Threefold Rotational Symmetry Research Articles

Design and Robustness Evaluation of Valley Topological Elastic Wave Propagation in a Thin Plate with Phononic Structure

Based on the concept of band topology in phonon dispersion, we designed a topological phononic crystal in a thin plate for developing an efficient elastic waveguide. Despite that various topological phononic structures have been actively proposed, a quantitative design strategy of the phononic band and its robustness assessment in an elastic regime are still missing, hampering the realization of topological acoustic devices. We adopted a snowflake-like structure for the crystal unit cell and determined the optimal structure that exhibited the topological phase transition of the planar phononic crystal by changing the unit cell structure. The bandgap width could be adjusted by varying the length of the snow-side branch, and a topological phase transition occurred in the unit cell structure with threefold rotational symmetry. Elastic waveguides based on edge modes appearing at interfaces between crystals with different band topologies were designed, and their transmission efficiencies were evaluated numerically and experimentally. The results demonstrate the robustness of the elastic wave propagation in thin plates. Moreover, we experimentally estimated the backscattering length, which measures the robustness of the topologically protected propagating states against structural inhomogeneities. The results quantitatively indicated that degradation of the immunization against the backscattering occurs predominantly at the corners in the waveguides, indicating that the edge mode observed is a relatively weak topological state.

Strain effects on high-harmonic generation in monolayer hexagonal boron nitride

Based on the time-dependent density functional theory, we theoretically investigate the influence of mechanical strains on the high-order harmonic generation (HHG) in the monolayer hexagonal boron nitride (hBN) crystal. We show that mechanical strains can largely modify the band structure and facilitate the harmonic emission. Compared to uniaxial strains, we find that biaxial strains may enhance the HHG yield significantly, and the HHG spectroscopy generated by a linearly polarized laser is closely related to the symmetry of the deformed hBN. Moreover, when driven by a circularly polarized laser, we find that the appearance of the 3n-order harmonics manifests the restoration of the three-fold rotational symmetry. Our results will be useful in controlling the HHG spectroscopy and probing lattice deformations in crystals.

Loop currents in AV3Sb5 kagome metals: Multipolar and toroidal magnetic orders

Experiments in the recently discovered vanadium-based kagome metals have suggested that their charge-ordered state displays not only bond distortions, characteristic of a ``real" charge density-wave (rCDW), but also time-reversal symmetry-breaking, typical of loop currents described by an ``imaginary" charge density-wave (iCDW). Here, we combine density-functional theory, group-theory, and phenomenological modeling to investigate the complex charge-ordered states that arise from interactions between the low-energy van Hove singularities present in the electronic structure of $A$V$_3$Sb$_5$. We find two broad classes of mixed iCDW-rCDW configurations: triple-$\mathbf{Q}$ iCDW, triple-$\mathbf{Q}$ rCDW order, dubbed $3\mathbf{Q}$-$3\mathbf{Q}$, and double-$\mathbf{Q}$ iCDW, single-$\mathbf{Q}$ rCDW order, dubbed $2\mathbf{Q}$-$1\mathbf{Q}$. Moreover, we identify seven different types of iCDW order, stemming from the different vanadium-orbital and kagome-sublattice structures of the two pairs of van Hove singularities present above and below the Fermi level. While the $2\mathbf{Q}$-$1\mathbf{Q}$ states trigger an orthorhombic distortion that breaks the threefold rotational symmetry of the kagome lattice, the $3\mathbf{Q}$-$3\mathbf{Q}$ states induce various types of subsidiary uniform magnetic orders, from conventional ferromagnetism to magnetic octupolar, magnetic toroidal, and even magnetic monopolar order. We show that these exotic orders display unique magneto-striction, magneto-electric, and magneto-electric-striction properties that can be probed experimentally to identify which iCDW state is realized in these compounds. We briefly discuss the impact of an out-of-plane modulation of the charge order and the interplay between these complex charge-ordered states and superconductivity.

