sqrt{(1+F_0^s)/3} \approx \sqrt{F_0^s/3}$ up to a numerical factor, confirming that zero sound and first sound merge at strong coupling.

Derivation of the first sound velocity from hydrodynamics

Step 1: In the hydrodynamic regime ($\omega\tau \ll 1$), the distribution function is a local equilibrium Fermi-Dirac function characterized by local density $n(\mathbf{r}, t)$, velocity $\mathbf{u}(\mathbf{r}, t)$, and temperature $T(\mathbf{r}, t)$.

Step 2: The linearized continuity and Euler equations are:

$$\frac{\partial(\delta n)}{\partial t} + n_0\,\nabla\cdot\mathbf{u} = 0$$

$$m^* n_0\,\frac{\partial\mathbf{u}}{\partial t} + \nabla(\delta P) = 0$$

Step 3: The pressure change is related to the density change through the compressibility: $\delta P = (\partial P/\partial n)_s\,\delta n$. For the Fermi liquid:

$$\frac{\partial P}{\partial n}\bigg|_s = \frac{n}{m^*}\cdot\frac{1 + F_0^s}{N^*(0)/n} = \frac{\hbar^2 k_F^2}{3m^{*2}}(1 + F_0^s)m^*$$

Step 4: Combining the equations gives a wave equation $\partial^2(\delta n)/\partial t^2 = c_1^2\,\nabla^2(\delta n)$ with:

$$\boxed{c_1^2 = \frac{v_F^{*2}}{3}(1 + F_0^s) = \frac{v_F^2}{3}\cdot\frac{1 + F_0^s}{(m^*/m)^2}}$$

Derivation of the $T^4$ attenuation of zero sound

Step 1: In the nearly collisionless regime, include the collision integral as a small perturbation. The zero sound mode acquires a complex frequency $\omega = \omega_0 - i\alpha$, where $\alpha$ is the attenuation rate.

Step 2: The attenuation arises from quasiparticle collisions that damp the coherent Fermi surface oscillation. The collision rate scales as $1/\tau \propto T^2$ (from the Pauli blocking phase space argument). The attenuation is second order in the collision rate:

$$\alpha \propto \frac{\omega}{(\omega\tau)^2}$$

Step 3: At fixed frequency $\omega$, using $\tau \propto T^{-2}$:

$$\boxed{\alpha \propto \frac{1}{\tau^2} \propto T^4}$$

This $T^4$ temperature dependence was confirmed experimentally by ultrasonic attenuation measurements in liquid He-3. It should be contrasted with the $T^2$ attenuation of first sound (which is limited by viscosity, $\eta \propto T^{-2}$, giving $\alpha_1 \propto \omega^2\eta/\rho c_1^3 \propto T^{-2}$ at fixed frequency — first sound actually propagates better at lower temperatures).

Derivation 2: Zero Sound Dispersion from the Transport Equation

We derive zero sound starting from the full linearized Landau kinetic equation and systematically reducing it to the eigenvalue equation for the zero sound velocity.

Step 1: The linearized Landau kinetic equation (transport equation) for quasiparticle distribution deviations is:

$$\frac{\partial \delta n}{\partial t} + \mathbf{v}_k \cdot \nabla_r \delta n + \frac{\partial n^0}{\partial \epsilon}\mathbf{v}_k \cdot \nabla_r\!\left(\sum_{k'} f_{kk'}\delta n_{k'}\right) = 0$$

where $\delta n = \delta n_k(\mathbf{r}, t)$ is the deviation from the equilibrium distribution, $\mathbf{v}_k = \nabla_k \epsilon_k / \hbar$ is the quasiparticle group velocity, and $f_{kk'}$ is the Landau interaction function. We have set the collision integral to zero (collisionless regime, $\omega\tau \gg 1$).

Step 2: Assume a plane wave form for the disturbance:

$$\delta n_k(\mathbf{r}, t) = \delta \tilde{n}_k\, e^{i(qx - \omega t)}$$

Choose $\mathbf{q} = q\hat{x}$ along the x-axis. Substituting into the kinetic equation, each spatial gradient $\nabla_r$ brings down a factor of $iq\hat{x}$ and the time derivative gives $-i\omega$:

$$-i\omega\,\delta\tilde{n}_k + iq\,v_k\cos\theta\,\delta\tilde{n}_k + iq\,v_k\cos\theta\,\frac{\partial n^0}{\partial \epsilon}\sum_{k'} f_{kk'}\delta\tilde{n}_{k'} = 0$$

where $\theta$ is the angle between $\mathbf{v}_k$ and $\hat{x}$ (i.e., the polar angle on the Fermi surface relative to $\mathbf{q}$).

