Fourfold Rotational Symmetry Research Articles

Anisotropic spin fluctuations in detwinned FeSe.

Superconductivity in FeSe emerges from a nematic phase that breaks four-fold rotational symmetry in the iron plane. This phase may arise from orbital ordering, spin fluctuations or hidden magnetic quadrupolar order. Here we use inelastic neutron scattering on a mosaic of single crystals of FeSe, detwinned by mounting on a BaFe2As2 substrate to demonstrate that spin excitations are most intense at the antiferromagnetic wave vectors QAF = (±1, 0) at low energies E = 6-11 meV in the normal state. This two-fold (C2) anisotropy is reduced at lower energies, 3-5 meV, indicating a gapped four-fold (C4) mode. In the superconducting state, however, the strong nematic anisotropy is again reflected in the spin resonance (E = 3.6 meV) at QAF with incommensurate scattering around 5-6 meV. Our results highlight the extreme electronic anisotropy of the nematic phase of FeSe and are consistent with a highly anisotropic superconducting gap driven by spin fluctuations.

Fast reflective optic-based rotational anisotropy nonlinear harmonic generation spectrometer

We present a novel Rotational Anisotropy Nonlinear Harmonic Generation (RA-NHG) apparatus based primarily upon reflective optics. The data acquisition scheme used here allow for fast accumulation of RA-NHG traces, mitigating low frequency noise from laser drift, while permitting real-time adjustment of acquired signals with significantly more data points per unit angle rotation of the optics than other RA-NHG setups. We discuss the design and construction of the optical and electronic components of the device and present example data taken on a GaAs test sample at a variety of wavelengths. The RA-second harmonic generation data for this sample show the expected four-fold rotational symmetry across a broad range of wavelengths, while those for RA-third harmonic generation exhibit evidence of cascaded nonlinear processes possible in acentric crystal structures.

Intertwined Magnetic and Nematic Orders in Semiconducting KFe_{0.8}Ag_{1.2}Te_{2}.

Superconductivity in the iron pnictides emerges from metallic parent compounds exhibiting intertwined stripe-type magnetic order and nematic order, with itinerant electrons suggested to be essential for both. Here we use x-ray and neutron scattering to show that a similar intertwined state is realized in semiconducting KFe_{0.8}Ag_{1.2}Te_{2} (K_{5}Fe_{4}Ag_{6}Te_{10}) without itinerant electrons. We find that Fe atoms in KFe_{0.8}Ag_{1.2}Te_{2} form isolated 2×2 blocks, separated by nonmagnetic Ag atoms. Long-range magnetic order sets in below T_{N}≈35 K, with magnetic moments within the 2×2 Fe blocks ordering into the stripe-type configuration. A nematic order accompanies the magnetic transition, manifest as a structural distortion that breaks the fourfold rotational symmetry of the lattice. The nematic orders in KFe_{0.8}Ag_{1.2}Te_{2} and iron pnictide parent compounds are similar in magnitude and in how they relate to the magnetic order, indicating a common origin. Since KFe_{0.8}Ag_{1.2}Te_{2} is a semiconductor without itinerant electrons, this indicates that local-moment magnetic interactions are integral to its magnetic and nematic orders, and such interactions may play a key role in iron-based superconductivity.

Polarized Raman spectroscopy study of metallic (Sr1−xLax)3Ir2O7 : A consistent picture of disorder-interrupted unidirectional charge order

We have used rotational anisotropic polarized Raman spectroscopy to study the symmetries, the temperature and the doping dependence of the charge ordered state in metallic $(Sr_{1-x}La_{x})_{3}Ir_{2}O_{7}$. Although the Raman probe size is greater than the charge ordering length, we establish that the charge ordering breaks the fourfold rotational symmetry of the underlying tetragonal crystal lattice into twofold, as well as the translational symmetry, and forms short-range domains with $90^{\circ}$ rotated charge order wave vectors, as soon as the charge order sets in below $T_{CO} = \sim$ 200K and across the doping-induced insulator metal transition. We observe that this charge order mode frequency remains nearly constant over a wide temperature range and up to the highest doping level. These above features are highly reminiscent of the ubiquitous unidirectional charge order in underdoped high-$T_C$ copper-oxide-based superconductors (cuprates). We further resolve that the charge order damping rate diverges when approaching $T_{CO}$ from below and increases significantly as increasing the La doping level, which resembles the behaviors for a disorder-interrupted ordered phase and has not been observed for the charge order in cuprates.

