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arXiv:2403.07897 · The European Physical Journal B

Entanglement detection in postquench nonequilibrium states: thermal Gibbs vs. generalized Gibbs ensemble

Ferenc Iglói, Csaba Zoltán Király

Published Jun 1, 2024• 1 citation
arXivHTMLPDF

Abstract

Abstract We use entanglement witnesses related to the entanglement negativity of the state to detect entanglement in the XY chain in the postquench states in the thermodynamic limit after a quench when the parameters of the Hamiltonian are changed suddenly. The entanglement negativity is related to correlations, which in the postquench stationary state are described by a generalized Gibbs ensemble, in the ideal case. If, however, integrability breaking perturbations are present, the system is expected to thermalize. Here, we compare the nearest-neighbor entanglement in the two circumstances. Graphical Abstract Postquench states after a sudden quench protocol $$(h_0, \gamma ) \rightarrow (h, \gamma )$$ ( h 0 , γ ) → ( h , γ ) in the XY chain.

DOI
10.1140/epjb/s10051-024-00715-2

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•Dec 21, 2023•0 citations
Abstract We use entanglement witnesses related to the entanglement negativity of the state to detect entanglement in the XY chain in the postquench states in the thermodynamic limit after a quench when the parameters of the Hamiltonian are changed suddenly. The entanglement negativity is related to correlations, which in the postquench stationary state are described by a generalized Gibbs ensemble, in the ideal case. If, however, integrability breaking perturbations are present, the system is expected to thermalize. Here, we compare the nearest-neighbor entanglement in the two circumstances. Graphical Abstract Postquench states after a sudden quench protocol (h0,γ)→(h,γ)(h_0, \gamma ) \rightarrow (h, \gamma )(h0​,γ)→(h,γ) ( h 0 , γ ) → ( h , γ ) in the XY chain.
We use entanglement witnesses to detect entanglement in the XYXYXY chain in thermal equilibrium and determine the temperature bound below which the state is detected as entangled. We consider the entanglement witness based on the Hamiltonian. Such a witness detects a state as entangled if its energy is smaller than the energy of separable states. We also consider a family of entanglement witnesses related to the entanglement negativity of the state. We test the witnesses in infinite and finite systems. We study how the temperature bounds obtained are influenced by a quantum phase transition or a disorder line in the ground state. Very strong finite-size corrections are observed in the ordered phase due to the presence of a quasidegenerate excitation. We also study the postquench states in the thermodynamic limit after a quench when the parameters of the Hamiltonian are changed suddenly. In the case of the Ising model, we find that the mixed postquench state is detected as entangled by the two methods if the parameters of the Hamiltonian before and after the quench are close to each other. We find that the two witnesses give qualitatively similar results, showing that energy-based entanglement witnesses are efficient in detecting the nearest-neighbor entanglement in spin chains in various circumstances. For other XY models, we find that the negativity-based witnesses also detect states in some parameter regions where the energy-based witness does not, in particular, if the quench is performed from the paramagnetic phase to the ferromagnetic phase and vice versa. The domains in parameter space corresponding to postquench states detected as entangled by the energy-based witness have been determined analytically, which stresses further the utility of our method.
We study the propagation of entanglement after quantum quenches in the nonintegrable paramagnetic quantum Ising spin chain. Tuning the parameters of the system, we observe a sudden increase in the entanglement production rate, which we show to be related to the appearance of new quasiparticle excitations in the postquench spectrum. We argue that the phenomenon is the nonequilibrium version of the well-known Gibbs paradox related to mixing entropy and demonstrate that its characteristics fit the expectations derived from the quantum resolution of the paradox in systems with a nontrivial quasiparticle spectrum.
Integrable quantum systems evolving from non-equilibrium initial states do not thermalize to conventional Gibbs ensembles (GE). Instead, at long times they relax to generalized Gibbs ensembles (GGEs), which incorporate the full set of local and quasi-local conserved quantities. While GGEs have been extensively studied in the literature, a quantitative analytic characterization of how different they are from ordinary GEs is still lacking. In this work, we address this question by employing the concept of athermality, which we define within quantum resource theory as the relative entropy between a given state and the closest thermal state. By means of integrability techniques we compute the athermality for several quantum quenches in paradigmatic integrable models, including the free XY spin chain, the interacting Lieb-Liniger model, the XXZ spin chain, and the harmonic chain. We find that often the athermality becomes anomalously small when the post-quench Hamiltonian is critical in its ground state, despite probing physics at a finite energy density. We also prove that it systematically develops a singularity at criticality, which is inherited from the entropy of the GGE.
Quantum quenches in the Dicke model were studied both in the thermodynamic limit and the finite systems. For the integrable situation in the thermodynamic limit, the generalized Gibbs ensemble can effectively describe the energy-level occupations for the quench within the normal phase, but it fails for the quench to the superradiant phase. For the finite systems which are considered non-integrable, the post quench systems were studied by comparing with the thermal ensembles. The canonical ensembles are directly available for the quench within the normal phase. With the increasing of the target coupling strength over the equilibrium phase transition critical point, sudden changes take place for the effective temperature and the distance to the thermal ensembles. The thermalization was also studied by comparing with the results of the microcanonical ensembles.
We consider dynamic (non-equal-time) correlation functions of local observables after a quantum quench. We show that, in the absence of long-range interactions in the final Hamiltonian, the dynamics is determined by the same ensemble that describes static (equal-time) correlations. For many integrable models, static correlation functions of local observables after a quantum quench relax to stationary values, which are described by a generalized Gibbs ensemble. The same generalized Gibbs ensemble then determines dynamic correlation functions, and the basic form of the fluctuation dissipation theorem holds, although the absorption and emission spectra are not simply related as in the thermal case. For quenches in the transverse field Ising chain, we derive explicit expressions for the time evolution of dynamic order parameter correlators after a quench.
In a recent study concerning the multipartite entanglement structure in states satisfying the eigenstate thermalization hypothesis (ETH), a measure known as the quantum Fisher information (QFI) was observed to detect differences in the entanglement content between thermal states of the same temperature; thus establishing a new, distinct way to distinguish between thermal states of different types (pure or mixed). In this work, we try to extend the previous analysis to the class of integrable models–i.e., systems which do not thermalize in the conventional way. We start this investigation with the simplest case, that of noninteracting systems. We derive exact analytic expressions of the QFI for a state described by the generalized Gibbs ensemble (GGE) and for a quenched pure state. We then show through various examples that this hierarchy of entanglement content persists