Step 3: Project onto the Fermi surface. Near $T = 0$, $-\partial n^0/\partial\epsilon \approx \delta(\epsilon - \epsilon_F)$, so the relevant quasiparticles lie on the Fermi surface with $|\mathbf{v}_k| = v_F$. Write $\delta\tilde{n}_k = (-\partial n^0/\partial\epsilon)\,\nu(\theta)$ and define $s = \omega/(v_F q)$. Dividing through by $iq\,v_F\cos\theta\,(-\partial n^0/\partial\epsilon)$:

$$\frac{s - \cos\theta}{\cos\theta}\,\nu(\theta) = F_0^s\,\langle\nu\rangle$$

where we have used the $\ell = 0$ approximation $N^*(0)\,f_{kk'} \to F_0^s$ and defined the angular average $\langle\nu\rangle = \int (d\Omega'/4\pi)\,\nu(\theta')$. Solving for $\nu(\theta)$:

$$\nu(\theta) = \frac{F_0^s\,\cos\theta}{s - \cos\theta}\,\langle\nu\rangle$$

Step 4: Self-consistency — integrate both sides over $d\Omega/(4\pi)$ using $u = \cos\theta$:

$$\langle\nu\rangle = F_0^s\,\langle\nu\rangle\int_{-1}^{1}\frac{du}{2}\,\frac{u}{s - u}$$

For a nontrivial solution ($\langle\nu\rangle \neq 0$), we require the eigenvalue equation:

$$1 = F_0^s\int_{-1}^{1}\frac{du}{2}\,\frac{u}{s - u}$$

Rewriting the integrand as $u/(s-u) = -1 + s/(s-u)$ and integrating:

$$\int_{-1}^{1}\frac{du}{2}\,\frac{u}{s - u} = \frac{1}{2}\left[-2 + s\ln\!\left|\frac{s+1}{s-1}\right|\right]$$

Step 5: This yields the transcendental equation for $s$:

$$1 = \frac{F_0^s}{2}\left[s\ln\!\left(\frac{s+1}{s-1}\right) - 2\right]$$

Step 6: Existence for any $F_0^s > 0$. Define the right-hand side as $g(s) = (F_0^s/2)[s\ln((s+1)/(s-1)) - 2]$. As $s \to 1^+$, $\ln((s+1)/(s-1)) \to +\infty$, so $g(s) \to +\infty$. As $s \to \infty$, expand $s\ln((s+1)/(s-1)) \approx 2 + 2/(3s^2) + \cdots$, giving $g(s) \to F_0^s/(3s^2) \to 0^+$. Since $g(s)$ is continuous and strictly decreasing for $s > 1$, for any $F_0^s > 0$ there is exactly one solution $s > 1$. This proves zero sound exists for all repulsive interactions.

Step 7: Limiting cases.

$$F_0^s \ll 1:\quad s \approx 1 + 2\,e^{-2/F_0^s - 2} \quad(\text{exponentially close to } 1)$$

$$F_0^s \gg 1:\quad s \approx \sqrt{F_0^s/3} \quad(\text{matches first sound at strong coupling})$$

The zero sound dispersion is therefore $\omega = s(F_0^s)\,v_F\,q$, linear in $q$ with a velocity that depends on the Landau parameter. The mode is undamped because $s > 1$ places it outside the particle-hole continuum.

Derivation 3: First Sound vs Zero Sound Crossover

We derive the crossover condition between the collisionless (zero sound) and hydrodynamic (first sound) regimes, compare their velocities, and analyze attenuation in both limits.

Step 1: Crossover condition. The collision integral $I[\delta n]$ has a characteristic rate $1/\tau$. The transport equation with collisions reads:

$$(-\omega + v_F q\cos\theta)\,\nu(\theta) + v_F q\cos\theta\,F_0^s\,\langle\nu\rangle = -\frac{i}{\tau}\,\nu(\theta)$$

The collision term is negligible when $\omega\tau \gg 1$ (zero sound regime) and dominant when $\omega\tau \ll 1$ (first sound regime). The crossover occurs at:

$$\omega\tau \sim 1$$

Since $\tau \propto T^{-2}$ (Pauli blocking), at fixed frequency this crossover occurs at a characteristic temperature $T^*$ where $\omega\tau(T^*) = 1$.

Step 2: First sound velocity with $F_1^s$ correction. When higher Landau parameters are included, the hydrodynamic sound velocity receives corrections. The general result for first sound including the $\ell = 1$ Landau parameter $F_1^s$ is:

$$c_1 = v_F\sqrt{\frac{(1+F_0^s)(1+F_1^s/3)}{3}}$$

The $F_1^s$ parameter enters because it renormalizes the effective mass: $m^*/m = 1 + F_1^s/3$. For liquid He-3, $F_1^s$ is significant (of order 6–15 depending on pressure), making this correction important.

Step 3: Proof that $c_0 > c_1$ always. The zero sound velocity $c_0 = s\,v_F$ satisfies $s > 1$. The first sound velocity (with $F_1^s = 0$ for simplicity) is $c_1 = v_F\sqrt{(1+F_0^s)/3}$. We need to show $s > \sqrt{(1+F_0^s)/3}$.

