Microscopic Quasiparticles and Fermi-Liquid Self-Energy

A microscopic Fermi liquid is more than a Green function with a visible peak. Near each regular point of the interacting Fermi surface, the retarded propagator must be controlled asymptotically by a quasiparticle pole on the appropriate continuation, producing a narrow resonance with finite residue, a smooth dispersion, and damping parametrically smaller than the energy or temperature being resolved. The same self-energy must be paired with the correct vertex limits: those limits turn pole data into Landau interactions, conserved response, and current backflow. This page derives that dictionary for a translationally invariant, normal, single-band Fermi liquid in two or three spatial dimensions, using the site’s inherited natural-unit convention. It does not prove that an arbitrary strongly coupled system is a Fermi liquid.

Required background. Landau theory supplies the quasiparticle energy functional. Dyson equations fix the self-energy convention, and Ward-consistent vertices explain why response cannot be inferred from a dressed propagator alone.

Linearizing the pole at the interacting Fermi surface

Interacting surface and local expansion

Write $\xi_{\mathbf k}=\varepsilon_{\mathbf k}-\mu$, with $\omega=0$ at the interacting chemical potential, and use the retarded convention

$$G^R(\mathbf k,\omega) = \frac{1}{\omega-\xi_{\mathbf k}-\Sigma^R(\mathbf k,\omega)}, \qquad \Sigma^R=\Sigma'+i\Sigma''.$$

At zero temperature, a regular point $\mathbf k_F$ of the interacting Fermi surface obeys

$$\xi_{\mathbf k_F}+\Sigma'(\mathbf k_F,0)=0, \qquad \Sigma''(\mathbf k_F,0)=0,$$

and has a finite, nonzero residue. The first equation alone is not enough: it can also locate a broadened crossing at nonzero temperature, and matrix-valued propagators or Green-function zeros require a more careful eigenvalue analysis. Nor does this local definition establish the volume enclosed by the surface; that is a separate theorem with additional hypotheses.

Let $\widehat{\mathbf n}_F$ point outward from the filled region and define the signed normal displacement

$$k_\perp=\widehat{\mathbf n}_F\mathbin{\cdot}(\mathbf k-\mathbf k_F), \qquad v_{F,\perp}=\widehat{\mathbf n}_F\mathbin{\cdot} \boldsymbol\nabla_{\mathbf k}\varepsilon_{\mathbf k}\big|_{\mathbf k_F}.$$

Every derivative carrying a subscript $F$ below is evaluated at $(\mathbf k_F,\omega=0)$. Taylor expansion of the inverse propagator gives

$$\begin{aligned} (G^R)^{-1} &\simeq \left(1-\partial_\omega\Sigma'_F\right)\omega -\left(v_{F,\perp}+\partial_{k_\perp}\Sigma'_F\right)k_\perp -i\Sigma''(\mathbf k,\omega) \\ &=Z^{-1}\left(\omega-v_F^*k_\perp\right) -i\Sigma''(\mathbf k,\omega), \end{aligned}$$

where

$$Z=\frac{1}{1-\partial_\omega\Sigma'_F}, \qquad v_F^*=Z\left(v_{F,\perp}+\partial_{k_\perp}\Sigma'_F\right).$$

Thus, to leading order in a narrow-width expansion,

$$G^R(\mathbf k,\omega) \simeq \frac{Z}{\omega-E_{\mathbf k}+i\gamma_{\mathbf k}} +G^R_{\mathrm{inc}}, \qquad E_{\mathbf k}=v_F^*k_\perp, \qquad \gamma_{\mathbf k}=-Z\Sigma''(\mathbf k,E_{\mathbf k})\ge 0.$$

Exact pole and pole mass

When the appropriate continuation contains a pole, its exact position and residue solve

$$z_p-\xi_{\mathbf k}-\Sigma^R(\mathbf k,z_p)=0, \qquad Z_p=\left[1-\partial_z\Sigma^R(\mathbf k,z_p)\right]^{-1}$$

on the appropriate analytic continuation. Away from the surface, $Z_p$ can be complex. The real-axis formulas above are the controlled leading approximation when $\Sigma^R$ varies little across the width; terms such as $\partial_\omega\Sigma''$ then enter at higher order.

