Six-fold Rotational Symmetry Research Articles

The Relation Between Cardinal Axes, Spatial Cells and Navigation Performance

Abstract Obtaining unbiased spatial information is a prerequisite for accurate spatial orientation, navigation, and spatial memory. Common sources for obtaining such spatial information are 2D maps and 3D visualizations of space. However, the exact way external spatial representations are integrated into a cognitive map is still not fully understood. Currently, there is much research on the cells forming the spatial map in the hippocampal and parahippocampal cortices. In interaction with other cell types, grid cells appear to play a dominant role in the formation of the cognitive spatial map. Grid cells are characterized by repeated firing fields arranged in a sixfold rotational symmetry. Using a path integration task, we investigated whether artificial spatial elements mirroring the sixfold rotational symmetry can improve navigation performance. This would be an indication that grid cells respond to these artificial elements. In the reported study, we investigated the influence of artificial cardinal axes arranged with 60° (matched to grid cell symmetry) and 30° (mismatched to grid cell symmetry) angles in a reduced 3D spatial environment. Although, the results are not conclusive, they do indicate a trend that participants benefit from a 60° angle in trials with moderate difficulty. Thus, we found the first evidence that mirroring their sixfold rotational symmetry with artificial cardinal axes may trigger grid cells. Future studies should investigate this connection in more detail and a virtual 3D environment shown by head-mounted displays. This might lead to a more detailed insight into the neuropsychological mechanisms underlying the transfer from maps and 3D environments in the cognitive map and how this can be optimized.

Superconductivity and nematic order in a new titanium-based kagome metal CsTi3Bi5 without charge density wave order

The cascade of correlated topological quantum states in the newly discovered vanadium-based kagome superconductors, AV3Sb5 (A = K, Rb, and Cs), with a Z2 topological band structure has sparked immense interest. Here, we report the discovery of superconductivity and electronic nematic order in high-quality single-crystals of a new titanium-based kagome metal, CsTi3Bi5, that preserves the translation symmetry, in stark contrast to the charge density wave superconductor AV3Sb5. Transport and magnetic susceptibility measurements show superconductivity with an onset superconducting transition temperature Tc of approximately 4.8 K. Using the scanning tunneling microscopy/spectroscopy and Josephson scanning tunneling spectroscopy, we demonstrate that the single crystals of CsTi3Bi5 exhibit two distinct superconducting gaps. Furthermore, the superconducting gaps break the six-fold crystal rotational symmetry down to two-fold. At low energies, we find that the quasiparticle interference patterns exhibit rotational-symmetry-breaking C2 patterns, revealing a nematic ordered normal state with the same nematic direction as in the superconducting state. Our findings uncover a novel superconducting state in CsTi3Bi5 and provide new insights for the intrinsic electron liquid crystal phases in kagome superconductors.

Real-Space Imaging of Intrinsic Symmetry-Breaking Spin Textures in a Kagome Lattice.

The electronic orders in kagome materials have emerged as a fertile platform for studying exotic quantum states, and their intertwining with the unique kagome lattice geometry remains elusive. While various unconventional charge orders with broken symmetry is observed, the influence of kagome symmetry on magnetic order has so far not been directly observed. Here, using a high-resolution magnetic force microscopy, it is, for the first time, observed a new lattice form of noncollinear spin textures in the kagome ferromagnet in zero magnetic field. Under the influence of the sixfold rotational symmetry of the kagome lattice, the spin textures are hexagonal in shape and can further form a honeycomb lattice structure. Subsequent thermal cycling measurements reveal that these spin textures transform into a non-uniform in-plane ferromagnetic ground state at low temperatures and can fully rebuild at elevated temperatures, showing a strong second-order phase transition feature. Moreover, some out-of-plane magnetic moments persist at low temperatures, supporting the Kane-Mele scenario in explaining the emergence of the Dirac gap. The observations establish that the electronic properties, including both charge and spin orders, are strongly coupled with the kagome lattices.

