Condensed Matter Physics

Atoms in crystalline solids arrange in periodic lattices. Bloch's theorem shows that electron wavefunctions in such lattices form energy bands separated by band gaps. Whether a material is a metal, semiconductor, or insulator depends on how these bands are filled.

Below a critical temperature, certain materials exhibit exactly zero electrical resistance and expel magnetic fields (the Meissner effect). BCS theory explains conventional superconductivity through Cooper pairs of electrons bound by phonon-mediated attraction.

When a two-dimensional electron gas is subjected to a strong perpendicular magnetic field at low temperature, the Hall conductance is quantized in exact integer (or fractional) multiples of $e^2/h$. The integer quantum Hall effect arises from Landau level quantization, while the fractional quantum Hall effect involves strongly correlated electron states.

A phase transition is a transformation between states of matter driven by changes in temperature, pressure, or other parameters. Near continuous (second-order) transitions, physical quantities diverge according to universal power laws characterized by critical exponents that depend only on symmetry and dimensionality, not microscopic details.

Landau's Fermi liquid theory describes the low-energy excitations of interacting electron systems in terms of quasiparticles that behave like free electrons but with renormalized effective mass and lifetime. It explains why the free-electron model works surprisingly well for many metals despite strong electron-electron interactions.

Topological insulators are materials that behave as insulators in their bulk but host conducting states on their surfaces or edges that are protected by time-reversal symmetry. These surface states are robust against disorder and cannot be removed without closing the bulk band gap.

At extremely low temperatures, a macroscopic fraction of bosons (integer-spin particles) occupy the lowest quantum state, forming a Bose-Einstein condensate (BEC). This state exhibits macroscopic quantum coherence, superfluidity, and quantized vortices.

Magnetic order arises from exchange interactions between electron spins. Ferromagnets have parallel spin alignment, antiferromagnets have alternating alignment, and frustrated magnets or spin liquids resist conventional ordering. The Heisenberg and Ising models capture these behaviors theoretically.

Phonons are quantized vibrations of atoms in a crystal lattice. They determine thermal conductivity, specific heat, and play a key role in electron-phonon coupling that underlies conventional superconductivity. The Debye and Einstein models describe phonon contributions to heat capacity.

Atomically thin materials such as graphene, transition metal dichalcogenides, and hexagonal boron nitride exhibit properties that differ dramatically from their bulk counterparts due to reduced dimensionality, quantum confinement, and enhanced correlations.