From the zero sound equation, $s$ satisfies $1 = (F_0^s/2)[s\ln((s+1)/(s-1)) - 2]$. For the first sound value $s_1 = \sqrt{(1+F_0^s)/3}$, evaluate $g(s_1) = (F_0^s/2)[s_1\ln((s_1+1)/(s_1-1)) - 2]$. One can verify that $g(s_1) > 1$ for all $F_0^s > 0$. Since $g(s)$ is monotonically decreasing, the root $s$ must satisfy $s > s_1$, hence:

$$c_0 > c_1 \quad \text{for all } F_0^s > 0$$

Step 4: Attenuation of zero sound. Including collisions perturbatively, the zero sound mode acquires a complex frequency $\omega = \omega_0 - i\gamma$. The damping rate is:

$$\gamma_{\text{zero}} \propto \frac{\omega}{(\omega\tau)^2}$$

The spatial attenuation coefficient (amplitude decay per unit length) is $\alpha_0 = \gamma/c_0$, so:

$$\alpha_{\text{zero}} \propto \frac{q}{(\omega\tau)^2} \propto q\,T^4$$

Step 5: Attenuation of first sound. In the hydrodynamic regime, sound attenuation arises from viscosity $\eta$ and thermal conductivity $\kappa$. The dominant contribution in Fermi liquids is viscous:

$$\alpha_{\text{first}} = \frac{\omega^2}{2\rho c_1^3}\left(\frac{4}{3}\eta + \zeta\right)$$

Since the shear viscosity of a Fermi liquid scales as $\eta \propto \tau \propto T^{-2}$:

$$\alpha_{\text{first}} \propto \omega^2\,T^{-2}$$

Summary: zero sound attenuation grows as $T^4$ (worsens with temperature), while first sound attenuation decreases as $T^{-2}$ (improves with temperature). At the crossover $\omega\tau \sim 1$, the attenuation is maximal — neither regime propagates well, and the sound is strongly damped.

Derivation 4: Spin Zero Sound

We derive the spin-antisymmetric collective mode (paramagnon) from the spin channel of the transport equation and show why spin zero sound is overdamped in He-3.

Step 1: Spin channel of the transport equation. The Landau interaction has both spin-symmetric and spin-antisymmetric components. For spin-$1/2$ quasiparticles, write:

$$f_{kk'} = f^s_{kk'} + f^a_{kk'}\,\boldsymbol{\sigma}\cdot\boldsymbol{\sigma}'$$

Density fluctuations ($\delta n_\uparrow + \delta n_\downarrow$) couple to $f^s$ and give the zero sound discussed above. Spin-density fluctuations ($\delta n_\uparrow - \delta n_\downarrow$) couple to $f^a$. The spin-density deviation $\delta m_k = \delta n_{k\uparrow} - \delta n_{k\downarrow}$ satisfies:

$$\frac{\partial \delta m_k}{\partial t} + \mathbf{v}_k \cdot \nabla_r \delta m_k + \frac{\partial n^0}{\partial \epsilon}\mathbf{v}_k \cdot \nabla_r\!\left(\sum_{k'} f^a_{kk'}\delta m_{k'}\right) = 0$$

Step 2: Eigenvalue equation. Following the same plane wave ansatz and projection onto the Fermi surface as in the density channel, with $\delta m_k = (-\partial n^0/\partial\epsilon)\,\mu(\theta)\,e^{i(qx - \omega t)}$:

$$\mu(\theta) = \frac{F_0^a\,\cos\theta}{s_a - \cos\theta}\,\langle\mu\rangle$$

where $s_a = \omega/(v_F q)$ for the spin mode. Self-consistency gives the identical transcendental equation with $F_0^s$ replaced by $F_0^a$:

$$1 = \frac{F_0^a}{2}\left[s_a\ln\!\left(\frac{s_a+1}{s_a-1}\right) - 2\right]$$

Step 3: Condition for propagating spin zero sound. Exactly as in the density channel, a real solution $s_a > 1$ exists if and only if $F_0^a > 0$. This requires a repulsive exchange interaction.

Step 4: The case of He-3 ($F_0^a < 0$). In liquid He-3, the exchange interaction is attractive: $F_0^a \approx -0.7$ (at zero pressure). Since $F_0^a < 0$, there is no real solution for $s_a > 1$. The spin collective mode has $s_a < 1$, placing it inside the particle-hole continuum, where it is Landau-damped. The result is an overdamped spin fluctuation (paramagnon) rather than a propagating mode.

Step 5: Paramagnon propagator. For $F_0^a < 0$ and $|F_0^a| < 1$ (Pomeranchuk stability), the spin susceptibility takes the diffusive form:

$$\chi^{-1}(q, \omega) = \chi_0^{-1}\!\left(1 + F_0^a\right) - i\frac{\omega}{D_s q^2}$$

where $D_s$ is the spin diffusion constant. Near the Stoner instability ($F_0^a \to -1$), the static susceptibility is enhanced by a factor $1/(1 + F_0^a)$ (Stoner enhancement), and the paramagnon spectral weight is concentrated at low frequencies.

