Exploring Quantum Many Body Systems with Modern Numerical Tools

In this course the programming language of choice is Julia. (As interactive as Python, but as fast as C.) No previous experience with Julia is assumed and we will pick up the syntaxes along the way. Of course, some familiarity with at least one programming language (Python, C, C++, Fortran, etc.) would be a big plus.

Topics ( 1st semester )

  • Introduction

  • Gentle introduction to Julia language

    • REPL, Pluto / Jupyter NoteBooks, and Scripts
    • program flow, style guide
    • basic data types, struct, etc.
    • mutables and immutables: arrays, tuples
    • iterators and generators
    • scope and typeof
    • type instability
    • tips on writing efficient code

  • Basic skills of applied computational physics

    • differentiation: finite difference, complex-step, ect.
    • integration: Gaussian quadrature
    • Monte Carlo Methods I
    • differential equations I
    • minimization
    • Linear Algebra I
    • fast Fourier transform

  • Quantum mechanics I

    • solution to the Schroedinger equations: bound states
    • momentum grid (matrix Hamiltonian) method
    • scattering problems: hard spheres, Born approximations, VPA etc.

  • Thermal field theory I

    • thermal Green’s functions
    • spectral functions
    • screening masses
    • optical potentials

  • Ginzburg-Landau theory

    • order parameter and theory of phase transitions
    • Legendre transform and thermodynamic functions
    • effective models of quarks and gluons
    • density functional theory

  • Spin models

    • Ising model
    • Potts model
    • Lanczos Algorithm VS Guided Random Walk

  • Simulating a Pure Gauge System

    • Z(2) gauge theory
    • non-abelian $SU(N_c)$ gauge theory

Further topics ( 2nd semester )

  • Further skills of applied computational physics

    • Iterations
    • Interpolations
    • Linear Algebra (II)
    • Monte Carlo Methods (II)
    • Further skills in solving ODE

  • Quantum mechanics II

    • scattering theory and resonances
    • dispersion relations
    • complex planes: poles, roots, cuts, and Riemann sheets
    • coupled channel models

  • Quantum field theory

    • the path integral
    • $\phi^4$ theory
    • zeta function regularization
    • non-abelian gauge theories
    • fermion fields and the Dirac equations
    • dynamical generation of mass and symmetry breaking

  • Thermal field theory II

    • real-time approaches
    • collective effects in medium
    • mean free time and transport coefficients

  • Thermal description of an interacting gas

    • Density of States (DoS)
    • N-body phase spaces
    • classical and quantum cluster / virial expansions
    • quasi-particle properties
    • interacting pion gas
    • molecular dynamics

  • Random Matrix Theory

    • SU(N), Haar measure and all that
    • Lie Algebra, root system
    • Matrix models

  • Special Topics (depending on time and interest)

    • percolation and universality
    • constituent quark model
    • additions of angular momenta
    • introduction to Schwinger Dyson equations
    • confinement models of QCD
    • non-perturbative methods in QFT
    • S-matrix treatment of non-ideal gases
    • neural networks