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Quantum phase transitions happen when ground state properties of a quantum system can be fundamentally changed by a small variation in an external field. They provide some of the most striking examples of the richness of many-body quantum physics. The most popular examples of quantum phase transitions include a ferromagnet-paramagnet transition
in spin systems and a Mott insulator-superfluid transition in cold atom systems. I will discuss two recent developments in the theory of quantum phase transitions.
Firstly, I will present the fidelity approach to quantum phase transitions, where some equilibrium information about a quantum phase transition is obtained from the overlap between two ground states. This is a simple approach allowing for finding both the critical point separating two phases and the critical exponent of the system describing divergence of the correlation length near the critical point.
Secondly, I will present a simple approach to non-equilibrium quantum phase transitions, i.e., quantum phase transitions happening as a result of finite-rate driving of the system from one phase to another. Such driving results in creation of topological defects during the symmetry-breaking transitions (vortices, kinks, etc.). The theory that I will describe
relates the number of excitations to the quench rate and the equilibrium properties of the system (critical exponents).
I will provide several examples of physical systems where these results might be studied.