Role of electromagnetic gauge-field fluctuations in the selection between chiral and nematic superconductivity

Motivated by the observation of nematic superconductivity in several systems, we revisit the problem of the leading pairing instability of two-component unconventional superconductors on the triangular lattice -- such as $(p_{x},\,p_{y})$-wave and $(d_{x^{2}-y^{2}},\,d_{xy})$-wave. Such a system has two possible superconducting states: the chiral state (e.g. $p+ip$ or $d+id$), which breaks time-reversal symmetry, and the nematic state (e.g. $p+p$ or $d+d$), which breaks the threefold rotational symmetry of the lattice. Weak-coupling calculations generally favor the chiral over the nematic superconducting state, raising the question of what mechanism can stabilize the latter. Here, we show that the electromagnetic field fluctuations can play a crucial role in selecting between these two states. Specifically, we derive and analyze the effective free energy for the two-component superconducting order parameter after integrating out the gauge-field fluctuations, which is formally justified if the spatial order parameter fluctuations can be neglected. A non-analytic cubic term arises, as in the case of a conventional $s$-wave superconductor. However, unlike the latter, the cubic term depends on the relative phase and on the relative amplitudes between the two order parameter components, in such a way that it generally favors the nematic state. This result is a direct consequence of the fact that the stiffness of the superconducting order parameter is not isotropic. Competition with the quartic term, which favors the chiral state, leads to a renormalized phase diagram in which the nematic state displaces the chiral state over a wide region in the parameter space. We analyze the stability of the fluctuation-induced nematic phase, generalize our results to tetragonal lattices, and discuss their applicability to candidate nematic superconductors, including twisted bilayer graphene.

Continuous amplitude control of second harmonic waves from the metasurfaces through interference paths

Nonlinear plasmonic metasurfaces represent a promising platform for generating and controlling the multiple degrees of freedom of harmonic waves at the subwavelength scale. Among various functionalities, the amplitude control of the harmonic waves is usually achieved by varying the fundamental frequency and the pumping power. However, these kinds of methods lack the ability to control the energy distributions of the nonlinear waves at different diffraction orders. Here, we report the continuous intensity control of the harmonic waves from the plasmonic metasurfaces through interference paths. The metasurfaces consist of geometric phase controlled gold meta-atoms with threefold rotational symmetry. By controlling the relative phase of two neighboring metasurface paths, the intensity of the diffracted second harmonic waves can be artificially modulated. Our findings suggest that the conventional interference technique represents a powerful route for controlling the radiation of the nonlinear waves from sub-wavelength meta-atoms and may have great applications in developing ultra-compact nonlinear optical sources.

Epitaxial Growth of SnS2 Ribbons on a Au-Sn Alloy Seed Film Surface.

Two-dimensional metal chalcogenide film has attracted considerable interest for its use as an emerging device material for nanoelectronics. The film has been synthesized by various methods such as chemical vapor deposition, molecular beam epitaxy, and thermal vapor sulfurization. In this study, we took a new approach to synthesize tin disulfide (SnS2) by using Au-Sn alloy film as a metal seed deposited on a sapphire substrate. Multilayer SnS2 films were formed in a ribbon shape on (111)-oriented Au film and were aligned in the directions of threefold rotational symmetry. Furthermore, the SnS2 was found to have an epitaxial relationship relative to the Au film and the sapphire substrate as follows: SnS2[2̅110] || Au[101̅] || Al2O3[011̅0]. The segregation of grains consisting of a Sn-rich phase on the Au film surface and the subsequent sulfurization of these grains can be the key to the epitaxial SnS2 ribbon formation.

Moment constraints in beams with discrete and continuous rotational symmetry

This study presents a general analysis of how the transverse rotational symmetry of a beam imposes equality constraints among transverse beam moments. Efficient analytic methods are developed to derive symmetry-imposed constraints among $k$th order moments in beams with continuous or $n$-fold rotational symmetry for arbitrary $k$ and $n$. The formalism also enables one to construct symmetry arguments based on how discrete and continuous rotational symmetries manifest themselves differently in terms of beam moments. Three case studies on beams with continuous, twofold, and threefold rotational symmetries are conducted. We prove that, regardless of their triangulation, beams with threefold rotational symmetry (e.g., from electron cyclotron resonance ion source) always have the same rms properties as beams with cylindrical symmetry. These counterintuitive results derived purely from symmetry considerations have clarified beam dynamics at the Facility for Rare Isotope Beams.

