Landau Fermi Liquid Theory

Stability condition:

The effective mass must be positive for the Fermi liquid to be stable. This requires$F_1^s > -3$. When $F_1^s \to -3$, the effective mass diverges and the quasiparticle description breaks down, signalling a quantum phase transition (Pomeranchuk instability in the $l=1$ channel).

Complete Structure of the Energy Functional

The Landau energy functional is the central object of Fermi liquid theory. We present here a detailed derivation that carefully distinguishes between bare quasiparticle energies, the interaction function, and the resulting renormalized quantities. The total energy of the interacting Fermi system, expanded to second order in deviations $\delta n_{\mathbf{k}\sigma}$ from the ground-state distribution, reads:

Here $E_0$ is the exact ground-state energy of the fully interacting system,$\epsilon_{\mathbf{k}}^0 = \delta E/\delta n_{\mathbf{k}\sigma}\big|_{n^{(0)}}$ is the bare quasiparticle energy (the first variational derivative evaluated at the ground state), and the Landau interaction function is the second variational derivative:

The factor of $V$ (system volume) ensures that $f$ has dimensions of energy times volume and remains finite in the thermodynamic limit. The truncation at second order is exact for low-energy excitations because the density of excited quasiparticles scales as $T/\epsilon_F$ at low temperatures.

Definition of $f_{\mathbf{k}\mathbf{k}'}$ and Spin-Channel Decomposition

For a rotationally invariant system with spin-$\frac{1}{2}$ quasiparticles, the interaction function on the Fermi surface depends only on the angle $\theta$ between $\hat{\mathbf{k}}$ and$\hat{\mathbf{k}}'$. It can be decomposed into spin-symmetric and spin-antisymmetric channels:

Equivalently, the parallel-spin and antiparallel-spin interaction functions are:

Each channel is expanded in Legendre polynomials:

The dimensionless Landau parameters are obtained by multiplying by the density of states at the Fermi surface:

where $N^*(0) = m^* k_F / (\pi^2 \hbar^2)$ is the renormalized density of states per unit volume (for both spin species). The $F_\ell^{s,a}$ are the physical quantities that enter all thermodynamic and transport relations. In practice, only $F_0^s$, $F_0^a$, and$F_1^s$ are needed for most observables, as higher harmonics are small in typical Fermi liquids.

Rigorous Derivation of $m^*/m = 1 + F_1^s/3$

The relation between the effective mass and the $\ell = 1$ Landau parameter is not a model assumption but an exact consequence of Galilean invariance. We present here the complete derivation, emphasizing each logical step.

Step 1 — Exact current operator. In a Galilean-invariant system of $N$ particles with bare mass $m$, the exact particle current density operator is:

This follows from $\hat{\mathbf{P}} = m\,d\hat{\mathbf{R}}_{\text{CM}}/dt$ and is an exact identity independent of interactions. In second quantization for a translationally invariant system:

Step 2 — Quasiparticle current. Within Landau theory, the same current must be computable from quasiparticle degrees of freedom. The quasiparticle group velocity is:

The first term is the free quasiparticle velocity; the second is the backflow correction arising from the interaction-induced redistribution of quasiparticles. The quasiparticle current is:

Step 3 — Galilean boost. Consider boosting the entire system by an infinitesimal velocity $\delta\mathbf{v}$, corresponding to shifting the Fermi sphere by$\delta\mathbf{q} = m\,\delta\mathbf{v}/\hbar$. The deviation from equilibrium is:

This is a purely $\ell = 1$ (dipolar) deformation. The bare-particle current for this state is exactly $\mathbf{j} = n\,\delta\mathbf{v}$, where $n = k_F^3/(3\pi^2)$ is the particle density.

