Band Touching Points Research Articles

Nodal line fermions in magnetic oxides

We show by first-principles calculations that the Dirac nodal-line semimetal phase can co-exist with the ferromagnetic order at room temperature in chromium dioxide, a widely used material in magnetic tape applications, under small tensile hydrostatic strains. An ideally flat Dirac nodal ring close to the Fermi energy is placed in the reflection-invariant boundary of the Brillouin zone perpendicular to the magnetic order, and is topologically protected by the unitary mirror symmetry of the magnetic group $D_{4h}(C_{4h})$, which quantizes the corresponding Berry phase into integer multiples of $\pi$. The symmetry-dependent topological stability is demonstrated through showing that only the topologically protected nodal ring can persistently exist under small anisotropic stains preservingthe symmetry $D_{4h}(C_{4h})$, while the other seeming band touching points are generically gapped. Our work provides a practical platform for the investigation of novel physics and potential applications of the Dirac nodal-line and drumhead fermions, in particular those related to ferromagnetic properties

Pressure-Induced Topological Phase Transitions in CdGeSb2 and CdSnSb2.

Using first-principles calculations, we study the occurrence of topological quantum phase transitions (TQPTs) as a function of hydrostatic pressure in CdGeSb2 and CdSnSb2 chalcopyrites. At ambient pressure, both materials are topological insulators, having a finite band gap with inverted order of Sb-s and Sb-p x,p y orbitals of valence bands at the Γ point. Under hydrostatic pressure, the band gap reduces, and at the critical point of the phase transition, these materials turn into Dirac semimetals. Upon further increasing the pressure beyond the critical point, the band inversion is reverted, making them trivial insulators. This transition is also captured by the Lüttinger model Hamiltonian, which demonstrates the critical role played by pressure-induced anisotropy in frontier bands in driving the phase transitions. These theoretical findings of peculiar coexistence of multiple topological phases provide a realistic and promising platform for experimental realization of the TQPTs.

Experimental Observation of a Generalized Thouless Pump with a Single Spin.

Adiabatic cyclic modulation of a one-dimensional periodic potential will result in quantized charge transport, which is termed the Thouless pump. In contrast to the original Thouless pump restricted by the topology of the energy band, here we experimentally observe a generalized Thouless pump that can be extensively and continuously controlled. The extraordinary features of the new pump originate from interband coherence in nonequilibrium initial states, and this fact indicates that a quantum superposition of different eigenstates individually undergoing quantum adiabatic following can also be an important ingredient unavailable in classical physics. The quantum simulation of this generalized Thouless pump in a two-band insulator is achieved by applying delicate control fields to a single spin in diamond. The experimental results demonstrate all principal characteristics of the generalized Thouless pump. Because the pumping in our system is most pronounced around a band-touching point, this work also suggests an alternative means to detect quantum or topological phase transitions.

Band structure and unconventional electronic topology of CoSi

Semimetals with certain crystal symmetries may possess unusual electronic structure topology, distinct from that of the conventional Weyl and Dirac semimetals. Characteristic property of these materials is the existence of band-touching points with multiple (higher than two-fold) degeneracy and nonzero Chern number. CoSi is a representative of this group of materials exhibiting the so-called ‘new fermions’. We report on an ab initio calculation of the electronic structure of CoSi using density functional methods, taking into account the spin–orbit interactions. The linearized Hamiltonian, describing the anisotropic electronic structure of CoSi near the Γ point is derived. The topological features of band-touching nodes with four- and six-fold degeneracy located at the Γ and R points in the first Brillouin zone are analysed using the linearized Hamiltonians and first principle calculations. In particular, we show, using the non-Abelian Berry curvature, that these band-touching points carry topological charges of , which change signs at certain values of parameters of the Hamiltonians. We describe the resulting Fermi arc surface states and their spin texture. We also discuss the influence of many body corrections on the electronic band structure and the topological properties of CoSi.

Floquet band structure of a semi-Dirac system

In this work we use Floquet-Bloch theory to study the influence of circularly and linearly polarized light on two-dimensional band structures with semi-Dirac band touching points, taking the anisotropic nearest neighbor hopping model on the honeycomb lattice as an example. We find circularly polarized light opens a gap and induces a band inversion to create a finite Chern number in the two-band model. By contrast, linearly polarized light can either open up a gap (polarized in the quadratically dispersing direction) or split the semi-Dirac band touching point into two Dirac points (polarized in the linearly dispersing direction) by an amount that depends on the amplitude of the light. Motivated by recent pump-probe experiments, we investigated the non-equilibrium spectral properties and momentum-dependent spin-texture of our model in the Floquet state following a quench in absence of phonons, and in the presence of phonon dissipation that leads to a steady-state independent of the pump protocol. Finally, we make connections to optical measurements by computing the frequency dependence of the longitudinal and transverse optical conductivity for this two-band model. We analyze the various contributions from inter-band transitions and different Floquet modes. Our results suggest strategies for optically controlling band structures and experimentally measuring topological Floquet systems.

