Session AA02: V: Topological Band Structures: Chern Insulators, Topological Insulators and the Quantum Hall Effect

Universal topological marker

We elaborate that for topological insulators and topological superconductors described by Dirac models in any dimension and symmetry class, the topological order can be mapped to lattice sites by a universal topological marker. Deriving from a recently discovered momentum-space universal topological invariant, we introduce a topological operator that consists of alternating projectors to filled and empty lattice eigenstates and the position operators, multiplied by the Dirac matrices that are omitted in the Hamiltonian. The topological operator projected to lattice sites yields the topological marker, whose form is explicitly constructed for every topologically nontrivial symmetry class from 1D to 3D. The off-diagonal elements of the topological operator yields a nonlocal topological marker, which decays with a correlation length that diverges at topological phase transitions, and represents a Wannier state correlation function. Various prototype examples, including Su-Schrieffer-Heeger model, Majorana chain, Chern insulators, Bernevig-Hughes-Zhang model, 2D chiral and helical p-wave superconductors, lattice model of 3He B-phase, and 3D time-reversal symmetric topological insulators, etc, are employed to demonstrate the ubiquity of our formalism.   

Chern dartboard insulator: sBZ topology and skyrmion multipoles

Topology has played a crucial rule in many physical systems, often associated with the nontrivial gapless surface states. The paradigmatic example is the Chern number defined in the Brillouin zone (BZ) which leads to the robust gapless edge states. In this work, we extend the notion of Chern number to the reduced Chern number defined in a fraction of the total BZ. By introducing the idea of sub-Brillouin zone (sBZ) topology, we explicitly construct a family of models with trivial Chern number, yet hosting quantized reduced Chern number in the sBZ. These Chern dartboard insulators are protected by mirror symmetries and belong to a new class of delicate topology. Interestingly, by going into the pseudospin textures, the (anti)skyrmions appear inside the sBZs which are directly related to the sBZ topology. In addition, we observe the gapless edge states in all these systems, in particular with the Möbius fermions and midgap corner states in certain cases. Remarkably, Chern dartboard insulators are noncompact atomic insulators if C₂ symmetry is present. We suggest the photonic systems to realize the Chern dartboard phases and the corresponding gapless edge states. Our work opens the fruitful area to explore the sBZ topology and its nontrivial surface responses in topological systems.   

Hyperbolic Chern Insulators

The Haldane model transforms the semi-metallic graphene into a Chern insulator by introducing complex-valued hopping terms t₂ e^±iφ between the second-nearest neighbors. It is an open question whether the Chern phases are generally present in hyperbolic lattices, recently realized experimentally in circuit QED and topolectrical circuitry. In this work, we theoretically generalize the Haldane model to a large number of regular {p, q} hyperbolic lattices, tessellated by p-sided polygons with q polygons meeting at each site, satisfying (p − 2)(q − 2) > 4. We numerically generate the finite-sized lattices on the Poincaré disk, with p ranging from 6 to 10 and q from 3 to 5, on which we construct the tight-binding hyperbolic Haldane models. Induced gaps are observed in all {p,3} hyperbolic Haldane models, and the computation of real-space Chern numbers indicate nontrivial Chern phases at these gaps. As q increases from 3 to 5, eigenstates appear in the induced gaps, trivializing the real-space Chern numbers. We attribute the absence of nontrivial Chern phases in models with larger q to the stronger negative curvature, which hinders the formation of cyclotron orbits. Furthermore, we discover a universality in the density-of-states as a function of hopping phase φ, such that their qualitative features depend only on p.   