Step 6: Connection to paramagnon-mediated pairing. The exchange of virtual paramagnons generates an effective quasiparticle interaction. Since paramagnons carry spin, this interaction is attractive in the spin-triplet, odd-parity (p-wave) channel:

$$V_{\text{eff}}(q, \omega) \sim \frac{(f^a)^2\,\chi(q, \omega)}{1 + F_0^a}$$

Near the ferromagnetic instability, the large Stoner enhancement factor makes this interaction strong enough to produce p-wave Cooper pairing. This mechanism, proposed by Layzer and Fay (1971) and by Anderson and Brinkman (1973), is now understood to be responsible for the triplet superfluidity of He-3 (the A and B phases). The same physics — spin fluctuation exchange near a magnetic instability — has been invoked to explain unconventional superconductivity in Sr$_2$RuO$_4$, heavy-fermion compounds, and organic superconductors.

Applications

Applications of Zero Sound and Collective Modes

1. Experimental observation in He-3 by Abel, Anderson, and Wheatley (1966). The first and most celebrated application of zero sound theory is its direct experimental verification in liquid He-3. Abel, Anderson, and Wheatley used ultrasonic pulse-echo techniques at frequencies of 15–45 MHz to measure both the velocity and attenuation of sound as a function of temperature. They observed the predicted 5% increase in velocity as the mode character changed from first sound ($c_1 = v_F\sqrt{(1+F_0^s)/3}$) to zero sound ($c_0 = s\,v_F$), the characteristic peak in attenuation at the crossover $\omega\tau \sim 1$, and the frequency scaling of the crossover temperature. These measurements yielded precise values of the Landau parameter $F_0^s \approx 10.8$ (at zero pressure) and confirmed the $T^4$ attenuation law. Subsequent experiments by Roach and Ketterson (1976) extended the measurements to high pressures, mapping out $F_0^s$ across the entire phase diagram.

2. Phonon-like collective modes in ultracold Fermi gases. Ultracold atomic Fermi gases ($^6$Li, $^{40}$K) confined in optical traps provide a tunable platform for studying zero sound. By adjusting the scattering length via Feshbach resonances, experimenters can continuously vary the interaction strength and observe the collisionless-to-hydrodynamic transition. In the weakly interacting BCS regime, the collective breathing and quadrupole oscillation modes of the trapped gas behave as zero sound at low temperatures and cross over to hydrodynamic modes as the temperature or interaction strength is increased. Measurements of the oscillation frequencies and damping rates by the groups of Grimm, Ketterle, Thomas, and Jin have confirmed Fermi liquid predictions and provided model-independent determinations of Landau parameters in these clean, disorder-free systems.

3. Zero sound in quark-gluon plasma. At densities exceeding several times nuclear saturation density (as may exist in the cores of neutron stars), quarks are expected to form a degenerate Fermi liquid interacting via the strong force. Zero sound in such quark matter has been studied theoretically using both perturbative QCD and holographic (AdS/CFT) methods. The color Fermi liquid supports density oscillations analogous to zero sound, with velocities determined by the quark-quark forward scattering amplitude. These modes could influence the bulk viscosity and neutrino emissivity of dense quark matter, affecting the cooling curves and gravitational wave signatures of neutron star mergers. Holographic models of strongly coupled Fermi liquids (via gauge-gravity duality) have provided non-perturbative predictions for zero sound dispersion and attenuation.

4. Acoustic plasmons in layered electron systems. In layered two-dimensional electron systems (e.g., semiconductor quantum wells, graphene multilayers, and high-$T_c$ cuprate superconductors), the interplay between intralayer Coulomb repulsion and interlayer tunneling produces a hierarchy of collective modes. The in-phase oscillation is the usual optical plasmon, but the out-of-phase (acoustic) branch has a linear dispersion $\omega \propto q$ at long wavelengths — effectively a zero-sound-like mode. These acoustic plasmons have been observed in GaAs double quantum wells via inelastic light scattering and in van der Waals heterostructures via near-field optical microscopy. Their dispersion and damping provide information about interlayer coupling and quasiparticle interactions, directly connecting to the Landau Fermi liquid framework.

5. Spin fluctuations and unconventional superconductivity. The paramagnon mechanism — spin fluctuation exchange near a magnetic instability — provides the leading candidate pairing interaction for numerous unconventional superconductors. In heavy-fermion compounds (CeCoIn$_5$, UPt$_3$, UBe$_{13}$), the proximity to an antiferromagnetic quantum critical point generates strong, nearly critical spin fluctuations that mediate d-wave or p-wave pairing. In the iron-based superconductors (e.g., BaFe$_2$As$_2$ family), antiferromagnetic spin fluctuations centered at the nesting wave vector $\mathbf{Q} = (\pi, 0)$ are believed to drive $s^{\pm}$ pairing symmetry. In Sr$_2$RuO$_4$ and organic superconductors, ferromagnetic spin fluctuations (paramagnons) favor triplet pairing. In all these cases, the spin fluctuation spectrum — directly descended from Landau's spin zero sound analysis — determines the symmetry, magnitude, and temperature dependence of the superconducting gap.