Symmetry Incorporated Cost-Effective Architectures for Two-Dimensional Digital Filters

Professor Fettweis as far back as 1977 published a paper generalizing McClellan transformation to obtain circular symmetry in 2-D and spherical, hyper-spherical symmetries in multidimensional digital filters [1]. This survey paper presents stateof-the-art two-dimensional (2-D) VLSI digital filter architectures possessing various symmetries in the filter magnitude response. Preceding the symmetry structures, a generalized formulation is given that allows the derivation of various new 2-D VLSI filter structures of any order without global broadcast. Following this, two types (namely, Type 1 [20] and Type 3 [21], [25], [26]) of cost-effective 2-D magnitude symmetry filter architectures possessing diagonal, four-fold rotational, quadrantal, and octagonal symmetries with reduced number of multipliers are given. By combining the identities of the Types-1 and 3 symmetry filter structures, multimode 2-D symmetry filters which enable the above four symmetry modes are discussed. The Type-1 and Type-3 multimode filters can result in a 65.3% cost reduction in terms of number of multipliers compared with the sum of the multipliers of the four individual Type-1 symmetry filter structures studied in this paper. Furthermore, Type-3 has shorter critical path than Type-1 multimode filter. The paper is concluded with the presentation of a 2-D filter design example and a corresponding structure.

Modeling of quasicrystal lattices with a 4-fold symmetry axis

The construction method of a quasilattice with a four-fold rotational symmetry axis is proposed. The described method is based on the recurrent generation of the initial group of lattice points, which are a set of vertices of a square. The aperiodic crystal reciprocal lattice modeling algorithm is analyzed. Used modeling technique is compared with conventional projection approach. The orthogonal basis of a four-dimensional hypercubic lattice is proposed. This lattice produces two-dimensional quasicrystal with a four-fold symmetry axis after it projection on a flat surface. It is shown that the indexation of diffraction pattern of similar quasiperiodic structures can be carry out using 3 integer indexes, which is analogous to the indexing system proposed by Cahn for application to icosahedral quasicrystals.

Polarization-Independent Narrowband Near-Perfect Absorption Based on One-Dimension Embedded Aluminum Grating

In the past, the polarization-insensitive absorption is realized mainly employing the two-dimensional periodic structure of fourfold rotational symmetry, which greatly increases the manufacturing complexity and cost. Here, we present the numerical design of a polarization-independent near-perfect absorber incorporating one-dimensional embedded aluminum grating. The absorption peaks near 99% at 520-nm wavelength for different polarization angles are achieved. The underlying mechanism associated with the resonance is attributed to the magnetic polariton resonance and the cavity-mode resonance for TM and TE polarization, respectively, and further explained by the inductor-capacitor circuit model and the eigen equation of cavity mode. Furthermore, the effects of geometrical parameters of the nano-cavity on the absorption performance are discussed. The proposed structure may provide a new way to achieve polarization-independent absorption with one-dimensional meta-surface, which has broad applications in plasmonic sensors, photo-detectors, and so on.

Novel topological insulators from crystalline symmetries

We discuss recent advances in the study of topological insulators protected by spatial symmetries by reviewing three representative, theoretical examples. In three dimensions (3D), these states of matter are generally characterized by the presence of gapless boundary states at surfaces that respect the protecting spatial symmetry. We discuss the appearance of these topological states in both crystals with negligible spin–orbit coupling and a fourfold rotational symmetry, as well as in mirror-symmetric crystals with sizable spin–orbit interaction characterized by the so-called mirror Chern number. Finally, we also discuss similar topological crystalline states in one-dimensional (1D) insulators, such as nanowires or atomic chains, with mirror symmetry. There, the prime physical consequence of the non-trivial topology is the presence of quantized end charges.