Step 4 — Interaction correction selects $F_1^s$. Substituting $\delta n$ into the quasiparticle velocity and performing the angular integration over $\hat{\mathbf{k}}'$, the orthogonality of Legendre polynomials ensures that only the $\ell = 1$ harmonic of$f^s(\theta)$ contributes (the antisymmetric channel $f^a$ drops out because the boost is spin-independent):

Step 5 — Equating the two expressions for the current. Setting$\mathbf{j}_{\text{bare}} = \mathbf{j}_{\text{qp}}$ and using $N^*(0) = m^* k_F/(\pi^2\hbar^2)$:

Canceling $\hbar k_F$ and inverting:

This result is a rigorous theorem for any Galilean-invariant Fermi liquid. It reflects the fact that when a quasiparticle moves through the medium, it drags along a backflow of surrounding quasiparticles, effectively increasing its inertia. The $F_1^s$ parameter quantifies this backflow. For liquid He-3,$F_1^s \approx 6.05$ gives $m^*/m \approx 3.02$, in excellent agreement with specific heat measurements. Crucially, this relation does not hold in crystalline solids where Galilean invariance is broken by the lattice — there the band effective mass and Landau parameters are independent quantities.

Compressibility: $\kappa/\kappa_0 = (m^*/m)/(1 + F_0^s)$

The isothermal compressibility is defined as $\kappa = -\frac{1}{V}\frac{\partial V}{\partial P}\big|_T = \frac{1}{n^2}\frac{\partial n}{\partial \mu}\big|_T$. To evaluate $\partial n/\partial \mu$ in the Landau framework, we consider a uniform change in the chemical potential $\mu \to \mu + \delta\mu$ and compute the resulting change in density.

Step 1 — Self-consistency condition. When the chemical potential shifts by $\delta\mu$, the quasiparticle energy also shifts through the interaction term. A quasiparticle at the Fermi surface sees a renormalized energy:

For a uniform compression (isotropic, spin-independent perturbation), $\delta n_{\mathbf{k}\sigma} = \delta(k - k_F)\,\delta k_F$ is independent of angle. The angular integration selects only the $\ell = 0$ component:

Step 2 — Equilibrium condition. At equilibrium, the Fermi surface adjusts so that$\tilde{\epsilon}_{k_F} = \mu + \delta\mu$. The change in density is$\delta n = N^*(0)\,\delta\tilde{\epsilon}$, but the self-consistency requires:

Therefore $\partial n/\partial\mu = N^*(0)/(1 + F_0^s)$. Using $N^*(0)/N_0(0) = m^*/m$ and$\kappa_0 = N_0(0)/n^2$:

A repulsive interaction in the $\ell = 0$ symmetric channel ($F_0^s > 0$) stiffens the system against compression. Conversely, $F_0^s \to -1$ drives the compressibility to infinity — the system becomes mechanically unstable and undergoes spinodal decomposition (phase separation).

Liquid Helium-3: The Canonical Fermi Liquid

Liquid He-3 remains the purest realization of Landau's theory. At temperatures below approximately 100 mK, the quasiparticle description becomes quantitatively accurate, with a mean free path scaling as $\ell \propto T^{-2}$ due to the phase-space restriction on quasiparticle scattering. The experimentally determined Landau parameters (Greywall 1983) are:

These parameters yield: effective mass $m^*/m = 1 + F_1^s/3 \approx 3.02$, compressibility ratio$\kappa/\kappa_0 = 3.02/11.07 \approx 0.27$ (the liquid is much stiffer than the free gas due to strong repulsion), and spin susceptibility ratio $\chi/\chi_0 = 3.02/0.33 \approx 9.15$(dramatically enhanced by the nearly ferromagnetic exchange interaction).

The prediction and observation of zero sound — a collisionless collective oscillation of the Fermi surface shape propagating at velocity $c_0 \approx v_F\sqrt{F_0^s/3}$ in the collisionless regime $\omega\tau \gg 1$ — provided the most dramatic confirmation of the theory. The crossover from first sound (hydrodynamic, $\omega\tau \ll 1$) to zero sound occurs near 50 mK and was first observed by Abel, Anderson, and Wheatley (1966).