Historical Context

Landau's 1957 Prediction

Zero sound was predicted by Lev Landau in his landmark 1957 paper “The Theory of a Fermi Liquid” (Zh. Eksp. Teor. Fiz. 32, 59). This was the third and final paper in his Fermi liquid trilogy (1956–1957), which established the theoretical framework for interacting fermion systems. The prediction was remarkable: Landau showed that even in the collisionless regime (where ordinary sound cannot propagate because there are no collisions to maintain local equilibrium), a collective density oscillation could still exist, sustained by the coherent mean-field interaction between quasiparticles on the Fermi surface. He derived the full integral equation for the mode, showed that it has a solution with phase velocity exceeding $v_F$ for any repulsive interaction, and pointed out that this mode should be observable in liquid He-3 at sufficiently low temperatures.

The Race to Observe Zero Sound

Landau's prediction sparked an experimental race that lasted nearly a decade. The challenge was formidable: observing zero sound requires $\omega\tau \gg 1$, meaning the quasiparticle collision time must exceed the oscillation period. Since $\tau \propto T^{-2}$ in a Fermi liquid, this demanded extremely low temperatures — below about 100 mK for typical ultrasonic frequencies of 10–50 MHz. In the late 1950s, such temperatures were at the frontier of experimental capability. Several groups attempted the measurement, including Keen, Matthews, and Wilks at Oxford (1963), who observed anomalous ultrasonic absorption in He-3 but could not definitively identify zero sound. The key difficulty was separating the zero sound signal from first sound and from viscous damping effects near the crossover.

The Abel–Anderson–Wheatley Experiment (1966)

The definitive experimental confirmation came in 1966, when William Abel, Allan Anderson, and John Wheatley at the University of California, San Diego, observed the transition from first sound to zero sound in liquid He-3 (Phys. Rev. Lett. 17, 74). Using ultrasonic pulse-echo techniques at frequencies of 15.4, 25.6, and 45.5 MHz, they tracked both the sound velocity and attenuation as the temperature was lowered through the crossover region. They observed three key signatures predicted by Landau's theory: (1) an increase in sound velocity of approximately 5% as the mode transitioned from first to zero sound, (2) a peak in attenuation at the crossover temperature where $\omega\tau \sim 1$, and (3) the frequency dependence of the crossover temperature, confirming the $\omega\tau$ scaling. Their crossover temperatures ranged from about 15 mK to 50 mK depending on frequency, achieved using the Pomeranchuk cooling technique (which exploits the anomalous entropy of solid He-3). This experiment is regarded as one of the most elegant confirmations of Landau's Fermi liquid theory.

Theoretical Developments and Spin Fluctuations

The spin channel analogue — spin zero sound and paramagnon modes — was developed theoretically by Doniach and Engelsberg (1966) and Berk and Schrieffer (1966), who showed that strong paramagnon fluctuations near the Stoner instability could mediate p-wave superconducting pairing. This mechanism is now believed to be responsible for the triplet superfluidity of He-3 (discovered by Osheroff, Richardson, and Lee in 1972) and has been proposed for several unconventional superconductors. The formal connection between paramagnon exchange and p-wave pairing was made precise by Anderson and Brinkman (1973), who showed that the ABM state (A-phase of superfluid He-3) is stabilized by spin fluctuation feedback.

Modern Cold Atom Realizations

Beginning in the 2000s, the advent of ultracold atomic Fermi gases opened a new chapter in the study of zero sound. Experiments with trapped $^6$Li and $^{40}$K atoms near Feshbach resonances achieved tunable interaction strengths, enabling systematic studies of the collisionless-to-hydrodynamic crossover. Groups led by Grimm (Innsbruck), Ketterle (MIT), Thomas (Duke), and Jin (JILA) observed collective oscillation modes — including breathing modes, quadrupole modes, and scissors modes — whose frequencies and damping rates matched Fermi liquid predictions in the weakly interacting regime and displayed the zero-to-first sound crossover as the collision rate was varied by tuning temperature or interaction strength. These cold atom experiments provide the cleanest modern realization of Landau's zero sound, free from the complications of lattice effects, impurities, and electromagnetic interactions present in condensed matter systems.

Python Simulation: Zero Sound Velocity and Crossover

Numerical solution of the zero sound integral equation. The four panels show: (1) zero sound vs first sound velocity as a function of $F_0^s$, (2) their ratio, (3) the crossover as a function of $\omega\tau$, and (4) collisional attenuation.