Bosonic topological excitations from the instability of a quadratic band crossing

We investigate the interaction-driven instability of a quadratic band crossing arising for ultracold bosonic atoms loaded into a two-dimensional optical lattice. We consider the case when the degenerate point becomes a local minimum of both crossing energy bands such that it can support a stable Bose-Einstein condensate. Repulsive contact interaction among the condensed bosons induces a spontaneously time-reversal symmetry broken superfluid phase and a topological gap is opened in the excitation spectrum. We propose two concrete realizations of the desired quadratic band crossing in lattices with either fourfold or sixfold rotational symmetries via suitable tuning of the unit cell leading to reduced Brillouin zones and correspondingly folded bands. In either case, topologically protected edge excitations are found for a finite system.

Imaging Anomalous Nematic Order and Strain in Optimally Doped BaFe_{2}(As,P)_{2}.

We present the strain and temperature dependence of an anomalous nematic phase in optimally doped BaFe_{2}(As,P)_{2}. Polarized ultrafast optical measurements reveal broken fourfold rotational symmetry in a temperature range above T_{c} in which bulk probes do not detect a phase transition. Using ultrafast microscopy, we find that the magnitude and sign of this nematicity vary on a 50-100 μm length scale, and the temperature at which it onsets ranges from 40K near a domain boundary to 60K deep within a domain. Scanning Laue microdiffraction maps of local strain at room temperature indicate that the nematic order appears most strongly in regions of weak, isotropic strain. These results indicate that nematic order arises in a genuine phase transition rather than by enhancement of local anisotropy by a strong nematic susceptibility. We interpret our results in the context of a proposed surface nematic phase.

Higher-order topological insulators.

Three-dimensional topological (crystalline) insulators are materials with an insulating bulk but conducting surface states that are topologically protected by time-reversal (or spatial) symmetries. We extend the notion of three-dimensional topological insulators to systems that host no gapless surface states but exhibit topologically protected gapless hinge states. Their topological character is protected by spatiotemporal symmetries of which we present two cases: (i) Chiral higher-order topological insulators protected by the combination of time-reversal and a fourfold rotation symmetry. Their hinge states are chiral modes, and the bulk topology is -classified. (ii) Helical higher-order topological insulators protected by time-reversal and mirror symmetries. Their hinge states come in Kramers pairs, and the bulk topology is -classified. We provide the topological invariants for both cases. Furthermore, we show that SnTe as well as surface-modified Bi2TeI, BiSe, and BiTe are helical higher-order topological insulators and propose a realistic experimental setup to detect the hinge states.

Ferromagnetic Weyl fermions in CrO2

Ferromagnetic topological semimetals due to their band topology co-existing with intrinsic magnetization exerted important influences on early study of topological fermions. However, they have not been observed in experiments up to now. In this work, we propose that rutile CrO$_{2}$, a widely used half-metallic ferromagnetic material in magnetic recording taps, exhibits unexpected ferromagnetic topological features. Using first-principles calculations and symmetry analysis, we reveal that rutile CrO$_2$ hosts the triple nodal points in the absence of spin orbital coupling (SOC). By taking into account of SOC, each triple nodal point splits into two Weyl points, which are located on the magnetic axis with four-fold rotational symmetry. Notably, the Fermi arcs and accompanying quasiparticle interference patterns are clearly visible, which facilitate experimental observations. In addition, we find that another experimentally synthesized CrO$_{2}$ in CaCl$_2$ structure, also hosts the topologically nontrivial ferromagnetic phase. Our findings substantially advance the experimental realization of ferromagnetic topological semimetals. More importantly, room-temperature time-reversal-breaking Weyl fermions in CrO$_{2}$ may result in promising applications related to the topological phenomena in industry.

