In the present thesis, the properties of the entanglement spectra of the ground state of graphene bilayers Bernal-stacked is analytically investigated. In addition, the entanglement spectra of the ground state of the honeycomb lattice in the presence of superconductivity instabilities are analytically studied. We consider not only the eigenvalues of the reduced density matrix that form the entanglement spectrum, but also, its eigenstates. From these eigenstates, we construct topological quantities, such as the Berry phase and the Chern number, in order to investigate the topological properties of the entanglement spectrum. In the first part of this thesis, we present an analytical study of the graphene bilayers, mainly focusing on the effects of the trigonal warping. When the term causing the trigonal warping is neglected, the entanglement Hamiltonian obtained by tracing out one layer shows a proportionality with the energetic Hamiltonian of the remaining monolayer graphene, in the limit of strongly coupled layers. We demonstrate that this proportionality leads to an agreement of the topological quantities of these Hamiltonians. Moreover, we demonstrate that the entanglement spectrum of graphene bilayers with a trigonal warping spectrum clearly differs in the geometric shape from the energy spectrum of the remaining monolayer graphene. However, there is an agreement of the topological quantities such as Berry phase contributions to Chern numbers. In the second part of this thesis, we give a detailed study of graphene in the presence of superconductivity instabilities, mainly considering the s-wave and the chiral dx2−y2 +idxy state.We investigate the relationship between the entanglement and energy spectrum, making use of the concepts of the Chern number constructed from eigenstates of the entanglement Hamiltonian. We demonstrate that the entanglement and remaining subsystem Hamiltonians can have different topologies. These findings are illustrated by considering the entanglement Hamiltonian of the ground state of graphene with dx2−y2 + idxy superconductivity obtained by tracing out one spin direction. Our investigations are based on closed analytical expressions for the full eigensystem in the entire Brillouin zone of bilayer graphene with a trigonally warped spectrum and graphene with superconductivity instabilities.
We study Haldane's honeycomb lattice model and a bilayer generalization thereof from the perspective of edge states, entanglement spectra, and Wannier function behavior. For the monolayer model, we obtain the zigzag edge states analytically, and identify the edge state crossing point
kc with where the
f=1/2 entanglement occupancy and the half-odd-integer Wannier centers occur. A continuous interpolation between the entanglement states and the Wannier states is introduced. We then construct a bilayer model by Bernal stacking two monolayers coupled by interlayer hopping. We analyze a particular limit of this model where an extended symmetry, related to inversion, is present. The band topology then reveals a break-down of the correspondence between edge and entanglement spectrum, which is analyzed in detail, and compared with the inversion-symmetric Z2 topological insulators which show a similar phenomenon.