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Non equilibrium dynamics of quantum systems : the Loschmidt echo
Eric Vernier (SISSA, Trieste)
The non-equilibrium dynamics of quantum many-body systems has attracted a large interest over the last decade, prompted by formidable advances in cold-atomic experiments. While much progress has been done in understanding the relaxation mechanisms of physical observables and the characterization of the stationary state following, for instance, a quantum quench (where an isolated system is let evolve after one or several parameters have been suddenly changed), very few analytical results exist about the full time dynamics despite the existence of prototypical integrable models. Indeed, the time dynamics involves contributions for arbitrarily excited eigenstates of the Hamiltonian, making calculations prohibitively difficult.
In this talk I will present some progress made recently in this direction (based on the preprint arXiv:1611.06126), namely an analytical computation of the Loschmidt echo, which measures the overlap between the state of the system at a given time and its initial state, for various types of quenches in the Heisenberg XXZ spin chain. The latter has attracted a renewed interest recently in the context of dynamical phase transitions, which it signals through its non-analyticities as a function of the time. Using a reformulation of the problem in terms of an auxiliary boundary quantum transfer matrix and using an infinite set of functional relations, we write the Loschmidt echo as the solution of an infinite set of Non Linear Integral Equations, which allows for its exact determination at arbitrarily large time. This method overcomes the time limitations experienced by numerical approaches, and may serve as a basis for the computation of other physical observables.