The quantum Ising chain for beginners

TL;DR

This paper introduces fundamental techniques for analyzing quantum Ising chains, including transformations and correlation calculations, aimed at students and non-experts to understand ground states, excitations, and entanglement.

Contribution

It provides a beginner-friendly review of methods like Jordan-Wigner transformation and their applications to quantum Ising chains, including disordered systems.

Findings

Analysis of ground state and excitations

Methods for calculating dynamical quantities

Insights into entanglement entropy

Abstract

We present here various techniques to work with clean and disordered quantum Ising chains, for the benefit of students and non-experts. Starting from the Jordan-Wigner transformation, which maps spin-1/2 systems into fermionic ones, we review some of the basic approaches to deal with the superconducting correlations that naturally emerge in this context. In particular, we analyse the form of the ground state and excitations of the model, relating them to the symmetry-breaking physics, and illustrate aspects connected to calculating dynamical quantities, thermal averages, correlation functions and entanglement entropy. A few problems provide simple applications of the techniques.