Entanglement structures and topology

Entanglement is one of the most intriguing features of quantum mechanics. It is the manifestation of a particular type of correlations that is encoded in the wave function, and is the most important ingredient behind the concept of quantum computation. Moreover, entanglement also plays a pivotal role in describing phases of quantum matter at very low temperatures and understanding their fundamental properties. Hence, in the last decades the characterization of entanglement has become a leading direction in the research of quantum many-body systems. One fascinating property of certain phases of quantum matter is the presence of topological order. These phases elude the conventional description of phase transitions based on symmetry breaking and local order parameters, and they feature a particular form of long-range entanglement. Indeed, one possible way to identify topological order is via the topological entropy, which is a particular measure of entanglement. However, a detailed study of the entanglement structure actually reveals much more information, and was shown to carry a fingerprint of the underlying topological order. The main goal of this project is to explore the real-space structure of entanglement in the ground states of paradigmatic quantum many-body systems with topological properties. The focus of our investigations is the so-called entanglement Hamiltonian, which gives the total characterization of entanglement between a subsystem and its environment. In fact, the setup is completely analogous to the one in statistical mechanics, where a subsystem immersed in a reservoir is known to be described by a thermal ensemble with respect to the physical Hamiltonian. The key question is then, whether there is any relation between the entanglement Hamiltonian and the physical one. Quite remarkably, the answer is positive for a large class of many-body ground states, where the entanglement structure is effectively described by a thermal state with a space-dependent temperature. In other words, one obtains an entanglement Hamiltonian with a local structure. The crucial question to be addressed in this project is how this locality is affected by the presence of topological order in the system. Our primary targets are quantum Hall states, which are the most well known systems displaying topological properties, and have a high experimental relevance. Furthermore, we shall also explore topological defects in one-dimensional quantum chains, which support nontrivial topological phases. Apart from gaining novel theoretical insight on the interplay of topology and entanglement, the results could contribute to the development of tomographic protocols in the quantum simulation of topological states.