Quantum multicriticality in disordered Weyl semimetals

In electronic band structure of solid state material, two band touching points with linear dispersion appear in pair in the momentum space. When they annihilate with each other, the system undergoes a quantum phase transition from three-dimensional Weyl semimetal (WSM) phase to a band insulator phase such as Chern band insulator (CI) phase. The phase transition is described by a new critical theory with a `magnetic dipole' like object in the momentum space. In this paper, we reveal that the critical theory hosts a novel disorder-driven quantum multicritical point, which is encompassed by three quantum phases, renormalized WSM phase, CI phase, and diffusive metal (DM) phase. Based on the renormalization group argument, we first clarify scaling properties around the band touching points at the quantum multicritical point as well as all phase boundaries among these three phases. Based on numerical calculations of localization length, density of states and critical conductance distribution, we next prove that a localization-delocalization transition between the CI phase with a finite zero-energy density of states (zDOS) and DM phase belongs to an ordinary 3D unitary class. Meanwhile, a localization-delocalization transition between the Chern insulator phase with zero zDOS and a renormalized Weyl semimetal (WSM) phase turns out to be a direct phase transition whose critical exponent $\nu=0.80\pm 0.01$. We interpret these numerical results by a renormalization group analysis on the critical theory.

Influence of hopping self-energy effects and quasiparticle degradation on the antiferromagnetic ordering in the bilayer honeycomb Hubbard model

We study the Hubbard model on the AB-stacked bilayer honeycomb lattice with a repulsive onsite interaction U in second order perturbation theory and in self-consistent random phase approximation. We determine the changes in the antiferromagnetic magnetic ordering tendencies due to the real and imaginary parts of the selfenergy at the band crossing points. In particular we present an estimate for the threshold value U* below which the magnetic order is endangered by the splitting of the quadratic band touching points into four Dirac points by an interaction-induced interlayer skew hopping. For most of the parameter space however, the quasiparticle degradation by the frequency-dependence of the sublattice-diagonal selfenergies and the Dirac-cone steepening are more essential for suppressing the AF ordering tendencies considerably. Our results might help to understand to understand the energy scales obtained in renormalization group treatments of the same model and shed light on recent quantum Monte Carlo investigations about the fate of the magnetic ordering down to lower U.

Topological gapless phase in Kitaev model on square lattice

We study the topological feature of gapless states in the fermionic Kitaev model on a square lattice. There are two types of gapless states which are topologically trivial and nontrivial. We show that the topological gapless phase lives in a wide two-dimensional parameter region and are characterized by two vertices of an auxiliary vector field de-fined in the two-dimensional momentum space, with opposite winding numbers. The isolated band touching points, as the topological defects of the field, move, emerge, and disappear as the parameters vary. The band gap starts to open only at the merg-ing points, associated with topologically trivial gapless states. The symmetry protect-ing the topological gapless phase and the robustness under perturbations are also discussed.

Two-dimensional topological semimetal state in a nanopatterned semiconductor system

We propose the creation of a two-dimensional topological semimetal in a semiconductor artificial lattice with triangular symmetry. An in-plane magnetic field drives a quantum phase transition between the topological insulating and topological semimetal phases. The topological semimetal is characterized by robust band touching points which carry quantized Berry flux and edge states which terminate at the band touching points. The topological phase transition is predicted to occur at magnetic fields $\sim 4\text{T}$ in high mobility GaAs artificial lattices, and can be detected via the anomalous behaviour of the edge conductance.

Floquet states in (LaNiO3)2/(LaAlO3)N heterostructures grown along the (111) direction

Using Floquet-Bloch theory we study the effect of circularly and linearly polarized light on the electronic structure of (LaNiO$_3$)$_2$/(LaAlO$_3$)$_N$ heterostructure grown along the (111) direction. In equilibrium, a tight-binding fit to the first principles band structure shows that nearest-neighbor hopping plays a dominant role while second-neighbor hopping breaks the particle-hole symmetry and determines the finer band features. The four bands of the LaNiO$_3$ bilayer exhibit both quadratic band touching points and Dirac points. By varying the amplitude of the incident light, one can independently tune the first and second-neighbor hopping for fixed frequency, which leads to considerable control over the Floquet band structure. We investigate this control in detail, and study how the quadratic and Dirac band touchings are influenced by the polarization and intensity of the light. We derive effective 2-band Hamiltonians (for both quadratic and Dirac band touching points) that accurately captures these results. We further study an extended model which explicitly includes oxygen $p$-orbitals and compare the results to the effective model that contains only the nickel $d$-orbitals. We conclude with a computation of the frequency dependent optical Hall conductivity using the full four band model and analyze the various inter-band contributions of the Floquet modes.

