Fourfold Degeneracy Research Articles

Floquet topological phases with fourfold-degenerate edge modes in a driven spin-1/2 Creutz ladder

Floquet engineering has the advantage of generating new phases with large topological invariants and many edge states by simple driving protocols. In this work, we propose an approach to obtain Floquet edge states with fourfold degeneracy and even-integer topological characterizations in a spinful Creutz ladder model, which is realizable in current experiments. Putting the ladder under periodic quenches, we found rich Floquet topological phases in the system, which belong to the symmetry class CII. Each of these phases is characterized by a pair of even integer topological invariants $(w_{0},w_{\pi}) \in 2\mathbb{Z} \times 2 \mathbb{Z}$, which can take arbitrarily large values with the increase of driving parameters. Under the open boundary condition, we further obtain multiple quartets of topological edge states with quasienergies zero and $\pi$ in the system. Their numbers are determined by the bulk topological invariants $(w_{0},w_{\pi})$ due to the bulk-edge correspondence. Finally, we propose a way to dynamically probe the Floquet topological phases in our system by measuring a generalized mean chiral displacement. Our findings thus enrich the family of Floquet topological matter, and put forward the detection of their topological properties.

The Mechanism of Formation of Nanostructures of Hexagonal Symmetry on Surfaces of Metals by a Sequence of Doubled Ultrashort Pulses of Radiation of Orthogonal Polarization

Recent experimental results on formation of ordered surface nanostructures of hexagonal symmetry on metals under the action of a series of scanned doubled collinear femtosecond pulses of orthogonal polarization and variable time delay are analyzed. It is demonstrated that the appearance of structures is related to the formation of a dynamic grating by the first pulse and excitation of two pairs of surface plasmon–polaritons by the second pulse in directions symmetric with respect to its vector of polarization and forming an angle of π/3 with each other. This leads to the formation of corresponding dynamic gratings and fourfold degeneracy of the first grating. Three dynamic gratings symmetric with respect to the surface normal form a residual surface relief of hexagonal symmetry that can be observed experimentally.

S=12 kagome Heisenberg antiferromagnet revisited

We examine the perennial quantum spin-liquid candidate $S=1/2$ Heisenberg antiferromagnet on the kagome lattice. Our study is based on achieving Lanczos diagonalization of the Hamiltonian on a $48$ site cluster in sectors with dimensions as a large as $5 \times 10^{11}$. The results reveal novel intricate structures in the low-lying energy spectrum. These structures by no means unambiguously support an emerging consensus of a $\mathbb{Z}_2$ spin liquid ground state, but instead appear compatible with several scenarios, including four-fold topological degeneracy, inversion symmetry breaking and a combination thereof. We discuss finite-size effects, such as the apparent absence of ETH, and note that while considerably reduced, some are still present for the largest cluster. Finally, we observe that an XXZ model in the Ising limit reproduces remarkably well the most striking features of finite-size spectra.

Moiré-pattern fluctuations and electron-phason coupling in twisted bilayer graphene

In twisted bilayer graphene, long-wavelength lattice fluctuations on the scale of the moir\'e period are dominated by phason modes, i.e., acoustic branches of the incommensurate lattice resulting from coherent superpositions of optical phonons. In the limit of small twist angles, these modes describe the sliding motion of stacking domain walls separating regions of partial commensuration. The resulting soliton network is a soft elastic manifold, whose reduced rigidity arises from the competition between intralayer (elastic) and interlayer (adhesion) forces governing lattice relaxation. Shear deformations of the beating pattern dominate the electron-phason coupling to the leading order in ${t}_{\ensuremath{\perp}}/t$, the ratio between interlayer and intralayer hopping parameters. This coupling lifts the layer degeneracy of the Dirac cones at the corners of the moir\'e Brillouin zone, which could explain the observed fourfold (instead of eightfold) Landau level degeneracy. Electron-phason scattering gives rise to a linear-in-temperature contribution to the resistivity that increases with decreasing twist angle due to the reduction of the stiffness of the soliton network. This contribution alone, however, seems to be insufficient to explain the huge enhancement of the resistivity of the normal state close to the magic angle.

