Quasiparticles

Phase-space argument for quasiparticle decay (Landau, 1956)

Consider a quasiparticle with momentum $\mathbf{k}_1$ and energy $\epsilon_1 = E_F + \delta$ (with $\delta > 0$) above the Fermi sea. It can scatter off a particle below the Fermi surface, creating a particle-hole pair.

Step 1: By Fermi's golden rule, the decay rate is:

subject to the Pauli constraints: $\mathbf{k}_2$ must be occupied (below $E_F$), while $\mathbf{k}_3$ and $\mathbf{k}_4$ must be empty (above $E_F$).

Step 2: Energy conservation gives $\epsilon_1 + \epsilon_2 = \epsilon_3 + \epsilon_4$. Define the excitation energies relative to $E_F$: $\xi_i = \epsilon_i - E_F$. Then $\xi_1 + \xi_2 = \xi_3 + \xi_4$ with the constraints:

Step 3: Count the phase space. Choose $\xi_3$ freely in the range $[0, \delta + |\xi_2|]$. Then $\xi_4 = \delta + \xi_2 - \xi_3$, and we need $\xi_4 > 0$, so $\xi_3 < \delta + \xi_2$. Also $\xi_2 < 0$ and $\xi_2 > -\delta$ (since $\xi_4 = \delta + \xi_2 - \xi_3$ must be positive and $\xi_3 > 0$).

Step 4: Integrating over the available phase space (assuming constant density of states and matrix element near $E_F$):

Step 5: Therefore the scattering rate scales as:

This is the celebrated result: the quasiparticle lifetime diverges as $\tau \propto (\epsilon - E_F)^{-2}$ at the Fermi surface. Since the excitation energy itself is $\sim |\epsilon - E_F|$, the ratio $\hbar/(\tau \cdot |\epsilon - E_F|) \to 0$ as $\epsilon \to E_F$, guaranteeing that quasiparticles are well-defined near the Fermi surface.

Derivation of the quasiparticle residue $Z_k$ from the Dyson equation

Step 1: Start from the Dyson equation for the retarded Green's function:

Step 2: Near the quasiparticle pole at $\omega = \tilde{\epsilon}_k$, expand the real part of the self-energy to first order:

Step 3: Use the pole condition $\tilde{\epsilon}_k - \epsilon_k - \text{Re}\,\Sigma(\mathbf{k}, \tilde{\epsilon}_k) = 0$ to write:

Step 4: Treating $\text{Im}\,\Sigma$ as small near $E_F$, the Green's function becomes:

where the quasiparticle weight is:

and the decay rate is $\Gamma_k = -2Z_k\,\text{Im}\,\Sigma(\mathbf{k}, \tilde{\epsilon}_k)$. Since $\partial\text{Re}\,\Sigma/\partial\omega < 0$ for repulsive interactions, we get $0 < Z_k < 1$: interactions transfer spectral weight from the coherent peak to an incoherent background.

Derivation of the momentum distribution discontinuity $\Delta n_{k_F} = Z_{k_F}$

Step 1: The momentum distribution is given by:

where the integral runs over occupied states (negative frequencies measured from the chemical potential).

Step 2: At $T = 0$, for $k$ just below $k_F$, the quasiparticle pole at $\tilde{\epsilon}_k < 0$ is included in the integral, contributing $Z_k$ to $n_k$. For $k$ just above $k_F$, the pole at $\tilde{\epsilon}_k > 0$ is excluded.

Step 3: The incoherent background $G_\text{inc}$ varies smoothly across $k_F$. Therefore the discontinuity in $n_k$ is precisely:

This discontinuity is the defining hallmark of a Fermi liquid: it persists even in the presence of interactions, though it is reduced from the free-gas value of 1 to $Z_{k_F} < 1$. Its vanishing ($Z_{k_F} \to 0$) signals the breakdown of the Fermi liquid state.

