Session X28: Topological Semimetals: Theory

Weyl Phases in a Three Dimensional Network Model

We study the topological properties of 3D ``Floquet'' band structures, defined using unitary evolution matrices rather than Hamiltonians. Such band structures can be realized in coherent-wave networks or lattices subjected to time-periodic drives. Previously, 2D Floquet band structures have been shown to exhibit unusual topological behaviors such as topologically-nontrivial zero-Chern-number phases. Here, we analyze the Floquet band structure of a 3D network model, which exhibits an Floquet analogue of a Weyl phase. The surface states exhibit topologically-protected ``Fermi'' arcs, similar to the recently-discovered Weyl semi-metals; however, the Weyl points in different quasi-energy gaps are related by a particle-hole symmetry which is unique to the Floquet system. By tuning the coupling parameters of the network, we can drive a transition between conventional insulator, weak topological insulator, and Weyl phases. Finally, we discuss the possibility of realizing this model using custom-designed electromagnetic networks.   

Magnetic response in three-dimensional nodal semimetals

We study the magnetic response in various three-dimensional gapless systems, including Dirac and Weyl semimetals and a line-node semimetal. We show that the susceptibility is decomposed into the orbital term, the spin term and also the spin-orbit cross term which is caused by the spin-orbit interaction. We show that the orbital susceptibility logarithmically diverges at the band touching energy in the point-node case, while it exhibits a stronger delta-function singularity in the line node case. The spin-orbit cross term is shown to be paramagnetic in the electron side while diamagnetic in the hole side, in contrast with other two terms which are both even functions in Fermi energy. The spin-orbit cross term in the nodal semimetal is found to be directly related to the chiral surface current induced by the topological surface modes.   

Quasi-Topological Electromagnetic Response of Line-node Semimetals

Topological semimetals are gapless states of matter which have robust surface states and interesting electromagnetic responses. We consider the electromagnetic response of gapless phases in $3+1$-dimensions with line nodes. We show through a layering approach that an intrinsic $2$-form ${\cal{B}}_{\mu\nu}$ emerges in the effective response field theory that is determined by the geometry and energy-embedding of the nodal lines. This 2-form is shown to be simply related to the charge polarization and orbital magnetization of the sample. We conclude by discussing the relevance for recently proposed materials and heterostructures with line-node fermi-surfaces.   

Multipolar orders and quantum criticality of a three-dimensional parabolic semimetal

Motivated by the observation of multipolar ordering in many heavy fermion compounds and 227 pyrochlore iridates, we investigate the phase diagram of an interacting, three dimensional parabolic semimetal as a paradigmatic toy model for studying the interplay among electronic correlations, topology and quantum critical phenomena. The generic forms of the local order parameters and quartic interactions are constructed according to the irreducible representations of octahedral point group symmetry. Through a renormalization group analysis, we elucidate the competition between time-reversal symmetric quadrupolar and time-reversal symmetry breaking octopular ordered phases for sufficiently strong interactions. We show that the quadrupolar ordering can give rise to a correlated topological insulator phase, while the octupolar order generically leads to a Weyl semimetal phase. The quantum phase transitions between the semimetal and the broken symmetry phases are controlled by non-Gaussian, itinerant quantum critical points.   

Topological `Luttinger' invariants protected by crystal symmetry in semimetals

Luttinger's theorem is a fundamental result in the theory of interacting Fermi systems: it states that the volume inside the Fermi surface is left invariant by interactions, if the number of particles is held fixed. Although this is traditionally justified using perturbation theory, it can be viewed as arising from a momentum balance argument that examines the response of the ground state to the insertion of a single flux quantum [M. Oshikawa, {\it Phys. Rev. Lett.} {\bf 84}, 3370 (2000)]. This reveals that the Fermi sea volume is a topologically protected quantity. Extending this approach, I show that spinless or spin-rotation-preserving fermionic systems in non-symmorphic crystals possess generalized topological `Luttinger invariants' that can be nonzero even in cases where the Fermi sea volume vanishes. A nonzero Luttinger invariant then forces energy bands to touch, leading to semimetals whose gaplessness is rooted in topology; opening a gap without symmetry breaking automatically triggers fractionalization. The existence of these invariants is linked to the inability of non-symmorphic crystals to host band insulating ground states except at special fillings. I exemplify the use of these new invariants by showing that they distinguish various classes of semimetals.   