Emergent Haldane phase in an alternating-bond Z3 parafermion chain

The Haldane phase represents one of the most important symmetry-protected states in modern physics. This state can be realized using spin-1 and $\mathrm{spin}\ensuremath{-}\frac{1}{2}$ Heisenberg models and bosonic particles. Here we explore the emergent Haldane phase in an alternating bond ${\mathbb{Z}}_{3}$ parafermion chain, which is different from the previous proposals from fundamental statistics and symmetries. We show that this emergent phase can also be characterized by a modified long-range string order, as well as fourfold degeneracy in the ground-state energies and entanglement spectra. This phase is protected by both the charge conjugate and parity symmetry, and the edge modes are shown to satisfy parafermionic statistics in which braiding of the two edge modes yields a $\frac{2\ensuremath{\pi}}{3}$ phase. This model also supports rich phases, including a topological ferromagnetic parafermion (FP) phase, trivial paramagnetic parafermion phase, classical dimer phase, and gapless phase. The boundaries of the FP phase are shown to be gapless and critical with central charge $c=4/5$. Even in the topological FP phase, it is also characterized by long-range string order; thus we observe a drop of string order across the phase boundary between the FP phase and the Haldane phase. This work opens a new way for finding of exotic topological phases with parafermions.

Pseudospins and topological edge states for fundamental antisymmetric Lamb modes in snowflakelike phononic crystal slabs.

The topological transport of Lamb wave in phononic crystal slabs provides a great potential in reinforcing nondestructive testing, high sensitivity sensing, and information processing. In this paper, the authors investigate the pseudospins edge states of fundamental antisymmetric Lamb waves in a snowflakelike phononic slab. Significantly, the fourfold Dirac degeneracy for antisymmetric Lamb mode is accidentally formed at the Γ point with the critical angle of the snowflakelike holes, which does not require the folding of the lattices. Meanwhile, based on the rotating-scatterer mechanism, the mirror symmetry is broken and the topological multipole phase transitions are well induced during the gradual change of the scattering strength among the scatterers with the rotation angle. The topologically protected edge states and its unidirectional robust propagation are further demonstrated. The proposed topological phononic slabs will be a more hopeful option to apply in engineering practices.

Ferromagnetic Mott state in Twisted Graphene Bilayers at the Magic Angle.

We address the effective tight-binding Hamiltonian that describes the insulating Mott state of twisted graphene bilayers at a magic angle. In that configuration, twisted bilayers form a honeycomb superlattice of localized states, characterized by the appearance of flat bands with fourfold degeneracy. After calculating the maximally localized superlattice Wannier wave functions, we derive the effective spin model that describes the Mott state. We suggest that the system is an exotic ferromagnetic Mott insulator, with well-defined experimental signatures.

Fractional charged edge states in ladder topological insulators

We propose a model of two-leg ladder topological insulator in which the spin–orbit couplings are presented in both intra-chain and inter-chain interactions. The topological phase supports four fractional charged edge states localized at opposite ends of the ladder, which belongs to the chiral symplectic (CII) class protected by time-reversal symmetry and chiral symmetry. In our model, the presence of time-reversal and chiral symmetry generates fourfold degeneracy for the edge states, and the two edge states with same chirality at one end of the ladder each carries half charge. In contrast to the two edge states spatially localized at one end of the ladder being not distinguished, these two edge states can be detected by the momentum density. The experimental scheme for realizing our model with cold atoms in optical lattice is discussed. By introducing a magnetic field in the x direction, the system is driven from CII class to AIII class. In AIII class, there exist two distinct topological phases that exhibit four degenerate edge states and two degenerate edge states in the gap, respectively. As same as the system in CII class, each edge state carries a half charge in AIII class.

