Aspects of Entanglement Negativity in One Dimensional Critical Systems

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 quanti er. 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 de nition in terms of the density matrix describing the quantum state. Since the de nition 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 con guration 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, A1 and A2, 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 A1 or A2. 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 eld 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 eld theories. Hence, throughout the thesis we will usually consider conformal eld 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 brie y reviewing some of the main topics in the literature of quantum many-body systems and quantum eld 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 de ne the entanglement negativity and stress its main advantages, as well as its drawbacks. After reviewing some basic facts of conformal eld theories, we will give some technical details on the computation of entanglement entropy and logarithmic negativity in general quantum eld 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 di erent 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 eld. 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 eld theory of the free fermion. Again, the computation of the negativity could not be accomplished due to technical di culties 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 Rényi 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, speci cally the harmonic chain in its ground state and out of equilibrium, while in Chaps. 4 and 5 we will consider fermionic systems, speci cally the XX and Ising spin chains, and the tight-binding model. Finally, we will draw some conclusions and discuss some open problems.