Many-Body Quantum Systems & Condensed Matter
Second quantization, lattice models, symmetry breaking, phase transitions, entanglement, tensor networks, topological order, and numerical methods. Expanded with original in-page technical chapters, mathematical proofs, independently checked calculation baselines and directly embedded credited original instructor videos where relevant.
Published syllabus
Course Units
Fock space and second quantization
Fock space assembles sectors with different particle numbers by taking direct sums of symmetrized or antisymmetrized tensor powers of the one-particle space. Creation operators add occupation to modes, while number operators count quanta. This language captures many-particle sectors without tracking artificial particle labels.
Fermions, bosons, and exchange algebra
Bosonic mode operators obey commutators and allow unbounded mode occupancy; fermionic operators obey anticommutators and enforce zero or one fermion per mode. Exchanging identical particles is different from the exchange of two distinguishable subsystems, and the normalization of (anti)symmetrized states depends on orbital overlap.
Hubbard and Heisenberg lattice models
The Hubbard model balances hopping t and on-site repulsion U on a lattice. At strong repulsion and half filling, virtual hopping can yield an effective antiferromagnetic Heisenberg exchange with scale J≈4t²/U on appropriate lattices. This result requires controlled perturbation in t/U and the relevant filling and geometry.
Mean-field theory and broken symmetry
Mean-field theory replaces certain fluctuating operator products with averages and self-consistency conditions. It often captures broken-symmetry order parameters but may misrepresent critical exponents or strong quantum fluctuations. One should identify the decoupling channel, possible competing orders and whether a unique minimum is physically trustworthy.
Quantum phases and phase transitions
Quantum phases are distinguished through ground-state properties, excitation structure, symmetry and long-range entanglement. A continuous transition can develop a diverging correlation length and a closing gap, but first-order transitions and topological changes have different signatures. Finite-size scaling is essential when interpreting spectra from small simulated lattices.
Correlation functions and response
Equal-time and dynamical correlation functions quantify how disturbances at one point relate to another. Linear response connects susceptibility to commutators of operators evaluated in equilibrium. Spectral functions can reveal collective modes, continua and damping, but experimental probes couple to specific operators and selection rules.
Entanglement entropy and area laws
Bipartite entanglement entropy measures how nonfactorizable a pure many-body state is across a spatial cut. Gapped one-dimensional ground states often obey an area law, while critical chains have logarithmic violations controlled by conformal structure. Entanglement entropy depends on the chosen partition and cannot directly substitute for an ordinary thermodynamic entropy in general.
Tensor networks and matrix product states
A matrix product state factors many-body wavefunction amplitudes into chains of low-rank tensors. Bond dimension controls the maximum bipartite Schmidt rank and the entanglement that a state can represent. Canonical forms simplify norms and truncation. DMRG searches within this ansatz class but must be checked against increasing bond dimension and finite-size effects.
Topological order
Topological order is a phase not characterized by an ordinary local symmetry-breaking order parameter. Examples feature anyonic excitations, topology-dependent ground-state degeneracy and long-range entanglement. The toric code provides commuting stabilizers whose excitations have mutual anyonic statistics. Ground-state degeneracy depends on global boundary topology under the ideal model.
Quantum Hall physics
Landau levels quantize kinetic motion of charged particles in a magnetic field. Integer Hall plateaus are connected to filled topological bands and disorder localization; fractional Hall states arise from strongly correlated electron fluids and support fractionally charged excitations. Filling factor alone does not uniquely specify the state without interaction and symmetry data.
Exact diagonalization and MPS numerics
Exact diagonalization constructs a finite Hamiltonian matrix and extracts spectra and eigenstates with controlled algebraic accuracy. Hilbert-space dimension grows exponentially with site count, making symmetry reduction and sparse methods essential. Matrix product states use entanglement structure to access larger one-dimensional systems, but convergence in bond dimension does not replace thermodynamic finite-size analysis.
Research frontiers in many-body physics
Current frontiers in many-body physics include nonequilibrium phases, constrained dynamics, fracton order, unconventional superconductivity and analog quantum simulators. Meaningful progress requires a sharply stated Hamiltonian and observable, then a comparison among analytic limits, tensor-network predictions and experiment. Research labels such as 'nonergodic' or 'topological' require measurable diagnostics.