I will present various techniques to work with quantum Ising chains. Starting from the Jordan-Wigner transformation, which maps spin-1/2 systems into fermionic ones, we will review some of the basic approaches to deal with the superconducting correlations that naturally emerge in this context. A few problems illustrating important aspects of the physics of this model will provide simple applications of the techniques.
More in detail, the course will consist of:
- A crash course in scientific programming with python and with jupyter notebooks
- The Jordan-Wigner transformation applied to one-dimensional spin-1/2 models
- Free-fermion models (Ising/XY)
- Translationally invariant and disordered Nambu formalism
- Time-dependent Bogoljubov-de Gennes equations and calculation of dynamical quantities
- Calculating thermal averages
- Calculating correlation functions involving Jordan-Wigner strings
- Calculating the entanglement entropy
The course will be strongly ``hands-on based'': collaborating in small groups, we should eventually construct a python library to calculate many properties of Ising spin chains.