These notes follow the lecture 'Solid-State Theory' taught by Prof. Sigrist in the spring semester of 2023. For the original version of my notes, please visit the exam collection of the VMP here. However, since the script handed out my Prof. Sigrist is of impressive quality, I decided to convert them to markdown (as used on this website) and implement annotations and footnotes seamlessly. All credit goes to Prof. Sigrist.

Table of Contents

Solid state physics, also known as condensed matter physics, investigates the properties of materials and systems with many degrees of freedom. As the largest subfield of physics, it bridges fundamental scientific questions and technological applications, underpinning modern industrialised civilisation.

This field primarily examines phenomena at room temperature or below, corresponding to energy scales much smaller than one Hartree ($e^2/a_B$). Key length scales include the system size (sample size) and the spatial extent of electronic wave functions. This contrasts with high-energy physics, which seeks to probe unknown high-energy theories using the standard model as a phenomenological framework. In solid state physics, the starting point is a well-defined "high-energy" Hamiltonian, with the challenge being to describe low-energy properties through effective, phenomenological theories.

At its core, solid state physics revolves around the Coulomb interaction, where electrons interact with each other and with ions. Microscopic formulations of these interactions often operate at energy scales too high to directly explain low-energy phenomena. Thus, reduced or effective theories are critical to understanding the "emergent physics" that governs condensed matter systems.

One central goal of solid state physics is characterising a system’s ground state. However, measurable properties often arise from excited states, making the concept of 'elementary excitations' essential. Here, the ground state functions as an effective vacuum, with excitations treated as particles within this framework. According to P. W. Anderson, two fundamental principles underlie the field:

1. Adiabatic Continuity: Complex systems can often be replaced by simpler models with equivalent low-energy properties, as they can be smoothly transformed into one another without altering qualitative behaviour. For example, Landau's Fermi liquid theory demonstrates how strongly interacting electrons at low energies behave like non-interacting Fermions with renormalised parameters.
2. Spontaneously Broken Symmetry: Phase transitions into states with qualitatively different properties often involve symmetry breaking. Examples include magnetically ordered states, which break rotational and time-reversal symmetry, and superconducting states, which break global gauge symmetry. Symmetry breaking simplifies theoretical descriptions and offers insight into material properties

These principles, combined with the study of emergent behaviour and excitations, form the foundation of solid state physics, connecting its theoretical richness to practical applications.

Table of Contents

1 Electrons in the Periodic Crystal
2 Semiconductors
3 Metals
4 Itinerant Electrons in a Magnetic Field
5 Landau's Theory of Fermi Liquids
6 Transport Properties of Metals
7 Magnetism in Metals
8 Magnetism of Localised Moments
9 Identical Quantum Particles - Formalism of Second Quantisation