Small Organic Molecules with Anchoring Adsorption Enhance Corrosion Inhibition of Zinc Electrodes in ZnSO4 Medium

AbstractIn this study, trimercapto‐S‐triazine trisodium salt (TMT) with threefold rotational symmetry as a corrosion inhibitor was introduced into 2 mol ⋅ L−1 ZnSO4 medium. By electrochemical measurements, the corrosion inhibition efficiency of the zinc electrodes was 92.52 % when the concentration of TMT corrosion inhibitor was 2 mmol ⋅ L−1. Based on the microstructural characterization and theoretical analysis of Zn electrodes, it was confirmed that S and N atoms of TMT corrosion inhibitor serve as alternately zinc‐philic adsorption sites with sixfold rotational symmetry for the formation of intermolecular complexes. Adsorption energies of TMT on Zn surfaces were calculated and indicated that the bonds of Zn−N and Zn−S promoted anchoring adsorption on sixfold symmetric (002) plane of zinc. Meanwhile, TMT corrosion inhibitor can adsorb Zn2+ from the electrolyte and directionally deposit them on the surface of Zn (002), effectively avoiding the occurrence of dendrites. This study changes the common knowledge that organic corrosion inhibitors must form hydrophobic films to have corrosion inhibition performance, and provides a new approach for using small organic molecules with anchoring adsorption as metal corrosion inhibitors.

Three-dimensional hidden phase probed by in-plane magnetotransport in kagome metal CsV3Sb5 thin flakes

Transition metal compounds with kagome structure have been found to exhibit a variety of exotic structural, electronic, and magnetic orders. These orders are competing with energies very close to each other, resulting in complex phase transitions. Some of the phases are easily observable, such as the charge density wave (CDW) and the superconducting phase, while others are more challenging to identify and characterize. Here we present magneto-transport evidence of a new phase below ~ 35 K in the kagome topological metal CsV3Sb5 (CVS) thin flakes between the CDW and the superconducting transition temperatures. This phase is characterized by six-fold rotational symmetry in the in-plane magnetoresistance (MR) and is connected to the orbital current order in CVS. Furthermore, the phase is characterized by a large in-plane negative magnetoresistance, which suggests the existence of a three-dimensional, magnetic field-tunable orbital current ordered phase. Our results highlight the potential of magneto-transport to reveal the interactions between exotic quantum states of matter and to uncover the symmetry of such hidden phases.

Photoinduced Floquet higher-order Weyl semimetal in C6 symmetric Dirac semimetals

Abstract Topological Dirac semimetals are a parent state from which other exotic topological phases of matter, such as Weyl semimetals and topological insulators, can emerge. In this study, we investigate a Dirac semimetal possessing sixfold rotational symmetry and hosting higher-order topological hinge Fermi arc states, which is irradiated by circularly polarized light. Our findings reveal that circularly polarized light splits each Dirac node into a pair of Weyl nodes due to the breaking of time-reversal symmetry, resulting in the realization of the Weyl semimetal phase. This Weyl semimetal phase exhibits rich boundary states, including two-dimensional surface Fermi arc states and hinge Fermi arc states confined to six hinges. Furthermore, by adjusting the incident direction of the circularly polarized light, we can control the degree of tilt of the resulting Weyl cones, enabling the realization of different types of Weyl semimetals.

Emergence of the isotropic Kitaev honeycomb lattice α− RuCl3 and its magnetic properties

We present a comprehensive investigation of the crystal and magnetic structures of the van der Waals antiferromagnet α− RuCl3 using single crystal x-ray and neutron diffraction. The crystal structure at room temperature is a monoclinic ( C2/m ). However, with decreasing temperature, a remarkable first-order structural phase transition is observed, leading to the emergence of a rhombohedral ( R3ˉ ) structure characterized by three-fold rotational symmetry forming an isotropic honeycomb lattice. On further cooling, a zigzag-type antiferromagnetic order develops below TN=6∼6.6 K. The critical exponent of the magnetic order parameter was determined to be β=0.11(1) , which is close to the two-dimensional Ising model. Additionally, the angular dependence of the magnetic critical field of the zigzag antiferromagnetic order for the polarized ferromagnetic phase reveals a six-fold rotational symmetry within the ab− plane. These findingsreflect the symmetry associated with the Ising-like bond-dependent Kitaev spin interactions and underscore the universality of the Kitaev interaction-dominated antiferromagnetic system.

Isotropic exchange-bias in twinned epitaxial Co/Co3O4 bilayer

Exchange bias (EB) is a unidirectional anisotropy caused by interface coupling between a ferromagnet and an antiferromagnet. It causes a preferential direction of magnetization in the ferromagnet, which manifests as a shift of the hysteresis loop along the magnetic field axis. Here, we demonstrate a large EB of over 1000 Oe at 20 K in a twinned Co(111)/Co3O4(111) thin film epitaxially grown on sapphire(0001) with sixfold rotational lattice symmetry, which is among the highest values reported for Co/Co1−yO systems. In such systems, the effect intensity is largest along the magnetic easy axes, which usually results in an anisotropy of the EB in epitaxial interfaces. However, we observed identical EB values for 0°, 15°, and 30° angles between the magnetic field and the nearest Co[002] magnetic easy axes. The measurements imply a relaxation of the magnetization to the nearest easy axis, suggesting increasingly isotropic EB fields with higher orders of rotational lattice symmetry.