For an anisotropic surface the velocity statement is vectorial,

$$\mathbf v_F^* =Z\left[ \boldsymbol\nabla_{\mathbf k}\varepsilon_{\mathbf k} +\boldsymbol\nabla_{\mathbf k}\Sigma'(\mathbf k,0) \right]_F.$$

For an isotropic parabolic band, $v_F=k_F/m$ and $v_F^*=k_F/m^*$, so the pole mass is

$$\frac{m^*}{m} = \frac{1-\partial_\omega\Sigma'_F} {1+(m/k_F)\partial_k\Sigma'_F}.$$

Consequently, $m^*/m=1/Z$ requires $\partial_{k_\perp}\Sigma'_F=0$; tangential momentum dependence need not vanish. Vertex backflow does not alter this pole-derivative formula; it supplies a separate relation between the same $m^*$ and the Landau interaction when the system has the required symmetries.

Residue, width, and lifetime are different data

With the spectral convention $A=-2\operatorname{Im}G^R$, the coherent part is

$$A_{\mathrm{coh}}(\mathbf k,\omega) = \frac{2Z\gamma_{\mathbf k}} {(\omega-E_{\mathbf k})^2+\gamma_{\mathbf k}^2}.$$

The leading Lorentzian integrates to $Z$ rather than one. For an exact canonical scalar Green function, the sum rule $\int d\omega\,A/(2\pi)=1$ implies that the remaining positive spectral weight tends to $1-Z$ in the zero-temperature, $\mathbf k\to\mathbf k_F$ pole limit. Away from that limit, a finite-width resonance has no unique exact positive “coherent area,” and the complex $Z_p$ need not equal the real narrow-width $Z$. The residue measures overlap between the microscopic field and the long-lived excitation; it is not the fraction carried by every observable.

The factor-of-two convention is worth making explicit because both $\gamma$ and $2\gamma$ are often called “the linewidth.”

Quantity	How it is read from the pole	Physical meaning	Do not identify it with
Residue $Z$	Coefficient of the narrow real-axis pole	Coherent spectral weight in the Fermi-surface limit	The generally complex $Z_p$, $m/m^*$, or a conserved charge
Renormalized velocity $\mathbf v_F^*$	Gradient of $E_{\mathbf k}$	Motion of the pole along the dispersion	Current carried after backflow
Pole damping $\gamma$	$z_p=E-i\gamma$	Spectral half-width at half maximum	The full width or a transport rate
Spectral full width	$2\gamma$	Full width at half maximum of the Lorentzian	A current-relaxation rate
Amplitude-decay time	$G^R(t)\propto e^{-\gamma t}$	$\tau_{\mathrm{amp}}=1/\gamma$	An occupation lifetime
Occupation lifetime	$\tau_{\mathrm{occ}}^{-1}=2\gamma$ in the weak-damping kinetic limit	Decay of a quasiparticle population	$\tau_{\mathrm{tr}}^{-1}$

The chapter’s surface-to-observables map uses $\Gamma$ for the pole-width datum. Whenever another source uses $\Gamma$, check whether it means the HWHM $\gamma$, the FWHM $2\gamma$, or the occupation rate before comparing formulas.

Why Pauli phase space makes the pole long lived

Consider a particle with energy $E>0$ above a zero-temperature Fermi surface. Its simplest decay creates two particles above the surface and one hole below it. If the density of states and the scattering amplitude are smooth, the radial energy part of the cut self-energy contains

$$\begin{aligned} I(E) &= \int_{-E}^{0}d\xi_h \int_0^\infty d\xi_1\,d\xi_2\, \delta(E+\xi_h-\xi_1-\xi_2) \\ &= \int_0^E dx\,(E-x) =\frac{E^2}{2}. \end{aligned}$$

