Microscopic Quasiparticles and Fermi-Liquid Self-Energy

A Landau quasiparticle is realized microscopically when the retarded Green function has a pole near the interacting Fermi surface whose width is small compared with its excitation energy. Frequency and momentum derivatives of the real self-energy determine the residue and velocity separately; phase space makes the width quadratic at low energy in an ordinary Fermi liquid.

Required background. Use Landau theory, Dyson equations, and Ward-consistent vertices.

Quasiparticle pole expansion

With

$$G^R(\mathbf k,\omega)= \frac{1}{\omega-\xi_{\mathbf k}-\Sigma^R(\mathbf k,\omega)},$$

define the interacting surface by $\xi_{\mathbf k_F}+\operatorname{Re}\Sigma^R(\mathbf k_F,0)=0$. Expanding normal to it gives

$$G^R\simeq\frac{Z}{\omega-v_F^*k_\perp+i\Gamma},$$

where

$$Z^{-1}=1-\left.\partial_\omega\operatorname{Re}\Sigma^R\right|_F, \qquad v_F^*=Z\left(v_F+\left.\partial_{k_\perp}\operatorname{Re}\Sigma^R\right|_F\right), \qquad \Gamma=-Z\operatorname{Im}\Sigma^R\big|_{\omega=v_F^*k_\perp}.$$

The signs follow from the retarded convention, for which $\operatorname{Im}\Sigma^R\le0$ on a stable particle branch. The pole is sharp only if $\Gamma/|\omega|\to0$.

Lifetime and vertices

Pauli blocking restricts two-body decay to a shell of thickness set by $\max(|\omega|,T)$. In a regular three-dimensional Fermi liquid,

$$-\operatorname{Im}\Sigma^R(\mathbf k_F,\omega,T) \propto \omega^2+\pi^2T^2.$$

Two dimensions can carry logarithmic corrections. This single-particle lifetime is not the transport lifetime: forward collisions broaden a spectral line but relax current inefficiently Shankar 1994, § VI, pp. 157–177.

The zero-transfer and zero-frequency limits of the four-point vertex do not commute. Ward identities connect the appropriate limit to $\partial_\omega\Sigma$ and current backflow; dropping the vertex while retaining a momentum-dependent self-energy can violate charge conservation. The standard microscopic construction is presented in Baym and Pethick 1991, ch. 3.

Independent checks

The spectral sum rule integrates the coherent pole plus incoherent background to one, so $0<Z\le1$ in the conventional single-band setting does not mean missing probability. A Galilean-invariant calculation must reproduce $m^*/m=1+F_1^s/3$ after including the vertex. A broad maximum without $\Gamma/|\omega|\to0$ is not a quasiparticle pole.

Exercises

Show why a momentum-independent self-energy can still renormalize the mass.

Solution

If $\partial_k\Sigma=0$, then $v_F^*=Zv_F$. For a parabolic isotropic band, $v_F^*=k_F/m^*$ and $v_F=k_F/m$, so $m^*/m=1/Z$. In a general system momentum dependence and vertex backflow alter this simple relation.

References

• Gordon Baym and Christopher Pethick, Landau Fermi-Liquid Theory: Concepts and Applications, Wiley-VCH (1991), ch. 3, doi:10.1002/9783527617159.
• Ramamurti Shankar, “Renormalization-Group Approach to Interacting Fermions,” Reviews of Modern Physics 66 (1994) 129–192, § VI, doi:10.1103/RevModPhys.66.129.