Higher-order Topological Hyperbolic Lattices

A hyperbolic lattice allows for any p-fold rotational symmetry, in stark contrast to a two-dimensional crystalline material, where only twofold, threefold, fourfold or sixfold rotational symmetry is permitted. This unique feature motivates us to ask whether the enriched rotational symmetry in a hyperbolic lattice can lead to any new topological phases beyond a crystalline material. Here, by constructing and exploring tight-binding models in hyperbolic lattices, we theoretically demonstrate the existence of higher-order topological phases in hyperbolic lattices with eight-fold, twelve-fold, sixteen-fold or twenty-fold rotational symmetry, which is not allowed in a crystalline lattice. Since such models respect the combination of time-reversal symmetry and p-fold (p=8, 12, 16 or 20) rotational symmetry, p zero-energy corner modes are protected. For the hyperbolic {8,3} lattice, we find a gapped, a gapless and a reentrant gapped higher-order topological hyperbolic phases. The reentrant phase arises from finite-size effects, which open the gap of edge states while leave the gap of corner modes unchanged. Our results thus open the door to studying higher-order topological phases in hyperbolic lattices.   

Higher-Order Topological Phases Hidden in the Quantum Spin Hall Insulators

Topological materials burgeoned with the discovery of the quantum spin Hall insulators (QSHIs). Since their discovery, QSHIs have been viewed as being Z₂ topological insulators. This commonly held viewpoint, however, hides the far richer nature of the QSHI state. Unlike the Z₂ topological insulator, which hosts gapless boundary states protected by the time-reversal symmetry, the QSHI does not support gapless edge states because the spin-rotation symmetry breaks down in real systems. Here, we demonstrate that QSHIs hide higher-order topological insulator phases through two exemplar systems. We first consider the Kane-Mele model under an external field and show that it carries an odd spin Chern number C_s = 1. The model is found to host gapless edge states in the absence of Rashba spin-orbit coupling (SOC). But, a gap opens up in the edge spectrum when SOC is included, and the system turns into a higher-order topological insulator with in-gap corner states emerging in the spectrum of a nanodisk. We also discuss monolayer α-Sb as an example of a real material, which is time-reversal symmetric, and show that it carries an even spin Chern number C_s = 2. This unique phase of α-Sb has been taken to be topologically trivial because of its gapped edge spectrum. We show it supports in-gap corner states and hosts a higher-order topological phase. Our study unveiled the presence of higher-order topological phases that have so far remained hidden in the QSHI phase. Furthermore, our work suggests that the high spin Chern number insulators provide a new basis for developing materials platforms for efficient spintronic devices.

    

Engineering Supersymmetric Potentials on the Edge of a Quantum Spin Hall System: Emergence of Volkov-Pankratov States and Construction of Reflectionless Potentials

Volkov-Pankratov (VP) states are a family of sub-gap states which appear at the smooth interface ( domain wall ) which separates two distinct gapped  topological states of matter. We exploit known results from supersymmetry quantum mechanics to study the emergence of such states in the edge spectrum of the quantum spin Hall  edge state when exposed to a smoothly varying mass term (Zeeman field) that switches sign at a given spatial point, thereby inducing band-inversion. Both the VP states at non-zero energy and the zero energy Jackiw-Rebbi mode stay localized at the interface, however the spin texture of the VP states turns out to be endowed with periodic windings in real space which is controllable with an electric field to which the Jackiw-Rebbi mode is insensitive. Moreover, the VP states exhibit an intriguing interplay between the electric and the magnetic field with a collapse of the entire spectrum when they are equal. Our theoretical predictions are confirmed with a BHZ model lattice simulation¹. We further take a tan hyperbolic type mass term (which is experimentally more realizable) and show the existence of the in-gap bound states. This type of potential also has the interesting property of reflectionless transmission when certain conditions are met. Using this idea, one can engineer such potentials in the BHZ lattice model in a quantum transport simulation to study the above the gap transmission which is very close to unity. This may have a potential application to engineer a device with lossless transmission². 