This simple triangle is the origin of the power: Pauli blocking leaves two independent low-energy integrations after energy conservation is imposed. For a regular three-dimensional Fermi surface, the remaining angular integral is finite, and the retarded self-energy has the leading form

$$-\Sigma''(\mathbf k_F,\omega,T) = C_{\mathbf k_F}\left(\omega^2+\pi^2T^2\right) +o(\omega^2+T^2), \qquad C_{\mathbf k_F}\ge0.$$

The coefficient has dimensions of inverse energy and contains the renormalized scattering amplitude and local Fermi-surface geometry; it is positive when an allowed scattering channel has nonzero amplitude and can vanish in a free or kinematically protected limit. It is not universal. The quadratic zero-temperature law and its perturbative control are developed in Luttinger 1961, pp. 942–949, while the thermal energy integrals are analyzed in Chubukov and Maslov 2012, § II.

In two dimensions the angular integral is marginal. For a regular surface with nonsingular short-range or screened interactions, the generic leading limits are

$$\begin{aligned} -\Sigma''(\mathbf k_F,\omega,0) &\sim C_2\,\omega^2\ln\frac{\Lambda}{\lvert\omega\rvert}, \\ -\Sigma''(\mathbf k_F,0,T) &\sim C_2\,\pi^2T^2\ln\frac{\Lambda}{T}, \qquad C_2\ge0. \end{aligned}$$

Only to logarithmic accuracy may these be compressed into an expression involving $C_2(\omega^2+\pi^2T^2)\ln[\Lambda/\max(\lvert\omega\rvert,T)]$; nonlogarithmic terms and the full crossover function contain more information Chubukov and Maslov 2012, § III.A. The logarithm still permits a quasiparticle because $E\ln(\Lambda/E)\to0$.

The sharpness test must use the correct order of limits. At $T=0$,

$$\frac{\gamma(E)}{\lvert E\rvert}\longrightarrow0 \qquad(E\to0).$$

At fixed $T>0$, taking $E\to0$ first would make this ratio ill posed. Thermally active excitations instead require $\gamma(\mathbf k_F,0,T)/T\to0$ as $T\to0$, or, in a joint scaling limit, $\gamma\ll\max(\lvert E\rvert,T)$. Scale separation is necessary but not sufficient: no competing singularity may change $G^R$ appreciably across the width, and the self-energy must be smooth on that scale.

The pole lifetime is not a transport lifetime. Near-forward finite-transfer collisions can broaden a spectral line while changing current only weakly because transport vertices supply angular weights such as $1-\cos\theta$. In a clean Galilean-invariant fluid, electron–electron collisions cannot relax the uniform electric current at all. On a lattice, Umklapp, momentum overlap, disorder, and the full response vertex determine whether a finite dc relaxation rate results.

Forward limits and Ward matching

Two vertices enter this discussion, and they should not share a symbol:

• $\Lambda^\mu$ is the three-leg electromagnetic vertex that couples a fermion to an external probe.
• $\Gamma^{(4)}$ is the antisymmetrized four-fermion vertex that becomes the Landau interaction or forward scattering amplitude in different limits.

Probe-vertex limits

For unit charge, the exact Ward–Takahashi identity is

$$\Omega\Lambda^0(p+q/2,p-q/2) -\mathbf q\mathbin{\cdot}\boldsymbol\Lambda(p+q/2,p-q/2) =G^{-1}(p+q/2)-G^{-1}(p-q/2).$$

Take $\mathbf q\to0$ first while $\Omega\ne0$, and then take $\Omega\to0$. This homogeneous, or dynamic, density limit gives

$$\Lambda_\omega^0=\partial_\omega G^{-1}=1-\partial_\omega\Sigma, \qquad Z\Lambda_\omega^0\big|_F=1.$$

The field overlap $Z$ is therefore canceled by the density vertex: the conserved quasiparticle charge is not reduced to $Z$. Reverse the order—take $\Omega\to0$ first and then the longitudinal $\mathbf q\to0$ limit—and obtain

$$\boldsymbol\Lambda_q=-\boldsymbol\nabla_{\mathbf k}G^{-1} =\mathbf v_{\mathbf k}+\boldsymbol\nabla_{\mathbf k}\Sigma.$$

This identity fixes the longitudinal vertex. A gauge-invariant response may additionally require contact or diamagnetic terms, and current backflow requires the Fermi-surface Bethe–Salpeter equation; neither is supplied by the one-particle pole alone.