Interacting Weyl fermions: Phases, phase transitions, and global phase diagram

We study the effects of short-range interactions on a generalized three-dimensional Weyl semimetal, where the band touching points act as the (anti)monopoles of Abelian Berry curvature of strength $n$. We show that any local interaction has a \emph{negative} scaling dimension $-2/n$. Consequently all Weyl semimetals are stable against weak short-range interactions. For sufficiently strong interactions, we demonstrate that the Weyl semimetal either undergoes a first order transition into a band insulator or a continuous transition into a symmetry breaking phase. A translational symmetry breaking axion insulator and a rotational symmetry breaking semimetal are two prominent candidates for the broken symmetry phase. At the one-loop order, the correlation length exponent for continuous transitions is $\nu=n/2$, indicating their non-Gaussian nature for any $n>1$. We also discuss the scaling of the thermodynamic and transport quantities in general Weyl semimetals as well as inside the broken symmetry phases.

Topological gapless phases in nonsymmorphic antiferromagnets

Topologically protected fermionic quasiparticles occur in metals with band degeneracy as a consequence of band structure topology. Here we unveil topological semimetal and metal phases in a variety of non-symmorphic collinear antiferromagnets with glide reflection symmetry, a combination of mirror and half-lattice translation. We find gapless phases with Dirac points having multiple symmetry-protection as well as electronic structures with triple and quadruple band-crossing points. Glide semimetal is shown to be converted into a topological phase with non-trivial $\mathbb{Z}_2$ topological charges at the Dirac points due to inversion and time-inversion symmetry combination. More striking is the emergence of a hidden non-unitary relation between the states in the glide sectors that provide a general mechanism to get multiple band touching points. The split Fermi points uncover a $\mathbb{Z}_2$ protection that drives the changeover of the multiple-degenerate gapless phase into a topological metal built from their connection through distinct Fermi lines. Besides a new perspective of ordered states in complex materials, our findings indicate that novel topological gapless phases and edge states may occur in a wide class of magnetic systems.

Importance of spin-orbit coupling in layered organic salts

We investigate the spin-orbit coupling (SOC) effects in $\ensuremath{\alpha}$- and $\ensuremath{\kappa}$-phase BEDT-TTF and BEDT-TSF organic salts. Contrary to the assumption that SOC in organics is negligible due to light C, S, and H atoms, we show the relevance of such an interaction in a few representative cases. In the weakly correlated regime, SOC manifests primarily in the opening of energy gaps at degenerate band touching points. This effect becomes especially important for Dirac semimetals such as $\ensuremath{\alpha}\ensuremath{-}{(\mathrm{ET})}_{2}{\mathrm{I}}_{3}$. Furthermore, in the magnetic insulating phase, SOC results in additional anisotropic exchange interactions, which provide a compelling explanation for the puzzling field-induced behavior of the quantum spin-liquid candidate $\ensuremath{\kappa}\ensuremath{-}{(\mathrm{ET})}_{2}{\mathrm{Cu}}_{2}{(\mathrm{CN})}_{3}$. We conclude by discussing the importance of SOC for the description of low-energy properties in organics.

Quadratic band touching points and flat bands in two-dimensional topological Floquet systems

In this work we theoretically study, using Floquet-Bloch theory, the influence of circularly and linearly polarized light on two-dimensional band structures with Dirac and quadratic band touching points, and flat bands, taking the nearest neighbor hopping model on the kagome lattice as an example. We find circularly polarized light can invert the ordering of this three band model, while leaving the flat-band dispersionless. We find a small gap is also opened at the quadratic band touching point by 2-photon and higher order processes. By contrast, linearly polarized light splits the quadratic band touching point (into two Dirac points) by an amount that depends only on the amplitude and polarization direction of the light, independent of the frequency, and generally renders dispersion to the flat band. The splitting is perpendicular to the direction of the polarization of the light. We derive an effective low-energy theory that captures these key results. Finally, we compute the frequency dependence of the optical conductivity for this 3-band model and analyze the various interband contributions of the Floquet modes. Our results suggest strategies for optically controlling band structure and interaction strength in real systems.

