Session F13: Topological Insulators: Theory I

Homological Order in Three and Four dimensions: Wilson Algebra, Entanglement Entropy and Twist Defects

We investigate homological orders in two, three and four dimensions by studying $Z_k$ toric code models on simplicial, cellular or in general differential complexes. The ground state degeneracy is obtained from Wilson loop and surface operators, and the homological intersection form. We compute these for a series of closed 3 and 4 dimensional manifolds and study the projective representations of mapping class groups (modular transformations). Braiding statistics between point and string excitations in (3+1)-dimensions or between dual string excitations in (4+1)-dimensions are topologically determined by the higher dimensional linking number, and can be understood by an effective topological field theory. An algorithm for calculating entanglemnent entropy of any bipartition of closed manifolds is presented, and its topological signature is completely characterized homologically. Extrinsic twist defects (or disclinations) are studied in 2,3 and 4 dimensions and are shown to carry exotic fusion and braiding properties.   

Weyl points and line nodes in gapless gyroid photonic crystals

Weyl points and line nodes are three-dimensional linear point- and line-degeneracies between two bands. In contrast to Dirac points, which are their two-dimensional analogues, Weyl points are stable in the momentum space and the associated surface states are predicted to be topologically non-trivial. However, Weyl points are yet to be discovered in nature. Here, we report photonic crystals, based on the double-gyroid structures, exhibiting frequency-isolated Weyl points with complete phase diagrams by breaking the parity and time-reversal symmetries. The surface states associated with the non-zero Chern numbers are demonstrated. Line nodes are also found in similar geometries; the associated surface states are shown to be at bands. Our results are based on realistic ``numerical experiments'' with true predictive power and should be readily experimentally realizable at both microwave and optical frequencies.   

The optical conductivity of quasicrystals: evidence of a Weyl semimetal with 3D Dirac spectrum

The optical conductivity of quasicrystals is characterized by an absence of the Drude peak and a conductivity that rises linearly over a wide range of frequencies. The absence of the Drude peak has been attributed to a pseudogap at the Fermi surface but a detailed explanation of the linear behavior has not been found. This unusual behavior is seen in all icosahedral quasicrystal families and their periodic approximants. A simple model that assumes that the entire Fermi surface is gapped, with the exception at a finite set of Dirac points, fits the data. There is no evidence of a semiconducting gap in any of the materials suggesting that the massless Dirac spectrum is protected by topology leading to a Weyl semimetal. The model gives rise to a linear conductivity with only one parameter, the Fermi velocity. In accord with this picture decagonal quasicrystals should have a frequency independent conductivity, without a Drude peak. This is in accord with the experimental data as well.   

Chiral magnetic effect in Weyl semimentals and insulators

It has been proposed recently, on the basis of field-theoretical considerations, that the effective electromagnetic action of Weyl semimetals (as well as the closely related Weyl insulators) contains an axion term with the $\theta$-angle dependent on time $t$ or spatial position ${\bf r}$. If correct this would lead to a number of novel observable phenomena, such as the chiral magnetic effect, whereby an applied uniform magnetic field induces a dissipationless bulk current ${\bf j}\propto {\bf B}$. In this work we construct a simple lattice model for a Weyl semimental (insulator) and use it to explicitly test for the chiral magnetic effect by means of numerical techniques combined with analytical calculations. We discuss possible ways to engineer a suitable material in layered nanostructures and comment on the physical observability of the effect.   

Axion field theory, chiral anomaly and anomalous non-dissipative transport properties of (3+1)-dimensional Weyl semi-metals and superconductors

From a direct calculation of the anomalous Hall conductivity and an effective electromagnetic action obtained via Fujikawa's chiral rotation technique, we conclude that an axionic field theory with a non-quantized coefficient describes the electromagnetic response of the (3+1)-dimensional Weyl semi-metal. The coefficient is proportional to the momentum space separation of the Weyl nodes. Akin to the Chern-Simons field theory of quantum Hall effect, the axion field theory violates gauge invariance in the presence of the boundary, which is cured by the chiral anomaly of the surface states via the Callan-Harvey mechanism. A direct linear response calculation also establishes an anomalous thermal Hall effect and a Wiedemann-Franz law. But, thermal Hall conductivity does not directly follow from the well known formula for the (3+1)-dimensional gravitational chiral anomaly. By calculating the gravitational chiral anomaly at finite temperature we show the existence of a new term, which correctly accounts for the thermal Hall effect in (3+1)-dimensional Weyl materials, topological insulators and topological superconductors.   

Friedel oscillations due to Fermi arcs in Weyl semimetals

Weyl semimetals harbor unusual surface states known as Fermi arcs, which are essentially disjoint segments of a two-dimensional Fermi surface. We describe a prescription for obtaining Fermi arcs of arbitrary shape and connectivity by stacking alternate two-dimensional electron and hole Fermi surfaces and adding suitable interlayer coupling. Using this prescription, we compute the local density of states--a quantity directly relevant to scanning tunneling microscopy--on a Weyl semimetal surface in the presence of a point scatterer and present results for a particular model that is expected to apply to pyrochlore iridate Weyl semimetals. For thin samples, Fermi arcs on opposite surfaces conspire to allow nested backscattering, resulting in strong Friedel oscillations on the surface. These oscillations die out as the sample thickness is increased and Fermi arcs from the opposite surface retreat and weak oscillations, due to scattering between the top surface Fermi arcs alone, survive. The surface spectral function, accessible to photoemission experiments, is also computed. In the thermodynamic limit, this calculation can be done analytically and separate contributions from the Fermi arcs and the bulk states can be seen.   