Four-point limits and Landau parameters

The two four-point limits are likewise distinct. Superscript names vary across the literature, so the order of limits is the definition:

$$\begin{aligned} \Gamma^\omega &\equiv \lim_{\Omega\to0}\lim_{\mathbf q\to0}\Gamma^{(4)}(p,p';q), &&\frac{v_F^*\lvert\mathbf q\rvert}{\lvert\Omega\rvert}\to0, \\ \Gamma^q &\equiv \lim_{\mathbf q\to0}\lim_{\Omega\to0}\Gamma^{(4)}(p,p';q), &&\frac{\lvert\Omega\rvert}{v_F^*\lvert\mathbf q\rvert}\to0. \end{aligned}$$

They differ by an on-shell particle–hole contribution. With the sign convention that repulsion is positive in the density channel,

$$f_{\sigma\sigma'}(\mathbf k_F,\mathbf k_F') =Z_{\mathbf k_F}Z_{\mathbf k_F'} \Gamma^\omega_{\sigma\sigma'}(\mathbf k_F,\mathbf k_F'),$$

whereas the forward scattering amplitude comes from $\Gamma^q$, also denoted $\Gamma^k$. In the chapter’s isotropic three-dimensional harmonic convention,

$$A_\ell^{s,a} = \frac{F_\ell^{s,a}} {1+F_\ell^{s,a}/(2\ell+1)}.$$

The Pitaevskii–Landau identities follow by differentiating Dyson’s equation and using these exact vertex equations. Structurally, they establish that

• $Z^{-1}$ and the momentum derivative of $\Sigma$ contain integrals over all internal energies;
• the difference between the two products of Green functions in the $q$ and $\omega$ limits is supported on the Fermi surface; and
• the Bethe–Salpeter equation converts $\Gamma^\omega$ into $\Gamma^q$.

Thus $Z$ need not be generated by a thin low-energy shell even though the difference of forward limits is a Fermi-surface effect. A modern derivation with explicit normalization is given in Chubukov, Klein, and Maslov 2018, § II.A, pp. 2–5, Open arXiv version.

Galilean boost identity

In an isotropic, rotationally and Galilean-invariant three-dimensional continuum, let $N(0)$ be the total quasiparticle density of states including both spin projections, take $f^s=(f_{\uparrow\uparrow}+f_{\uparrow\downarrow})/2$, and define

$$f^s(\cos\theta) =\frac{1}{N(0)} \sum_{\ell=0}^\infty F_\ell^sP_\ell(\cos\theta).$$

With this convention, the current or boost identity becomes

$$\frac{k_F}{m} =v_F^*\left(1+\frac{F_1^s}{3}\right), \qquad \frac{m^*}{m}=1+\frac{F_1^s}{3}.$$

The first equation says that quasiparticle motion plus the induced motion of the surrounding Fermi sea carries the bare current. It is not a second contribution to the pole residue, and the displayed relation does not hold on a lattice. Normalizations that expand the interaction in $(2\ell+1)P_\ell$ absorb the factor of three into their definition of $F_1$; observables agree after translation.

