Many-Body Physics in Solids - Wyatt's Notes

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.

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.

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.

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

• Confusing the Thomas-Fermi screening length with the Debye length (the former applies to degenerate electrons, the latter to classical plasmas).
• Assuming that the Hubbard model $U/t$ parameter fully determines the physics; lattice geometry and filling also matter.
• Treating quasiparticles as real particles: they are emergent excitations that can decay, and their effective mass can be negative.
• Neglecting that the Kondo effect requires a magnetic impurity in a non-magnetic host; in a magnetic host the physics is different.
• Confusing the Mott insulator (driven by electron correlations) with a band insulator (driven by filled bands and a band gap).

flowchart TD
    A[14_Many Body Physics In Solids] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

Many-body physics studies how enormous numbers of interacting particles produce emergent phenomena. In a metal, electrons interact via Coulomb forces, but the collective behavior is described by quasiparticles with modified mass and lifetime. The electron gas behaves like a nearly free Fermi liquid, with interactions creating a sea of virtual excitations. Phonons are collective vibrations that behave as particles, mediating interactions between electrons. This framework explains superconductivity, magnetism, and the metal-insulator transition. The key insight is that complex many-body problems can often be mapped to simpler single-particle problems with renormalized parameters.

• Crystal Structures: The lattice geometry and crystal symmetry determine the electronic band structure and the validity of the tight-binding model.

• Lattice Vibrations and Phonons: Electron-phonon coupling leads to Cooper pairing in conventional superconductors and affects quasiparticle properties.

• Semiconductors: The Fermi liquid concept and quasiparticle picture build on the semiconductor band structure framework.

• Magnetism in Solids: The Hubbard model and exchange interactions connect many-body physics to magnetic ordering in solids.

• Calculus

• Linear Algebra

• Vector Calculus

• Quantum Computing

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Ensure you have mastered the prerequisite material before attempting this advanced content.