How do we uniquely identify a quantum phase, given its ground state wave-function? This is a key question for many body theory especially when we consider phases like topological insulators, that share the same symmetry but differ at the level of topology. The entanglement spectrum has been proposed as a ground state property that captures characteristic edge excitations. Here we study the entanglement spectrum for topological band insulators. We first show that insulators with topological surface states will necessarily also have protected modes in the entanglement spectrum. Surprisingly, however, the converse is not true. Protected entanglement modes can also appear for insulators without physical surface states, in which case they capture a more elusive property. This is illustrated by considering insulators with only inversion symmetry. Inversion is shown to act in an unusual way, as an antiunitary operator, on the entanglement spectrum, leading to this protection. The entanglement degeneracies indicate a variety of different phases in inversion symmetric insulators, and these phases are argued to be robust to the introduction of interactions.How do we uniquely identify a quantum phase, given its ground state wave-function? This is a key question for many body theory especially when we consider phases like topological insulators, that share the same symmetry but differ at the level of topology. The entanglement spectrum has been proposed as a ground state property that captures characteristic edge excitations. Here we study the entanglement spectrum for topological band insulators. We first show that insulators with topological surface states will necessarily also have protected modes in the entanglement spectrum. Surprisingly, however, the converse is not true. Protected entanglement modes can also appear for insulators without physical surface states, in which case they capture a more elusive property. This is illustrated by considering insulators with only inversion symmetry. Inversion is shown to act in an unusual way, as an antiunitary operator, on the entanglement spectrum, leading to this protection. The entanglement degeneracies indicate a variety of different phases in inversion symmetric insulators, and these phases are argued to be robust to the introduction of interactions.

arXiv:0909.3119Sep 17, 20099 citations

Topology in Correlated Quantum Insulators

Gil Young Cho

Feb 11, 20200 citations

Relating the entanglement spectrum of noninteracting band insulators to their quantum geometry and topology

Markus Legner, Titus Neupert

We study the entanglement spectrum of noninteracting band insulators, which can be computed from the two-point correlation function, when restricted to one part of the system. In particular, we analyze a type of partitioning of the system that maintains its full translational symmetry, by tracing over a subset of local degrees of freedom, such as sublattice sites or spin orientations. The corresponding single-particle entanglement spectrum is the band structure of an entanglement Hamiltonian in the Brillouin zone. We find that the hallmark of a nontrivial topological phase is a gapless entanglement spectrum with an ``entanglement Fermi surface.'' Furthermore, we derive a relation between the entanglement spectrum and the quantum geometry of Bloch states contained in the Fubini-Study metric. The results are illustrated with lattice models of Chern insulators and ${\mathbb{Z}}_{2}$ topological insulators.We study the entanglement spectrum of noninteracting band insulators, which can be computed from the two-point correlation function, when restricted to one part of the system. In particular, we analyze a type of partitioning of the system that maintains its full translational symmetry, by tracing over a subset of local degrees of freedom, such as sublattice sites or spin orientations. The corresponding single-particle entanglement spectrum is the band structure of an entanglement Hamiltonian in the Brillouin zone. We find that the hallmark of a nontrivial topological phase is a gapless entanglement spectrum with an ``entanglement Fermi surface.'' Furthermore, we derive a relation between the entanglement spectrum and the quantum geometry of Bloch states contained in the Fubini-Study metric. The results are illustrated with lattice models of Chern insulators and ${\mathbb{Z}}_{2}$ topological insulators.

arXiv:1306.4008Physical Review BSep 9, 201329 citations

{\mathbb{Z}}_{2}

topological insulators.