Skeleton diagrams and conserving checks

The general conserving-approximation construction becomes a sharp Fermi-liquid test here. A diagrammatic approximation earns the Baym–Kadanoff, or $\Phi$-derivable, conservation guarantee only when its self-energy and response vertex come from the same functional. Other constructions may satisfy conservation identities by a different proof. In modern notation, choose closed two-particle-irreducible skeleton diagrams built with the fully dressed $G$ and the declared bare interactions,

$$\Phi[G]=\text{selected closed skeletons}, \qquad \Sigma(1,2)=\frac{\delta\Phi}{\delta G(2,1)}, \qquad I=\frac{\delta\Sigma}{\delta G}.$$

Dyson’s equation must be solved self-consistently, and the same kernel $I$ must be used in the Bethe–Salpeter response. Diagrammatically, cutting every inequivalent propagator line in a retained $\Phi$ diagram must reproduce the retained self-energy diagrams with the correct signs and symmetry factors. For an instantaneous Hermitian two-body interaction, Hartree–Fock is real and static, so it can shift or distort the surface but cannot produce a decay rate. The complete antisymmetrized second-Born cut—including direct, exchange, and interference terms—generates the two-particle–one-hole phase space and must give the nonnegative decay rate $-\Sigma''\ge0$ on the stable particle branch; an isolated cut need not be positive by itself.

For an approximation intended to describe a Fermi liquid, check all of the following rather than relying on its name:

1. Stationarity and closure: every internal line uses the same converged $G$, and the response uses $I=\delta\Sigma/\delta G$.
2. Causality and spectrum: $\Sigma''\le0$ for the retarded particle branch, $A\ge0$, the spectral sum rule is satisfied, and the pole is narrow on its claimed scale.
3. Conserved response: the continuity equation, compressibility relation, and longitudinal sum rules agree when computed through independent routes.
4. Symmetry checks: the regulator and truncation preserve the claimed symmetry; a Galilean calculation reproduces the $F_1^s$ mass identity.
5. Controlled limits: weak coupling, known solvable limits, cutoff stability, and branch stability are reproduced.

The Luttinger–Ward stationary functional and its skeleton construction were introduced in Luttinger and Ward 1960, pp. 1422–1423. The conserving result requires a symmetry-invariant functional, full self-consistency, and the compatible response kernel Baym 1962, pp. 1395–1397. Those hypotheses guarantee the associated macroscopic conservation laws, not Fermi-liquid existence, crossing symmetry, spectral positivity, a unique physical branch, or quantitative accuracy at strong coupling. At finite truncation, internal propagators and higher vertices need not satisfy every Ward–Takahashi identity; the consistent external response must be constructed with the associated Bethe–Salpeter kernel van Hees and Knoll 2002, §§ II–III, Open arXiv version. One-shot and partially self-consistent variants do not inherit the complete theorem. Skeleton series can even select an unphysical branch in strongly correlated models Kozik, Ferrero, and Georges 2015.

A microscopic checkpoint

The following local model is not a universal self-energy; it is a compact test of the definitions. After absorbing the constant surface shift into $\mu$, take

$$\Sigma^R(\mathbf k,\omega,T) = \left(1-Z_0^{-1}\right)\omega +\alpha v_Fk_\perp -i\frac{C}{E_{\mathrm{FL}}} \left(\omega^2+\pi^2T^2\right),$$

where $E_{\mathrm{FL}}>0$ is the crossover scale controlling the expansion, $Z_0$, $\alpha$, and $C$ are dimensionless, $0<Z_0\le1$, $C>0$, and $\lvert\omega\rvert,T\ll E_{\mathrm{FL}}$. Direct substitution yields

$$Z=Z_0, \qquad v_F^*=Z_0(1+\alpha)v_F, \qquad \gamma(E,T) =\frac{Z_0C}{E_{\mathrm{FL}}} \left(E^2+\pi^2T^2\right).$$

At $T=0$, $\gamma/\lvert E\rvert=(Z_0C/E_{\mathrm{FL}})\lvert E\rvert\to0$. At the surface, $\gamma(0,T)/T=Z_0C\pi^2T/E_{\mathrm{FL}}\to0$. The free limit is $Z_0\to1$, $\alpha\to0$, and $C\to0$. Positivity requires $1+\alpha>0$ for the chosen outward normal; otherwise the local dispersion has changed orientation or the assumed branch has become unstable.

This checkpoint extracts pole data only. To predict compressibility, spin response, zero sound, or electrical transport, one must also specify the compatible four-point and probe vertices. The chapter’s Fermi-liquid claim table records which limit, observable, and falsifying signature belong to each such claim.

