Many-Body Physics in Solids

14.1 Electron—Electron Interactions: Screening

In a metal, the Coulomb interaction between electrons is screened by the other electrons. The Thomas—Fermi screening wavevector:

$q_{\text{TF}^2 = \frac{e^2 g(\varepsilon_F)}{\varepsilon_0} = \frac{4k_F}{\pi a_0}}$

Where $a_0 = 4\pi\varepsilon_0\hbar^2/(m_e e^2)$ is the Bohr radius. The screened potential:

$V_{\text{scr}(r) = \frac{e^2}{4\pi\varepsilon_0 r}\,e^{-q_{\text{TF} r}}}$

The screening length $\lambda_{\text{TF} = 1/q_{\text{TF} \sim 0.5}}$ Å in metals (about one atomic spacing), meaning the Coulomb interaction is very short-ranged.

14.2 The Hubbard Model

The Hubbard model captures the competition between kinetic energy (delocalisation) and on-site Coulomb repulsion (localisation):

$\hat{H} = -t\sum_{\langle i,j\rangle,\sigma}\hat{c}_{i\sigma}^\dagger\hat{c}_{j\sigma} + U\sum_i \hat{n}_{i\uparrow}\hat{n}_{i\downarrow}$

Where $t$ is the hopping integral and $U$ is the on-site repulsion energy.

Limiting cases:

• $U \ll t$: Weakly correlated metal (well-described by band theory)
• $U \gg t$: Mott insulator (each site has exactly one electron, localised by strong repulsion)
• $U/t \sim 1$: Strongly correlated regime (intermediate coupling), relevant for transition metal oxides

The half-filled Hubbard model on a bipartite lattice has a metal—insulator transition at $U_c \sim W$ (bandwidth). For $U > U_c$The system is a Mott insulator even though band theory predicts a metal.

14.3 Quasiparticles and Fermi Liquid Theory

Landau”s Fermi liquid theory (1956) states that the low-energy excitations of an interacting Fermi system can be described as quasiparticles --- weakly interacting fermions with renormalised parameters:

• Effective mass $m^*$ (renormalised by interactions)
• Residual quasiparticle—quasiparticle interactions described by Landau parameters $F_l$

The quasiparticle lifetime goes as $\tau \propto 1/(\varepsilon - \varepsilon_F)^2$So quasiparticles near the Fermi surface are well-defined (long-lived).

Experimental signatures of Fermi liquid behaviour:

• $C_V = \gamma T$ with enhanced $\gamma \propto m^*$
• $\rho = \rho_0 + AT^2$ (quadratic temperature dependence)
• Pauli-like susceptibility $\chi$ (temperature-independent)

Non-Fermi liquid behaviour occurs in many correlated systems (cuprates near optimal doping, heavy fermion materials near quantum critical points) where $\rho \propto T$ (linear) or other anomalous power laws are observed.

14.4 Kondo Effect

When a magnetic impurity (e.g., a single Fe atom) is embedded in a non-magnetic metal, the impurity spin is screened by conduction electrons at low temperature, forming a singlet state. This is the Kondo effect.

The Kondo temperature $T_K$ sets the energy scale:

$k_B T_K \sim D\,e^{-1/(N(E_F)J)}$

Where $D$ is the bandwidth and $J$ is the exchange coupling. Below $T_K$:

• The resistivity increases logarithmically: $\Delta\rho \propto \ln(T_K/T)$
• The impurity moment is fully screened
• A narrow resonance (Kondo resonance) appears at the Fermi level in the density of states