Inverse design of quantum spin hall-based phononic topological insulators

We propose a computational methodology to perform inverse design of quantum spin hall effect (QSHE)-based phononic topological insulators. We first obtain two-fold degeneracy, or a Dirac cone, in the band structure using a level set-based topology optimization approach. Subsequently, four-fold degeneracy, or a double Dirac cone, is obtained by using zone folding, after which breaking of translational symmetry, which mimics the effect of strong spin-orbit coupling and which breaks the four-fold degeneracy resulting in a bandgap, is applied. We use the approach to perform inverse design of hexagonal unit cells of C6 and C3 symmetry. The numerical examples show that a topological domain wall with two variations of the designed metamaterials exhibit topologically protected interfacial wave propagation, and also demonstrate that larger topologically-protected bandgaps may be obtained with unit cells based on C3 symmetry.

Dual-band acoustic topological insulator based on honeycomb lattice sonic crystal

Based on honeycomb-lattice sonic crystals with gear-like scatterers, we study and design a pseudospin-dependent dual-band acoustic topological insulator. Compared with cylindrical scatterers with only a single tunable structure parameter (radius), there exist four tunable parameters for the gear scatterer, which enables the sonic crystal to realize four-fold accidental degeneracy at two different frequencies simultaneously. By changing structure parameters of the gear-like scatterers, we can obtain topological phase transitions between two sonic crystals. Based on this, we design acoustic topological waveguides based on two honeycomb-lattice sonic crystals with different topological phases, and introduce two kinds of defects (a lattice disorder and a bend) into the topological waveguide near the domain wall. Numerical simulations show that pseudospin edge states almost immune to two types of defects and can pass through the topological waveguides with negligible backscatterings. Compared with the results for the topological waveguide without defects, the measured transmission spectra are almost unchanged with the two types of defects, which further experimentally verify the robustness of pseudospin-dependent edge states. Additionally, by keeping the structure of the sonic crystals unchanged, we can also obtain another four-fold accidental degenerate Dirac point and the corresponding topological sound phase transitions in the high-frequency region. The simulations show that there also exists a pair of edge states in the overlapped bulk bandgap of the two sonic crystals in the high-frequency region. It is worth noting that the tiny gap between two edge states is larger than that in the low-frequency region, which may arise from the greater difference between the distributions of pressure eigenfunction of two sonic crystals. The proposed dual-band acoustic topology insulator has potential applications in multi-band sound communication and sound information processing.

Transition between Electron Localization and Antilocalization and Manifestation of the Berry Phase in Graphene on a SiC Surface

It is shown that the transport properties of graphitized silicon carbide are controlled by a surface graphene layer heavily doped with electrons. In weak magnetic fields and at low temperatures, a negative magnetoresistance is observed due to weak localization. A crossover in the magnetoresistance from weak localization to weak antilocalization (the latter is the manifestation of the isospin in graphene) is observed for the first time in samples of this kind at elevated temperatures. A pronounced pattern of Shubnikov–de Haas oscillations is observed in strong magnetic fields (up to 30 T). This pattern demonstrated fourfold carrier spectrum degeneracy due to the double spin and double valley degeneracies. Also, the manifestation of the Berry phase is observed. The effective electron mass is estimated to be m* = 0.08m0, which is characteristic of graphene with a high carrier concentration.

