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 gapl
ess 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.
We study three dimensional insulators with inversion symmetry, in which other point group symmetries, such as time reversal, are generically absent. Their band topology is found to be classified by the parities of occupied states at time reversal inv
ariant momenta (TRIM parities), and by three Chern numbers. The TRIM parities of any insulator must satisfy a constraint: their product must be +1. The TRIM parities also constrain the Chern numbers modulo two. When the Chern numbers vanish, a magneto-electric response parameterized by theta is defined and is quantized to theta= 0, 2pi. Its value is entirely determined by the TRIM parities. These results may be useful in the search for magnetic topological insulators with large theta. A classification of inversion symmetric insulators is also given for general dimensions. An alternate geometrical derivation of our results is obtained by using the entanglement spectrum of the ground state wave-function.
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 top
ology. 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.
Iron telluride (FeTe), a relative of the iron based high temperature superconductors, displays unusual magnetic order and structural transitions. Here we explore the idea that strong correlations may play an important role in these materials. We argu
e that the unusual orders observed in FeTe can be understood from a picture of correlated local moments with orbital degeneracy, coupled to a small density of itinerant electrons. A component of the structural transition is attributed to orbital, rather than magnetic ordering, introducing a strongly anisotropic character to the system along the diagonal directions of the iron lattice. Double exchange interactions couple the diagonal chains leading to the observed ordering wavevector. The incommensurate order in samples with excess iron arises from electron doping in this scenario. The strong anisotropy of physical properties in the ordered phase should be detectable by transport in single domains. Predictions for ARPES, inelastic neutron scattering and hole/electron doping studies are also made.
In contrast to charge vortices in a superfluid, spin vortices in a ferromagnetic condensate move inertially (if the condensate has zero magnetization along an axis). The mass of spin vortices depends on the spin-dependent interactions, and can be mea
sured as a part of experiments on how spin vortices orbit one another. For Rb87 in a 1 micron thick trap m_v is about 10^-21 kg.
Condensates of atoms with spins can have vortices of several types; these are related to the symmetry group of the atoms ground state. We discuss how, when a condensate is placed in a small magnetic field that breaks the spin symmetry, these vortices
may form bound states. Using symmetry classification of vortex-charge and rough estimates for vortex interactions, one can show that some configurations that are stable at zero temperature can decay at finite temperatures by crossing over energy barriers. Our focus is cyclic spin 2 condensates, which have tetrahedral symmetry.
We present a systematic study of how vortices in superfluid films interact with the spatially varying Gaussian curvature of the underlying substrate. The Gaussian curvature acts as a source for a geometric potential that attracts (repels) vortices to
wards regions of negative (positive) Gaussian curvature independently of the sign of their topological charge. Various experimental tests involving rotating superfluid films and vortex pinning are first discussed for films coating gently curved substrates that can be treated in perturbation theory from flatness. An estimate of the experimental regimes of interest is obtained by comparing the strength of the geometrical forces to the vortex pinning induced by the varying thickness of the film which is in turn caused by capillary effects and gravity. We then present a non-perturbative technique based on conformal mappings that leads an exact solution for the geometric potential as well as the geometric correction to the interaction between vortices. The conformal mapping approach is illustrated by means of explicit calculations of the geometric effects encountered in the study of some strongly curved surfaces and by deriving universal bounds on their strength.