Heavy Fermion Systems: CeAl$_3$, UPt$_3$, and Beyond

Heavy fermion compounds represent Fermi liquid theory taken to its extreme. In these materials, localized $f$-electrons hybridize with conduction electrons through the Kondo effect, forming composite quasiparticles with effective masses $m^*/m \sim 100$–$1000$.

CeAl$_3$ was one of the first identified heavy fermion metals (Andres, Graebner, Ott 1975), with a Sommerfeld coefficient$\gamma \approx 1600\;\text{mJ}\,\text{mol}^{-1}\text{K}^{-2}$ — roughly 1000 times larger than in copper. The $T^2$ resistivity below about 0.3 K confirms Fermi liquid behavior, with the coefficient $A$ in $\rho = \rho_0 + AT^2$ satisfying the Kadowaki-Woods ratio $A/\gamma^2 \approx 1.0 \times 10^{-5}\;\mu\Omega\,\text{cm}\,(\text{mol}\,\text{K}/\text{mJ})^2$.

UPt$_3$ exhibits heavy Fermi liquid behavior with$\gamma \approx 450\;\text{mJ}\,\text{mol}^{-1}\text{K}^{-2}$ and undergoes unconventional superconductivity at $T_c \approx 0.53$ K with two distinct superconducting phases in the$H$–$T$ phase diagram — strong evidence for a multi-component order parameter arising from the heavy quasiparticles. De Haas–van Alphen measurements directly map the heavy Fermi surface, confirming $m^*/m \approx 180$ on certain orbits.

The universal Kadowaki-Woods scaling $A/\gamma^2 \approx \text{const}$ across different heavy fermion compounds demonstrates that the scattering rate and specific heat both scale with the same quasiparticle density of states, as predicted by Fermi liquid theory. Deviations from this scaling (as in CeCu$_6$ near its quantum critical point) signal the breakdown of the Fermi liquid paradigm.

Normal State of Superconductors

The BCS theory of superconductivity is built upon the assumption that the normal state above$T_c$ is a well-defined Fermi liquid. The electron-phonon interaction renormalizes the quasiparticle effective mass through $m^*/m_b = 1 + \lambda$, where $\lambda$ is the electron-phonon coupling constant and $m_b$ is the band mass. Typical values are$\lambda \approx 0.43$ for aluminum, $\lambda \approx 1.55$ for lead, and$\lambda \approx 1.6$ for mercury.

The quasiparticle residue $Z = 1/(1 + \lambda)$ is directly measurable through tunneling spectroscopy: the tunneling density of states in the superconducting state reveals the energy-dependent self-energy $\Sigma(\omega)$, from which $\lambda$ and the spectral function$\alpha^2F(\omega)$ are extracted by McMillan-Rowell inversion. These experiments provide the most direct microscopic validation of the Fermi liquid picture in metals.

In the cuprate high-temperature superconductors, the situation is more complex. The optimally doped materials show a $T$-linear resistivity (the "strange metal" phase) that violates the Fermi liquid prediction of $T^2$ scattering, while the overdoped side recovers conventional Fermi liquid behavior. Understanding this crossover remains one of the central open problems in condensed matter physics.

Nuclear Matter

The interior of atomic nuclei and the outer core of neutron stars constitute dense Fermi liquids of nucleons. The nuclear Landau parameters are determined from nuclear structure data and nucleon-nucleon scattering phase shifts. A distinctive feature of nuclear matter is that the effective mass isreduced below the bare nucleon mass: $m^*/m \approx 0.7$–$0.8$, implying a negative $F_1^s$.

The nuclear symmetry energy — the energy cost per nucleon of converting protons to neutrons — is directly related to the isospin-antisymmetric Landau parameter $F_0^a$ through:

This quantity is a critical input for neutron star equation-of-state calculations and determines the proton fraction, cooling rate, and maximum mass of neutron stars. Giant dipole and quadrupole resonances in nuclei provide experimental access to the nuclear Landau parameters, connecting nuclear structure physics to the astrophysics of compact objects.

