Questions: Fermi Liquid Theory

Question 1

Fermi liquid theory relies on 'adiabatic continuity' between the non-interacting and interacting ground states. What does this mean physically?

• A. The interactions can be turned on quickly without changing the physics
• B. If you imagine slowly turning on the electron-electron interaction starting from the free Fermi gas, the quantum numbers (k, spin) of each state are preserved — the states deform continuously without any level crossings or phase transitions, creating a one-to-one mapping between free-electron states and quasiparticle states
• C. The interacting system has exactly the same energy levels as the free Fermi gas
• D. Adiabatic continuity means the system is always in thermal equilibrium

Answer: (B)

Adiabatic continuity is the foundation of Fermi liquid theory. As interactions are turned on infinitely slowly, each free-electron state evolves into a quasiparticle state with the same quantum numbers (momentum, spin) but renormalized energy and effective mass. The crucial requirement is that no phase transition occurs during this process — if interactions drive an instability (superconductivity, magnetism, Mott transition), adiabatic continuity breaks down and Fermi liquid theory fails.

Question 2

The quasiparticle lifetime in a Fermi liquid scales as τ ∝ 1/(E - E_F)^2. What is the physical origin of this energy dependence?

• A. It follows from the uncertainty principle applied to the quasiparticle energy
• B. Phase space restriction: a quasiparticle at energy E above E_F can only scatter into empty states above E_F and create particle-hole pairs, and the available phase space for both the final state and the particle-hole pair scales as (E - E_F), giving a scattering rate proportional to (E - E_F)^2
• C. The Coulomb interaction itself becomes weaker at energies close to E_F
• D. Crystal symmetry protects quasiparticles near E_F from scattering

Answer: (B)

This is the key result. At the Fermi surface (E = E_F), there are zero available final states for scattering because all states below are filled (Pauli exclusion). An excitation at energy ε above E_F has a phase space for scattering that is proportional to ε for the excited electron's final state and ε for the particle-hole pair it can create, giving τ^{-1} ∝ ε^2. This means quasiparticles become infinitely long-lived as E → E_F, which is why the Fermi liquid description is internally consistent: the entities it calls quasiparticles are well-defined precisely where the theory is applied.

Question 3

Fermi liquid theory predicts that the electronic specific heat of a metal is linear in temperature, C = γT, just as for a free Fermi gas, but with a renormalized coefficient γ ∝ m*/m.

• T. True
• F. False

Answer: True

The linear-T specific heat is a hallmark of a Fermi liquid. The renormalized coefficient γ = (π²/3)k_B² g*(E_F) reflects the quasiparticle density of states g* ∝ m* at the Fermi level. In heavy-fermion systems, m*/m can be 100-1000, producing enormously enhanced specific heat coefficients — but the functional form C ∝ T is preserved. Deviations from linearity (such as C ∝ T log T) signal non-Fermi-liquid behavior.

Question 4

Under what conditions does Fermi liquid theory break down?

Model answer: Fermi liquid theory breaks down when adiabatic continuity fails — when interactions drive the system through a phase transition or qualitative change. Key examples include: (1) Superconductivity, where Cooper pairing opens a gap and the Fermi surface is destroyed. (2) Magnetic ordering, where spontaneous symmetry breaking creates a new ground state. (3) Mott transitions, where strong correlations localize electrons despite a partially filled band. (4) One-dimensional systems, where the Fermi surface reduces to two points and any interaction, no matter how weak, destroys the quasiparticle picture (Luttinger liquid behavior). (5) Quantum critical points, where divergent fluctuations produce non-Fermi-liquid scaling (e.g., resistivity ∝ T instead of T²).

The remarkable thing about Fermi liquid theory is how robust it is: it works for most metals despite interaction strengths comparable to the kinetic energy. Its failures define some of the most interesting problems in condensed matter: unconventional superconductors, heavy fermions, quantum criticality, and strongly correlated systems.