Landau Parameters

Derivation of the Pomeranchuk stability conditions

The Pomeranchuk conditions follow from requiring that the energy cost of any infinitesimal Fermi surface deformation be positive.

Step 1: Consider a general deformation of the Fermi surface parametrized by:

where $\nu_\sigma(\hat{\mathbf{k}})$ is the deformation amplitude. Expand in spherical harmonics: $\nu_\sigma(\hat{\mathbf{k}}) = \sum_{lm} \nu_{lm}^\sigma\,Y_{lm}(\hat{\mathbf{k}})$.

Step 2: The energy change to second order in $\delta n$ is:

Step 3: The kinetic energy contribution is $\sum_{\mathbf{k}\sigma} \epsilon_k^{(0)}\,\delta n_{\mathbf{k}\sigma} = N^*(0)\sum_{lm\sigma} |\nu_{lm}^\sigma|^2 / 2$. The interaction term decomposes into angular momentum channels using the addition theorem for spherical harmonics:

Step 4: Decomposing into spin-symmetric ($\nu^s = \nu_\uparrow + \nu_\downarrow$) and spin-antisymmetric ($\nu^a = \nu_\uparrow - \nu_\downarrow$) channels, the total energy change becomes:

Step 5: For the Fermi liquid to be stable, $\delta E > 0$ for all possible deformations. This requires each coefficient to be positive:

When any condition is violated, the corresponding Fermi surface deformation lowers the energy, signaling a spontaneous symmetry breaking (Pomeranchuk instability).

Derivation of the first sound velocity $c_1$ from the Landau parameters

Step 1: First sound is an adiabatic density wave in the hydrodynamic (collision-dominated) regime. The velocity is $c_1^2 = (\partial P / \partial \rho)_s = n\,\partial\mu/\partial n$ at constant entropy per particle.

Step 2: From the compressibility result, $\partial\mu/\partial n = (1 + F_0^s)/N^*(0)$. Using $N^*(0) = m^* k_F / (\pi^2\hbar^2)$ and $n = k_F^3 / (3\pi^2)$:

Step 3: Recognizing $v_F^* = \hbar k_F / m^*$ as the renormalized Fermi velocity:

Or equivalently, in terms of the bare Fermi velocity:

For liquid He-3 at zero pressure: $F_0^s \approx 10$, $F_1^s \approx 6$, giving $c_1 \approx v_F \cdot \sqrt{11/3}/3 \approx 183$ m/s, in excellent agreement with experiment.

Derivation of the Wilson ratio

Step 1: The Wilson ratio is defined as:

Step 2: Substitute the Fermi liquid results: $\chi = \chi_0 \cdot (m^*/m)/(1 + F_0^a)$ and $\gamma = \gamma_0 \cdot m^*/m$. The ratio is:

Step 3: For the free Fermi gas, $(\pi^2 k_B^2 / 3\mu_B^2)(\chi_0/\gamma_0) = 1$, so:

The Wilson ratio isolates the spin-antisymmetric interaction: $R_W > 1$ for negative $F_0^a$ (enhanced spin fluctuations), and $R_W \to \infty$ at the Stoner instability $F_0^a \to -1$. For He-3, $R_W \approx 1/(1 - 0.67) \approx 3$. In heavy-fermion metals, $R_W$ values of 2–5 are common, indicating strong exchange enhancement. A Wilson ratio of exactly 2 characterizes the single-impurity Kondo effect.

Derivation 2: Forward Scattering Amplitude and Landau Parameters

The Landau interaction function can be connected rigorously to microscopic scattering amplitudes. This derivation shows how the phenomenological parameters arise from the second functional derivative of the energy.

Step 1: The total energy of the Fermi liquid is a functional of the quasiparticle distribution $n_k$. The Landau interaction function is defined as the second functional derivative:

This quantity represents the change in energy of quasiparticle $k$ when quasiparticle $k'$ is added to the system. Physically, it is the forward scattering amplitude for two quasiparticles on the Fermi surface.

Step 2: For a rotationally invariant system, $f_{kk'}$ depends only on the angle $\theta$ between $\mathbf{k}$ and $\mathbf{k}'$ (both on the Fermi surface). We decompose the scattering amplitude using Legendre polynomials:

The orthogonality of Legendre polynomials allows extraction of each component:

Step 3: Define dimensionless Landau parameters by multiplying by the density of states at the Fermi level:

where $N(0) = m^* k_F / (\pi^2 \hbar^2)$ in three dimensions. The dimensionless parameters $F_\ell$ measure the interaction strength relative to the kinetic energy scale at the Fermi surface.

