After successful completion of the course, students are able to:

1) Identify relevant mathematical structures and concepts in condensed matter problems, e.g. eivenvectors, eigenvalues and other linear algebra structures, correlation functions, inital value and boundary value problem of differential equations, Fourier transform.

2) Use discrete and continuous symmetries to assess the correctness of numerical results and to simplify physical problems

3) Construct Fock space of fermionic problems and apply it to simple systems with few degrees of freedom

4) Use Fourier transforms both in space and time/frequency domain

5) Understand basic approximations used in condensed matter physics, e.g. strong and weak coupling expansion, large-N expansion, embedding, stochastic simulations

6) Calculate magnetic susceptibility of 6-site Hubbard molecule as a function of temperature

Foundations of quantum theory: causal evolution and measurement, observables, wavefunctions, Hermitean operators, evolution operator and Hamiltonian

Schrodinger equation, stationary states, time evolution, Hemitean and unitary operators

Basic statistical physics, Boltzmann factor

Symmetry and eigenvalue degeneracy

Quantum theory of many particles: fermions and bosons, Pauli principle, commutation relations, Pauli principle

Lattice models, translational symmetry

Exact diagonalization, construction of Fock space and fermionic operators

Correlation functions and linear response to external perturbations

Non-interacting electrons in periodic solids, Bloch theorem

The concepts will be developed step by step on a pilot problem (6-site cluster). We will develop a code to calculate the discussed observables during the coure.

The course will take place in **online** format of **prerecorded lectures** + weekly zoom **discussion session. **This is because I am leaving TU Wien and will not physically present on a regular basis.

The language of the course is English.

A short 'Vorbesprechung' will take palce **in person** on October 3, 14:30 in my office in Freihaus (We'll meet in front of the yellow elevator on 9th floor)