A bstract Quantum one-dimensional systems in their ordered phase admit kinks as elementary excitations above their symmetry-broken vacua. While the scattering properties of the kinks resemble those of quasiparticles, they have distinct locality features that are manifest in their entanglement content. In this work, we study the entanglement entropy of kink excitations. We first present detailed calculations for specific states of a spin-1/2 chain to highlight the salient features of these excitations. Second, we provide a field-theoretic framework based on the algebraic relations between the twist fields and the semilocal fields associated with the excitations, and we compute the Rényi entropies in this framework. We obtain universal predictions for the entropy difference between the excited states with a finite number of kinks and the symmetry-broken ground states, which do not depend on the microscopic details of the model in the limit of large regions. Finally, we discuss some consequences of the Kramers-Wannier duality, which relates the ordered and disordered phases of the Ising model, and we explain why, counterintuitively, no explicit relations between those phases are found at the level of entanglement.
A bstract Quantum one-dimensional systems in their ordered phase admit kinks as elementary excitations above their symmetry-broken vacua. While the scattering properties of the kinks resemble those of quasiparticles, they have distinct locality features that are manifest in their entanglement content. In this work, we study the entanglement entropy of kink excitations. We first present detailed calculations for specific states of a spin-1/2 chain to highlight the salient features of these excitations. Second, we provide a field-theoretic framework based on the algebraic relations between the twist fields and the semilocal fields associated with the excitations, and we compute the Rényi entropies in this framework. We obtain universal predictions for the entropy difference between the excited states with a finite number of kinks and the symmetry-broken ground states, which do not depend on the microscopic details of the model in the limit of large regions. Finally, we discuss some consequences of the Kramers-Wannier duality, which relates the ordered and disordered phases of the Ising model, and we explain why, counterintuitively, no explicit relations between those phases are found at the level of entanglement.
We investigate the quantum entanglement content of quasiparticle excitations in extended many-body systems. We show that such excitations give an additive contribution to the bipartite von Neumann and Rényi entanglement entropies that takes a simple, universal form. It is largely independent of the momenta and masses of the excitations and of the geometry, dimension, and connectedness of the entanglement region. The result has a natural quantum information theoretic interpretation as the entanglement of a state where each quasiparticle is associated with two qubits representing their presence within and without the entanglement region, taking into account quantum (in)distinguishability. This applies to any excited state composed of finite numbers of quasiparticles with finite de Broglie wavelengths or finite intrinsic correlation length. This includes particle excitations in massive quantum field theory and gapped lattice systems, and certain highly excited states in conformal field theory and gapless models. We derive this result analytically in one-dimensional massive bosonic and fermionic free field theories and for simple setups in higher dimensions. We provide numerical evidence for the harmonic chain and the two-dimensional harmonic lattice in all regimes where the conditions above apply. Finally, we provide supporting calculations for integrable spin chain models and other interacting cases without particle production. Our results point to new possibilities for creating entangled states using many-body quantum systems.
Quantum systems display counterintuitive features, escaping our everyday-life intuition, whose origin is related to entanglement. This phenomenon arises when different parts of the same systems display correlations, which cannot be understood via classical mechanics, and they are dubbed as quantum. To characterize the correlations above, it is necessary to find good entanglement measures which are able to capture quantitatively the quantum features of a given system. This motivates the investigation and the computation of quantities such as the von Neumann entropy of regions, which provides information about how a system is entangled. Our focus is quantum one-dimensional systems close to their critical point, where quantum correlations are enhanced, and a characterization via Quantum Field Theory (QFT) can be provided. In particular, we aim to understand the role of symmetries and localized defects on the entanglement properties. Those aspects display an intimate connection. First, via replica trick, the entropy is expressed via a replica permutation symmetry. Then, single defects break symmetries that are present in the corresponding homogeneous systems. Due to these reasons, the characterization of symmetry actions on subsystems can be viewed as a unifying framework for quantifying entanglement. We split this thesis into two parts. The first part is devoted to the symmetry-resolved entanglement. We first compute it in massive integrable field theories, giving analytical predictions for free fermions/bosons and the 3-state Potts model. For instance, we characterize the universal corrections to area law via form factor bootstrap. We also study excited states of field theories, and we characterize both low-energy states of Conformal Field Theory (CFT) and quasi-particle states. The second part deals with quantum correlations across defects. We provide a CFT framework to compute Rényi entropy and negativity in the ground state of conformal junctions, giving analytical predictions for free fermions and bosons. We find logarithmic growth of the entanglement measures above, and we relate the universal prefactor to the boundary conditions, encoded in a boundary scattering matrix. Then, we focus on the entanglement dynamics, and we study the time evolution of correlations for two paradigmatic classes of quench protocols. For a global quench, we find a linear growth of entropy across the defect whose slope is computed both in CFT, for thermal states, and for free fermions, via a modification of the quasi-particle picture. For a domain wall melting protocol, also known as inhomogeneous quench, we find that, while local observables are easily understood via hydrodynamics and local generalized Gibbs ensembles, the same is not true for the entropy. In contrast, by employing the quasi-particle picture, we provide precise predictions for the dynamics of entanglement, revealing the influence of long-range correlations between points that are specular with respect to the defect.