Validity and breakdown

The derivation assumes a normal phase, a smooth self-energy, a finite residue, a nonsingular low-energy interaction amplitude, and a regular Fermi-surface patch. It can fail or change form near a van Hove point, a nested or flat patch, an unscreened singular interaction, a critical boson, a collective-mode threshold, elastic disorder, or an intervening ordered phase. In one dimension, the generic interacting fixed point is a Luttinger liquid rather than a Landau Fermi liquid.

Even within two or three dimensions, a finite-order perturbative or skeleton calculation does not establish adiabatic continuity at strong coupling. The strongest justified statement is conditional: if the normal state has a regular pole and nonsingular renormalized vertices, the displayed expansion, Ward matching, and phase-space laws reconstruct Landau quasiparticles and identify the checks that can falsify that reconstruction.

Common pitfalls

Calling any narrow-looking peak a quasiparticle. Resolution broadening, a threshold cusp, or overlapping poles can imitate a Lorentzian. Locate the complex pole or demonstrate a controlled narrow-width expansion, then vary numerical or experimental resolution.

Using “lifetime” without a convention. State whether the reported number is $\gamma$, $2\gamma$, $\tau_{\mathrm{amp}}$, $\tau_{\mathrm{occ}}$, or a channel-specific transport time.

Setting $m^*/m=1/Z$ in general. Momentum dependence of $\Sigma'$ changes the pole mass. Backflow supplies a separate current identity; it does not repair an omitted momentum derivative.

Taking the forward limit without its path. The dynamic and static limits differ by on-shell particle–hole propagation. Name the order of limits and distinguish $\Lambda^\mu$ from $\Gamma^{(4)}$.

Calling an approximation conserving because it dresses $G$. A dressed bubble with an unrelated bare vertex is not a conserving response. Derive and use the compatible kernel, then verify the relevant continuity and sum-rule identities.

Exercises

1. Pole factorization and the factor of two

Set $\alpha=0$ in the checkpoint self-energy. Derive $Z$, $v_F^*$, the spectral FWHM, the amplitude-decay time, and the occupation lifetime.

Solution

The real inverse propagator is $Z_0^{-1}\omega-v_Fk_\perp$, so factoring out $Z_0^{-1}$ gives $Z=Z_0$ and $v_F^*=Z_0v_F$. On shell,

$$\gamma=\frac{Z_0C}{E_{\mathrm{FL}}} \left(E^2+\pi^2T^2\right).$$

The denominator is $\omega-E+i\gamma$, so the spectral HWHM is $\gamma$ and its FWHM is $2\gamma$. Fourier transformation gives a retarded amplitude proportional to $e^{-\gamma t}$, hence $\tau_{\mathrm{amp}}=1/\gamma$. In the simple weak-damping kinetic limit, the optical theorem gives $\tau_{\mathrm{occ}}^{-1}=-2Z\Sigma''(E)=2\gamma$, so $\tau_{\mathrm{occ}}=1/(2\gamma)$.

2. Pauli-blocked phase space

Evaluate the zero-temperature two-particle–one-hole energy integral $I(E)$ used above. Explain why this argument alone does not prove the two-dimensional result.

Solution

Put $x=-\xi_h$. For a fixed $x\in[0,E]$, energy conservation leaves $\xi_1+\xi_2=E-x$, whose allowed interval has length $E-x$. Therefore

$$I(E)=\int_0^E(E-x)\,dx=\frac{E^2}{2}.$$

This counts radial energy phase space. In three dimensions a regular angular integral multiplies it by a finite coefficient. In two dimensions collinear and backscattering regions make that angular integral marginal and generate the logarithm; the radial triangle by itself cannot see this geometry.

3. Ward limits for a derivative self-energy

Let $\Sigma'(\mathbf k,\omega)=\lambda\omega+\alpha\xi_{\mathbf k}$ near the surface. Find the dynamic density vertex, the static longitudinal current vertex, $Z$, and $\mathbf v_F^*$. When is a bare probe vertex consistent?

