Session B31: Topological Insulators: Semimetals and Interactions

Adler-Bell-Jackiw anomaly in Weyl semi-metals: Application to Pyrochlore Iridates

Weyl semimetals are three dimensional analogs of graphene where the energy of the excitations are a linear function of their momentum. Pyrochlore Iridates are conjectured to be examples of such a system, with the low energy physics described by twenty four Weyl nodes. An intriguing possibility is that these materials provide a physical realization of the Adler-Bell-Jackiw anomaly. In this talk we report on the properties of pyrochlore iridates in an applied magnetic field. We find that the dispersion of the lowest landau level depends on the direction of the applied magnetic field. As a consequence the magneto-conductivity in an electric field, applied parallel to the magnetic field is highly anisotropic, providing a detectable signature of the semi-metallic state.   

Dirac Semimetal in Three Dimensions

In a Dirac semimetal the conduction and valence bands contact at discrete points in the Brillouin zone, dispersing linearly away from these critical points with the low energy physics described by a four band Dirac Hamiltonian. In 2D this situation is realized in graphene in the absence of spin-orbit coupling. 3D Dirac semimetals are predicted to exist at the phase transition between a topological insulator and an ordinary insulator when inversion symmetry is preserved. Here we show that 3D Dirac points can also be protected by crystallographic symmetries in particular space groups and enumerate the criteria necessary to identify these groups. As an example of a Dirac semimetal, we present calculations for $\beta$-Cristobalite ${\rm BiO_2}$ which exhibits Dirac points at the three symmetry related $X$ points of the FCC Brillouin zone. We find that $\beta$-Cristobalite ${\rm BiO_2}$ is metastable, so it can be physically realized as a 3D analog to graphene. We provide a systematic approach that includes crystallographic symmetry arguments and physical and chemical considerations to identify other such materials and rule out possible alternatives such as HgTe. This would greatly expand the range of applications that take advantage of properties arising from Dirac points.   

Semi-metal and Topological Insulator in Perovskite Iridates

The two-dimensional (2D) layered perovskite Sr$_2$IrO$_4$ was proposed to be a spin-orbit (SO) Mott insulator, where the effect of Hubbard interaction is amplified on a narrow J$_{eff}$=$\frac{1}{2}$ band due to strong spin-orbit coupling. On the other hand, the three-dimensional (3D) orthorhombic perovskite SrIrO$_3$ remains metallic. We construct a tight-binding model for SrIrO$_3$ to understand the physical origin of the metallic behaviour and study possible metal-insulator transitions. In particular, we identify possible perturbations that turn the material into a topological insulator.   

Topological Insulators and Semimetals with Point Group Symmetries

In this work, we study the theory of topological phases in systems with point group symmetries (PGSs) in one-, two- and three-dimension. The systems we study in general do not require time-reversal symmetry, and hence may be realized in both non-magnetic and magnetic materials. We show that a point group symmetry introduces new quantum numbers which reveal themselves in the entanglement spectrum as mid-gap states. PGSs also define a series of topological semimetals, in which the band touching points are protected by certain symmetries. We apply our theory to 3D ferromagnetic semimetal HgCr$_2$Se$_4$ which possesses a double-vortex band crossing protected by $C_4$ rotation symmetry.   

Diamagnetism of Weyl semimetals

We present a study of the diamagnetic orbital response in a Weyl semimetal, the recently discovered gapless topological phase of matter. Weyl semimetal is a three-dimensional (3D) material, characterized by the presence of isolated Dirac (Weyl) point nodes in its band structure. It can be thought of as the closest 3D analog of graphene. It is known from graphene studies that two-dimensional (2D) Dirac fermions have a highly nontrivial singular diamagnetic response to an applied perpendicular magnetic field, reflecting the quantum critical nature of the ground state of undoped graphene. Here we investigate the analogous orbital response of 3D Dirac fermions in a Weyl semimetal to an applied magnetic field. As in 2D graphene, we find strong signatures of quantum criticality in the diamagnetic response of 3D Weyl semimetal. In particular, we find that the orbital susceptibility has a characteristic logarithmic dependence on the applied field, deviation of the chemical potential from the charge-neutral position and temperature. This unusual diamagnetic response can be used for experimental characterization of Weyl semimetals.   

