The quantum Ising chain for beginners

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.

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• Uncertain Quantum Critical Metrology: From Single to Multi Parameter Sensing
• Taming quantum systems: A tutorial for using shortcuts-to-adiabaticity, quantum optimal control, and reinforcement learning
• Entanglement Transition due to particle losses in a monitored fermionic chain
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• Long-range entanglement and topological excitations
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• Time-dependent Neural Galerkin Method for Quantum Dynamics
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• Geodesic Algorithm for Unitary Gate Design with Time-Independent Hamiltonians
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