Landau's Program: 1956–1958

Landau's theory of the Fermi liquid was developed in three landmark papers that progressively expanded the scope of the framework:

1956: The first paper (Zh. Eksp. Teor. Fiz. 30, 1058; JETP 3, 920) introduced the quasiparticle concept and the energy functional formalism. Landau derived the thermodynamic identities — specific heat, compressibility, and susceptibility — in terms of the interaction function. The physical motivation was liquid He-3, which had recently been liquefied and whose properties were anomalous compared to classical liquid theory.

1957: The second paper (Zh. Eksp. Teor. Fiz. 32, 59; JETP 5, 101) extended the theory to non-equilibrium phenomena and predicted the existence of zero sound — a collisionless collective mode of the Fermi surface. This was a genuinely new prediction with no classical analogue: in contrast to ordinary sound, which requires frequent collisions to maintain local equilibrium, zero sound propagates through the coherent forward scattering of quasiparticles and exists only in the collisionless regime $\omega\tau \gg 1$.

1958: The third paper (Zh. Eksp. Teor. Fiz. 35, 97; JETP 8, 70) treated the charged Fermi liquid — the electron gas in metals. Here Landau showed how the long-range Coulomb interaction modifies the formalism: the $\ell = 0$ Landau parameter acquires a Coulomb contribution that is screened at finite wavevector (Thomas-Fermi screening), while higher harmonics are essentially short-range. This paper established the connection between Fermi liquid theory and the established theory of metals by Sommerfeld and Bloch.

Adiabatic Continuity: The Foundation

The concept of adiabatic continuity is the logical cornerstone of Fermi liquid theory. The argument proceeds as follows: start with a non-interacting Fermi gas with a well-defined ground state (filled Fermi sea) and sharp quasiparticle excitations. Now imagine turning on the interactions infinitely slowly (adiabatically). If no phase transition (level crossing) occurs during this process, then:

(i) The ground state evolves continuously into the interacting ground state. (ii) Each single-particle excitation $|\mathbf{k}\sigma\rangle$ evolves into a quasiparticle excitation with the samequantum numbers ($\mathbf{k}$, $\sigma$). (iii) The one-to-one correspondence between free-particle states and quasiparticle states preserves the Fermi surface topology and Luttinger's theorem ($k_F$ is unchanged by interactions). (iv) The quasiparticle residue$Z_{k_F}$ — the overlap between the bare electron and the dressed quasiparticle — satisfies $0 < Z_{k_F} \leq 1$.

This mapping is guaranteed to work in three dimensions because the quasiparticle decay rate vanishes as$\text{Im}\,\Sigma \propto (\epsilon - \epsilon_F)^2$ near the Fermi surface — the quasiparticles are long-lived enough to be well-defined. In one dimension, the phase space argument fails (the decay rate only vanishes linearly) and adiabatic continuity breaks down, leading instead to a Luttinger liquid. In two dimensions, the situation is marginal: logarithmic corrections appear, but the Fermi liquid generally survives (except at van Hove singularities or in the presence of nesting).

Migdal's Theorem and Its Consequences

Arkady Migdal (1957) proved that the momentum distribution function$n(\mathbf{k})$ of an interacting Fermi system retains a discontinuity at the Fermi surface:

where $Z_{k_F}$ is the quasiparticle residue. This result has two profound implications. First, it provides a sharp, experimentally testable criterion for the existence of a Fermi liquid: if the momentum distribution (measurable via Compton scattering or angle-resolved photoemission) shows a discontinuity, quasiparticles exist. Second, it connects the phenomenological Landau theory to the microscopic Green's function formalism through:

Migdal's theorem also establishes that vertex corrections to the electron-phonon coupling are suppressed by $\mathcal{O}(\omega_D/\epsilon_F) \sim \mathcal{O}(\sqrt{m/M})$, where$\omega_D$ is the Debye frequency and $M$ the ion mass. This result is essential for the validity of Eliashberg theory of strong-coupling superconductivity and explains why the quasiparticle picture survives even when the electron-phonon coupling constant $\lambda$ is of order unity.