Magnetic Phase Diagrams and Magnetization Plateaus of the Spin-1/2 Antiferromagnetic Heisenberg Model on a Square-Kagome Lattice with Three Nonequivalent Exchange Interactions

Magnetization plateaus in quantum spin systems emerges in two-dimensional frustrated systems such as kagome lattice. The spin-1/2 antiferromagnetic Heisenberg model on square-kagome lattice is also appropriate for the study of magnetization plateau. Motivated by recent experimental finding of such a square-kagome lattice with nonequivalent three bonds, we investigate the phase diagrams and magnetization plateaus of the lattice by the exact diagonalization method. In addition to previously reported the 1/3 and 2/3 plateaus in the model with two equivalent bonds, we find a new 2/3 plateau whose magnetic structure is characterized by spontaneously broken four-fold rotational symmetry. The plateau appears only in the case of three nonequivalent bonds. We propose the possibility of finding plateaus including the new one.

Topology-optimized dual-polarization Dirac cones

We apply a large-scale computational technique, known as topology optimization, to the inverse design of photonic Dirac cones. In particular, we report on a variety of photonic crystal geometries, realizable in simple isotropic dielectric materials, which exhibit dual-polarization and dual-wavelength Dirac cones. We demonstrate the flexibility of this technique by designing photonic crystals of different symmetry types, such as ones with four-fold and six-fold rotational symmetry, which possess Dirac cones at different points within the Brillouin zone. The demonstrated and related optimization techniques could open new avenues to band-structure engineering and manipulating the propagation of light in periodic media, with possible applications in exotic optical phenomena such as effective zero-index media and topological photonics.

Polarization-maintaining reflection-mode THz time-domain spectroscopy of a polyimide based ultra-thin narrow-band metamaterial absorber

This paper reports the design, the microfabrication and the experimental characterization of an ultra-thin narrow-band metamaterial absorber at terahertz frequencies. The metamaterial device is composed of a highly flexible polyimide spacer included between a top electric ring resonator with a four-fold rotational symmetry and a bottom ground plane that avoids misalignment problems. Its performance has been experimentally demonstrated by a custom polarization-maintaining reflection-mode terahertz time-domain spectroscopy system properly designed in order to reach a collimated configuration of the terahertz beam. The dependence of the spectral characteristics of this metamaterial absorber has been evaluated on the azimuthal angle under oblique incidence. The obtained absorbance levels are comprised between 67% and 74% at 1.092 THz and the polarization insensitivity has been verified in transverse electric polarization. This offers potential prospects in terahertz imaging, in terahertz stealth technology, in substance identification, and in non-planar applications. The proposed compact experimental set-up can be applied to investigate arbitrary polarization-sensitive terahertz devices under oblique incidence, allowing for a wide reproducibility of the measurements.

Ordered defects in Fe1−xS generate additional magnetic anisotropy symmetries

Non-stoichiometric monoclinic 4C pyrrhotite (Fe7S8), a ferrimagnetic monosulfide that has been intensively used as a remanence carrier to infer the magnetization of the Earth's crust and extraterrestrial materials, exhibits a characteristic low-temperature transition accompanied by complex modifications in anisotropy and magnetization. We demonstrate that the magnetic rotational symmetry of the 4C pyrrhotite is critically affected by the order of the defective Fe-sites, and this in turn is a key to decipher the physics behind the low-temperature transition. Our torque experiments and numerical simulations show an emergent four-fold rotational symmetry in the c-plane of the 4C pyrrhotite at T < 30 K. This symmetry breaking associated with the transition is caused by the competitive interaction of two inherently hexagonal systems generated by two groups of Fe-sites with different local anisotropy fields that stem from the vacancy arrangement in the 4C stacking sequence, and it is triggered by changes in the spin orbit coupling due to the overlap of Fe-Fe electron orbitals at low-temperature. This mechanism provides a new explanation for the magnetic transition in 4C pyrrhotite at low temperature and could also cast light on non-trivial magnetic phenomena in defective systems.