Epitaxial growth and characterization of magnesium gallate (MgGa2O4) thin films by pulsed laser deposition

In this work, crystalline MgGa2O4 thin films were grown on c-plane sapphire substrates using the pulsed laser deposition (PLD) technique in a broad range of temperatures and oxygen pressures. The parameters for the crystalline growth were within the temperature range of 300 ℃ to 700 ℃ and the pressure range of 1 × 10-1 to 1 × 10-3 Torr. The acquired XRD patterns confirmed the film growth along the [111] preferential crystal orientation in the lattice and the rocking curve measurement of the prominent (222) plane showed an increasing trend of the crystallinity with growth temperature and pressure. Furthermore, the XRD phi (φ) scan demonstrated the six-fold rotational symmetry of the MgGa2O4 films and the epitaxial relationship of 30° between the film and sapphire substrate. The XPS spectra confirmed the presence of +2 and +3 oxidation state for Mg and Ga respectively, in the films and vacancies that increase with decreasing oxygen partial pressure. The direct bandgap of MgGa2O4 films was obtained to be ∼5.27 ± 0.03 eV by analyzing the UV-Vis absorbance spectra. The film surface exhibited a granular morphology with an increasing trend in grain size as the temperature increased. The refractive index and thickness of these films were in the range of ∼1.90 ± 0.02 and ∼70 ± 2.0 nm, respectively determined from the fitted spectroscopic ellipsometry data. The surface roughness was obtained from AFM images and found to be in good agreement with those obtained from the analysis of the ellipsometry data.

Visualizing symmetry-breaking electronic orders in epitaxial Kagome magnet FeSn films

Kagome lattice hosts a plethora of quantum states arising from the interplay of topology, spin-orbit coupling, and electron correlations. Here, we report symmetry-breaking electronic orders tunable by an applied magnetic field in a model Kagome magnet FeSn consisting of alternating stacks of two-dimensional Fe3Sn Kagome and Sn2 honeycomb layers. On the Fe3Sn layer terminated FeSn thin films epitaxially grown on SrTiO3(111) substrates, we observe trimerization of the Kagome lattice using scanning tunneling microscopy/spectroscopy, breaking its six-fold rotational symmetry while preserving the translational symmetry. Such a trimerized Kagome lattice shows an energy-dependent contrast reversal in dI/dV maps, which is significantly enhanced by bound states induced by Sn vacancy defects. This trimerized Kagome lattice also exhibits stripe modulations that are energy-dependent and tunable by an applied in-plane magnetic field, indicating symmetry-breaking nematicity from the entangled magnetic and charge degrees of freedom in antiferromagnet FeSn.

Design and mechanical properties of 3D circular curve transversal-isotropic auxetic structure

In this work, the 3D circular curve transversal-isotropic auxetic structure (3D-CCTAS) is designed based on a six-fold rotation symmetry in space and combines circular-curve and re-entrant hexagonal structures. Based on energy methods to obtain the expressions of Young's modulus and Poisson's ratio for 3D-CCTAS in transverse and longitudinal directions and numerical simulations by imposing periodic boundary conditions on the unit cell. Young's modulus and Poisson's ratio of 3D-CCTAS are investigated by numerical simulation and theoretical analysis to study the influence of structural parameters (angle of the circular curve section θ, rod width b, and rod thickness t). Both numerical simulation and theoretical analysis are also verified by the uniaxial compression experiment of 3D-CCTAS, for which the results show satisfactory concordance. Significantly, the transverse isotropy of 3D-CCTAS is determined by the spatial specificity of the structure, and therefore 3D-CCTAS exhibits transverse isotropy no matter how much θ, t, or b is changed.

Nematicity-enhanced superconductivity in systems with a non-Fermi liquid behavior

We explore the interplay between nematicity (spontaneous breaking of the sixfold rotational symmetry), superconductivity, and non-Fermi liquid behavior in partially flat-band (PFB) models on the triangular lattice. A key result is that the nematicity (Pomeranchuk instability), which is driven by many-body effect and stronger in flat-band systems, enhances superconducting transition temperature in a systematic manner on the T c dome. There, a plausible -wave symmetry, in place of the conventional -wave, governs the nematicity-enhanced pairing with a sharp rise in the T c dome on the filling axis. When the sixfold symmetry is spontaneously broken, the pairing interaction is shown to become stronger with more compact pairs in real space than when the symmetry is enforced. These are accompanied by a non-Fermi character of electrons in the PFBs with many-body interactions.