Step 1: Fermi's Golden Rule with Pauli Blocking

A quasiparticle in state $\mathbf{k}_1$ with energy $\epsilon_1$ scatters off a filled state $\mathbf{k}_2$ into empty states $\mathbf{k}_3, \mathbf{k}_4$. At finite temperature, the occupation factors are Fermi functions $f(\epsilon) = 1/(e^{(\epsilon - E_F)/k_BT} + 1)$. The scattering rate is:

The Pauli factors enforce: $\mathbf{k}_2$ occupied, $\mathbf{k}_3$ and $\mathbf{k}_4$ empty. At $T = 0$ these become sharp step functions, recovering the zero-temperature result.

Step 2: Converting to Energy Integrals

Assuming a constant density of states $N(0)$ at the Fermi level and a contact interaction $|V|^2 \to U^2$, the momentum sums convert to energy integrals. Define $\xi_i = \epsilon_i - E_F$. After using energy conservation to eliminate $\xi_4 = \xi_1 + \xi_2 - \xi_3$:

Step 3: Phase-Space Evaluation at Finite T

The key integral to evaluate is the thermal convolution of three Fermi factors. Using the identity for products of Fermi and Bose functions, one can show:

For a quasiparticle above the Fermi level ($\xi_1 > 0$), in the limit $\xi_1 \gg k_BT$, the denominator approaches 2, and we obtain:

This is the celebrated result: the scattering rate has two additive contributions — one from the excitation energy $(\epsilon - E_F)^2$ and one from thermal smearing $(\pi k_BT)^2$. Both arise from the same phase-space restriction imposed by the Pauli principle.

Step 4: Why States Near E_F Are Long-Lived

At $T = 0$, a quasiparticle at energy $\epsilon$ above $E_F$ has decay rate $\Gamma \propto (\epsilon - E_F)^2$. The ratio of the decay rate to the excitation energy is:

This means that as we approach the Fermi surface, the quasiparticle becomes arbitrarily well-defined: its energy uncertainty (width) vanishes faster than the energy itself. This is the fundamental reason Landau's Fermi liquid theory works — quasiparticles are asymptotically exact eigenstates at $E_F$.

Physically, the phase-space restriction is simple: to decay, the quasiparticle must find a partner below $E_F$ and scatter into two states above $E_F$. Energy conservation confines all participants to a shell of thickness $\sim |\epsilon - E_F|$ around the Fermi surface. Two independent energy integrations over this shell produce the $\delta^2$ scaling.

Step 5: Derivation of T$^2$ Resistivity

The electrical resistivity of a Fermi liquid is controlled by the quasiparticle scattering rate at the Fermi surface. Setting $\epsilon = E_F$ (transport involves states near $E_F$), the relevant rate is:

Using the Drude formula $\rho = m^* / (ne^2 \tau_{\text{tr}})$, this gives the characteristic Fermi liquid resistivity:

Here $\rho_0$ is the residual resistivity from impurity scattering and $A$ is the Kadowaki–Woods coefficient. Experimentally, the $T^2$ resistivity is the transport hallmark of a Fermi liquid. The Kadowaki–Woods ratio $A/\gamma^2$(where $\gamma$ is the Sommerfeld coefficient of specific heat) is approximately universal across many correlated metals:

for heavy-fermion compounds, confirming the common origin of enhanced specific heat and enhanced resistivity from the renormalized quasiparticle mass $m^*$.

Step 1: Spectral Representation of the Green's Function

The retarded Green's function satisfies the Dyson equation:

where $\Sigma = \Sigma' + i\Sigma''$ is the complex self-energy, with $\Sigma' = \text{Re}\,\Sigma$ and $\Sigma'' = \text{Im}\,\Sigma$. The spectral function is defined as:

Step 2: Deriving the Lorentzian Form

Writing $G^R$ explicitly:

Taking the imaginary part (note $\Sigma'' < 0$ for the retarded function):

Therefore:

This is a generalized Lorentzian. If $\Sigma$ were frequency-independent, it would be an exact Lorentzian of width $|\Sigma''|$ centered at the shifted energy $\epsilon_k + \Sigma'$. The frequency dependence of $\Sigma$ distorts the line shape and is essential for the quasiparticle picture.

