Session Y29: Topological Insulators: Theory II

Gauge-discontinuity contributions to the Chern-Simons orbital magnetoelectric coupling

We propose a new method for calculating the Chern-Simons orbital magnetoelectric coupling, conventionally parametrized in terms of a phase angle $\theta$. We propose to relax the periodicity condition in one direction ($k_z$) so that a gauge discontinuity is introduced on a 2D $\mathbf{k}$ plane normal to $k_z$. The total $\theta$ response then has contributions from both the integral of the Chern-Simons 3-form over the 3D bulk BZ and the gauge discontinuity expressed as a 2D integral over the $\mathbf{k}$ plane. Sometimes the boundary plane may be further divided into subregions by 1D ``vortex loops" which make a third kind of contribution to the total $\theta$, expressed as a combination of Berry phases around the vortex loops. The total $\theta$ thus consists of three terms which can be expressed as integrals over 3D, 2D and 1D manifolds. When time-reversal symmetry is present and the gauge in the bulk BZ is chosen to respect this symmetry, both the 3D and 2D integrals vanish; the entire contribution then comes from the vortex-loop integral, which is either 0 or $\pi$ corresponding to the $\mathbb{Z}_2$ classification of 3D time-reversal invariant insulators. We demonstrate our method by applying it to the Fu-Kane-Mele model with an applied staggered Zeeman field.   

Bulk-boundary correspondence in (3+1)-dimensional topological phases

We discuss (2+1)-dimensional gapless surface theories of bulk (3+1)-dimensional topological phases, such as the BF theory at level K, and its generalization. In particular, we put these theories on a flat (2+1) dimensional torus $\mathbb T^3$ parameterized by its modular parameters, and compute the partition functions obeying various twisted boundary conditions. We show the partition functions are transformed into each other under $SL(3,\mathbb Z)$ modular transformations, and furthermore establish the bulk-boundary correspondence in (3+1) dimensions by matching the modular $\mathcal S$ and $\mathcal T$ ? matrices computed from the boundary field theories with those computed in the bulk. We also propose the three-loop braiding statistics can be studied by constructing the modular $\mathcal S$ and $\mathcal T$? matrices from an appropriate boundary field theory.   

Classification of interacting fermionic phases by dimensional reduction

Topological phases of noninteracting fermions are classified in each spatial dimension according to their symmetry class, in a periodic way [1]. When including interactions, however, this classification can be modified. It was first shown that in one-dimensional chains, the $\mathds{Z}$ classification of the BDI symmetry class is reduced to $\mathds{Z}_8$ [2]. That is, every group of 8 Majorana states at the edge of a BDI chain can be gapped out through a suitable interaction, despite preserving its fundamental symmetries. In this work, we present a dimensional reduction argument to derive the role of interactions in the classification of fermionic symmetry protected topological phases. For symmetry classes classified by a $\mathds{Z}$ invariant in odd dimensions, we propose a general $n$-particle quartic interaction that renders the system topologically trivial. We argue that all phases characterized by a topological invariant smaller than $n$ in the noninteracting limit remain topologically distinct once interactions are included, thereby reducing the noninteracting $\mathds{Z}$ classification to $\mathds{Z}_n$. [1] Ryu, S., \emph{et. al.}, NJP 12, 065010 (2010); [2] Fidkowski, L. and Kitaev, A., PRB 81, 134509 (2010).   

Dynamical Axion Field in a Magnetic Topological Insulator Superlattice

We propose that the dynamical axion field can be realized in a magnetic topological insulator superlattice or a topological paramagnetic insulator. The magnetic fluctuations of these systems produce a pseudoscalar field which has an axionic coupling to the electromagnetic field, and thus it gives a condensed-matter realization of the axion electrodynamics. Compared to the previously proposed dynamical axion materials where a long range antiferromagnetic order is required, the systems proposed here have the advantage that only an uniform magnetization or a paramagnetic state is needed for the dynamic axion. We further propose several experiments to detect such a dynamical axion field.   

Surface theorem for the Chern-Simons magnetoelectric coupling

The magnetoelectric response $\alpha_{ij}=\partial M_j/\partial {\cal E}_i$ of insulators has an isotropic geometric contribution, $\alpha_{ij}^{\rm CS}=(\theta e^2/2\pi h)\delta_{ij}$. For crystals that respect neither inversion nor time-reversal symmetry the Chern-Simons (CS) axion coupling $\theta$ can take arbitrary values, which however can only be determined modulo $2\pi$ from bulk calculations. Once an insulating surface termination is specified it becomes possible to resolve the quantum of indeterminacy, as with the spontaneous electric polarization. We prove this "surface theorem" by considering the $\theta$ coupling of a finite slab from the viewpoint of the hybrid Wannier representation. Each Wannier sheet carries a Chern number, and tiling up the periodic sheet structure close to the surface and counting the leftover Chern amount gives the excess quantized surface anomalous Hall conductivity (AHC). We illustrate these ideas for a tight-binding model consisting of Haldane-model layers with alternating Chern numbers. For appropriate choices of the interlayer couplings, this model realizes an adiabatic pump of CS axion coupling. Over a pumping cycle, one quantum of surface AHC gets transferred from the bottom to the top surface, changing $\theta$ by $2\pi$.   