Realizing symmetry-guaranteed pairs of bound states in the continuum in metasurfaces

Bound states in the continuum (BICs) have received significant attention for their ability to enhance light-matter interactions across a wide range of systems, including lasers, sensors, and frequency mixers. However, many applications require degenerate or nearly degenerate high-quality factor (Q) modes, such as spontaneous parametric down conversion, non-linear four-wave mixing, and intra-cavity difference frequency mixing for terahertz generation. Previously, degenerate pairs of bound states in the continuum (BICs) have been created by fine-tuning the structure to engineer the degeneracy, yielding BICs that respond unpredictably to structure imperfections and material variations. Instead, using a group theoretic approach, we present a design paradigm based on six-fold rotational symmetry (C6) for creating degenerate pairs of symmetry-protected BICs, whose frequency splitting and Q-factors can be independently and predictably controlled, yielding a complete design phase space. Using a combination of resonator and lattice deformations in silicon metasurfaces, we experimentally demonstrate the ability to tune mode spacing from 2 nm to 110 nm while simultaneously controlling Q-factor.

Rotationally Symmetrical Spoof-Plasmon Antenna for Polarization-Independent Radiation Enhancement

Plasmon antennas allow subwavelength confinement and enhancement of electromagnetic fields at the ``hotspot'' where the radiation efficiency of emitters can be substantially enhanced. Such enhancement, however, is often polarization dependent. Consequently, the radiation behaviors (e.g., radiation pattern and polarization states) of the emitter placed at the hotspot are also modified significantly. Enhancing the radiation efficiency without altering the original radiation pattern and polarization state of the emitter is highly desired for many sought-after applications involving chiral emitters but remains a challenging task, especially at low frequencies. To this end, spoof-plasmon antennas with fourfold and sixfold rotational symmetries are designed and realized experimentally. These plasmon antennas support polarization-independent localized plasmon resonances, which can significantly enhance the local density of photonic states at the structural center without changing the polarization state of the emitter. As a typical example, the structure with sixfold rotational symmetry is coupled with a half-wave dipole antenna. The measurement results show that the far-field radiation pattern of the dipole antenna is maintained with the radiation efficiency enhanced by more than 2 orders of magnitude, irrespective of the dipole orientation.

Three-state nematicity and magneto-optical Kerr effect in the charge density waves in kagome superconductors

The kagome lattice provides a fascinating playground to study geometrical frustration, topology and strong correlations. The newly discovered kagome metals AV$_3$Sb$_5$ (A=K, Rb, Cs) exhibit various interesting phenomena including topological band structure and superconductivity. Nevertheless, the nature of the symmetry breaking in the CDW phase is not yet clear, despite the fact that it is crucial to understand whether the superconductivity is unconventional. In this work, we perform scanning birefringence microscopy and find that six-fold rotation symmetry is broken at the onset of the CDW transition temperature in all three compounds. Spatial imaging and angle dependence of the birefringence show a universal three nematic domains that are 120$^\circ$ to each other. We propose staggered CDW orders with a relative $\pi$ phase shift between layers as a possibility to explain the three-state nematicity in AV$_3$Sb$_5$. We also perform magneto-optical Kerr effect and circular dichroism measurements on all three compounds, and the onset of the both signals is at the CDW transition temperature, indicating broken time-reversal symmetry and the existence of the long-sought loop currents in the CDW phase. Our work strongly constrains the nature of the CDWs and sheds light on possible unconventional superconductivity in AV$_3$Sb$_5$.

Temperature-driven reorganization of electronic order in CsV3Sb5

We report x-ray diffraction studies of the electronic ordering instabilities in the kagome material ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$ as a function of temperature and applied magnetic field. Our zero-field measurements between 10 and 120 K reveal an unexpected reorganization of the three-dimensional electronic order in the bulk of ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$: At low temperatures, a $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ superstructure modulation due to electronic order is observed, which upon warming changes to a $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}4$ superstructure at 60 K. The electronic order-order transition discovered here involves a change in the stacking of electronically ordered ${\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$ layers, which coincides with anomalies previously observed in magnetotransport measurements. This implies that the temperature-dependent three-dimensional electronic order plays a decisive role for transport properties, which are related to the Berry curvature of the V bands. We also show that the bulk electronic order in ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$ breaks the sixfold rotational symmetry of the underlying $P6/mmm$ lattice and perform a crystallographic analysis of the $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ phase. The latter yields two possible superlattices, namely a staggered star-of-David and a staggered inverse star-of-David structure. Applied magnetic fields up to 10 T have no effect on the x-ray diffraction signal. This, however, does not rule out time-reversal symmetry breaking in ${\mathrm{CsV}}_{3}{\mathrm{Sb}}_{5}$.