Ferromagnetic interactions between transition-metal impurities in topological and 3D Dirac semimetals

The magnitude of ferromagnetic coupling driven by inter-band (Bloembergen-Rowland - BR) and intra-band (Ruderman-Kittel-Kasuya-Yoshida - RKKY) spin polarization is evaluated within $kp$ theory for topological semimetals Hg$_{1-x}$Mn$_x$Te and Hg$_{1-x}$Mn$_x$Se as well as for 3D Dirac semimetal (Cd$_{1-x}$Mn$_x$)$_3$As$_2$. In these systems Mn$^{2+}$ ions do not introduce any carriers. Since, however, both conduction and valence bands are built from anion $p$-type wave functions, hybridization of Mn $d$ levels with neighboring anion $p$ states leads to spin-dependent $p-d$ coupling of both electrons and holes to localized Mn spins, resulting in sizable inter-band spin polarization and, thus in large BR interactions. We demonstrate that this ferromagnetic coupling, together with antiferromagnetic superexchange, elucidate a specific dependence of spin-glass freezing temperature on $x$, determined experimentally for these systems. Furthermore, by employing a multi-orbital tight-binding method, we find that superexchange becomes ferromagnetic when Mn is replaced by Cr or V. Since Cr should act as an isoelectronic impurity in HgTe, this opens a road for realization of ferromagnetic topological insulators based on (Hg,Cr)Te.   

Gyrotropic magnetic effect in Weyl semimetals

The transport current ${\bf J}$ induced in a clean metal by a magnetic field ${\bf B}$ is shown to be equivalent to the low-frequency limit of natural optical activity (optical gyrotropy). For a generic multiband Hamiltonian, there is a simple expression for $\alpha_{ij}= J_i/B_j$ in terms of the intrinsic magnetic moment (orbital plus spin) of the Bloch electrons on the Fermi surface. This ``gyrotropic magnetic effect'' (GME) is fundamentally different from the chiral magnetic effect (CME) driven by the chiral anomaly, which is only nonzero away from equilibrium and is governed by the Berry curvature. The two effects are compared for a minimal model of a Weyl semimetal. We discuss a simple semiclassical picture of the GME and its the possible experimental observation by measuring the rotary power of low-symmetry materials like SrSi$_2$.   

Optical conductivity of disordered Weyl semimetals in collisionless regime at zero temperature

Weyl semimetals have recently attracted considerable attention as prime examples of topologically nontrivial gapless states of quantum matter. They have been experimentally found and the chiral anomaly, which represents their hallmark feature, has been measured. In this work, we study transport in the disordered Weyl semimetals using the Kubo formalism. We consider point-like impurity potentials, which are irrelevant in the renormalization-group sense, and compute the corresponding leading correction to the collisionless conductivity at zero temperature. As a result, we find that all eight possible types of the point-like disorder potentials give rise to a correction to the real part of the optical conductivity in the clean limit, which is universal up to a sign. Consequently, the dielectric constant of a Weyl material receives a disorder correction which is linear in frequency. Finally, we discuss some experimental consequences of our findings.   

Current at a distance and resonant transparency in Weyl semimetals

Surface Fermi arcs are the most prominent manifestation of the topological nature of Weyl semimetals. In the presence of a static magnetic field oriented perpendicular to the sample surface, their existence leads to unique inter-surface cyclotron orbits. We propose two experiments which directly probe the Fermi arcs: a magnetic field dependent non-local DC voltage and sharp resonances in the transmission of electromagnetic waves at frequencies controlled by the field. We show that these experiments are insensitive to small momentum scattering and do not rely on quantum mechanical phase coherence, which renders them far more robust and experimentally accessible than quantum effects. We also comment on the applicability of these ideas to Dirac semimetals.   

