Abstract In this paper we study the increment of the entanglement entropy and of the (replica) logarithmic negativity in a zero-density excited state of a free massive bosonic theory, compared to the ground state. This extends the work of two previous publications by the same authors. We consider the case of two disconnected regions and find that the change in the entanglement entropy depends only on the combined size of the regions and is independent of their connectivity. We subsequently generalize this result to any number of disconnected regions. For the replica negativity we find that its increment is a polynomial with integer coefficients depending only on the sizes of the two regions. The logarithmic negativity turns out to have a more complicated functional structure than its replica version, typically involving roots of polynomials on the sizes of the regions. We obtain our results by two methods already employed in previous work: from a qubit picture and by computing four-point functions of branch point twist fields in finite volume. We test our results against numerical simulations on a harmonic chain and find excellent agreement.
Abstract In this paper we study the increment of the entanglement entropy and of the (replica) logarithmic negativity in a zero-density excited state of a free massive bosonic theory, compared to the ground state. This extends the work of two previous publications by the same authors. We consider the case of two disconnected regions and find that the change in the entanglement entropy depends only on the combined size of the regions and is independent of their connectivity. We subsequently generalize this result to any number of disconnected regions. For the replica negativity we find that its increment is a polynomial with integer coefficients depending only on the sizes of the two regions. The logarithmic negativity turns out to have a more complicated functional structure than its replica version, typically involving roots of polynomials on the sizes of the regions. We obtain our results by two methods already employed in previous work: from a qubit picture and by computing four-point functions of branch point twist fields in finite volume. We test our results against numerical simulations on a harmonic chain and find excellent agreement.
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.
Abstract We evaluate the entanglement entropy of a single connected region in excited states of one-dimensional massive free theories with finite numbers of particles, in the limit of large volume and region length. For this purpose, we use finite-volume form factor expansions of branch-point twist field two-point functions. We find that the additive contribution to the entanglement due to the presence of particles has a simple “qubit” interpretation, and is largely independent of momenta: it only depends on the numbers of groups of particles with equal momenta. We conjecture that at large momenta, the same result holds for any volume and region lengths, including at small scales. We provide accurate numerical verifications.
In this thesis we study entanglement in one-dimensional critical quantum many-body systems and in particular we will focus on disconnected regions. Given a system in a pure state, to quantify the amount of entanglement between a multicomponent subsystem and the rest of the full system we can use as an entanglement measure the renowned entanglement entropy. However, if we are interested in the entanglement shared among the disconnected regions, the entanglement entropy fails to be a good quantifier. The reason is that the state of the subsystem is in general mixed once the rest is traced out, and the entanglement entropy is a good measure of entanglement only for pure states. A good measure of entanglement in mixed states is the logarithmic negativity, which is the quantitative version of Peres' criterion of separability. The main advantage of the negativity with respect to other entanglement measures is the simplicity of its definition in terms of the density matrix describing the quantum state. Since the definition does not require any variational calculus, it is much more easily computable than any other entanglement measure, and therefore we can obtain some results also in complicated settings such as many-body systems. We will be mostly interested in the configuration where the full system is divided into two parts, A and B. If we do not have access to the degrees of freedom in B, we can describe subsystem A through its reduced density matrix, where the degrees of freedom in B have been traced out. The residual subsystem in A will in general be left in a mixed state. Subsystem A is then divided again into two parts, A_1 and A_2, and we will be interested in the entanglement shared by these two components. Knowing the full density matrix, the logarithmic negativity can be easily computed from the eigenvalues of its partial transpose with respect to the degrees of freedom living in one of the two subsystems A_1 or A_2. However, the full density matrix of a many-body state is in general unaccessible, even numerically, since the size of the matrix grows exponentially with the size of the system. If we concentrate on critical systems whose low-energy physics can be described by a quantum field theory, we can resort to its powerful tools to compute the entanglement properties. In particular the main tool is the replica trick. The entanglement entropy is obtained from the moments of the reduced density matrix, while the negativity can be computed from the moments of the partial transpose. However, even the computation of the integer-order moments is not an easy task, and exact analytical results are known only for the simplest quantum field theories. Hence, throughout the thesis we will usually consider conformal field theories, which have an enhanced set of symmetries and therefore allow for some exact computations. In the Introduction, we will try to frame this work by briefly reviewing some of the main topics in the literature of quantum many-body systems and quantum field theories where entanglement plays a crucial role. We will also describe what are the main features that are requested to a `good' measure of entanglement, and we will review some of the main measures that have been considered so far in the context of quantum information. We will define the entanglement negativity and stress its main advantages, as well as its drawbacks. After reviewing some basic facts of conformal field theories, we will give some technical details on the computation of entanglement entropy and logarithmic negativity in general quantum field theories, which will be needed in the rest of the thesis. In Chap. 2 we will start the study of the entanglement