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 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.
The time evolution of the entanglement entropy is a key concept to understand the structure of a nonequilibrium quantum state. In a large class of models, such evolution can be understood in terms of a semiclassical picture of moving quasiparticles spreading the entanglement throughout the system. However, it is not yet known how the entanglement splits between the sectors of an internal local symmetry of a quantum many-body system. Here, guided by the examples of conformal field theories and free-fermion chains, we show that the quasiparticle picture can be adapted to this goal, leading to a general conjecture for the charged entropies whose Fourier transform gives the desired symmetry-resolved entanglement Sn(q). We point out two physically relevant effects that should be easily observed in atomic experiments: a delay time for the onset of Sn(q) which grows linearly with $|\mathrm{\ensuremath{\Delta}}q|$ (the difference between the charge q and its mean value) and an effective equipartition when $|\mathrm{\ensuremath{\Delta}}q|$ is much smaller than the subsystem size.
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.
Entanglement and symmetries are two pillars of our understanding of modern Physics, but surprisingly the study of the interplay between these two fundamental concepts has become the subject of an intense research area only in the last few years. In particular, the resolution of entanglement in the various symmetry sectors of a quantum many-body system with a global conserved charge gained a lot of interest within the entanglement community. As shown in some experimental papers, the study of this symmetry resolved entanglement is necessary for a better understanding of the non-equilibrium dynamics of quantum many-body systems. This Thesis is devoted to the study of the symmetry resolution of the entanglement measures in free fermionic systems and Conformal Field Theories. It is organised as follows. In Chap. 1 we give an overview of the main definitions and concepts related to the entanglement and its measures in quantum mechanics and quantum field theory. Then, we specialize to the case of conformal field theories, to derive expressions of the entanglement entropy and negativity in all the cases that will be examined in the second part of the Thesis. We finally illustrate the main free-fermionic techniques that will be used to perform the analytical as well as numerical lattice calculations. In Chap. 2, we consider a 1-dimensional system of free fermions hopping on a lattice, in which the conserved U(1) charge is the total particle number. We evaluate the symmetry resolved R'enyi and entanglement entropies determining exactly all the non-universal constants and logarithmic corrections to the scaling, thanks to the so-called Fisher-Hartwig conjecture. In Chap. 3, we study the symmetry resolved entanglement entropies in 1-dimensional systems in presence of boundaries, considering the cases of an interval starting from the boundary and disconnected from it. We evaluate the charged and symmetry resolved entropies for conformal invariant theories and a semi-infinite chain of free fermions, finding entanglement equipartition. Using a generalization of the lattice calculations performed in Chap. 2, we also evaluate the first term breaking equipartition for the free fermionic chain. In Chap. 4, we study the symmetry resolution of the entanglement negativity in free fermionic systems, which admits a decomposition in terms of the charge imbalance between the two subsystems. We use conformal field theory to derive the universal behavior of the charge imbalance resolved negativity for a free Dirac field at finite temperature and size. For the lattice version of the Dirac filed, we also determine exactly the model-dependent terms using the Fisher-Hartwig conjecture. We find that at leading order the charge imbalance resolved negativity verifies equipartition. For the lattice model, we also evaluate the first corrections to this equipartition. In Chap. 5, we study the time evolution of the symmetry resolved entanglement entropy and a suitably defined symmetry resolved mutual information in a system of free fermions after a quench. In the context of free fermions and conformal field theories, we are able to provide analytical results for the time-dependent charged and resolved entropies, and shed light on two new physical phenomena. First, the symmetry resolved R'enyi entropies start evolving with a calculable time delay that depends on the charge sector. Second, there is an effective equipartition of the entanglement in the limit of large subsystem size. For free fermions, we also find that effective equipartition is verified by the mutual information. We expect that these phenomena occur in more general quantum systems, as they can be understood from the quasiparticle picture for the spreading of entanglement. In the Appendix, we give an overview of conformal field theory, reporting the main concepts and formulas that are needed for a more complete understanding of the previous Chapters.
Abstract We consider the effect of projective measurements on the quench dynamics of the bipartite entanglement entropy in one-dimensional free fermionic systems. In our protocol, we consider projective measurements of a U (1) conserved charge, the particle number, on some large subsystem, and study the entanglement entropies between the same subsystem and its complement. We compare the dynamics emanating from two classes of initial states, one which is an eigenstate of the charge and another which is not. Moreover, we consider the effects of a single measurement as well as multiple which are periodically performed. Using the quasiparticle picture, we obtain analytic expressions for the behaviour of the entanglement, which admit a transparent physical interpretation. In general, we find that measurements introduce two distinct types of corrections to the entanglement, which can be interpreted separately as classical and quantum contributions. The classical contribution is independent of the measurement outcome and scales logarithmically with the variance of the charge distribution. In contrast, the quantum contribution depends on the specific measurement outcome and can be significant for individual realisations; however, it becomes negligible when averaged over all possible outcomes. Our expressions reduce to previously known results for symmetry-resolved entanglement and full counting statistics in some relevant limits, and are confirmed by an exact calculation performed on the Néel initial state.
We investigate symmetry-resolved entanglement in non-relativistic quantum field theories, including complex Lifshitz scalar chains and Lifshitz fermionic models. Using charged moments and the correlator method, we compute symmetry-resolved Renyi and von Neumann entropies and analyze their dependence on subsystem size, charge, mass, and the dynamical exponent z. Our results reveal distinct features of non-relativistic entanglement. In Lifshitz scalar theories, approximate equipartition among charge sectors emerges in the large-z regime, with configurational entropy dominating, whereas Lifshitz fermionic models exhibit genuine equipartition only in the relativistic limit, with fluctuation entropy prevailing. These findings highlight a rich interplay between conserved charges, subsystem size, mass, and dynamical scaling, and provide a framework to explore operationally accessible entanglement in non-relativistic systems. Our study offers insights relevant to experimental platforms such as cold atom setups and mesoscopic systems, where particle-number-resolved measurements can probe symmetry-resolved entanglement.
A bstract We investigate symmetry-resolved entanglement in non-relativistic quantum field theories, including complex Lifshitz scalar chains and Lifshitz fermionic models. Using charged moments and the correlator method, we compute symmetry-resolved Rényi and von Neumann entropies and analyze their dependence on subsystem size, charge, mass, and the dynamical exponent z . Our results reveal distinct features of non-relativistic entanglement. In Lifshitz scalar theories, approximate equipartition among charge sectors emerges in the large- z regime, with configurational entropy dominating, whereas Lifshitz fermionic models exhibit genuine equipartition only in the relativistic limit, with fluctuation entropy prevailing. These findings highlight a rich interplay between conserved charges, subsystem size, mass, and dynamical scaling, and provide a framework to explore operationally accessible entanglement in non-relativistic systems. Our study offers insights relevant to experimental platforms such as cold atom setups and mesoscopic systems, where particle-number-resolved measurements can probe symmetry-resolved entanglement.