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We study three aspects of work statistics in the context of the fluctuationtheorem for the quantum spin chains by numerical methods based onmatrix-product states. First, we elaborate that the work done on the spin-chainby a sudden quench can be used to characterize the quantum phase transitions(QPT). We further obtain the numerical results to demonstrate its capability ofcharacterizing the QPT of both Landau-Ginzbrug types, such as the Ising chain,or topological types, such as the Haldane chain. Second, we propose to use thefluctuation theorem, such as Jarzynski's equality, which relates the real-timecorrelator to the ratio of the thermal partition functions, as a benchmarkindicator for the numerical real-time evolving methods. Third, we study thepassivity of ground and thermal states of quantum spin chains under some cyclicimpulse processes. We show that the passivity of thermal states and groundstates under the hermitian actions are ensured by the second laws andvariational principles, respectively, and also verify it by numericalcalculations. Besides, we also consider the passivity of ground states undernon-hermitian actions, for which the variational principle cannot be applied.Despite that, we find no violation of passivity from our numerical results forall the cases considered in both Ising-like and Haldane-like chains.

Work statistics for Quantum Spin Chains: characterizing quantum phase  transitions, benchmarking time evolution, and examining passivity of quantum  states