Many-body physics fundamentals

Define creation/annihilation operators for bosons and fermions and derive their canonical (anti)commutation relations.

Express single-particle and two-particle operators in the second-quantized formalism.

Transform operators and states between different single-particle bases (e.g., position and momentum).

Define normal ordering for bosonic and fermionic operators.

State and apply Wick's theorem to evaluate ground state expectation values of operator products.

Formulate the Hamiltonian for the non-interacting electron gas (Jellium model) in second quantization.

Construct the ground state (Fermi sea) and calculate its energy and pressure.

Derive the single-particle density of states in 1D, 2D, and 3D.

Calculate key properties of the Fermi sea: Fermi energy, Fermi wavevector, and average kinetic energy.

Apply Wick's theorem to compute correlation functions in the non-interacting ground state.

Explain the concept of a hole excitation relative to the Fermi sea.

Explain the physical motivation for the Jellium model and the problem of the bare Coulomb interaction.

Derive the Hartree-Fock equations for the interacting electron gas using the variational principle on a Slater determinant.

Provide a physical interpretation of the direct (Hartree) and exchange (Fock) terms.

Solve the Hartree-Fock equations for the homogeneous electron gas to find the single-particle dispersion.

Calculate the ground state energy in the Hartree-Fock approximation and derive the exchange energy.

Define the concept of a quasiparticle within the Hartree-Fock approximation.

Explain the limitations of Hartree-Fock theory, focusing on the absence of correlation effects.

Define the single-particle Green's function (propagator) in the time and frequency domains.

Calculate the Green's function for the non-interacting electron gas.

Derive the Lehmann representation of the Green's function and interpret its physical content.

Define the spectral function and relate it to the Lehmann representation.

Demonstrate how the spectral function provides access to the single-particle density of states and quasiparticle properties.

Use the equation of motion method to derive the Green's function for simple interacting models.

Develop the perturbation expansion for the Green's function in the interaction picture.

Apply Wick's theorem to time-ordered products to handle the interaction term.

Derive the Feynman rules in frequency-momentum space for the interacting electron gas.

Define the self-energy and its role in dressing the propagator.

Derive the Dyson equation, which resums the perturbation series in terms of the proper self-energy.

Demonstrate that the first-order self-energy diagrammatically reproduces the Hartree-Fock approximation.

Distinguish between connected and disconnected diagrams and state the linked-cluster theorem.

Define the polarization bubble as the lowest-order diagram for the susceptibility.

Derive the Random Phase Approximation (RPA) by summing the geometric series of polarization bubbles.

Calculate the RPA dynamic dielectric function and the statically screened Coulomb interaction (Thomas-Fermi screening).

Identify plasmons as collective charge excitations corresponding to the poles of the inverse dielectric function.

Calculate the ground state correlation energy of the electron gas within RPA.

Explain the physical meaning of screening and its importance in metallic systems.

Derive the Kubo formula, which relates linear response functions to equilibrium correlation functions.

Relate the charge and current correlation functions to conductivity and the dielectric function.

State and interpret the fluctuation-dissipation theorem, connecting response to spontaneous fluctuations.

Outline the foundational assumptions of Landau's Fermi Liquid Theory and the concept of quasiparticles.

Define Landau parameters and relate them to physical observables like effective mass, compressibility, and spin susceptibility.

Discuss the success and breakdown of Fermi Liquid Theory.

Define the imaginary-time (Matsubara) Green's function for thermal averages.

Derive the periodic/antiperiodic boundary conditions in imaginary time for bosons/fermions.

Transform the Matsubara Green's function to discrete Matsubara frequencies.

Establish the Feynman rules at finite temperature.

Perform Matsubara frequency summations using contour integration techniques.

Describe the process of analytic continuation to obtain real-frequency spectral functions from imaginary-time data.

Calculate thermodynamic potentials from the finite-temperature Green's function.

Analyze the Hubbard model as a paradigm for competition between kinetic energy and on-site repulsion.

Solve the Hubbard model in the atomic limit (zero hopping) and identify the Mott insulating state.

Derive the Heisenberg model as the effective low-energy theory of the Hubbard model at half-filling in the large-U limit.

Introduce spin waves (magnons) as the elementary excitations of the Heisenberg ferromagnet.

Apply the Holstein-Primakoff transformation to map spin operators to bosonic operators.

Diagonalize the bosonic Hamiltonian to find the magnon dispersion relation.

Quantize lattice vibrations to obtain phonons and derive the electron-phonon interaction Hamiltonian.

Explain how phonon-mediated attraction between electrons can overcome Coulomb repulsion.

Analyze the Cooper instability of the Fermi sea in the presence of an attractive interaction.

Construct the Bardeen-Cooper-Schrieffer (BCS) variational ground state wavefunction.

Derive the BCS gap equation and solve for the superconducting energy gap.

Describe the Bogoliubov quasiparticle spectrum and its relation to the energy gap.

Relate the BCS theory to key experimental signatures like the Meissner effect and specific heat jump.

Implement the exact diagonalization (ED) method for small quantum systems like the Hubbard chain.

Formulate a quantum problem for the Variational Monte Carlo (VMC) method and optimize a trial wavefunction.

Explain the origin and implications of the fermion sign problem in Quantum Monte Carlo (QMC).

Outline the principles of Density Functional Theory (DFT) via the Hohenberg-Kohn theorems and the Kohn-Sham scheme.

Describe the Dynamical Mean-Field Theory (DMFT) approach, mapping a lattice problem to a self-consistent impurity problem.

Explain the structure of Matrix Product States (MPS) and their application in the Density Matrix Renormalization Group (DMRG) for 1D systems.