Lifting Of Degeneracy Research Articles

Coexistence of orbital degeneracy lifting and superconductivity in iron-based superconductors

In contrast to conventional superconducting (SC) materials, superconductivity in high-temperature superconductors (HTCs) usually emerges in the presence of other fluctuating orders with similar or higher energy scales, thus instigating debates over their relevance for the SC pairing mechanism. In iron-based superconductors (IBSCs), local orbital fluctuations have been proposed to be directly responsible for the structural phase transition and closely related to the observed giant magnetic anisotropy and electronic nematicity. However, whether superconductivity can emerge from, or even coexist with orbital fluctuations, remains unclear. Here we report the angle-resolved photoemission spectroscopy (ARPES) observation of the lifting of symmetry-protected band degeneracy, and consequently the breakdown of local tetragonal symmetry in the SC state of Li(Fe1-xCox)As. Supported by theoretical simulations, we analyse the doping and temperature dependences of this band-splitting and demonstrate an intimate connection between ferro-orbital correlations and superconductivity.

Rotational modes in a phononic crystal with fermion-like behavior

The calculated band structure of a two-dimensional phononic crystal composed of stiff polymer inclusions in a soft elastomer matrix is shown to support rotational modes. Numerical calculations of the displacement vector field demonstrate the existence of modes whereby the inclusions and the matrix regions between inclusions exhibit out of phase rotations but also in phase rotations. The observation of the in-phase rotational mode at low frequency is made possible by the very low transverse speed of sound of the elastomer matrix. A one-dimensional block-spring model is used to provide a physical interpretation of the rotational modes and of the origin of the rotational modes in the band structure. This model is analyzed within Dirac formalism. Solutions of the Dirac-like wave equation possess a spinor part and a spatio-temporal part. The spinor part of the wave function results from a coupling between the senses (positive or negative) of propagation of the wave. The wave-number dependent spinor-part of the wave function for two superposed waves can impose constraints on the integral of the spatio-temporal part that are reflected in a fermion-like lifting of degeneracy in the phonon band structure associated with in-phase rotations.

Acoustics concepts you can demonstrate with a coffee mug

The simple coffee mug can be used to illustrate many important concepts in acoustics. Adding water to the mug shows the effect of mass on the frequency of oscillation. Striking on the handle of the mug compared to striking the side of the mug shows the vibrational mode degeneracy lifting. Adding instant coffee or hot chocolate mix to a mug full of water heated in a microwave oven shows the effect of bubble density on the speed of sound. Each of these concepts are easily demonstrated and can also be used as starting points for student exploration in the laboratory.

Quantum phase diagram of the triangular-lattice XXZ model in a magnetic field.

The triangular lattice of S=1/2 spins with XXZ anisotropy is a ubiquitous model for various frustrated systems in different contexts. We determine the quantum phase diagram of the model in the plane of the anisotropy parameter and the magnetic field by means of a large-size cluster mean-field method with a scaling scheme. We find that quantum fluctuations break up the nontrivial continuous degeneracy into two first-order phase transitions. In between the two transition boundaries, the degeneracy-lifting results in the emergence of a new coplanar phase not predicted in the classical counterpart of the model. We suggest that the quantum phase transition to the nonclassical coplanar state can be observed in triangular-lattice antiferromagnets with large easy-plane anisotropy or in the corresponding optical-lattice systems.