Weyl Semimetal in a Topological Insulator Multilayer

We propose a simple realization of the three-dimensional (3D) Weyl semimetal phase, utilizing a multilayer structure, composed of identical thin films of a magnetically-doped 3D topological insulator (TI), separated by ordinary-insulator spacer layers. We show that the phase diagram of this system contains a Weyl semimetal phase of the simplest possible kind, with only two Dirac nodes of opposite chirality, separated in momentum space, in its bandstructure. This Weyl semimetal has a finite anomalous Hall conductivity, chiral edge states, and occurs as an intermediate phase between an ordinary insulator and a 3D quantum anomalous Hall insulator. We discuss unusual transport properties of the Weyl semimetal, and in particular point out quantum critical-like scaling of the DC and optical conductivity.   

3D Weyl Semimetal in a Honeycomb Array of Topological Nano-wires

The Weyl semimetal phase has been recently introduced and suggested to exist in strongly correlated pyrochlore iridates as well as in the non-interacting layered Normal/Topological band insulator systems. This unusual phase has a number of interesting properties in the bulk and at the surface arising from the appearance of isolated point-like hedge-hog topological defects (known as Weyl-Dirac points) in the Bloch-state manifold. Here we discuss the possible emergence of this phase in a honeycomb arrangement of the parallel topological insulator nano-rods each exposed to a half-integer multiple of magnetic flux quantum. We consider direct hoping between rods as well as the electron-electron interaction between them. We discuss how the initially degenerate Weyl points can be separated in the Brillouin zone by various perturbations breaking either inversion or time-reversal symmetry.   

Charge transport in Weyl semimetals

Weyl semimetals are three-dimensional phases with band touchings, whose low-energy excitations are governed by the Weyl equation. They can be thought of as higher dimensional cousins of graphene. Recent theoretical work predicted certain pyrochlore iridates such as Y2Ir2O7 to be in this phase. We study charge transport in Weyl semimetals in the presence of Coulomb interactions or disorder at temperature T and compare our results to existing data on Y2Ir2O7 and Eu2Ir2O7. In the interacting clean limit, we determine the conductivity by solving a quantum Boltzmann equation within a ``leading log'' approximation and find it to be proportional to T, upto logarithmic factors arising from the flow of couplings. In the noninteracting disordered case, we compute the finite-frequency Kubo conductivity and show that it exhibits distinct behaviors for low and high frequencies compared to T. The behavior of Weyl semimetals in a magnetic field will also be briefly discussed.   

Magnetic Instability on the Surface of Topological Insulators

Gapless surface states that are protected by time reversal symmetry and charge conservation are among the manifestations of 3D topological insulators. In this work we study how electron-electron interaction may lead to spontaneous breaking of time reversal symmetry on surfaces of such insulators. We find that a critical interaction strength exists, above which the surface is unstable to spontaneous formation of magnetization, and study the dependence of this critical interaction strength on temperature and chemical potential.   

