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Register a new userBeyond Band Insulators: Topology of Semi-metals and Interacting Phases
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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.
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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.
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.
Recent works have proved the existence of symmetry-protected edge states in certain one-dimensional topological band insulators and superconductors at the gap-closing points which define quantum phase transitions between two topologically nontrivial phases. We show how this picture generalizes to multiband critical models belonging to any of the chiral symmetry classes AIII, BDI, or CII of noninteracting fermions in one dimension.
Ultracold fermions trapped in a honeycomb optical lattice constitute a versatile setup to experimentally realize the Haldane model [Phys. Rev. Lett. 61, 2015 (1988)]. In this system, a non-uniform synthetic magnetic flux can be engineered through laser-induced methods, explicitly breaking time-reversal symmetry. This potentially opens a bulk gap in the energy spectrum, which is associated with a non-trivial topological order, i.e., a non-zero Chern number. In this work, we consider the possibility of producing and identifying such a robust Chern insulator in the laser-coupled honeycomb lattice. We explore a large parameter space spanned by experimentally controllable parameters and obtain a variety of phase diagrams, clearly identifying the accessible topologically non-trivial regimes. We discuss the signatures of Chern insulators in cold-atom systems, considering available detection methods. We also highlight the existence of topological semi-metals in this system, which are gapless phases characterized by non-zero winding numbers, not present in Haldanes original model.
Within this paper we outline a method able to generate truly minimal basis sets which describe either a group of bands, a band, or even just the occupied part of a band accurately. These basis sets are the so-called NMTOs, Muffin Tin Orbitals of order N. For an isolated set of bands, symmetrical orthonormalization of the NMTOs yields a set of Wannier functions which are atom-centered and localized by construction. They are not necessarily maximally localized, but may be transformed into those Wannier functions. For bands which overlap others, Wannier-like functions can be generated. It is shown that NMTOs give a chemical understanding of an extended system. In particular, orbitals for the pi and sigma bands in an insulator, boron nitride, and a semi-metal, graphite, will be considered. In addition, we illustrate that it is possible to obtain Wannier-like functions for only the occupied states in a metallic system by generating NMTOs for cesium. Finally, we visualize the pressure-induced s to d transition.
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