Entanglement Spectroscopy and its Application to the Quantum Hall Effects

The entanglement spectroscopy, initially introduced by Li and Haldane in the context of the fractional quantum Hall effects, has stimulated an extensive range of studies. The entanglement spectrum is the spectrum of the reduced density matrix, when we partition the system into two. For many quantum systems, it unveils a unique feature: Computed from the bulk ground state wave function, the entanglement spectrum give access to the physics of edge excitations. Using this property, the entanglement spectroscopy has proved to be a highly valuable tool to diagnose topological ordering. These lectures intend to provide an overview of the entanglement spectroscopy, mainly in the context of the fractional quantum Hall effect. We introduce the basic concepts through the case of the quantum spin chains. We discuss the connection with the entanglement entropy and the matrix product state representation. We show how the entanglement spectrum can be computed for non-interacting topological phases and how it reveals the edge excitation from the ground state. We then present an extensive review of the entanglement spectra applied to the fractional quantum Hall phases, showing how much information is encoded within the ground state and how different partitions probe different type of excitations. Finally, we discuss the application of this tool to study the fractional Chern insulators.