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arXiv:2205.02924 · Journal of Statistical Mechanics Theory and Experiment

Exact hydrodynamic description of symmetry-resolved Rényi entropies after a quantum quench

Stefano Scopa, Dávid X. Horváth

Published Aug 1, 2022• 31 citations
arXivHTMLPDF

Abstract

Abstract We investigate the non-equilibrium dynamics of the symmetry-resolved Rényi entropies in a one-dimensional gas of non-interacting spinless fermions by means of quantum generalised hydrodynamics, which recently allowed to obtain very accurate results for the total entanglement in inhomogeneous quench settings. Although our discussion is valid for any quench setting accessible with quantum generalised hydrodynamics, we focus on the case of a quantum gas initially prepared in a bipartite fashion and subsequently let evolve unitarily with a hopping Hamiltonian. For this system, we characterise the symmetry-resolved Rényi entropies as function of time t and of the entangling position x along the inhomogeneous profile. We observe an asymptotic logarithmic growth of the charged moments at half system and an asymptotic restoration of equipartition of entropy among symmetry sectors with deviations which are proportional to the square of the inverse of the total entropy.

DOI
10.1088/1742-5468/ac85eb

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Abstract We investigate the non-equilibrium dynamics of the symmetry-resolved Rényi entropies in a one-dimensional gas of non-interacting spinless fermions by means of quantum generalised hydrodynamics, which recently allowed to obtain very accurate results for the total entanglement in inhomogeneous quench settings. Although our discussion is valid for any quench setting accessible with quantum generalised hydrodynamics, we focus on the case of a quantum gas initially prepared in a bipartite fashion and subsequently let evolve unitarily with a hopping Hamiltonian. For this system, we characterise the symmetry-resolved Rényi entropies as function of time t and of the entangling position x along the inhomogeneous profile. We observe an asymptotic logarithmic growth of the charged moments at half system and an asymptotic restoration of equipartition of entropy among symmetry sectors with deviations which are proportional to the square of the inverse of the total entropy.
Abstract The study of the entanglement dynamics plays a fundamental role in understanding the behaviour of many-body quantum systems out of equilibrium. In the presence of a globally conserved charge, further insights are provided by the knowledge of the resolution of entanglement in the various symmetry sectors. Here, we carry on the program we initiated in Parez et al (2021 Phys. Rev. B 103 L041104), for the study of the time evolution of the symmetry resolved entanglement in free fermion systems. We complete and extend our derivations also by defining and quantifying a symmetry resolved mutual information. The entanglement entropies display a time delay that depends on the charge sector that we characterise exactly. Both entanglement entropies and mutual information show effective equipartition in the scaling limit of large time and subsystem size. Furthermore, we argue that the behaviour of the charged entropies can be quantitatively understood in the framework of the quasiparticle picture for the spreading of entanglement, and hence we expect that a proper adaptation of our results should apply to a large class of integrable systems. We also find that the number entropy grows logarithmically with time before saturating to a value proportional to the logarithm of the subsystem size.
Abstract We consider the symmetry resolved Rényi entropies in the one dimensional tight binding model, equivalent to the spin-1/2 XX chain in a magnetic field. We exploit the generalised Fisher–Hartwig conjecture to obtain the asymptotic behaviour of the entanglement entropies with a flux charge insertion at leading and subleading orders. The contributions are found to exhibit a rich structure of oscillatory behaviour. We then use these results to extract the symmetry resolved entanglement, determining exactly all the non-universal constants and logarithmic corrections to the scaling that are not accessible to the field theory approach. We also discuss how our results are generalised to a one-dimensional free fermi gas.
Abstract The presence of a global internal symmetry in a quantum many-body system is reflected in the fact that the entanglement between its subparts is endowed with an internal structure, namely it can be decomposed as a sum of contributions associated to each symmetry sector. The symmetry resolution of entanglement measures provides a formidable tool to probe the out-of-equilibrium dynamics of quantum systems. Here, we study the time evolution of charge-imbalance-resolved negativity after a global quench in the context of free-fermion systems, complementing former works for the symmetry-resolved entanglement entropy. We find that the charge-imbalance-resolved logarithmic negativity shows an effective equipartition in the scaling limit of large times and system size, with a perfect equipartition for early and infinite times. We also derive and conjecture a formula for the dynamics of the charged Rényi logarithmic negativities. We argue that our results can be understood in the framework of the quasiparticle picture for the entanglement dynamics, and provide a conjecture that we expect to be valid for generic integrable models.
A bstract We investigate the non-equilibrium time evolution of symmetry-resolved entanglement entropy (SREE) following an inhomogeneous quench in a critical one-dimensional free fermion system. Using conformal field theory, we derive an exact expression for the SREE and analyze its behavior. We find that, at leading order in the long-time limit, the SREE grows logarithmically as log t . While the equipartition of entanglement holds at leading order, we identify subleading corrections that break it. Our numerical simulations corroborate the analytical predictions with excellent agreement.
The large-scale behaviour of entanglement entropy in finite-density states, in and out of equilibrium, can be understood using the physical picture of particle pairs. However, the full theoretical origin of this picture is not fully established yet. In this work, we clarify this picture by investigating entanglement entropy using its connection with the large-deviation theory for thermodynamic and hydrodynamic fluctuations. We apply the universal framework of Ballistic Fluctuation Theory (BFT), based the Euler hydrodynamics of the model, to correlation functions of branch-point twist fields, the starting point for computing Rényi entanglement entropies within the replica approach. Focusing on free fermionic systems in order to illustrate the ideas, we show that both the equilibrium behavior and the dynamics of Rényi entanglement entropies can be fully derived from the BFT. In particular, we emphasise that long-range correlations develop after quantum quenches, and accounting for these explain the structure of the entanglement growth. We further show that this growth is related to fluctuations of charge transport, generalising to quantum quenches the relation between charge fluctuations and entanglement observed earlier. The general ideas we introduce suggest that the large-scale behaviour of entanglement has its origin within hydrodynamic fluctuations.
Abstract The large-scale properties of homogeneous states after quantum quenches in integrable systems have been successfully described by a semiclassical picture of moving quasiparticles. Here we consider the generalisation for the entanglement evolution after an inhomogeneous quench in noninteracting systems in the framework of generalised hydrodynamics. We focus on the protocol where two semi-infinite halves are initially prepared in different states and then joined together, showing that a proper generalisation of the quasiparticle picture leads to exact quantitative predictions. If the system is initially prepared in a quasistationary state, we find that the entanglement entropy is additive and it can be computed by means of generalised hydrodynamics. Conversely, additivity is lost when the initial state is not quasistationary; yet the entanglement entropy in the large-scale limit can be exactly predicted in the quasiparticle picture, provided that the initial state is low entangled.