Universal Fluctuation-Driven Eccentricities in Proton-Proton, Proton-Nucleus, and Nucleus-Nucleus Collisions

We show that the statistics of fluctuation-driven initial-state anisotropies in proton-proton, proton-nucleus and nucleus-nucleus collisions is to a large extent universal. We propose a simple parametrization for the probability distribution of the Fourier coefficient $\varepsilon_n$ in harmonic $n$, which is in good agreement with Monte-Carlo simulations. Our results provide a simple explanation for the 4-particle cumulant of triangular flow measured in Pb-Pb collisions, and for the 4-particle cumulant of elliptic flow recently measured in p-Pb collisions. Both arise as natural consequences of the condition that initial anisotropies are bounded by unity. We argue that the initial rms anisotropy in harmonic $n$ can be directly extracted from the measured ratio $v_n\{4\}/v_n\{2\}$: this gives direct access to a property of the initial density profile from experimental data. We also make quantitative predictions for the small lifting of degeneracy between $v_n\{4\}$, $v_n\{6\}$ and $v_n\{8\}$. If confirmed by future experiments, they will support the picture that long-range correlations observed in p-Pb collisions at the LHC originate from collective flow proportional to the initial anisotropy.

Possible Evidence for Helical Nuclear Spin Order in GaAs Quantum Wires

We present transport measurements of cleaved edge overgrowth GaAs quantum wires. The conductance of the first mode reaches 2e(2)/h at high temperatures T≳10 K, as expected. As T is lowered, the conductance is gradually reduced to 1e(2)/h, becoming T independent at T≲0.1 K, while the device cools far below 0.1 K. This behavior is seen in several wires, is independent of density, and not altered by moderate magnetic fields B. The conductance reduction by a factor of 2 suggests lifting of the electron spin degeneracy in the absence of B. Our results are consistent with theoretical predictions for helical nuclear magnetism in the Luttinger liquid regime.

Charge Tuning of Nonresonant Magnetoexciton Phonon Interactions in Graphene

Far from resonance, the coupling of the G-band phonon to magnetoexcitons in single layer graphene displays kinks and splittings versus filling factor that are well described by Pauli blocking and unblocking of inter- and intra-Landau level transitions. We explore the nonresonant electron-phonon coupling by high-magnetic field Raman scattering while electrostatic tuning of the carrier density controls the filling factor. We show qualitative and quantitative agreement between spectra and a linearized model of electron-phonon interactions in magnetic fields. The splitting is caused by dichroism of left- and right-handed circular polarized light due to lifting of the G-band phonon degeneracy, and the piecewise linear slopes are caused by the linear occupancy of sequential Landau levels versus ν.

Field-induced chirality in the helix structure of Ho/Y multilayers

We study the net chirality in the spin helix structure of Ho/Y multilayers induced by an in-plane applied magnetic field. The lifting of degeneracy of the chiral symmetry was revealed by means of polarized neutron reflectometry. Three samples of different thicknesses of the Ho and Y layers were grown by molecular-beam epitaxy. The chiral states are degenerated upon zero field cooling below the critical temperature ${T}_{N}=115\ifmmode\pm\else\textpm\fi{}3$ K. The chirality parameter $\ensuremath{\gamma}$ rises during the field cooling procedure in the field range from 0 to 1 T and saturates at a value of $0.12\ifmmode\pm\else\textpm\fi{}0.01$. The chirality appears stepwise below ${T}_{N}$ and depends weakly on temperature. The phenomenon is interpreted in terms of the Dzyaloshinskii-Moriya interaction appearing at the interface between the Ho and Y layers.

Mode Beating Analysis by Sample Stretching and Wavelength Sweeping in a Few-Mode Fiber

We analyze the beating of coherently copropagating modes in a few-mode optical fiber using two complementary techniques, which involve respectively stretching the sample under examination and sweeping the wavelength for different polarizations of the incoming light. We show that analyzing the spectral composition of the intensity fluctuations allows for a direct measurement of the difference between the mode propagation constants and between their group delays, and we observe the lifting of the mode degeneracy, most possibly induced by residual anisotropies and birefringence. The outlined methodology can be used for the modal characterization of few-mode fibers employed in sensing and in spatially-multiplexed communications.

Analytical model for the coupling constant of a directional coupler in terms of slab waveguides

We present an analytical model for a generic two-wave directional coupler, in the conceptual frame of coupled single-mode planar (slab) waveguides. The modal relation of dispersion is expressed exactly under a matrix form. In the simplest symmetric configuration, the lift of degeneracy between the propagation constants of the even (slow) and odd (fast) supermodes is the exact image of the coupling constant.