Plasmons in Topological Insulators

A theory is presented for calculating the plasmon mode dispersion relation in three-dimensional topological insulators (TI). There are two-dimensional (2D) conducting surface states. The conducting states localized close to the surface of the semi-infinite slab have a well defined Dirac cone. The bulk energy gap is large and comparable with room temperature. We investigate plasmon excitations of those surface bound electrons in the long wavelength limit employing the random-phase approximation. Results from our calculations show that for a quasi-1DTI, the plasmon dispersion relation is given by $\omega_p \approx q \left({ 1- \omega_{0} \ln(q)}\right)$ where $\omega_0 = \frac{2 e^2}{\pi \epsilon_0} \frac{3}{10}$. On the other hand, for the conventional 1DEG, the plasmon dispersion satisfies $\omega_p \approx q \sqrt{-\omega_{0} \ln(q)}$, with $\omega_0 = 2n_{1D} e^2/\epsilon_0 m$ and $n_{1D}$ denoting the linear electron density. The plasmons in 1DTI are density-independent as they are in metallic armchair graphene nanoribbons but obey different dispersion relation. The material parameters we chose correspond to $\texttt{Bi}_2 \texttt{Te}_3$ crystals.   

Topological Insulators with electron-electron interactions

We consider the effect of interaction for the $3$ dimensional Topological insulators. We show that effectively the system is equivalent to non-interacting Topological Insulators in 4+1 dimensions.   

Actinide Topological Insulator Materials with Strong Interaction

Topological band insulators have recently been discovered in spin-orbit coupled two and three dimensional systems. In this work, we theoretically predict a class of topological Mott insulators where interaction effects play a dominant role. In actinide elements, simple rocksalt compounds formed by Pu and Am lie on the boundary of metal to insulator transition. We show that interaction drives a quantum phase transition to a topological Mott insulator phase with a single Dirac cone on the surface.   

Fractional quantum Hall effect and plasmons in topological insulators

I will discuss theoretical studies of the effect of Coulomb interactions at the topological insulator surface in the presence of a magnetic field. Coulomb interaction can cause composite fermion formation and the fractional quantum Hall effect. We predict the stability of the fractional quantum Hall effect by considering the form of the effective interparticle interaction: if it is sufficiently short ranged, then there will be composite fermion formation. We will also study plasmons and magnetoplsmons of the surface states both for an ideal topological insulator and for a topological insulator with bulk conduction.   

Half quantum spin Hall effect on the surface of weak topological insulators

We investigate interaction effects in three dimensional weak topological insulators (TI) with an even number of Dirac cones on the surface. We find that the surface states can be gapped by a surface charge density wave (CDW) order without breaking the time-reversal symmetry. In this sense, timereversal symmetry alone can not robustly protect the weak TI state in the presence of interactions. If the translational symmetry is additionally imposed in the bulk, a topologically non-trivial weak TI state can be obtained with helical edge states on the CDW domain walls. In other words, a CDW domain wall on the surface is topologically equivalent to the edge of a two-dimensional quantum spin Hall insulator. Therefore, the surface state of a weak topological insulator with translation symmetry breaking on the surface has a ``half quantum spin Hall effect,'' in the same way that the surface state of a strong topological insulator with time-reversal symmetry breaking on the surface has a ``half quantum Hall effect.'' The on-site and nearest neighbor interactions are investigated in the mean field level and the phase diagram for the surface states of weak topological insulators is obtained.   

2D symmetry protected topological orders and their protected gapless edge excitations

Topological insulators/superconductors with time reversal or particle-hole symmetry protected gapless edge excitations have been well characterized and classified in free fermion systems. However, it is not clear in general interacting boson or fermion systems, when such symmetry protected topological(SPT) orders exist with gapless edge excitations that are protected even against strong interactions. Here, we present a systematic construction of 2D interacting bosonic models with non-trivial SPT orders for any on-site symmetry of group $G$. We demonstrate the non-trivialness of the models by rigorously proving that the gapless edge excitations of the system is stable against any interaction as long as symmetry is not broken. We prove this result by developing the tool of matrix product unitary operator to study the nonlocal symmetry transformation on the edge degrees of freedom and revealing its relation to the non-trivial 3-cocycles of the symmetry group $G$. This relation between SPT orders and group cocycles has actually been established in 1D interacting systems and led to a complete classification of 1D SPT orders. We show here that this relation also extends to $>2$ spatial dimensions and possibly provides a (partial) classification of SPT orders in all interacting systems.