The Rényi entanglement entropies of quasiparticle excitations in the many-body gapped systems show a remarkable universal picture which is independent of the model, the quasiparticle momenta, and the connectedness of the subsystem. In this letter we calculate exactly the single-interval and double-interval Rényi entanglement entropies of quasiparticle excitations in the many-body gapped fermions, bosons, and XY chains and find additional contributions to the universal Rényi entanglement entropy in the excited states with quasiparticles of different momenta. The additional terms are different in different models, depend on the momentum differences of the quasiparticles, and are different for the single interval and the double interval. We derive the analytical Rényi entanglement entropy in the extremely gapped limit, matching perfectly the numerical results as long as either the intrinsic correlation length of the model or all the de Broglie wavelengths of the quasiparticles are small. When the momentum difference of any pair of distinct quasiparticles is small, the additional terms are non-negligible. We argue that the derived formulas can be applied to a wider range of models than those discussed here.
Abstract The excess entanglement resulting from exciting a finite number of quasiparticles above the ground state of a free integrable quantum field theory has been investigated quite extensively in the literature. It has been found that it takes a very simple form, depending only on the number of excitations and their statistics. There is now mounting evidence that such formulae also apply to interacting and even higher-dimensional quantum theories. In this paper we study the entanglement content of such zero-density excited states focusing on the symmetry resolved entanglement, that is on 1+1D quantum field theories that possess an internal symmetry. The ratio of charged moments between the excited and grounds states, from which the symmetry resolved entanglement entropy can be obtained, takes a very simple and universal form, which in addition to the number and statistics of the excitations, now depends also on the symmetry charge. Using form factor techniques, we obtain both the ratio of moments and the symmetry resolved entanglement entropies in complex free theories which possess U(1) symmetry. The same formulae are found for simple qubit states.
We study entanglement via the subsystem purity relative to bipartitions of arbitrary excited states in (1+1)-dimensional conformal field theory, equivalent to the scaling limit of one-dimensional quantum critical systems. We compute the exact subpurity as a function of the relative subsystem size for numerous excited states in the Ising and three-state Potts models. We find that it decays exponentially when the system and the subsystem sizes are comparable until a saturation limit is reached near half-partitioning, signaling that excited states are maximally entangled. The exponential behavior translates into extensivity for the second R\'enyi entropy. Since the coefficient of this linear law depends only on the excitation energy, this result shows an interesting relationship between energy and quantum information and elucidates the role of microscopic details.
We consider two measures of entanglement, the logarithmic negativity, and the entanglement entropy, between regions of space in excited states of many-body systems formed by a finite number of particle excitations. In Papers I and II [O. A. Castro-Alvaredo et al., J. High Energy Phys. 2018(10), 39; ibid., e-print arXiv:1904.01035 (2019)], it has been shown in one-dimensional free-particle models that, in the limit of large system’s and regions’ sizes, the contribution from the particles is given by the entanglement of natural qubit states, representing the uniform distribution of particles in space. We show that the replica logarithmic negativity and Rényi entanglement entropy of such qubit states are equal to the partition functions of certain graphs, formed out of Wick pairings, which encode the topology of the manifold induced by permutation twist fields. Using this new connection, we provide a general proof, in the massive free boson model, which the qubit result holds in any dimensionality and for any regions’ shapes and topology. The proof is based on clustering and the permutation-twist exchange relations and is potentially generalizable to other situations, such as lattice models, particle and hole excitations above generalized Gibbs ensembles, and interacting integrable models.