Zero Sound Velocity & Crossover Analysis

Python

Solves the Landau integral equation numerically for zero sound velocity s(F_0^s) and plots the crossover to first sound

script.py148 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
# numpy-only (no scipy needed)

# ──────────────────────────────────────────────────────
# Zero Sound Velocity s(F_0^s) and Crossover to First Sound
# Solving the Landau integral equation for zero sound
# ──────────────────────────────────────────────────────

def zero_sound_equation(s, F0s):
    if s <= 1.0:
        return -1e10
    integral = s * np.log((s + 1.0) / (s - 1.0)) - 2.0
    return (F0s / 2.0) * integral - 1.0

def bisect_solve(F0s, lo=1.0001, hi=None, tol=1e-10, maxiter=200):
    """Bisection root finder (replaces scipy.optimize.brentq)."""
    if hi is None:
        hi = 1.0 + 10.0 * F0s
    fa = zero_sound_equation(lo, F0s)
    fb = zero_sound_equation(hi, F0s)
    if fa * fb > 0:
        return np.nan
    for _ in range(maxiter):
        mid = 0.5 * (lo + hi)
        fm = zero_sound_equation(mid, F0s)
        if abs(fm) < tol or (hi - lo) < tol:
            return mid
        if fa * fm < 0:
            hi = mid
            fb = fm
        else:
            lo = mid
            fa = fm
    return 0.5 * (lo + hi)

# Solve for zero sound velocity s as function of F0s
F0s_values = np.linspace(0.01, 20.0, 500)
s_values = np.zeros_like(F0s_values)

for i, F0s in enumerate(F0s_values):
    s_values[i] = bisect_solve(F0s)

# First sound velocity: s_1 = sqrt((1 + F0s)/3) in units of v_F
s_first = np.sqrt((1.0 + F0s_values) / 3.0)

# ── Plotting ──
fig, axes = plt.subplots(2, 2, figsize=(14, 11))
fig.patch.set_facecolor('#0a0a1a')

for ax in axes.flat:
    ax.set_facecolor('#0a0a1a')
    for spine in ax.spines.values():
        spine.set_edgecolor('#334155')
    ax.tick_params(colors='#94a3b8')

# Panel 1: Zero sound velocity vs F0s
ax = axes[0, 0]
ax.plot(F0s_values, s_values, color='#3b82f6', linewidth=2.5, label='Zero sound s')
ax.plot(F0s_values, s_first, color='#ef4444', linewidth=2.0, linestyle='--', label='First sound $s_1$')
ax.axhline(y=1.0, color='#64748b', linewidth=0.8, linestyle=':')
ax.set_xlabel('$F_0^s$', color='#94a3b8', fontsize=13)
ax.set_ylabel('$s = \\omega / v_F k$', color='#94a3b8', fontsize=13)
ax.set_title('Zero Sound vs First Sound Velocity', color='white', fontsize=14)
ax.legend(fontsize=11, facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white')
ax.set_xlim(0, 20)
ax.set_ylim(0, 5)

# Panel 2: Ratio s_zero/s_first
ax = axes[0, 1]
ratio = s_values / s_first
ax.plot(F0s_values, ratio, color='#22c55e', linewidth=2.5)
ax.axhline(y=np.sqrt(3), color='#f59e0b', linewidth=1.0, linestyle='--', alpha=0.7,
           label=f'$\\sqrt{{3}}$ limit')
ax.axhline(y=1.0, color='#64748b', linewidth=0.8, linestyle=':')
ax.set_xlabel('$F_0^s$', color='#94a3b8', fontsize=13)
ax.set_ylabel('$s_0 / s_1$', color='#94a3b8', fontsize=13)
ax.set_title('Ratio of Zero to First Sound Speed', color='white', fontsize=14)
ax.legend(fontsize=11, facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white')
ax.set_xlim(0, 20)

# Panel 3: Crossover diagram (omega*tau dependence)
ax = axes[1, 0]
omega_tau = np.logspace(-2, 3, 500)
F0s_fixed = 5.0
# Interpolation: effective speed between first and zero sound
try:
    s_zero = bisect_solve(F0s_fixed, hi=60.0)
except ValueError:
    s_zero = 2.0
s_one = np.sqrt((1.0 + F0s_fixed) / 3.0)
# Simple crossover model: s(omega*tau) = s1 + (s0 - s1) * (omega*tau)^2 / (1 + (omega*tau)^2)
s_cross = s_one + (s_zero - s_one) * omega_tau**2 / (1.0 + omega_tau**2)
ax.semilogx(omega_tau, s_cross, color='#a855f7', linewidth=2.5)
ax.axhline(y=s_zero, color='#3b82f6', linewidth=1.0, linestyle='--', alpha=0.7, label=f'Zero sound: s={s_zero:.3f}')
ax.axhline(y=s_one, color='#ef4444', linewidth=1.0, linestyle='--', alpha=0.7, label=f'First sound: s={s_one:.3f}')
ax.axvline(x=1.0, color='#f59e0b', linewidth=1.0, linestyle=':', alpha=0.5)
ax.text(1.2, s_one + 0.05, '$\\omega\\tau = 1$', color='#f59e0b', fontsize=11)
ax.set_xlabel('$\\omega\\tau$', color='#94a3b8', fontsize=13)
ax.set_ylabel('$s = \\omega / v_F k$', color='#94a3b8', fontsize=13)
ax.set_title(f'Crossover: First $\\leftrightarrow$ Zero Sound ($F_0^s={F0s_fixed}$)', color='white', fontsize=14)
ax.legend(fontsize=10, facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white')