Magnetic field induced topological semimetals near the quantum critical point of pyrochlore iridates

Motivated by the recent experimental observation of anomalous magneto-transport properties near the Mott quantum critical point (QCP) of pyrochlore iridates, we study the generic topological band structure near QCP in the presence of magnetic field. We have found that the competition between different energy scales can generate various topological semi-metal phases near QCP. Here the central role is played by the presence of a quadratic band crossing (QBC) with four-fold degeneracy in the paramagnetic band structure. Due to the large band degeneracy and strong spin-orbit coupling, the degenerate states at QBC can show an anisotropic Zeeman effect as well as the conventional isotropic Zeeman effect. Through the competition between three different magnetic energy scales including the exchange energy between Ir electrons and two Zeeman energies, various topological semimetals can be generated near QCP. Moreover, we have shown that these three magnetic energy scales can be controlled by modulating the magnetic multipole moment (MMM) of the cluster of spins in a unit cell, which can couple to the intrinsic MMM of the degenerate states at QBC. We propose the general topological band structure under magnetic field achievable near QCP, which would facilitate the experimental discovery of novel topological semimetal states in pyrochlore iridates.

Quantum Transport and Band Structure Evolution under High Magnetic Field in Few-Layer Tellurene.

Quantum Hall effect (QHE) is a macroscopic manifestation of quantized states that only occurs in confined two-dimensional electron gas (2DEG) systems. Experimentally, QHE is hosted in high-mobility 2DEG with large external magnetic field at low temperature. Two-dimensional van der Waals materials, such as graphene and black phosphorus, are considered interesting material systems to study quantum transport because they could unveil unique host material properties due to the easy accessibility of monolayer or few-layer thin films at the 2D quantum limit. For the first time, we report direct observation of QHE in a novel low-dimensional material system, tellurene. High-quality 2D tellurene thin films were acquired from recently reported hydrothermal method with high hole mobility of nearly 3000 cm2/(V s) at low temperatures, which allows the observation of well-developed Shubnikov-de Haas (SdH) oscillations and QHE. A four-fold degeneracy of Landau levels in SdH oscillations and QHE was revealed. Quantum oscillations were investigated under different gate biases, tilted magnetic fields, and various temperatures, and the results manifest the inherent information on the electronic structure of Te. Anomalies in both temperature-dependent oscillation amplitudes and transport characteristics were observed that are ascribed to the interplay between the Zeeman effect and spin-orbit coupling, as depicted by the density functional theory calculations.

OGLE-2017-BLG-1130: The First Binary Gravitational Microlens Detected from Spitzer Only

Abstract We analyze the binary gravitational microlensing event OGLE-2017-BLG-1130 (mass ratio q ∼ 0.45), the first published case in which the binary anomaly was detected only by the Spitzer Space Telescope. This event provides strong evidence that some binary signals can be missed by observations from the ground alone but detected by Spitzer. We therefore invert the normal procedure, first finding the lens parameters by fitting the space-based data and then measuring the microlensing parallax using ground-based observations. We also show that the normal four-fold space-based degeneracy in the single-lens case can become a weak eight-fold degeneracy in binary-lens events. Although this degeneracy is resolved in event OGLE-2017-BLG-1130, it might persist in other events.

Direct observation of the orbital spin Kondo effect in gallium arsenide quantum dots

Besides the spin Kondo effect, other degrees of freedom can give rise to the pseudospin Kondo effect. We report a direct observation of the orbital spin Kondo effect in a series-coupled gallium arsenide (GaAs) double quantum dot device where orbital degrees act as pseudospin. Electron occupation in both dots induces a pseudospin Kondo effect. In a region of one net spin impurity, complete spectra with three resonance peaks are observed. Furthermore, we observe a pseudo-Zeeman effect and demonstrate its electrical controllability for the artificial pseudospin in this orbital spin Kondo process via gate voltage control. The fourfold degeneracy point is realized at a specific value supplemented by spin degeneracy, indicating a transition from the SU(2) to the SU(4) Kondo effect.