Self-consistent theory of electronic states in topological broken-gap quantum wells

Recently broken-gap quantum wells made of InAs/GaSb/AlSb have raised great interest as they may show a gate-tunable phase transition from a trivial phase to a topologically protected quantum spin Hall phase. We present a quantitative self-consistent theory of electronic states in such systems taking into account the charge transfer between different layers which can substantially modify the level structure including the phase boundary between the inverted and non-inverted regime. We also discuss spin effects and the unusual Landau fans in a quantizing magnetic field.   

Topological edge states in ultra thin Bi(110) puckered crystal lattice

We discuss the electronic structure of a 2-ML Bi(110) film with a crystal structure similar to that of black phosphorene. In the absence of Spin-Orbit coupling (SOC), the film is found to be a semimetal with two kinds of Dirac cones, which are classified by their locations in the Brillouin zone. All Dirac nodes are protected by crystal symmetry and carry non-zero winding numbers. When considering ribbons, along specific directions, projections of Dirac nodes serve as starting or ending points of edge bands depending on the sign of their carried winding number. After the inclusion of the SOC, all Dirac nodes are gapped out. Correspondingly, the edge states connecting Dirac nodes split and cross each other, and thus form a Dirac node at the boundary of the 1D Brillouin zone, which suggests that the system is a Quantum Spin Hall insulator. The nontrivial Quantum Spin Hall phase is also confirmed by counting the product of parities of the occupied bands at time-reversal invariant points.   

Detecting 2D symmetry-protected topological phases with the tensor-network method

Symmetry-protected topological (SPT) phases exhibit nontrivial order if symmetry is respected but are adiabatically connected to the trivial product phase if symmetry is not respected. However, unlike the symmetry breaking phase, there is no local order parameter for SPT phases. Here we employ a tensor-network method to compute the topological invariants characterized by the simulated modular S and T matrices proposed by Hung and Wen [PRB 89,075121 (2014)] to study a transition in a one-parameter family of wavefunctions which are Z2 symmetric. The studied wavefunctions are in some sense the SPT analog of Z2 topological states under a string tension. The numerically obtained S and T matrices are able to characterize the two different phases and identify the transition point.   

Analytical characterization of bulk-boundary separation for non-interacting fermionic Hamiltonians

In topological quantum matter the notions of bulk and boundary are closely intertwined by the Hamiltonian. For non-interacting systems, the bulk-boundary correspondence relates this phenomenon to topological properties of the single-particle Hamiltonian defined in momentum space, but, so far, no analytic, systematic approach has been put forward to investigate the edge modes themselves. We show how Schrodinger's equation for a confined system of independent fermions may be separated into a bulk and a boundary equation in a manner that depends critically on the nature of the Hamiltonian. The bulk equation may be solved in closed or near closed form, and the Brillouin zone associated to the infinitely extended system emerges naturally embedded in the full complex plane or higher dimensional analogue. The bulk equation determines uniquely all possible zero modes of the system, whereas the boundary equation selects those, if any, compatible with the prescribed boundary.   

Prediction of two-dimensional topological insulator by forming surface alloy on Au/Si(111) substrate

Two-dimensional (2D) topological insulators (TIs), which can be integrated into the modern silicon industry, are highly desirable for spintronics applications. Here, using first-principles electronic structure calculations, we show that the Au/Si(111)-root3 substrate can provide a new platform for hosting 2D-TIs obtained through the formation of surface alloys with a honeycomb pattern of adsorbed atoms. We systematically examined elements from groups III to VI of the periodic table at 2/3 monolayer coverage on Au/Si(111)-root3, and found that In, Tl, Ge, and Sn adsorbates result in topologically non-trivial phases with band gaps varying from zero to 72 meV. Our scanning tunneling microscopy and low-energy electron diffraction experiments confirm the presence of the honeycomb pattern when Bi atoms are deposited on Au/Si(111)-root3 in accord with our theoretical predictions. Our findings pave the way for using surface alloys as a potential new route for obtaining viable 2D-TI platforms.