# Panel 4: Attenuation of zero sound
ax = axes[1, 1]
F0s_range = np.array([1.0, 3.0, 6.0, 10.0, 15.0])
colors_att = ['#3b82f6', '#22c55e', '#f59e0b', '#ef4444', '#a855f7']
omega_tau_att = np.logspace(-1, 3, 300)
for F0s_val, col in zip(F0s_range, colors_att):
    try:
        s0 = bisect_solve(F0s_val, hi=100.0)
    except ValueError:
        continue
    # Attenuation: alpha ~ 1/(omega*tau)^2 in the collisionless regime
    # Landau damping vanishes for s > 1, attenuation from collisions ~ 1/(omega*tau)^2
    alpha_norm = 1.0 / omega_tau_att**2
    ax.loglog(omega_tau_att, alpha_norm, color=col, linewidth=2.0,
              label=f'$F_0^s={F0s_val:.0f}$, s={s0:.2f}')

ax.set_xlabel('$\\omega\\tau$', color='#94a3b8', fontsize=13)
ax.set_ylabel('Attenuation $\\alpha / \\alpha_0$', color='#94a3b8', fontsize=13)
ax.set_title('Zero Sound Attenuation', color='white', fontsize=14)
ax.legend(fontsize=9, facecolor='#1a1a2e', edgecolor='#334155', labelcolor='white')

plt.tight_layout(pad=2.5)
plt.savefig('zero_sound_analysis.png', dpi=130, bbox_inches='tight',
            facecolor=fig.get_facecolor())

print("Zero Sound Analysis Results")
print("=" * 50)
print(f"{'F0s':>6}  {'s_zero':>10}  {'s_first':>10}  {'ratio':>8}")
print("-" * 50)
for F0s_check in [0.5, 1.0, 2.0, 5.0, 10.0, 20.0]:
    try:
        sz = bisect_solve(F0s_check, hi=200.0)
    except ValueError:
        sz = float('nan')
    sf = np.sqrt((1.0 + F0s_check) / 3.0)
    print(f"{F0s_check:6.1f}  {sz:10.5f}  {sf:10.5f}  {sz/sf:8.4f}")
print()
print("Key physics:")
print("  - Zero sound: collisionless regime (omega*tau >> 1)")
print("  - First sound: hydrodynamic regime (omega*tau << 1)")
print("  - s_zero > s_first always (zero sound is faster)")
print("  - For large F0s: s_zero/s_first -> sqrt(3)")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran Implementation: Zero Sound Solver

High-precision computation of zero sound speed and attenuation for a range of $F_0^s$ values using bisection root finding.

Zero Sound Speed & Attenuation Calculator

Fortran

Solves for zero sound velocity and computes attenuation in the crossover region for different Landau parameter values

zero_sound_solver.f90131 lines

program zero_sound_solver
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  integer, parameter :: max_iter = 200
  integer, parameter :: n_F0s = 12
  real(dp), parameter :: pi = 3.141592653589793_dp
  real(dp), parameter :: tol = 1.0e-12_dp

real(dp) :: F0s_vals(n_F0s)
  real(dp) :: F0s, s, s_lo, s_hi, s_mid, f_lo, f_hi, f_mid
  real(dp) :: s_first, ratio, integral_val
  real(dp) :: omega_tau, alpha_att
  integer :: i, j, iter

! Landau parameter values to test
  data F0s_vals / 0.1_dp, 0.5_dp, 1.0_dp, 2.0_dp, 3.0_dp, 5.0_dp, &
                  7.0_dp, 10.0_dp, 15.0_dp, 20.0_dp, 50.0_dp, 100.0_dp /

write(*,'(A)') '============================================================='
  write(*,'(A)') '  Zero Sound Velocity Solver (Landau Fermi Liquid Theory)'
  write(*,'(A)') '============================================================='
  write(*,'(A)') ''
  write(*,'(A)') '  Solving: 1 = (F0s/2) * [s*ln((s+1)/(s-1)) - 2]'
  write(*,'(A)') '  for the zero sound speed ratio s = omega/(v_F * k)'
  write(*,'(A)') ''
  write(*,'(A6, A12, A12, A12, A10)') &
    'F0s', 's_zero', 's_first', 'ratio', 'iters'
  write(*,'(A)') '-------------------------------------------------------------'

do i = 1, n_F0s
    F0s = F0s_vals(i)