Bioinspired Co‐Design of Tactile Sensor and Deep Learning Algorithm for Human–Robot Interaction

Robots equipped with bionic skins for enhancing the robot perception capability are increasingly deployed in wide applications ranging from healthcare to industry. Artificial intelligence algorithms that can provide bionic skins with efficient signal processing functions further accelerate the development of this trend. Inspired by the somatosensory processing hierarchy of humans, the bioinspired co‐design of a tactile sensor and a deep learning‐based algorithm is proposed herein, simplifying the sensor structure while providing computation‐enhanced tactile sensing performance. The soft piezoresistive sensor, based on the carbon black‐coated polyurethane sponge, offers a continuous sensing area. By utilizing a customized deep neural network (DNN), it can detect external tactile stimulus spatially continuously. Besides, a novel data augmentation method is developed based on the sensor's hexagonal structure that has a sixfold rotation symmetry. It can significantly enhance the generalization ability of the DNN model by enriching the collected training data with generated pseudo‐data. The functionality of the sensor and the robustness of the proposed data augmentation strategy are verified by precisely recognizing five touch modalities, illustrating a well‐generalized performance, and providing a promising application prospect in human–robot interaction.

Observation of symmetry-protected corner states in breathing honeycomb topolectrical circuits

We report the experimental observation of second-order corner states in a two-dimensional breathing honeycomb topolectrical circuit with sixfold rotational symmetry C6 through voltage measurements. The topological corner states originate from the nontrivial bulk topology, which can be characterized by the topological invariant associated with the rotation eigenspectrum. We confirm two types of corner states, both originate from the C6 symmetry, while one of them is specially pinned to zero admittance because of the emerging chiral symmetry protection. We then examine the robustness of zero modes in the presence of next-nearest-neighbor hopping terms that destroy chiral symmetry but still preserve C6 symmetry. Our work provides a paradigm in circuit systems to study the exotic topological physics.

Theory of the charge density wave in AV3Sb5 kagome metals

The family of metallic kagome compounds $A$V$_3$Sb$_5$ ($A$=K, Rb, Cs) was recently discovered to exhibit both superconductivity and charge order. The nature of the charge-density wave (CDW) phase is presently unsettled, which complicates the interpretation of the superconducting ground state. In this paper, we use group-theory and density-functional theory (DFT) to derive and solve a phenomenological Landau model for this CDW state. The DFT results reveal three unstable phonon modes with the same in-plane momentum but different out-of-plane momenta, whose frequencies depend strongly on the electronic temperature. This is indicative of an electronically-driven CDW, stabilized by features of the in-plane electronic dispersion. Motivated by the DFT analysis, we construct a Landau free-energy expansion for coupled CDW order parameters with wave-vectors at the $M$ and $L$ points of the hexagonal Brillouin zone. We find an unusual trilinear term coupling these different order parameters, which can promote the simultaneous condensation of both CDWs even if the two modes are not nearly-degenerate. We classify the different types of coupled multi-$\bf{Q}$ CDW orders, focusing on those that break the sixfold rotational symmetry and lead to a unit-cell doubling along all three crystallographic directions, as suggested by experiments. We determine a region in parameter space, characterized by large nonlinear Landau coefficients, where these phases - dubbed staggered tri-hexagonal and staggered Star-of-David - are the leading instabilities of the system. Finally, we discuss the implications of our results for the kagome metals.

Geometry of the charge density wave in the kagome metal AV3Sb5

Kagom${\'e}$ lattice is a fertile platform for topological and intertwined electronic excitations. Recently, experimental evidence of an unconventional charge density wave (CDW) is observed in a Z2 kagom${\'e}$ metal AV$_{3}$Sb$_{5}$ (A= K, Cs, Rb). This observation triggers wide interests on the interplay between frustrated crystal structure and Fermi surface instabilities. Here we analyze the lattice effect and its impact on CDW in AV$_{3}$Sb$_{5}$. Based on published experimental data, we show that the CDW induced structural distortions is consistent with the theoretically predicted inverse star-of-David pattern, which preserves the $D_{6h}$ symmetry in the kagom${\'e}$ plane but breaks the sixfold rotational symmetry of the crystal due to the phase shift between kagom${\'e}$ layers. The coupling between the lattice and electronic degrees of freedom yields a weak first order structural transition without continuous change of lattice dynamics. Our result emphasizes the fundamental role of lattice geometry in proper understanding of unconventional electronic orders in AV$_{3}$Sb$_{5}$.