Quasiparticle Poles, Residues, and Lifetimes

A spectral peak defines a quasiparticle only when it is controlled by an isolated retarded pole with finite residue and a width small compared with the energy scale on which its dispersion and environment change. Residue, effective mass, spectral damping, occupation decay, and transport relaxation are different quantities. A broad maximum or threshold cusp is not promoted to a particle by fitting it with a Lorentzian. The original long-lived-excitation and interaction-function construction is given in Landau 1957, pp. 920–923, Open PDF.

Required background. Lehmann Representations and Spectral Functions in Matter fixes exact spectral weight, and Dyson Equations and Self-Energies defines the pole and residue.

Helpful background. Spectral Functions and Transport Peaks distinguishes low-frequency transport structure, while Transport Extraction, Inverse Problems, and Error Budgets treats resolution and reconstruction errors.

The pole criterion

For one diagonal normal-state channel,

G^R(\mathbf k,z)= \frac1{z-\xi_{\mathbf k}-\Sigma^R(\mathbf k,z)}.

A quasiparticle pole $z_{\mathbf k}=E_{\mathbf k}-i\gamma_{\mathbf k}$ solves the analytically continued denominator on the sheet connected to the retarded function. In the narrow-width approximation,

Z_{\mathbf k}= \left[1-\partial_\omega\operatorname{Re}\Sigma^R \right]^{-1}_{E_{\mathbf k}}, \qquad \gamma_{\mathbf k}=-Z_{\mathbf k}\operatorname{Im}\Sigma^R(\mathbf k,E_{\mathbf k}).

Near the pole,

A(\mathbf k,\omega) \simeq \frac{2Z_{\mathbf k}\gamma_{\mathbf k}} {(\omega-E_{\mathbf k})^2+\gamma_{\mathbf k}^2} +A_{\rm inc}.

The pole contribution integrates to $Z_{\mathbf k}$ ; the incoherent part carries the remaining weight. A valid quasiparticle description requires at least

\gamma_{\mathbf k}\ll \min(E_{\rm variation},E_{\rm separation}),

where the comparison scales include distance to a continuum edge, neighboring pole separation, and variation scale of $\Sigma$ . Near a Fermi surface, the useful criterion is typically $\gamma/|E|\to0$ as $E\to0$ .

Which lifetime?

With denominator $\omega-E+i\gamma$ , the retarded amplitude decays as $e^{-\gamma t}$ and the spectral full width at half maximum is $2\gamma$ . A nonequilibrium occupation often decays at rate $2\gamma$ in a simple weak-coupling kinetic limit. We therefore report the pole damping $\gamma$ and state separately which time-domain quantity is called a lifetime.

Transport relaxation weights collisions by their efficiency in degrading a current. For elastic scattering through angle $\theta$ ,

\frac1{\tau_{\rm sp}}\propto \int\mathrm d\Omega\,W(\theta), \qquad \frac1{\tau_{\rm tr}}\propto \int\mathrm d\Omega\,W(\theta)(1-\cos\theta).

Forward scattering can strongly broaden a single-particle state while barely relaxing momentum. Vertex corrections encode this distinction; it cannot be read from $\operatorname{Im}\Sigma$ alone.

Fermi-liquid checkpoint

For a conventional three-dimensional Fermi liquid close to its Fermi surface,

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

up to logarithmic or dimensional modifications. At $T=0$ , $\gamma/|\omega|\to0$ , so excitations become sharper as the Fermi surface is approached. At fixed nonzero $T$ , the lowest-frequency width remains $O(T^2)$ and the order of limits matters.

A finite $Z$ is useful but not sufficient: a pole can be too broad. Conversely, systems with vanishing $Z$ can retain other sharp collective excitations, but those are not Landau quasiparticles in the original single-particle channel.

Residue, velocity, and effective mass

The pole velocity is

\mathbf v^*= Z\left(\boldsymbol\nabla_{\mathbf k}\xi +\boldsymbol\nabla_{\mathbf k}\operatorname{Re}\Sigma\right).

Only when the self-energy is momentum independent does $m^*/m=Z^{-1}$ for a parabolic isotropic band. In a Galilean-invariant Fermi liquid, backflow and Landau parameters relate thermodynamic effective mass to current response; the relation is not the single-particle residue identity.

The zeroth spectral sum rule remains

Z_{\mathbf k} +\int\frac{\mathrm d\omega}{2\pi}A_{\rm inc}(\mathbf k,\omega)=1.

An approximation that places a pole of weight $Z>1$ in a positive diagonal spectrum must compensate with negative weight and therefore violates positivity.

Distinguishing poles from look-alikes

A threshold singularity can make a sharp asymmetric edge without an isolated pole.
Two unresolved poles can look like one broad peak.
Instrumental convolution can create an apparent width.
A maximum of a matrix-element-weighted intensity can shift away from the maximum of $A$ .
In finite volume, artificial broadening of delta functions is not a lifetime.

The minimum check varies resolution or broadening, locates the complex pole or demonstrates a controlled narrow-width expansion, and verifies spectral normalization.

Common pitfalls

Equating peak width with transport lifetime. Transport contains angular and vertex weights. Report the response channel.

Equating $Z$ with the coherent fraction of every observable. $Z$ is the pole weight of a named field; other probes include vertices and matrix elements.

Accepting a real-axis maximum as a pole. Inspect analytic continuation and continuum thresholds.

Exercises

Test the Fermi-liquid hierarchy

If $\gamma(E)=\alpha E^2/E_F$ at $T=0$ , find the condition for a sharp excitation.

Solution

The ratio is $\gamma/|E|=\alpha|E|/E_F$ . It tends to zero as $E\to0$ and is small when $|E|\ll E_F/\alpha$ , provided no nearer continuum or competing pole supplies a smaller comparison scale.

Compare two scattering lifetimes

For scattering confined to a narrow cone of angle $\theta_0\ll1$ , estimate $\tau_{\rm tr}/\tau_{\rm sp}$ .