Fundamentals of Condensed Matter

Working note — Part of my personal research notebook. Entries may be incomplete and are revised over time.

• physical system: a physical system consists of a large number of interacting particles (e.g., electrons, atoms, molecules) that form condensed phases of matter, such as solids, liquids, and exotic states like superconductors or Bose-Einstein condensates.
  • physical systems in Condensed Matter Physics:
    • A crystal lattice of atoms (e.g., a metal like copper or an insulator like diamond).
    • system of electrons in a semiconductor (e.g., silicon or gallium arsenide).
    • superconductor, where electrons form Cooper pairs and exhibit zero electrical resistance.
    • system of ultracold atoms in a Bose-Einstein condensate.
  • The physical system is characterized by its microscopic constituents (e.g., atoms, electrons) and their interactions (e.g., Coulomb interactions, phonon-mediated interactions).
• state of the physical system: described by the collective behavior of its constituent particles, which often leads to emergent phenomena that cannot be understood by studying individual particles alone.
  • state is typically described using:
    • wave functions or quantum states for systems governed by quantum mechanics (e.g., electrons in a crystal lattice).
    • order parameters that characterize the symmetry and phase of the system (e.g., magnetization in a ferromagnet, superconducting gap in a superconductor).
    • thermodynamic variables (e.g., temperature, pressure, entropy) for systems in thermal equilibrium.
  • states in Condensed Matter Physics:
    • crystalline state, where atoms are arranged in a periodic lattice.
    • superconducting state, where electrons form Cooper pairs and exhibit zero resistance.
    • ferromagnetic state, where electron spins are aligned, producing a net magnetization.
    • topological state, characterized by non-trivial topological invariants (e.g., topological insulators).
• physical law: physical laws are the fundamental principles and equations that govern the behavior of many-particle systems. These laws are derived from quantum mechanics, statistical mechanics, and electrodynamics, often adapted to describe collective phenomena.
  • physical laws in Condensed Matter Physics include
    • Schrodinger equation
    • Bloch theorem
    • Bardeen-Cooper-Scrieffer theory
    • Landau theory of phase tarnsitions
    • Fermi liquid theory
  • physical laws in Condensed Matter Physics are subject to conditions of applicability. For example:
    • The BCS theory applies to conventional superconductors but not to high-temperature superconductors.
    • The Fermi liquid theory applies to systems with weak interactions but breaks down in strongly correlated systems.