Advanced condensed matter physics

Construct Brillouin zones for common 2D and 3D lattices and identify high-symmetry points.

Set up and solve the tight-binding Hamiltonian for s-orbitals on 1D and 2D square lattices to derive band dispersions.

Derive the band structure of a 2D honeycomb lattice and analyze the emergence of Dirac points near the Fermi level.

Implement numerical diagonalization of a tight-binding Hamiltonian to compute and plot the band structure of a given lattice.

Calculate the electronic density of states (DOS) from a given band dispersion, identifying van Hove singularities.

Distinguish between metals, insulators, and semiconductors based on band filling and the calculated DOS.

Contrast the origins and optical consequences of direct and indirect bandgaps in semiconductors.

Derive the effective mass approximation from the band curvature near a band extremum.

Calculate intrinsic and extrinsic carrier concentrations in semiconductors as a function of temperature and doping.

Analyze the formation of a depletion region and built-in potential in a p-n junction by solving the Poisson equation.

Apply the Boltzmann transport equation in the relaxation time approximation to derive electrical and thermal conductivity.

Calculate the Seebeck coefficient and explain its physical origin in the context of thermoelectric devices.

Compute the Hall coefficient and magnetoresistance for single and multi-band systems.

Set up the dynamical matrix for a 1D diatomic chain and solve for the acoustic and optical phonon dispersion relations.

Generalize the concept of the dynamical matrix to a 3D monatomic crystal and identify transverse and longitudinal modes.

Apply second quantization to express the lattice Hamiltonian in terms of phonon creation and annihilation operators.

Calculate the phonon density of states and derive the general expression for lattice specific heat.

Compare the Einstein and Debye models for lattice specific heat, deriving the T³ dependence in the low-temperature limit for the latter.

Relate phonon dispersion curves to experimental data from inelastic neutron or X-ray scattering.

Derive the electron-phonon interaction Hamiltonian using the deformation potential approximation.

Derive the Fröhlich Hamiltonian for electron-longitudinal optical (LO) phonon coupling in polar materials.

Use Fermi's golden rule to calculate the scattering rate of electrons by phonons.

Derive the temperature dependence of phonon-limited resistivity in the high-temperature (T > Θ_D) and low-temperature (T < Θ_D) limits.

Explain the formation of a polaron and calculate the effective mass renormalization in the weak-coupling limit.

Analyze the Kohn anomaly in phonon dispersion as a signature of strong electron-phonon coupling.

Derive the frequency-dependent dielectric function for metals and doped semiconductors using the Drude model.

Calculate the optical absorption coefficient for a semiconductor with a direct bandgap using Fermi's golden rule.

Apply the Kramers-Kronig relations to connect the real and imaginary parts of the dielectric function.

Model an exciton as a hydrogenic bound state of an electron-hole pair and calculate its binding energy and Bohr radius.

Interpret photoluminescence spectra by identifying features corresponding to band-to-band, excitonic, and defect-related transitions.

Explain the physical mechanism of second harmonic generation (SHG) and its dependence on crystal symmetry.

Express single-particle and two-particle operators in the second quantization formalism for fermions.

Derive the Hartree-Fock equations from a variational principle applied to a many-electron Hamiltonian.

Define the single-particle Green's function and relate its poles to quasiparticle excitations.

Introduce the concept of dielectric screening and the reducible polarization bubble within linear response theory.

Derive the effective interaction within the Random Phase Approximation (RPA) by summing bubble diagrams.

Apply the RPA framework to derive the plasmon dispersion relation for a 3D electron gas.

Derive the Lehmann representation of the Green's function and connect the spectral function to the density of states.

Relate the single-particle spectral function to angle-resolved photoemission spectroscopy (ARPES) intensity maps.

Formulate the Feynman rules for diagrammatic perturbation theory of the Green's function in the imaginary-time formalism.

Calculate the first-order self-energy for electron-impurity and electron-electron interactions.

Use the Dyson equation to relate the bare and dressed Green's functions via the self-energy.

Extract quasiparticle energies and lifetimes from the real and imaginary parts of the self-energy.

Derive the Heisenberg exchange Hamiltonian as an effective low-energy model for interacting localized spins.

Apply mean-field theory to the Heisenberg model to derive the Curie-Weiss law for magnetic susceptibility.

Contrast the spin-wave dispersion in an antiferromagnet with that of a ferromagnet.

Explain the mechanism of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between magnetic impurities in a metal.

Describe the Kondo effect and its characteristic logarithmic temperature dependence of resistivity.

Derive the single-band Hubbard model from a tight-binding model with on-site Coulomb repulsion.

Analyze the Hubbard model in the atomic limit (t=0) to explain the formation of a Mott insulator.

Apply a mean-field approximation to the Hubbard model to derive the Stoner criterion for itinerant ferromagnetism.

Describe the concept of a Mott metal-insulator transition and its experimental signatures.

Formulate the self-consistent loop of Dynamical Mean-Field Theory (DMFT) by mapping the lattice model to a quantum impurity problem.

Interpret a typical DMFT spectral function for a metal, a correlated metal, and a Mott insulator.

Describe the phenomenology of superconductivity, including zero resistance, the Meissner effect, and flux quantization.

Apply Ginzburg-Landau theory to define the coherence length and penetration depth, and distinguish Type I and Type II superconductors.

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

Formulate the BCS ground state and derive the self-consistency equation for the superconducting gap.

Calculate the density of states in a BCS superconductor and relate it to scanning tunneling spectroscopy (STS) measurements.

Explain the mechanism of Andreev reflection at a normal-superconductor interface.

Contrast the pairing symmetry of unconventional d-wave superconductors with conventional s-wave pairing and describe experimental probes like ARPES.

Define the Berry phase and calculate it for a spin-1/2 in a rotating magnetic field.

Derive the Berry connection and Berry curvature for a general two-band model.

Calculate the Chern number as the integral of the Berry curvature over the Brillouin zone.

Solve for the Landau level spectrum of a 2D electron gas in a perpendicular magnetic field.

Explain the origin of quantized Hall conductivity in the Integer Quantum Hall Effect (IQHE) using the bulk-boundary correspondence.

Relate the quantized Hall conductivity to the sum of Chern numbers of occupied bands (TKNN formula).

Analyze the Haldane model on a honeycomb lattice as a prototype of a Chern insulator with a non-zero quantum Hall effect at zero magnetic field.

Explain how time-reversal symmetry leads to Kramers' degeneracy in systems with half-integer spin.

Define the Z₂ topological invariant for time-reversal invariant systems.

Analyze the Kane-Mele model to distinguish between a trivial insulator and a Quantum Spin Hall (QSH) insulator.

Describe the spin-momentum locking of helical edge states in a QSH insulator.

Explain the bulk-boundary correspondence for 2D and 3D topological insulators.

Describe the bulk band structure and surface Fermi arcs characteristic of a Weyl semimetal.

Contrast the stability and properties of Weyl and Dirac semimetals.