! Bisection method to find s > 1
    s_lo = 1.0001_dp
    s_hi = 1.0_dp + 10.0_dp * F0s

f_lo = evaluate_eqn(s_lo, F0s)
    f_hi = evaluate_eqn(s_hi, F0s)

! Extend upper bound if needed
    do while (f_hi < 0.0_dp .and. s_hi < 1.0e6_dp)
      s_hi = s_hi * 2.0_dp
      f_hi = evaluate_eqn(s_hi, F0s)
    end do

! Bisection
    iter = 0
    do j = 1, max_iter
      iter = iter + 1
      s_mid = 0.5_dp * (s_lo + s_hi)
      f_mid = evaluate_eqn(s_mid, F0s)
      if (abs(f_mid) < tol .or. (s_hi - s_lo) < tol) exit
      if (f_lo * f_mid < 0.0_dp) then
        s_hi = s_mid
        f_hi = f_mid
      else
        s_lo = s_mid
        f_lo = f_mid
      end if
    end do

s = s_mid
    s_first = sqrt((1.0_dp + F0s) / 3.0_dp)
    ratio = s / s_first

write(*,'(F6.1, F12.6, F12.6, F12.6, I6)') F0s, s, s_first, ratio, iter
  end do

write(*,'(A)') ''
  write(*,'(A)') '============================================================='
  write(*,'(A)') '  Attenuation in Crossover Region'
  write(*,'(A)') '============================================================='
  write(*,'(A)') ''
  write(*,'(A)') '  Collisional attenuation: alpha ~ k / (omega*tau)^2'
  write(*,'(A)') '  Regime:  omega*tau >> 1  =>  zero sound (low attenuation)'
  write(*,'(A)') '           omega*tau << 1  =>  first sound (hydrodynamic)'
  write(*,'(A)') ''

F0s = 5.0_dp
  s_lo = 1.0001_dp
  s_hi = 60.0_dp
  f_lo = evaluate_eqn(s_lo, F0s)
  do j = 1, max_iter
    s_mid = 0.5_dp * (s_lo + s_hi)
    f_mid = evaluate_eqn(s_mid, F0s)
    if (abs(f_mid) < tol) exit
    if (f_lo * f_mid < 0.0_dp) then
      s_hi = s_mid
    else
      s_lo = s_mid
      f_lo = f_mid
    end if
  end do
  s = s_mid
  s_first = sqrt((1.0_dp + F0s) / 3.0_dp)

write(*,'(A, F6.1)') '  F0s = ', F0s
  write(*,'(A, F10.6)') '  Zero sound speed s0 = ', s
  write(*,'(A, F10.6)') '  First sound speed s1 = ', s_first
  write(*,'(A)') ''
  write(*,'(A12, A16, A16)') 'omega*tau', 's_eff', 'alpha/alpha0'
  write(*,'(A)') '--------------------------------------------'

do i = -2, 4
    omega_tau = 10.0_dp ** (real(i, dp) * 0.5_dp)
    ! Crossover interpolation
    integral_val = s_first + (s - s_first) * omega_tau**2 / (1.0_dp + omega_tau**2)
    ! Attenuation
    if (omega_tau > 0.01_dp) then
      alpha_att = 1.0_dp / omega_tau**2
    else
      alpha_att = 1.0e4_dp
    end if
    write(*,'(ES12.3, F16.6, ES16.4)') omega_tau, integral_val, alpha_att
  end do

write(*,'(A)') ''
  write(*,'(A)') 'Physical interpretation:'
  write(*,'(A)') '  Zero sound = collisionless density oscillation of Fermi surface'
  write(*,'(A)') '  First sound = normal hydrodynamic sound in collision-dominated regime'
  write(*,'(A)') '  Crossover at omega*tau ~ 1'

contains

function evaluate_eqn(s_val, F0s_val) result(res)
    real(dp), intent(in) :: s_val, F0s_val
    real(dp) :: res
    res = (F0s_val / 2.0_dp) * (s_val * log((s_val + 1.0_dp) / (s_val - 1.0_dp)) - 2.0_dp) - 1.0_dp
  end function evaluate_eqn

end program zero_sound_solver

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Summary of Key Results

Quantity	Zero Sound	First Sound
Regime	$\omega\tau \gg 1$	$\omega\tau \ll 1$
Speed	$c_0 = s\,v_F$, $s > 1$	$c_1 = v_F\sqrt{(1+F_0^s)/3}$
Mechanism	Forward scattering (mean field)	Collisional equilibrium
Fermi surface	Anisotropic distortion	Rigid shift ($\ell=1$)
Attenuation	$\alpha \propto T^4$	$\alpha \propto T^2$ (viscous)
Existence	$F_0^s > 0$	$F_0^s > -1$

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