of several disjoint disconnected regions by considering the entanglement that the union of these regions share with the rest of the system. We will compute the integer-order Rényi entropies of the subsystem from which the entanglement entropy can be obtained through the replica limit. Unfortunately, the exact analytic continuation to real order is still out of reach and therefore the entanglement entropy cannot be computed exactly. However numerical extrapolations allow to obtain some accurate estimates. In the remaining chapters we will focus on the entanglement shared between two non complementary disjoint regions, through the computations of the entanglement negativity in different settings. In Chap. 3, we will consider a global quantum quench starting from a conformal boundary state and evolving through a conformal evolution. A general formula for the mutual information and the logarithmic negativity for two adjacent and disjoint intervals is given in the spacetime limit. In Chap. 4 we focus on the XY chain and we recover a formula for the partial transpose as a sum of four fermionic auxiliary Gaussian density matrices, by generalizing some previous results for the reduced density matrix of spin systems and for the partial transpose of pure fermionc Gaussian states. Even if the computation of the negativity is still out of reach, we can obtain formulas for the integer-order moments of the partial transpose in terms of the correlation matrices relative to the two components and to the region in between them. In Chap. 5 we will study the entanglement negativity for a free Dirac fermion field. Starting from the lattice results, we will obtain a path integral representation of the partial transpose which can be easily generalized to the continuum. With this representation of the partial transpose, we can construct all its integer-order moments and compute them for the simple conformal field theory of the free fermion. Again, the computation of the negativity could not be accomplished due to technical difficulties in the analytic continuation to real order. The same computation has been extended also to the modular invariant Dirac fermion and the Ising model, and the obtained formulas coincide with the ones already present in the literature. This analysis draws an interesting connection between some terms appearing in the formulas for the moments of the partial transpose (as well as of the R\'enyi entropies) found for the lattice models and the ones found for the corresponding theories describing their scaling limit. Whenever possible, all the analytical results will be checked against numerical calculations performed on simple free chain models. In Chaps. 2 and 3 we will consider bosonic Gaussian states, specifically the harmonic chain in its ground state and out of equilibrium, while in Chaps. 4 and 5 we will consider fermionic systems, specifically the XX and Ising spin chains, and the tight-binding model. Finally, we will draw some conclusions and discuss some open problems.
Abstract We consider the logarithmic negativity, a measure of bipartite entanglement, in a general unitary 1 + 1-dimensional massive quantum field theory, not necessarily integrable. We compute the negativity between a finite region of length r and an adjacent semi-infinite region, and that between two semi-infinite regions separated by a distance r . We show that the former saturates to a finite value, and that the latter tends to zero, as . We show that in both cases, the leading corrections are exponential decays in r (described by modified Bessel functions) that are solely controlled by the mass spectrum of the model, independently of its scattering matrix. This implies that, like the entanglement entropy (EE), the logarithmic negativity displays a very high level of universality, allowing one to extract information about the mass spectrum. Further, a study of sub-leading terms shows that, unlike the EE, a large- r analysis of the negativity allows for the detection of bound states.
Abstract In a recent paper we studied the entanglement content of zero-density excited states in complex free quantum field theories, focusing on the symmetry resolved entanglement entropy (SREE). By zero-density states we mean states consisting of a fixed, finite number of excitations above the ground state in an infinite-volume system. The SREE is defined for theories that possess an internal symmetry and provides a measure of the contribution to the total entanglement of each symmetry sector. In our work, we showed that the ratio of Fourier-transforms of the SREEs (i.e. the ratio of charged moments) takes a very simple and universal form for these states, which depends only on the number, statistics and symmetry charge of the excitations as well as the relative size of the entanglement region with respect to the whole system’s size. In this paper we provide numerical evidence for our formulae by computing functions of the charged moments in two free lattice theories: a 1D Fermi gas and a complex harmonic chain. We also extend our results in two directions: by showing that they apply also to excited states of interacting theories (i.e. magnon states) and by developing a higher dimensional generalisation of the branch point twist field picture, leading to results in (interacting) higher-dimensional models.
We report on a systematic approach for the calculation of the negativity in the ground state of a one-dimensional quantum field theory. The partial transpose of the reduced density matrix of a subsystem A = A 1 ∪ A 2 is explicitly constructed as an imaginary-time path integral and from this the replicated traces are obtained. The logarithmic negativity is then the continuation to n → 1 of the traces of the even powers. For pure states, this procedure reproduces the known results. We then apply this method to conformally invariant field theories (CFTs) in several different physical situations for infinite and finite systems and without or with boundaries. In particular, in the case of two adjacent intervals of lengths ℓ 1 ,ℓ 2 in an infinite system, we derive the result ∼ ( c /4)ln(ℓ 1 ℓ 2 /(ℓ 1 + ℓ 2 )), where c is the central charge. For the more complicated case of two disjoint intervals, we show that the negativity depends only on the harmonic ratio of the four end points and so is manifestly scale invariant. We explicitly calculate the scale invariant functions for the replicated traces in the case of the CFT for the free compactified boson, but we have not so far been able to obtain the n → 1 continuation for the negativity even in the limit of large compactification radius. We have checked all our findings against exact numerical results for the harmonic chain which is described by a non-compactified free boson.