TOPOLOGICAL PHOTONIC STATES

As exotic phenomena in optics, topological states in photonic crystals have drawn much attention due to their fundamental significance and great potential applications. Because of the broken time-reversal symmetry under the influence of an external magnetic field, the photonic crystals composed of magneto-optical materials will lead to the degeneracy lifting and show particular topological characters of energy bands. The upper and lower bulk bands have nonzero integer topological numbers. The gapless edge states can be realized to connect two bulk states. This topological photonic states originated from the topological property can be analogous to the integer quantum Hall effect in an electronic system. The gapless edge state only possesses a single sign of gradient in the whole Brillouin zone, and thus the group velocity is only in one direction leading to the one-way energy flow, which is robust to disorder and impurity due to the nontrivial topological nature of the corresponding electromagnetic states. Furthermore, this one-way edge state would cross the Brillouin center with nonzero group velocity, where the negative-zero-positive phase velocity can be used to realize some interesting phenomena such as tunneling and backward phase propagation. On the other hand, under the protection of time-reversal symmetry, a pair of gapless edge states can also be constructed by using magnetic–electric coupling meta-materials, exhibiting Fermion-like spin helix topological edge states, which can be regarded as an optical counterpart of topological insulator originating from the spin–orbit coupling. The aim of this article is to have a comprehensive review of recent research literatures published in this emerging field of photonic topological phenomena. Photonic topological states and their related phenomena are presented and analyzed, including the chiral edge states, polarization dependent transportation, unidirectional waveguide and nonreciprocal optical transmission, all of which might lead to novel applications such as one-way splitter, optical isolator and delay line. In addition, the possible prospect and development of related topics are also discussed.

Direct observation of substrate induced exciton in carbon nanotube

We have successfully measured the electroluminescence spectra of a single-walled carbon nanotube (CNT) grown to serpentine shape on quartz substrate. We observe two emission peaks: One locates at 0.85 eV and is identified as the usual E11 exciton peak, and the other locates at slightly higher energy of 0.94 eV with similar symmetrical line shape and comparable intensity. However, the extra peak is substantially wider and it broadens with increasing current at unusually faster speed. We show that the extra peak is not from interband transitions, and ascribe it to a type of exciton induced by the formation of substrate-CNT superlattice. The periodic surface potential of the substrate modulates the CNT band structure, causes degeneracy lifting and band flattering at the Brillouin zone, and generates the higher energy exciton. For confirmation, a similar device is fabricated using amorphous SiO2 substrate to avoid the formation of the superlattice. Indeed, the extra emission peak disappears.

Confined states in quantum dots defined within finite flakes of bilayer graphene: Coupling to the edge, ionization threshold, and valley degeneracy

We study quantum dots defined by external potentials within finite flakes of bilayer graphene using the tight-binding approach. We find that in the limit of large flakes containing zigzag edges the dot-localized energy levels appear within the energy continuum formed by extended states. As a consequence no ionization threshold for the carriers contained within the dot exists. For smaller flakes with zigzag boundaries the dot-localized energy levels appear interlaced with the energy levels outside the flake, so in a charging experiment the electrons will be added alternately to the dot area and to its neighborhood. We demonstrate that for flakes with armchair boundaries only, an energy window accessible uniquely to the dot-localized states is opened. Then a number of electrons can be added to the dot before the external states start to be occupied. We also discuss coupling of the dot-localized states to the edge states in the context of the valley degeneracy lifting. Moreover, we extract smooth envelope wave functions from the tight-binding solution and discuss their spatial symmetries. The coupling of the dot localized energy levels with reconstructed zigzag edges and atomic vacancies present within the layers is also considered.

