Entanglement in inhomogeneous quantum chains

Résumé

In this thesis we characterize the quantum entanglement properties of ground states of 1D inhomogeneous systems. In particular, we consider free fermion inhomogeneous chains and their corresponding spin versions obtained through the Jordan-Wigner transformation. Homogeneous models have been extensively studied, so they constitute a good reference when studying new physics due to inhomogeneity. For example, many ground states obey the so-called area law of entanglement entropy, which establishes that the amount of entanglement present between a subsystem and its environment is proportional to the measure of its boundary. Ground states of certain inhomogeneous systems constitute an example of a violation of the area law. A paradigmatic case is the so-called rainbow model describing an XX spin chain whose couplings decay exponentially from the center towards the edges of the chain. Its ground state is the so-called rainbow state because in the regime of strong inhomogeneity, the strong disorder Renormalization Group predicts a valence bond solid formed by concentric maximally entangled states (Bell pairs). The rainbow state exhibits long-range entanglement and violates (maximally) the area law. Moreover, the field theory describing the model in the weak inhomogeneity regime $h\ll1$ corresponds to a conformal field theory with central charge c=1. In this thesis we use this state as a starting point to explore the effects generated by inhomogeneity, characterizing the entanglement properties with various tools. In particular we study the Rényi entropies, the entanglement Hamiltonian, the entanglement spectrum and also the entanglement contour. The first four chapters are introductory and present the basic technology already presented in the literature. Thus, Chapter 1 is intended to give a general, non-technical introduction to the field. Chapter 2 introduces quantum entanglement and its characterization. Chapter 3 discusses in detail the characterization of entanglement in a homogeneous XX chain, and finally in Chapter 4 we introduce a family of inhomogeneous models and in particular the rainbow model. The following three chapters present the original content of this thesis. In Chapter 5 we explore the implications of adding a central defect with varying intensity in the rainbow model. In the strong inhomogeneity regime, we show that it is possible to generate transitions between phases with short (dimerized phases) and long (rainbow phase) entanglement just by changing the defect intensity. We calculate order parameters and single-body energies to support this fact. As far as the weak inhomogeneity regime is concerned, we show that the system can be described through a conformal field theory in curved space-time but with an effective central charge dependent on $\gamma$. In Chapter 6 we discuss the possibility of characterizing the critical phases in the framework of the so-called symmetry-protected topological phases. We show that a folding transformation converts long-range entanglement into short-range entanglement, allowing an efficient description in terms of matrix product states (MPS) and classification into symmetry-protected topological phases. Those states presenting a reflection symmetry with respect to the central coupling are topologically trivial, while those presenting a reflection symmetry with respect to the central site are non-trivial. In fact, we find a ground state belonging to the AIII phase of the classification of topological insulators whose paradigmatic representative is the Su-Schrieffer-Heeger model and a ground state of an inhomogeneous chain with SU(2) symmetry very similar to the AKLT state which belongs to the Haldane phase. Furthermore, we propose an extension of this result to chains with higher spin, finding a correspondence between the symmetry protection of gapped and gapless phases. In Chapter 7 we propose a rainbow Ising model with transverse magnetic field. In the strong inhomogeneity regime, the fundamental state is a rainbow state of Majorana fermions, i.e. a collection of SU(2)_2 singlets. The low inhomogeneity regime is described by a conformal theory with central charge c=1/2 in hyperbolic space-time. On the other hand, we consider a modification of the couplings of the Ising rainbow model adding a new parameter d. Thus, the weak inhomogeneity regime is described by a massive quantum field theory in the hyperbolic space-time. The high inhomogeneity regime is rich and varied. As a function of d we obtain states belonging to the D phase of topological invariants whose most representative model is the Kitaev chain. Likewise, we obtain states with a high and low range entanglement structure. Chapter 8 contains a discussion in English and Spanish of the work done and the results obtained in this thesis. Finally, a list of all references cited throughout the document concludes the thesis.