Solution

The two Ward limits give

$$\Lambda_\omega^0=1-\lambda, \qquad \boldsymbol\Lambda_q=(1+\alpha)\mathbf v_F.$$

Meanwhile,

$$Z=(1-\lambda)^{-1}, \qquad \mathbf v_F^*=Z(1+\alpha)\mathbf v_F.$$

Thus $Z\Lambda_\omega^0=1$, as required for conserved charge. A bare density or current vertex satisfies the corresponding Ward limit only if the associated self-energy derivative vanishes. In a full response, transverse and contact terms may impose additional conditions.

4. A minimal conserving closure

For a contact interaction, consider the Hartree skeleton

$$\Phi_H[G] =U\int_x G_\uparrow(x,x^+)G_\downarrow(x,x^+).$$

Differentiate it once and twice. What must be done before calling the approximation conserving, and why does it not produce a lifetime?

Solution

Functional differentiation gives

$$\Sigma_\uparrow=Un_\downarrow, \qquad \Sigma_\downarrow=Un_\uparrow, \qquad I_{\uparrow\downarrow}=I_{\downarrow\uparrow}=U, \qquad I_{\uparrow\uparrow}=I_{\downarrow\downarrow}=0.$$

Spacetime delta functions in these local kernels have been suppressed. The densities must be obtained from the same self-consistent dressed propagators, and response must use the kernel $U$ in the compatible Bethe–Salpeter equation. A one-shot Hartree shift or a dressed bubble without that kernel does not meet the conserving-functional hypotheses. Because the Hartree self-energy is real and frequency independent, $\Sigma''=0$ and it generates no collision phase space or finite lifetime.

Continue

Quasiparticle Poles, Residues, and Lifetimes develops the general analytic pole criterion. Fermi-Liquid Response and Zero Sound uses the forward vertex in a collisionless collective mode, while Luttinger’s Theorem, Fermi Volume, and Failure Modes asks the separate global question of Fermi volume.

References

• Baym, Gordon. “Self-Consistent Approximations in Many-Body Systems.” Physical Review 127 (1962): 1391–1401. doi:10.1103/PhysRev.127.1391.
• Chubukov, Andrey V., Avraham Klein, and Dmitrii L. Maslov. “Fermi-Liquid Theory and Pomeranchuk Instabilities: Fundamentals and New Developments.” Journal of Experimental and Theoretical Physics 127 (2018): 826–843. doi:10.1134/S1063776118110122. Open arXiv version.
• Chubukov, Andrey V., and Dmitrii L. Maslov. “First-Matsubara-Frequency Rule in a Fermi Liquid. I. Fermionic Self-Energy.” Physical Review B 86 (2012): 155136. doi:10.1103/PhysRevB.86.155136. Open arXiv version.
• Kozik, Evgeny, Michel Ferrero, and Antoine Georges. “Nonexistence of the Luttinger–Ward Functional and Misleading Convergence of Skeleton Diagrammatic Series for Hubbard-Like Models.” Physical Review Letters 114 (2015): 156402. doi:10.1103/PhysRevLett.114.156402.
• Luttinger, J. M. “Analytic Properties of Single-Particle Propagators for Many-Fermion Systems.” Physical Review 121 (1961): 942–949. doi:10.1103/PhysRev.121.942.
• Luttinger, J. M., and J. C. Ward. “Ground-State Energy of a Many-Fermion System. II.” Physical Review 118 (1960): 1417–1427. doi:10.1103/PhysRev.118.1417.
• van Hees, Hendrik, and Jörn Knoll. “Renormalization in Self-Consistent Approximation Schemes at Finite Temperature. III. Global Symmetries.” Physical Review D 66 (2002): 025028. doi:10.1103/PhysRevD.66.025028. Open arXiv version.

Further reading

• Baym, Gordon, and Christopher Pethick. Landau Fermi-Liquid Theory: Concepts and Applications. Weinheim: Wiley-VCH, 1991, ch. 3. doi:10.1002/9783527617159.