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Return amplitude after a quantum quench in the XY chain
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Abstract
We determine an exact formula for the transition amplitude between any two arbitrary eigenstates of the local z-magnetization operators in the quantum XY chain. We further use this formula to obtain an analytical expression for the return amplitude of fully polarized states and the N\'eel state on a ring of length L. Then, we investigate finite-size effects in the return amplitude: in particular quasi-particle interference halfway along the ring, a phenomenon that has been dubbed traversal~\cite{FE2016}. We show that the traversal time and the features of the return amplitude at the traversal time depend on the initial state and on the parity of L. Finally, we briefly discuss non-analyticities in time of the decay rates in the thermodynamic limit , which are known as dynamical phase transitions.
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We study the statistics of the work done on a quantum critical system by quenching a control parameter in the Hamiltonian. We elucidate the relation between the probability distribution of the work and the Loschmidt echo, a quantity emerging usually in the context of dephasing. Using this connection we characterize the statistics of the work done on a quantum Ising chain by quenching locally or globally the transverse field. We show that for local quenches starting at criticality the probability distribution of the work displays an interesting edge singularity.
We study the transition of a quantum system from a pure state to a mixed one, which is induced by the quantum criticality of the surrounding system E coupled to it. To characterize this transition quantitatively, we carefully examine the behavior of the Loschmidt echo (LE) of E modeled as an Ising model in a transverse field, which behaves as a measuring apparatus in quantum measurement. It is found that the quantum critical behavior of E strongly affects its capability of enhancing the decay of LE: near the critical value of the transverse field entailing the happening of quantum phase transition, the off-diagonal elements of the reduced density matrix describing S vanish sharply.
We show that the time dependence of correlation functions in an extended quantum system in d dimensions, which is prepared in the ground state of some Hamiltonian and then evolves without dissipation according to some other Hamiltonian, may be extracted using methods of boundary critical phenomena in d + 1 dimensions. For d = 1 particularly powerful results are available using conformal field theory. These are checked against those available from solvable models. They may be explained in terms of a picture, valid more generally, whereby quasiparticles, entangled over regions of the order of the correlation length in the initial state, then propagate classically through the system.
In this contribution we review the theory of integrability of quantum systems in one spatial dimension. We introduce the basic concepts such as the Yang-Baxter equation, commuting currents, and the algebraic Bethe ansatz. Quite extensively we present the treatment of integrable quantum systems at finite temperature on the basis of a lattice path integral formulation and a suitable transfer matrix approach (quantum transfer matrix). The general method is carried out for the seminal model of the spin-1/2 XXZ chain for which thermodynamic properties like specific heat, magnetic susceptibility and the finite temperature Drude weight of the thermal conductivity are derived.
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