    

Multiple non-trivial phases in quasi-2D dual topological insulators

Dual topological insulators (DTIs) have gapped bulk bands whose topology results in conducting edge states protected by time-reversal and crystalline symmetries, i.e., they host the topological insulator and topological crystalline insulator phases simultaneously. Here, we derive a k.p Hamiltonian for mirror-symmetric Na₂CdSn trilayers predicted to be DTIs by ab-initio calculations [1]. By fitting the k.p energy bands to our density functional theory calculations for Na₂CdSn, we obtain the parameters for our effective model. Our solutions for a semi-infinite plane geometry shows that Na₂CdSn hosts two pairs of edge states with quadratic dispersion relations. Varying the parameters, we also show a topological phase transition between such non-trivial phase and a second topological phase with a single pair of quadratic edge states. Our results contribute to the understanding of Na₂CdSn as DTIs, providing a new platform for their study and hopefully encoraging their experimental verification and near-future applications in novel quantum devices.

[1] N. Mao et. al., PRB 100, 205116 (2019).   

Adiabatic charge pump with a topological insulator nanowire device

Quantised charge pumping has been proposed in several confined nanostructures. Although most realisations exploit the Coulomb blockade effect, a different mechanism based on the Landau-Büttiker formalism allows the implementation of adiabatic charge pumping in non-interacting electronic systems, exploiting the constructive interference of the scattering states. The presence of Dirac-like states, protected from non-magnetic disorder, at the surface of topological insulators (TI) can potentially lead to an improvement in the design of charge pump devices.

This work theoretically studies the implementation of quantised adiabatic charge pumping in a device based on a TI nanowire. In contrast to the two-dimensional platform counterpart, in TI nanowires the bulk transport is highly suppressed, making these nanostructures potentially useful for exploiting their protected surface states. We show how adiabatic charge pumping can be implemented in a recently proposed device where the charge confinement is achieved via a radius variation along the nanowire [1]. The pumping mechanism involves the use of only electrostatic gates and avoids the use of strong local magnetic fields, that are experimentally difficult to work with. Limitations and possible extensions of the protocol are also presented.

[1] R. Saxena et al, Phys. Rev. B 106, 035407 (2022).   

Magnetization dynamics and spin-transport in a topological insulator/ferromagnetic metal bilayer system.

We have studied magnetization dynamics and spin-transport phenomenon in the topological insulator/ferromagnetic metal heterostructures here. The precessing magnetization of the metallic ferromagnet (N i80F e20) pumps spin current into the topological insulator (BiSbT e1.5Se1.5) layer due to a phenomenon called the spin-pumping effect. As a result of spin pumping, fast relaxation in the ferromagnet results in the broadening of ferromagnetic resonance linewidth (?H) and enhancement of the damping coefficient of a ferromagnet occurs. The calculated values of spin-transport parameters like spin-mixing conductance (g ↑↓ eff ), and spin current density (j 0 S) in our experiment confirm successful spin injection into the BiSbT e1.5Se1.5 layer. Low-temperature measurements have also been accomplished to understand the effect of the surface and/or bulk state of the topological insulator (TI) in the effective spin-electricity conversion. At low temperatures where the surface state of BiSbTe1.5Se1.5 dominates, the variation of ?H and damping coefficient due to spin pumping(αSP) are more prominent. It indicates a stronger surface state attribution to the dynamics of the ferromagnetic magnetization than the bulk state of the TI.   

Fate of topological character of an SSH ladder in the many-body limit

The Su-Schrieffer-Heeger (SSH) model in one dimension is known to exhibit a topological phase transition (TPT) as a function of hopping dimerization via a gap-closing point. It has been shown that when two such chains form a ladder, if the dimerization patterns coincide between the two chains, the TPT disappears due to hybridization. However, if the patterns are staggered, the TPT persists and the system also allows for a trivial phase. Here, we focus on the behavior of hardcore bosons in such ladder systems. Starting from the single-particle energy spectrum for both the dimerization patterns, we discuss the non-interacting phase diagrams in the many-body limit. Then we perform an analysis of interaction effects, with the prime focus being on the staggered configuration, where the attractive or repulsive nature of the interaction influences the topological critical point. Finally, as an experimentally relevant quantity, we present the behavior of interaction-induced topological charge pumping.