Effective hydrodynamic field theory and condensation picture of topological insulators

While many features of topological band insulators are commonly discussed at the level of singleparticle electron wave functions, such as the gapless Dirac spectrum at their boundary, it remains elusive to develop a hydrodynamic or collective description of fermionic topological band insulators in 3+1 dimensions. As the Chern-Simons theory for the 2+1-dimensional quantum Hall effect, such a hydrodynamic effective field theory provides a universal description of topological band insulators, even in the presence of interactions, and that of putative fractional topological insulators. In this paper, we undertake this task by using the functional bosonization. The effective field theory in the functional bosonization is written in terms of a two-form gauge field, which couples to a U(1) gauge field that arises by gauging the continuous symmetry of the target system (the U(1) particle number conservation). Integrating over the U(1) gauge field by using the electromagnetic duality, the resulting theory describes topological band insulators as a condensation phase of the U(1) gauge theory (or as a monopole condensation phase of the dual gauge field). The hydrodynamic description, and the implication of its duality, of the surface of topological insulators are also discussed.   

Emergence of a Chern-insulating state from a semi-Dirac dispersion

By combining first-principles calculations with Wannier-based tight-binding modeling, we demonstrate that a TiO$_2$/VO$_2$ heterostructure that was previously proposed as a prototypical semi-Dirac system becomes a Chern insulator (quantum anomalous Hall insulator) in the presence of spin-orbit coupling. We show that this occurs only when the semi-Dirac structure is of a special type that can be formed by the merging of three conventional Dirac points. Our results reveal how the nontrivial topology with nonzero Chern number emerges naturally from this kind of semi-Dirac structure, establishing a general scenario that provides a new route to the formation of Chern-insulating states in practical materials systems.   

Microscopic study of anomalous Hall effect edge states in topological insulator nanoribbons

Thin films of a magnetic topological insulator (TI) support gapless chiral states on their lateral surfaces, which give rise to the quantum anomalous Hall effect (QAHE) [1,2]. Despite progress in the experimental realization of the QAHE there are many open issues which require an explanation, including remnant longitudinal resistance and relatively imprecise Hall quantization compared to the ordinary QHE in a strong perpendicular magnetic field. These features can be linked to the presence of non-chiral dissipative states on the side walls of the sample. We develop a microscopic theory of side-wall states in ribbons of a magnetic TI. We show the emergence of the chiral edge states as a function of exchange field strength and ribbon width. We demonstrate the existence of non-chiral edge states, whose number depends on the system dimensions. In contrast to previous work [3], we find that the non-chiral states are always gapped, a finding which is supported by recent experiments assessing the precision and temperature dependence of QAHE [4]. Finally, we investigate the role of chemical disorder in equilibrating edge channels and in Hall quantization accuracy. 1.Yu, Science 329, 61(2010); 2.Chang, Science 340, 167(2013); 3.Wang, PRL 111, 086803(2013); 4.Chang, PRL 115, 057206(2015).   

Universal framework for identifying topological materials and its numerical implementation in Z2Pack software package

Band structure topology has drastic effects on many observable phenomena in solids, and thus is a fundamental characteristic of a material. We present general framework for identifying various topologies of band structures and introduce a public software package --Z2Pack -- for computing the associated topological invariants. Z2Pack works with first-principles calculations, tight-binding and k.p models. It can be used to identify both topological insulators and semimetals.   

Geometric Effect on Quantum Anomalous Hall State in Magnetic Topological Insulator

An intriguing observation on the quantum anomalous Hall (QAH) effect in a magnetic topological insulator (MTI) is the dissipative edge states. With the aid of non-equilibrium Green's functions,the QAH effect in an MTI with a three dimensional effective tight-binding model is studied.We predict that due to geometric structure in the third dimension $z$,the unideal contact between terminal leads and central scattering region induces the backscattering in the central Hall bar,as the function of split gates. Such backscattering leads to a nonzero longitudinal resistance and quantized Hall resistance, which would explain the dissipative edge states in experiments.A further numerical simulation prove above prediction as well.These results are rewarding on future experimental observations and transport calculations based on first principe.   