Topological Phases of Point Group Symmetric Weyl Superconductors

We study superconductivity in a Weyl semimetal with broken time-reversal symmetry and stabilized by a point-group symmetry. The resulting superconducting phase is characterized by topologically protected bulk nodes and surface states with Fermi arcs. We derive a phase diagram of possible superconducting phases which are distinguished by the number of bulk nodes and discuss novel properties of the corresponding surface states. We show how the topological behavior may be understood in terms of the properties of the parent Weyl semimetal at high-symmetry momenta.   

Excitonic Phases of Weyl Semi-Metals with Coulomb Interaction

Weyl semi-metals have an even number of nodes which are perfectly nested in the absence of a chiral chemical potential. For repulsive interactions these are susceptible to excitonic instabilities. The vanishing density of states requires that the coupling be larger than a critical value for the states to be realized. There are eight possible states in the particle-hole channel, only two of which gap out the weyl nodes for long range Coulomb interactions. The lowest energy state is the Charge Density Wave state, which is more stable than the ferromagnetic insulator that arises in the context of short range repulsion. The defects of the state, i.e. dislocations, have been shown in the literature, to carry gapless chiral modes.   

Excitonic Phases from Weyl Semi-Metals with short range interaction

Weyl semimetal, possibly realized in Pyrochlore irridates or supperlatice of 3D topological-normal insulators system, has strong spin orbit interactions leading to effective low energy described by massless linearly dispersing fermions. In the absence of interactions chirality is a conserved quantum number, protecting the semi-metallic physics against perturbations that are translationally invariant. We show that the interplay between short range repulsive interaction and topology yields a novel chiral excitonic insulator. The state is characterized by a complex vectorial order parameter leading to a gapping out of the Weyl nodes. An interesting feature is that it is ferromagnetic, with the phase of the order parameter determining the direction of the induced magnetic moment. The case of Coulomb interaction will be discussed by Huazhou Wei in his report.   

Probing the Chiral Anomaly via Nonlocal Transport in Weyl Semimetals

Weyl semimetals are three-dimensional analogs of graphene in which a pair of bands touch at points in momentum space, known as Weyl nodes. Electrons originating from a single Weyl node possess a definite topological charge, the chirality. Consequently, they exhibit the Adler-Jackiw-Bell anomaly, which in this condensed matter realization implies that application of parallel electric ($\mathbf{E}$) and magnetic fields ($\mathbf{B}$) pumps electrons between nodes of opposite chirality at a rate proportional to $\mathbf{E}\cdot\mathbf{B}$. We argue that this pumping is measurable via transport experiments, in the limit of weak internode scattering. Specifically, we show that injecting a current in a Weyl semimetal subject to an $\mathbf{E}\cdot\mathbf{B}$ term leads to nonlocal features in transport.   

Effective field theories for topological insulators by functional bosonization

Effective field theories that describe the dynamics of electric current for topological insulators in general dimension D = d+1 are discussed using the functional bosonization. For non-interacting topological insulators with a conserved U(1) charge and characterized by an integer topological invariant, we derive the BF-type topological field theories supplemented with the Chern-Simons (when D is odd) or the Axion term (when D is even). For topological insulators characterized by a Z2 topological invariant, their topological field theories are obtained by dimensional reduction. Building on this effective field theory description for non- interacting topological phases, we also discuss, following the spirit of the parton construction of the fractional quantum Hall effect, the putative ``fractional'' topological insulators and their possible effective field theories.   

Physics of three dimensional bosonic topological insulators I

We discuss physical properties of ``integer'' topological phases of bosons in D=3+1 dimensions, protected by internal symmetries like time reversal and/or charge conservation. These phases invoke interactions in a fundamental way but do not possess topological order and are bosonic analogs of free fermion topological insulators and superconductors. Here we develop a field theoretic description of several of these states and show that they possess unusual surface states, which if gapped, must either break the underlying symmetry, or develop topological order. In certain cases the topological phases are characterized by a quantized magneto-electric response $\theta$, which, somewhat surprisingly, is an odd multiple of $2\pi$. A surface theory in which vortices transform under a projective representation of the symmetry group is shown to capture these properties. A bulk field theory of these states is also identified, which furthermore predicts phases characterized by the thermal analog of the magneto-electric effect, that lie beyond the current cohomology description.   

Physics of three dimensional bosonic topological insulators II

We discuss physical properties of interacting boson/spin analogs of free fermion topological insulators and superconductors. We discuss general constraints on the surface theories of these phases, and their field theoretic descriptions. We illustrate some of the results in the context of quantum paramagnetic phases of spin systems. For the 3d states we describe the 2d surface either spontaneously breaks symmetry or is in a spin liquid phase. In the latter case the symmetry is realized in the surface spin liquid in a way that is forbidden in strictly two dimensional quantum magnets.   

Wilson-loop Classification of Inversion-Symmetric Topological Insulators and the Z$_2$ Crystalline Topological Insulator

In the context of translationally-invariant insulators, Wilson loops describe the adiabatic evolution of the ground state around a closed circuit in the Brillouin zone. We propose that the Wilson-loop eigenspectrum provides a complete characterization of (a) the inversion-symmetric topological insulator, and (b) the \textbf{Z}$_2$ crystalline topological insulator: time-reversal symmetric insulators with either C$_4$ or C$_6$ rotational symmetry, but with no spin-orbit coupling. For the 1D inversion-symmetric insulator, we formulate a \textbf{Z} Wilson-loop index, which is identifiable with the number of protected boundary modes in the entanglement spectrum. For the 2D inversion-symmetric insulator, we identify a \textbf{Z} relative-winding number, which is the inversion-analog of the first Chern class (for charge-conserving insulators). For the \textbf{Z}$_2$ crystalline topological insulator, we show how the \textbf{Z}$_2$ invariant can be extracted from the Wilson-loop eigenspectrum; this aids the identification of materials that realize this phase.