Topological aspects and transport properties of edge states in the multi-band superconductor Sr2RuO4

Motivated by the spin-triplet superconductor Sr2RuO4, we investigate the edge states by using a two-band tight-binding model with interorbital hybridization and spin-orbit coupling. In particular we focus on the topological aspects and the transport properties in the chiral spin-triplet superconducting phase. The Fermi surfaces correspond to the ones of Sr2RuO4 where the α and the β bands have hole- and electron-like characters, respectively. Although a full quasiparticle excitation gap is present in the bulk system, gapless edge states appear and affect both the spin and the charge currents. We study the interplay between electron- and hole-like particles in the formation of these edge states and edge currents. Topologically the two bands compensate for each other such that the edge state is not topologically protected. The repulsive interaction yields a spin polarization near the edges and lifts the degeneracy of the subgap edge states. The net spontaneous magnetization generated through both the edge charge current and the spin polarization may be small due to canceling effects. The lifting of the spin degeneracy could likely be detected by quasiparticle tunneling for the surface density of states.

Magnetic Intragap States and Mixed Parity Pairing at the Edge of Spin-Triplet Superconductors

We show that a spontaneous magnetic moment may appear at the edge of a spin-triplet superconductor if the system allows for pairing in a subdominant channel. To unveil the microscopic mechanism behind such an effect, we combine numerical solution of the Bogoliubov-de Gennes equations for a tight-binding model with nearest-neighbor attraction, and the symmetry based Ginzburg-Landau approach. We find that a potential barrier modulating the electronic density near the edge of the system leads to a nonunitary superconducting state close to the boundary where spin-singlet pairing coexists with the dominant triplet superconducting order. We demonstrate that the spin polarization at the edge appears due to the inhomogeneity of the nonunitary state and originates in the lifting of the spin degeneracy of the Andreev bound states.

Valley-polarized massive charge carriers in gapped graphene

The lifting of the fourfold degeneracy of the zeroth Landau level in graphene under high magnetic fields has been the subject of numerous experimental studies, and attributed to various mechanisms such as pure spin splitting, spin splitting combined with subsequent valley splitting, or the formation of a quantum Hall insulator. Unexplored, however, is the influence of an energy gap on the quantum Hall effect (QHE) states in graphene. Here we demonstrate, using measurements of the magnetoresistance of graphene antidot lattices (GALs) in magnetic fields up to 30 T and temperatures between 2 and 100 K, that gap opening in these samples is accompanied by valley polarization and a change from linear to parabolic band structure at low carrier energies. The emergence of a massive character of the carriers profoundly alters the transport characteristics of the zeroth Landau level, which manifests itself in a linear increase of the activated gap with magnetic field. Furthermore, samples of the highest quality display spin splitting on top of the valley splitting, albeit of significantly smaller magnitude.

Momentum-space spectroscopy for advanced analysis of dielectric-loaded surface plasmon polariton coupled and bent waveguides

We perform advanced radiation leakage microscopy of routing dielectric-loaded plasmonic waveguiding structures. By direct-plane imaging and momentum-space spectroscopy, we analyze the energy transfer between coupled waveguides as a function of gap distance and reveal the momentum distribution of curved geometries. Specifically, we observed a clear degeneracy lift of the effective indices for strongly interacting waveguides in agreement with coupled-mode theory. We use momentum-space representations to discuss the effect of curvature on dielectric-loaded waveguides. The experimental images are successfully reproduced by a numerical and an analytical model of the mode propagating in a curved plasmonic waveguide.