Step 3: Quasiparticle Pole and Residue $Z_k$

The quasiparticle pole occurs at the complex energy $\tilde{\omega}_k = \tilde{\epsilon}_k - i\Gamma_k/2$ where the denominator of $G^R$ vanishes. Expanding the self-energy around the real part of the pole, $\omega = \tilde{\epsilon}_k$:

Using the pole condition $\tilde{\epsilon}_k = \epsilon_k + \Sigma'(\mathbf{k}, \tilde{\epsilon}_k)$, the Green's function near the pole becomes:

where the quasiparticle residue is:

The residue $Z_k$ measures the overlap between the true quasiparticle state and a bare single-particle excitation. Since $\partial \Sigma'/\partial \omega < 0$for causal self-energies (Kramers–Kronig), we have $0 < Z_k \leq 1$. The missing weight $1 - Z_k$ resides in the incoherent part $G_{\text{inc}}$.

Step 4: Spectral Function Decomposition

The spectral function correspondingly decomposes into coherent and incoherent parts:

The sum rule $\int d\omega\, A(\mathbf{k}, \omega) = 1$ is maintained, with the coherent part carrying weight $Z_k$ and the incoherent background carrying $1 - Z_k$:

The incoherent background arises from multi-particle excitations (particle-hole pairs, plasmons, etc.) that share the same quantum numbers as the single-particle state. In ARPES measurements, the coherent peak is the sharp feature that disperses with $\mathbf{k}$, while the incoherent part forms a broad, largely featureless background.

Step 5: Kramers–Kronig Constraint on the Self-Energy

Causality of the retarded Green's function imposes Kramers–Kronig relations on the self-energy:

This guarantees that a negative $\Sigma''$ (damping) necessarily produces a negative slope $\partial \Sigma'/\partial \omega < 0$ near $E_F$, which in turn ensures $Z_k < 1$ and $m^* > m$. The real and imaginary parts of the self-energy are not independent — measuring one determines the other.

Statement of the Theorem

Let $n$ be the total electron density. The interacting Fermi surface is defined by the locus of points where the Green's function at zero frequency changes sign:

Luttinger's theorem asserts:

The volume enclosed by the interacting Fermi surface is identical to that of the non-interacting system, despite the self-energy shifting individual energy levels.

Step 1: Electron Number from the Green's Function

The total electron number can be expressed exactly using the imaginary-time Green's function:

where $\omega_n = (2n+1)\pi T$ are fermionic Matsubara frequencies. At $T = 0$, the Matsubara sum becomes an integral:

Step 2: Luttinger–Ward Identity

The key technical step uses the identity (integrating by parts along the imaginary axis):

Luttinger and Ward showed that the self-energy contribution to this integral vanishes identically at $T = 0$ due to a Ward identity connected to particle conservation. Specifically, defining the Luttinger–Ward functional $\Phi[G]$:

Step 3: Reducing to the Free Result

With the self-energy contribution vanishing, the electron number reduces to:

But the Fermi surface of the interacting system is defined by $G^{-1}(\mathbf{k}_F, 0) = 0$, not by $\epsilon_k = \mu$. The crucial point is that the integral counts the number of poles of $G$ enclosed in the lower half-plane, which equals the number of $\mathbf{k}$ points inside the interacting Fermi surface. Combining:

Step 4: Topological Argument

The modern understanding reveals that Luttinger's theorem has a topological character. The electron number can be written as a winding number of the Green's function:

where $\mathcal{C}$ encloses the lower half of the complex frequency plane. For each $\mathbf{k}$, this integral counts the number of zeros minus poles of $G(\mathbf{k}, \omega)$ inside $\mathcal{C}$, which is either 1 (inside the Fermi surface) or 0 (outside). Being an integer, this winding number cannot change under continuous deformations of $\Sigma$ — it is topologically protected.

This topological robustness explains why Luttinger's theorem holds to all orders in perturbation theory and is insensitive to the detailed form of the interaction. Its violation signals a true phase transition, such as the emergence of a Mott insulator (where $G$ develops additional zeros) or a fractionalized Fermi liquid (where topological order modifies the counting).