Low-energy Fermi-liquid quasiparticle lifetime (${\tau }^{-1}\propto (ϵ-\mu {)}^2+(\pi k_{B}T{)}^2$)

In a regular three-dimensional Fermi liquid, the Pauli exclusion principle leaves only a small set of final states available to a low-energy Landau quasiparticle. Writing $\xi =ϵ-\mu$, the leading decay rate has the scaleThe coefficient depends on the interaction and band structure.

At zero temperature, an excitation of energy $\xi >0$ creates an additional particle–hole pair. Conservation of energy leaves two small independent energy intervals, producing the quadratic phase-space factor. Its spectral width $\Gamma \sim \hslash /\tau$ then satisfies $\Gamma /|\xi |\to 0$ as $\xi \to 0$. This zero-temperature limit is the sense in which excitations become sharp at the Fermi surface. ^[ref]

At fixed nonzero temperature, the rate instead approaches a finite value proportional to $T^2$ as $\xi \to 0$. One must not infer a vanishing ratio $\Gamma /|\xi |$ in that different limit. A joint low-temperature, low-energy limit compares the width with the relevant excitation scale.

A particle loss rate can include the factor $1-f(\xi )$, whereas a spectral decay width also contains the complementary inverse collision process. Their sum has the regular quadratic low-energy scale ${\xi }^2+(\pi k_{B}T{)}^2$. Particle and hole decay must not be confused with an occupied-state loss probability.