Step 4: Including spin, we separate symmetric and antisymmetric channels. The physical meaning of the leading parameters is:

The forward scattering interpretation makes the connection to microscopic theory transparent: $f_\ell$ can in principle be computed from many-body perturbation theory (Feynman diagrams), quantum Monte Carlo, or extracted from experiment. The Landau framework guarantees that only these few numbers are needed for all low-energy physics.

Derivation 3: Stability Conditions (Pomeranchuk Criteria)

We derive the Pomeranchuk stability conditions $F_\ell^{s,a} > -(2\ell+1)$ for each angular momentum channel, showing that violation leads to spontaneous Fermi surface deformation.

Step 1: Consider an infinitesimal deformation of the equilibrium Fermi surface. Parametrize the local Fermi momentum as:

The corresponding change in the occupation function is $\delta n_{k\sigma} = -\delta(\epsilon_k - \mu) \, \hbar v_F \, \delta k_F(\hat{\mathbf{k}}, \sigma)$. Expand the deformation in spherical harmonics:

Step 2: The energy change has two contributions: kinetic energy (cost of moving quasiparticles away from the equilibrium Fermi surface) and interaction energy (from the Landau function). Computing to second order:

where $u^s = u^\uparrow + u^\downarrow$ and $u^a = u^\uparrow - u^\downarrow$ are the density and spin deformation amplitudes.

Step 3: Combining and using $F_\ell = N(0) f_\ell$, the total energy change in each channel becomes:

Step 4: Stability requires $\delta E > 0$ for all possible deformations. Since each $(\ell, m)$ channel is independent, we need every coefficient to be positive:

Step 5: Physical consequences of violation. When a Pomeranchuk condition is violated, the corresponding Fermi surface deformation lowers the total energy, triggering a phase transition:

Higher-order Pomeranchuk instabilities ($\ell \geq 2$) break rotational symmetry of the Fermi surface without changing the total density or magnetization. The $\ell = 2$ case produces an ellipsoidal Fermi surface — a "nematic" Fermi liquid that has been observed in several strongly correlated materials.

Derivation 4: Transport Equation and Collective Mode Dispersion

We derive the linearized Boltzmann transport equation for Landau quasiparticles and show how the Landau interaction term gives rise to collective modes.

Step 1: In kinetic theory, the quasiparticle distribution function $n_{k\sigma}(\mathbf{r}, t)$ obeys the Boltzmann equation. For small deviations from equilibrium, write $n_{k\sigma} = n_k^0 + \delta n_{k\sigma}$. The linearized transport equation is:

where $\mathbf{v}_k = \nabla_k \epsilon_k / \hbar$ is the quasiparticle group velocity and $I_{\text{coll}}$ is the collision integral.

Step 2: The crucial feature of Fermi liquid theory is the molecular field (or mean-field) term. The local quasiparticle energy is modified by the surrounding perturbation:

The molecular field $\delta\epsilon_{k\sigma} = \sum_{k'\sigma'} f_{k\sigma, k'\sigma'} \, \delta n_{k'\sigma'}$ acts as an additional restoring force on the quasiparticles. This self-consistent coupling between quasiparticle motion and the interaction is the origin of collective modes in Fermi liquids.

Step 3: In the collisionless regime ($\omega \tau \gg 1$), drop the collision integral and seek plane-wave solutions $\delta n_{k\sigma} \propto e^{i(\mathbf{q} \cdot \mathbf{r} - \omega t)}$. Restrict to the Fermi surface by writing:

Substituting into the linearized equation and using $\mathbf{v}_k = v_F \hat{\mathbf{k}}$ on the Fermi surface:

Step 4: Define $s = \omega / (q v_F)$ and $\cos\alpha = \hat{\mathbf{q}} \cdot \hat{\mathbf{k}}$. For the spin-symmetric, $\ell = 0$ channel with isotropic perturbation $\nu(\hat{\mathbf{k}}) = \nu(\alpha)$, the equation becomes:

Solving for $\nu(\alpha) = A \cos\alpha / (s - \cos\alpha)$ and requiring self-consistency gives the eigenvalue equation for zero sound:

which can also be written as $s \coth^{-1}(s) = 1 + 1/F_0^s$. Solutions with $s > 1$ (phase velocity exceeding the Fermi velocity) represent propagating zero sound modes. For large $F_0^s$, $s \to \sqrt{(1 + F_0^s)/3}$, recovering the first sound result.

Step 5: The spin-antisymmetric channel ($F_0^a$ replacing $F_0^s$) gives spin zero sound: a propagating spin-density wave. Including higher harmonics $F_\ell$ couples different angular momentum channels and produces a richer spectrum of collective modes. The general eigenvalue problem is an infinite matrix equation in the $\ell$-basis, which in practice is truncated at low $\ell$.