Syllabus

Condensed Matter courses

Relativistic and scalar relativistic pseudopotential theory
  • Norm conserving pseudopotentials
  • Ultrasoft pseudopotentials
  • Projector augmented-wave method
  • Dirac equation in spherical coordinates
  • Relativistic density functional theory
  • Scalar relativistic approximation
  • Fully relativistic projector augmented-wave method.
Strongly correlated electron systems
  • WHAT IS A STRONGLY CORRELATES SYSTEM?(1/2 lectures)
    • Generalities: Mott insulators and breakdown of band theory of solids
    • Which materials are strongly correlated?
    • d-orbitals: high Tc cuprates, manganites and many others
  • THE HUBBARD MODEL (2 lectures)
    • A reminder of the Hubbard model. Simple limits
    • Antiferromagnetism
      • Hartree-Fock approximation for the antiferromagnetic phase (*)
      • Strong-coupling: the t-J model
      • A little detour: Repulsion vs Repulsion: the BCS/BEC crossover and the antiferromagnetic phase of the Hubbard model
  • THE MOTT-HUBBARD TRANSITION (2 lectures)
    • Traditional methods
      • Strong-coupling methods: Hubbard I and Hubbard III (*)
      • The Gutzwiller wavefunction and the Brinkman-Rice transition
      • Slave bosons (*)
  • DYNAMICAL MEAN-FIELD THEORY (3 lectures)
    • Why a dynamical mean-field theory?
    • Deriving the DMFT equations: cavity method and the limit of infinite coordination
    • Mapping onto an Anderson Impurity Model
    • Solving the impurity model: exact numerical and approximate analytical tools
  • THE MOTT-HUBBARD TRANSITION IN DMFT (2 lectures)
    • Kondo physics and the Mott transition
    • Coexistence of insulating and metallic features
    • Antiferromagnetism
    • The phase diagram
  • BEYOND DYNAMICAL MEAN-FIELD THEORY(*)
    • Cluster methods: Cellular DMFT and Dynamical Cluster Approximation
    • d-wave superconductivity and the pseudogap.

(*) These topics might not be covered

Geometry and topology in electronic structure theory 
  • EARLY DISCOVERIES
    • Aharonov-Bohm
    • Conical intersections in molecules
    • Integer quantum Hall effect
  • GEOMETRY, PART I
    • Phases and distances in quantum mechanics
    • Connection, curvature, Berry phase
    • (First) Chern number
    • Quantum metric
    • NonAbelian theory
  • MANIFESTATIONS OF THE BERRY PHASE
    • Early discoveries reinterpreted
    • Anomalous Hall effect
    • Semiclassical transport (Niu)
    • Adiabatic approximation in a magnetic field
    • Quantum transport (Thouless)
    • Modern theory of polarization (independent electrons)
    • Polarization for a correlated wavefunction
  • QUANTUM METRIC
    • Marzari-Vanderbilt theory
    • Theory of the insulating state
    • Applications: band insulators
    • (integer) quantum Hall insulators
    • Anderson insulators
    • Mott insulators
    • Fractional quantum Hall effect
  • TOPOLOGY AND GEOMETRY, PART II
    • Some basic concepts in topology
    • Chern-Simons theory; second Chern number
  • TOPOLOGICAL INSULATORS
    • T-broken topological insulators (2d)
    • T-symmetric topological insulators (2 & 3d)
  • OTHER TOPICS
    • Modern theory of orbital magnetization
    • Magnetoelectrics: nontopological & topological responses

Courses in common with Statistical Physics

Statistical Physics of Cold Atoms
  •  introduction to superconductors: basic phenomena
  •  electrodynamics of superconductors: London equations, type-II superconductors- Ginzburg-Landau theory
  •  BCS-BEC crossover for dilute fermions
  •  fermions in optical lattices: Hubbard models
Lectures on “Many Body Physics out of Equilibrium”

The course is held by Dott. Alessandro Silva, who will follow the program below:

  • Why a non-equilibrium field theory is needed: failure of the fluctuation dissipation theorem, entropy production, and the Gell-Mann Low conjecture.
  • Open non-equilibrium systems: Keldysh Green’s functions, nonequilibrium distributions, Quantum kinetic equations and the Boltzmann limit.
  • Applications to transport in nanostructures: current-voltage characteristics in interacting quantum dots, adiabatic quantum pumping.
  • Closed non-equilibrium systems: quantum quenches, entropy production, statistics of the work, Jarzinsky equalities and Tasaki-Crooks fluctuation theorems.
  • Applications to quantum critical systems: the Kibble-Zurek mechanism for defect production, scaling, quenches through quantum critical points and exact solution for the Ising model

Courses in common with Statistical and Biological Physics

Advanced Sampling Techniques, course program:
  • General introduction to the “rare events” problem.
    Introduction to the concepts of free energy, rate constant, mean-first-passage time, separation of time scales, committor distribution, etc.
  • Computing the free energy in complex systems:
    1. Thermodynamic integration and umbrella sampling techniques. Ferrenberg and Swendsen method for computing the density of states and weighted-histogram analysis.
    2. Free energies from non-equilibrium processes: the Jarzynski equality for the irreversible work.
    3. History-dependent reconstruction of the free energy: metadynamics and Wand-Landau sampling.
  • Techniques for computing the rate constants:
    1. Classical transition state theory. Kramers theory and Bennett-Chandler method for computing the recrossing corrections.
    2. Methods for finding the saddle point in complex potential energy surfaces: nudged elastic band, eigenvalue following and the dimer method.
    3. Path integral formulation of the rare event problem: transition path sampling. Methods for computing the committor distribution (finite-temperature string, etc.).
Simulation of Biomolecules, course program:
  • Classical force fields.
  • Equations of motion and integration schemes
  • Molecular dynamics in various ensembles.
  • Microreversibility, Crooks and Jarzinski fluctuation theorems.
  • Selected topics on molecular dynamics.