Optical properties of a three-dimensional chiral metamaterial**Project supported by the National Natural Science Foundation of China (Grant No. 11304002), the Natural Science Foundation of Education Bureau of Anhui Province, China (Grant No. KJ2013A136), and the Natural Science Foundation of Anhui Province, China (Grant No. 1208085MA07).

A three-dimensional chiral metamaterial with four-fold rotational symmetry is designed, and its optical properties are investigated by numerical simulations. The results show that this chiral metamaterial has the following features: high polarization conversion, perfect circular dichroism, and asymmetric transmission of circularly polarized light. A comparison of the results of chiral metamaterials without and with weak coupling between the constituent nanostructures enables us to confirm that the optical properties of our proposed nanostructure are closely related to the coupling between the single nanoparticles. This means that the coupling between nanoparticles can enhance the polarization conversion, circular dichroism, and asymmetric transmission. Due to the excellent optical properties, our metamaterial might have potential applications in the development of future multi-functional optical devices.

Raman scattering study of the tetragonal magnetic phase in Sr1−xNaxFe2As2 : Structural symmetry and electronic gap

We use inelastic light scattering to study Sr$_{1-x}$Na$_x$Fe$_2$As$_2$ ($x\approx0.34$), which exhibits a robust tetragonal magnetic phase that restores the four-fold rotation symmetry inside the orthorhombic magnetic phase. With cooling, we observe splitting and recombination of an $E_g$ phonon peak upon entering the orthorhombic and tetragonal magnetic phases, respectively, consistent with the reentrant phase behavior. Our electronic Raman data reveal a pronounced feature that is clearly associated with the tetragonal magnetic phase, suggesting the opening of an electronic gap. No phonon back-folding behavior can be detected above the noise level, which implies that any lattice translation symmetry breaking in the tetragonal magnetic phase must be very weak.

Predicted Weyl fermions in magnetic GdBi and GdSb

Motivated by the chiral anomaly steering negative longitudinal magnetoresistance in GdBiPt under external magnetic field, we studied the electronic structures of GdBi with paramagnetism, antiferromagnetism and ferromagnetism by first-principles calculations with modified Becke and Johnson local density approximation plus Hubbard [Formula: see text]. Our calculated results reveal that paramagnetic GdBi is semiconducting, while the antiferromagnetic GdBi is a topological nontrivial compensation metal. We also predict the presence of a pair of Weyl fermions in ferromagnetic GdBi and GdSb. The band crossing along the direction of magnetization is protected by the fourfold rotation symmetry, and the topological charge associating with each [Formula: see text] band crossing point is [Formula: see text].

Local breaking of fourfold rotational symmetry by short-range magnetic order in heavily overdoped Ba(Fe1−xCux)2As2

We investigate Cu-doped Ba(Fe$_{1-x}$Cu$_x$)$_2$As$_2$ with transport, magnetic susceptibility, and elastic neutron scattering measurements. In the heavily Cu-doped regime where long-range stripe-type antiferromagnetic order in BaFe$_2$As$_2$ is suppressed, Ba(Fe$_{1-x}$Cu$_x$)$_2$As$_2$ (0.145 $\leq x \leq$ 0.553) samples exhibit spin-glass-like behavior in magnetic susceptibility and insulating-like temperature dependence in electrical transport. Using elastic neutron scattering, we find stripe-type short-range magnetic order in the spin-glass region identified by susceptibility measurements. The persistence of short-range magnetic order over a large doping range in Ba(Fe$_{1-x}$Cu$_x$)$_2$As$_2$ likely arises from local arrangements of Fe and Cu that favor magnetic order, with Cu acting as vacancies relieving magnetic frustration and degeneracy. These results indicate locally broken four-fold rotational symmetry, suggesting that stripe-type magnetism is ubiquitous in iron pnictides.