Переход между электронной локализацией и антилокализацией, а также проявление фазы Берри в графене на поверхности SiC

AbstractIt is shown that the transport properties of graphitized silicon carbide are controlled by a surface graphene layer heavily doped with electrons. In weak magnetic fields and at low temperatures, a negative magnetoresistance is observed due to weak localization. A crossover in the magnetoresistance from weak localization to weak antilocalization (the latter is the manifestation of the isospin in graphene) is observed for the first time in samples of this kind at elevated temperatures. A pronounced pattern of Shubnikov–de Haas oscillations is observed in strong magnetic fields (up to 30 T). This pattern demonstrated fourfold carrier spectrum degeneracy due to the double spin and double valley degeneracies. Also, the manifestation of the Berry phase is observed. The effective electron mass is estimated to be m * = 0 . 08 m _0, which is characteristic of graphene with a high carrier concentration.

Z4 parafermions in an interacting quantum spin Hall Josephson junction coupled to an impurity spin

$\mathbb{Z}_4$ parafermions can be realized in a strongly interacting quantum spin Hall Josephson junction or in a spin Hall Josephson junction strongly coupled to an impurity spin. In this paper we study a system that has both features, but with weak (repulsive) interactions and a weakly coupled spin. We show that for a strongly anisotropic exchange interaction, at low temperatures the system enters a strong coupling limit in which it hosts two $\mathbb{Z}_4$ parafermions, characterizing a fourfold degeneracy of the ground state. We construct the parafermion operators explicitly, and show that they facilitate fractional $e/2$ charge tunneling across the junction. The dependence of the effective low-energy spectrum on the superconducting phase difference reveals an $8\pi$ periodicity of the supercurrent.

Topological quantum phase transition from mirror to time reversal symmetry protected topological insulator

Topological insulators constitute a new phase of matter protected by symmetries. Time-reversal symmetry protects strong topological insulators of the Z2 class, which possess an odd number of metallic surface states with dispersion of a Dirac cone. Topological crystalline insulators are merely protected by individual crystal symmetries and exist for an even number of Dirac cones. Here, we demonstrate that Bi-doping of Pb1−xSnxSe (111) epilayers induces a quantum phase transition from a topological crystalline insulator to a Z2 topological insulator. This occurs because Bi-doping lifts the fourfold valley degeneracy and induces a gap at bar Gamma , while the three Dirac cones at the {bar{rm M}} points of the surface Brillouin zone remain intact. We interpret this new phase transition as caused by a lattice distortion. Our findings extend the topological phase diagram enormously and make strong topological insulators switchable by distortions or electric fields.

Weyl points and Dirac lines protected by multiple screw rotations

In three-dimensional noncentrosymmetric materials two-fold screw rotation symmetry forces electron’s energy bands to have Weyl points at which two bands touch. This is illustrated for space groups No. 19 (P212121) and No. 198 (P213), which have three orthogonal screw rotation axes. In the case of space groups No. 61 (Pbca) and No. 205 (Pa-3) that have extra inversion symmetry, Weyl points are promoted to four-fold degenerate line nodes in glide-invariant planes. The three-fold rotation symmetry present in the space groups No. 198 and No. 205 allows Weyl and Dirac points, respectively, to appear along its rotation axes in the Brillouin zone and generates four-fold and six-fold degeneracy at the Γ point and R point, respectively.

Single Nodal Loop of Accidental Degeneracies in Minimal Symmetry: Triclinic CaAs_{3}.

The existence of closed loops of degeneracies in crystals has been intimately connected with associated crystal symmetries, raising the following question: What is the minimum symmetry required for topological character, and can one find an example? Triclinic CaAs_{3}, in the space group P1[over ¯] with only a center of inversion, has been found to display, without need for tuning, a nodal loop of accidental degeneracies with topological character, centered on one face of the Brillouin zone that is otherwise fully gapped. The small loop is very flat in energy, yet is cut four times by the Fermi energy, a condition that results in an intricate repeated touching of inversion related pairs of Fermi surfaces at Weyl points. Spin-orbit coupling lifts the fourfold degeneracy along the loop, leaving trivial Kramers pairs. With its single nodal loop that emerges without protection from any point group symmetry, CaAs_{3} represents the primal "hydrogen atom" of nodal loop systems.