Elucidating the Nature of Pseudo Jahn–Teller Distortions in LixMnPO4: Combining Density Functional Theory with Soft and Hard X-ray Spectroscopy

A combination of soft and hard synchrotron-based spectroscopy with first-principles density functional theory within the GGA + U framework is used to investigate the distortion of the Mn local environment of LixMnPO4 as a function of electrochemical delithiation (x = 1.0, 0.75, 0.5, 0.25) and its effect on the electron and hole polaron formation. Analysis of the soft X-ray absorption spectroscopy (XAS) of the Mn L3,2-edges confirmed the evolution from the Mn2+ to the Mn3+ charge state as a two-phase reaction upon delithiation; the corresponding Mn K-edge extended X-ray fine structure measurements clearly revealed a splitting of the Mn–O nearest-neighbor distances with increasing Mn3+ character. In addition, the O K-edge absorption and emission spectra confirmed the corresponding orbital lifting of degeneracy accompanying the distortion of the MnO6 octahedra in the Mn3+ state. Our GGA + U calculations show that the distortion is not a strict Jahn–Teller distortion but is instead a preferential elongation of two of the equatorial Mn–O bonds (edge-sharing with the PO4), which results in a Mn–O–P induction driven hybridization of the unoccupied states (i.e., a pseudo Jahn–Teller distortion). Excellent agreement between the calculated electronic structure and our soft X-ray measurements of the electrochemically delithiated LixMnPO4 nanoparticles verifies the link between the preferential structural distortion and the resultant hybridization of the unoccupied 3d dxz and dx2–y2 orbitals. Our analysis of the corresponding calculated electron and hole polaron supports claims that the elongation of the equatorial bonds (edge-sharing with the PO4) in the Mn3+ charge state (i.e., the pseudo Jahn–Teller distortion) is responsible for increasing the activation energy for polaron migration and the formation energy of the electron (hole) lithium ion (vacancy) complex of the Mn olivine compared to the Fe olivine.

Berry phase mechanism for optical gyrotropy in stripe-ordered cuprates

Optical gyrotropy, the lifting of degeneracy between left and right circularly polarized light, can be generated by either time-reversal or chiral symmetry breaking. In the high-${T}_{c}$ superconductor La${}_{2\ensuremath{-}x}$Ba${}_{x}$CuO${}_{4}$ (LBCO), gyrotropy onsets at the same temperature as charge stripe order, suggesting that the rotation of the stripe direction from one plane to the next generates a helical pattern that breaks chiral symmetry. In order to further test this chiral stacking hypothesis it is necessary to develop an understanding of the physical mechanisms by which chirality generates gyrotropy. In this paper we show that, in chiral metals, optical gyrotropy is a consequence of Berry curvature in momentum space. We describe a physical picture showing that gyrotropy in chiral metals is closely related to the anomalous Hall effect in itinerant ferromagnets. We then calculate the magnitude of the gyrotropic response for a given Berry curvature using the semiclassical picture of anomalous velocity and Boltzmann transport theory. To connect this physical picture with experiment, we calculate the Berry curvature in two tight-binding models. The first model is motivated by the structure of LBCO and illustrates how gyrotropy is created when stripe perturbations are added to a simple cubic model. In the second model, we examine the dramatic enhancement of the gyrotropic coefficient when Rashba spin-orbit coupling is introduced. The magnitude of the rotation of polarization on reflection expected based on these models is calculated and compared with experimental data.

Broken Symmetry Quantum Hall States in Dual-Gated ABA Trilayer Graphene

ABA-stacked trilayer graphene is a unique 2D electron system with mirror reflection symmetry and unconventional quantum Hall effect. We present low-temperature transport measurements on dual-gated suspended trilayer graphene in the quantum Hall (QH) regime. We observe QH plateaus at filling factors ν = -8, -2, 2, 6, and 10, which is in agreement with the full-parameter tight binding calculations. In high magnetic fields, odd-integer plateaus are also resolved, indicating almost complete lifting of the 12-fold degeneracy of the lowest Landau level (LL). Under an out-of-plane electric field E(perpendicular), we observe degeneracy breaking and transitions between QH plateaus. Interestingly, depending on its direction, E(perpendicular) selectively breaks the LL degeneracies in the electron-doped or hole-doped regimes. Our results underscore the rich interaction-induced phenomena in trilayer graphene.