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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.
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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 Z_2 topological insulators.
The theory of topological insulators and superconductors has mostly focused on non-interacting and gapped systems. This review article discusses topological phases that are either gapless or interacting. We discuss recent progress in identifying gapless systems with stable topological properties (such as novel surface states), using Weyl semimetals as an illustration. We then review recent progress in describing topological phases of interacting gapped systems. We explain how new types of edge states can be stabilized by interactions and symmetry, even though the bulk has only conventional excitations and no topological order of the kind associated with Fractional Quantum Hall states.
A gas of strongly interacting spinless p-orbital fermionic atoms in 2D optical lattices is proposed and studied. Several interesting new features are found. In the Mott limit on a square lattice, the gas is found to be described effectively by an orbital exchange Hamiltonian equivalent to a pseudospin-1/2 XXZ model. For a triangular, honeycomb, or Kagome lattice, the orbital exchange is geometrically frustrated and described by a new quantum 120 degree model. We determine the orbital ordering on the Kagome lattice, and show how orbital wave fluctuations select ground states via the order by disorder mechanism for the honeycomb lattice. We discuss experimental signatures of various orbital ordering.
Topology is a central notion in the classification of band insulators and characterization of entangled many-body quantum states. In some cases, it manifests as quantized observables such as quantum Hall conductance. However, being inherently a global property depending on the entirety of the system, its direct measurement has remained elusive to local experimental probes in many cases. Here, we demonstrate that some topological band indices can be probed by resonant inelastic x-ray scattering. Specifically, for the paradigmatic Su-Schrieffer-Heeger and quadrupolar insulator models, we show that non-trivial band topology leads to distinct scattering intensity at particular momentum and energy. Our result establishes an incisive bulk probe for the measurement of band topology.
We uncover topological features of neutral particle-hole pair excitations of correlated quantum anomalous Hall (QAH) insulators whose approximately flat conduction and valence bands have equal and opposite non-zero Chern number. Using an exactly solvable model we show that the underlying band topology affects both the center-of-mass and relative motion of particle-hole bound states. This leads to the formation of topological exciton bands whose features are robust to nonuniformity of both the dispersion and the Berry curvature. We apply these ideas to recently-reported broken-symmetry spontaneous QAH insulators in substrate aligned magic-angle twisted bilayer graphene.
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