Generating controllable type-II Weyl points via periodic driving

Type-II Weyl semimetals are a novel gapless topological phase of matter discovered recently in 2015. Similar to normal (type-I) Weyl semimetals, type-II Weyl semimetals consist of isolated band touching points. However, unlike type-I Weyl semimetals which have a linear energy dispersion around the band touching points forming a three dimensional (3D) Dirac cone, type-II Weyl semimetals have a tilted cone-like structure around the band touching points. This leads to various novel physical properties that are different from type-I Weyl semimetals. In order to study further the properties of type-II Weyl semimetals and perhaps realize them for future applications, generating controllable type-II Weyl semimetals is desirable. In this paper, we propose a way to generate a type-II Weyl semimetal via a generalized Harper model interacting with a harmonic driving field. When the field is treated classically, we find that only type-I Weyl points emerge. However, by treating the field quantum mechanically, some of these type-I Weyl points may turn into type-II Weyl points. Moreover, by tuning the coupling strength, it is possible to control the tilt of the Weyl points and the energy difference between two Weyl points, which makes it possible to generate a pair of mixed Weyl points of type-I and type-II. We also discuss how to physically distinguish these two types of Weyl points in the framework of our model via the Landau level structures in the presence of an artificial magnetic field. The results are of general interest to quantum optics as well as ongoing studies of Floquet topological phases.

Strain-induced chiral magnetic effect in Weyl semimetals

We argue that strain applied to a time-reversal and inversion breaking Weyl semi-metal in a magnetic field can induce an electric current via the chiral magnetic effect. A tight binding model is used to show that strain generically changes the locations in the Brillouin zone but also the energies of the band touching points (tips of the Weyl cones). Since axial charge in a Weyl semi-metal can relax via inter-valley scattering processes the induced current will decay with a timescale given by the lifetime of a chiral quasiparticle. We estimate the strength and lifetime of the current for typical material parameters and find that it should be experimentally observable.

Topological Varma Superfluid in Optical Lattices.

Topological states of matter are peculiar quantum phases showing different edge and bulk transport properties connected by the bulk-boundary correspondence. While noninteracting fermionic topological insulators are well established by now and have been classified according to a tenfold scheme, the possible realization of topological states for bosons has not been explored much yet. Furthermore, the role of interactions is far from being understood. Here, we show that a topological state of matter exclusively driven by interactions may occur in the p band of a Lieb optical lattice filled with ultracold bosons. The single-particle spectrum of the system displays a remarkable parabolic band-touching point, with both bands exhibiting non-negative curvature. Although the system is neither topological at the single-particle level nor for the interacting ground state, on-site interactions induce an anomalous Hall effect for the excitations, carrying a nonzero Chern number. Our work introduces an experimentally realistic strategy for the formation of interaction-driven topological states of bosons.

Topological Superconductivity in Dirac Semimetals.

Dirac semimetals host bulk band-touching Dirac points and a surface Fermi loop. We develop a theory of superconducting Dirac semimetals. Establishing a relation between the Dirac points and the surface Fermi loop, we clarify how the nontrivial topology of Dirac semimetals affects their superconducting state. We note that the unique orbital texture of Dirac points and a structural phase transition of the crystal favor symmetry-protected topological superconductivity with a quartet of surface Majorana fermions. We suggest the possible application of our theory to recently discovered superconducting states in Cd_{3}As_{2}.

Spontaneous chiral symmetry breaking in bilayer graphene

Bilayer graphene and its thicker cousins with Rhombohedral stacking have attracted considerable attention because of their susceptibility to a variety of broken chiral symmetry states. Due to large density-of-states and quantized Berry phases near their gapless band touching points, each spin-valley flavor spontaneously transfers charge between layers to yield opening of energy gaps in quasiparticle spectra and spreading of momentum-space Berry curvatures. In this article we review the development of theories that predicted such chiral symmetry breaking and classified the possible topological many-body ground states, and the observations in recent experiments that are in reasonable agreement with these theories.

Constructing a Weyl semimetal by stacking one-dimensional topological phases

Topological semimetals in three-dimensions (e.g. Weyl semimetal) can be built by stacking two dimensional topological phases. The interesting aspect of such a construction is that even though the topological building blocks in the low dimension may be gapped, the higher dimensional semimetallic phase emerges as a gapless critical point of a topological phase transition between two distinct insulating phases. In this work, we extend this idea by constructing three-dimensional topological semimetallic phases akin to Weyl systems by stacking one-dimensional Aubry-Andre-Harper (AAH) lattice tight binding models with non-trivial topology. The generalized AAH model is a family of one dimensional tight binding models with cosine modulations in both hopping and onsite energy terms. In this paper, we present a two-parameter generalization of the AAH model that can access topological phases in three dimensions within a unified framework. We show that the $\pi$-flux state of this two-parameter AAH model manifests three dimensional topological semimetallic phases where the topological features are embedded in one dimension. The topological nature of the band touching points of the semimetallic phase in 3D is explicitly established both analytically and numerically from the 1D perspective. This dimensional reduction provides a simple protocol to experimentally construct the three dimensional Brillouin zone of the topological semimetallic phases using `legos' of simple 1D double well optical lattices. We also propose Zak phase imaging of optical lattices as a tool to capture the topological nature of the band touching points. Our work provides a theoretical connection between the commensurate AAH model in 1D and Weyl semimetals in 3D, and points toward practical methods for the laboratory realization of such three dimensional topological systems in atomic optical lattices.