Thermodynamic signatures of edge states in Topological Insulators

Topological insulators are states of matter distinguished by the presence of symmetry protected metallic boundary modes. These edge modes have been characterised in terms of transport and spectroscopic measurements, but a thermodynamic description has been lacking. The challenge arises because in conventional thermodynamics the potentials are required to scale linearly with extensive variables like volume, which does not allow for a general treatment of boundary effects. In this paper, we overcome this challenge with Hill thermodynamics. In this extension of the thermodynamic formalism, the grand potential is split into an extensive, conventional contribution, and the subdivision potential, which is the central construct of Hill's theory. For topologically non-trivial electronic matter, the subdivision potential captures measurable contributions to the density of states and the heat capacity: it is the thermodynamic manifestation of the topological edge structure. Furthermore, the subdivision potential reveals phase transitions of the edge even when they are not manifested in the bulk, thus opening a variety of new possibilities for investigating, manipulating, and characterizing topological quantum matter solely in terms of equilibrium boundary physics.   

Topological States of Heterostructures

Topological insulators (TIs) have exotic properties, such as having insulating behavior in the bulk and metallic states at the surface $^{\mathrm{[1]}}$. Observations of metallic states rely on the spin-orbit induced band inversion in bulk materials and are protected by time-reversal symmetry or crystal symmetry$^{\mathrm{\thinspace [2]}}$. These remarkable characteristics of TIs give rise to various applications from spintronics to quantum computers. In order to broaden the range of applications of TIs and make it more effective, an exploration of high quality heterostructures are required. Creating heterostructures of TIs has recently demonstrated to be advantageous for controlling electronic properties $^{\mathrm{[3]}}$. Inspired by these interesting properties, we have investigated the topological interface states of heterostructures. References [1] B. Yan, S-C. Zhang, \textit{Rep. Prog. Phys.}, \textbf{75}, 096501 (2012). [2] Y. Ando, \textit{J. Phys. Soc. Jpn.}, \textbf{82}, 102001 (2013). [3]K. Nakayama, K. Eto, Y. Tanaka, T. Sato, S. Souma, T. Takahashi, K. Segawa, Y. Ando, \textit{Phys. Rev. Lett.}, \textbf{109}, 236804 (2012)   

Detecting band inversions by measuring the environment: fingerprints of electronic band topology in bulk phonon linewidths

The interplay between topological phases of matter and dissipative baths constitutes an emergent research topic with links to condensed matter, photonic crystals, cold atomic gases and quantum information. While recent studies suggest that dissipative baths can induce topological phases in intrinsically trivial quantum materials, the backaction of topological invariants on dissipative baths is overlooked. By exploring this back action for a centrosymmetric Dirac insulator coupled to phonons, we show that the linewidths of bulk optical phonons can reveal electronic band inversions. This result is the first known example where topological phases of an open quantum system may be detected by measuring the bulk properties of the surrounding environment.   

Spin Generation Via Bulk Spin Current in Three Dimensional Topological Insulators

To date, charge transport and spin generation in three-dimensional topological insulators (3D TIs) are primarily modeled as a single-surface phenomenon. We propose a new mechanism of spin generation where the role of the insulating yet topologically non-trivial bulk becomes explicit: an external electric field creates a transverse pure spin current through the bulk of a 3D TI, which transports spins between the top and bottom surfaces and leads to spin accumulation on both. The surface spin density and charge current are then proportional to the spin relaxation time, which for a sufficiently high disorder level can be extended by nonmagnetic scattering analogous to the Dyakonov-Perel spin relaxation mechanism. This new spin generation mechanism suggests a distinct and practical strategy for the enhancement of surface spin polarization by increasing nonmagnetic impurity concentration. Numerical results obtained by coherent potential approximation (CPA) based on a 4-band lattice model confirm that this spin generation mechanism originates from the unique topological connection of the top and bottom surfaces and is absent in other two dimensional systems such as graphene, even though they possess a similar Dirac cone-type dispersion.   

Unconventional band structure for a periodically gated surface of a three dimensional Topological Insulator

The surface states of the three dimensional (3D) Topological Insulators are described by two-dimensional (2D) massless dirac equation. A gate voltage induced one dimensional potential barrier on such surface creates a discrete bound state in the forbidden region outside the dirac cone. Even for a single barrier it is shown such bound state can create electrostatic analogue of Shubnikov de Haas oscillation which can be experimentally observed for relatively smaller size samples. However when these surface states are exposed to a periodic arrangement of such gate voltage induced potential barriers, the band structure of the same got nontrivially modified. This is expected to significantly alters the properties of macroscopic system. We also suggest that in suitable limit the system may offer ways to control electron spin electrostatically which may be practically useful