Many-body physics is often built from lattice models such as the Hubbard Hamiltonian, Heisenberg spin models, and Bose-Hubbard systems. These capture competing kinetic, interaction, and exchange effects.

Correlation functions such as $\langle \hat O(x)\hat O(0)\rangle$ reveal collective behavior, characteristic lengths, and excitations. In critical systems, correlations often decay algebraically rather than exponentially.

Quantum phase transitions occur at zero temperature as a nonthermal control parameter is tuned. They are driven by quantum fluctuations and are characterized by gap closing and scaling laws.

In many-body theory, the field operator $\hat\psi(x)$ annihilates a particle at position $x$. Equal-time commutation or anticommutation relations encode bosonic or fermionic statistics and enable Green's-function methods.

Interactions can reorganize into quasiparticles such as phonons, magnons, polarons, and Bogoliubov quasiparticles. These excitations often behave as the effective degrees of freedom relevant to experiments.