Skyrmion crystal with integer and fractional skyrmion numbers in a nonsymmorphic lattice structure with the screw axis

The emergence of a magnetic skyrmion crystal in a nonsymmorphic lattice system with the screw symmetry is numerically investigated. By performing the simulated annealing for a layered spin model with the isotropic exchange interaction and the antisymmetric Dzyaloshinskii-Moriya interaction, and then constructing a low-temperature phase diagram, we reveal that the skyrmion crystal is stabilized in both zero and nonzero fields even without the threefold rotational symmetry in the two-dimensional plane. We show that a competition between the ferromagnetic interlayer exchange interaction and the layer- and momentum-dependent anisotropic interactions is a source of the skyrmion crystals in the presence of the screw axis. Moreover, we find two types of layer-dependent skyrmion crystals, which are characterized as a coexisting state of the skyrmion crystal and the spiral state, in the narrow field region. Our result provides a reference in the further search for skyrmion crystals in nonsymmorphic lattice systems.

Nonlinear vectorial holography with quad-atom metasurfaces

Vectorial optical holography represents a solution to control the polarization and amplitude distribution of light in the Fourier space. While vectorial optical holography has been experimentally demonstrated in the linear optical regime, its nonlinear counterpart, which can provide extra degrees of freedom of light-field manipulation through the frequency conversion processes, remains unexplored. Here, we experimentally demonstrate the nonlinear vectorial holography through the second harmonic generation process on a quad-atom plasmonic metasurface. The quad-atom metasurface consists of gold meta-atoms with threefold rotational symmetry. Based on the concept of nonlinear geometric phase, we can simultaneously manipulate the phase and amplitude of the left and right circularly polarized second harmonic waves generated from the quad-atom metasurface. By superposing the two orthogonal polarization components, the quad-atom metasurface can produce nonlinear holographic images with vectorial polarization distributions. The proposed metasurface platform may have important applications in vectorial polarization nonlinear optical source, high-capacity optical information storage, and optical encryption.

Berry curvature dipole senses topological transition in a moiré superlattice

Topological aspects of electron wavefunction play a crucial role in determining the physical properties of materials. Berry curvature and Chern number are used to define the topological structure of electronic bands. While Berry curvature and its effects in materials have been studied, detecting changes in the topological invariant, Chern number, is challenging. In this regard, twisted double bilayer graphene (TDBG) has emerged as a promising platform to gain electrical control over the Berry curvature hotspots and the valley Chern numbers of its flat bands. In addition, strain induced breaking of the three-fold rotation (C3) symmetry in TDBG, leads to a non-zero first moment of Berry curvature called the Berry curvature dipole (BCD), which can be sensed using nonlinear Hall (NLH) effect. We reveal, using TDBG, that the BCD detects topological transitions in the bands and changes its sign. In TDBG, the perpendicular electric field tunes the valley Chern number and the BCD simultaneously allowing us a tunable system to probe the physics of topological transitions. Furthermore, we find hysteresis of longitudinal and NLH responses with electric field that can be attributed to switching of electric polarization in moir\'e systems. Such a hysteretic response holds promise for next-generation Berry curvature-based memory devices. Probing topological transitions, as we show, can be emulated in other 3D topological systems.

Engineering the Dipole Orientation and Symmetry Breaking with Mixed-Dimensional Heterostructures.

Engineering of the dipole and the symmetry of materials plays an important role in fundamental research and technical applications. Here, a novel morphological manipulation strategy to engineer the dipole orientation and symmetry of 2D layered materials by integrating them with 1D nanowires (NWs) is reported. This 2D InSe –1D AlGaAs NW heterostructure example shows that the in‐plane dipole moments in InSe can be engineered in the mixed‐dimensional heterostructure to significantly enhance linear and nonlinear optical responses (e.g., photoluminescence, Raman, and second harmonic generation) with an enhancement factor of up to ≈12. Further, the 1D NW can break the threefold rotational symmetry of 2D InSe, leading to a strong optical anisotropy of up to ≈65%. These results of engineering dipole orientation and symmetry breaking with the mixed‐dimensional heterostructures open a new path for photonic and optoelectronic applications.

Phonon-induced rotation of the electronic nematic director in superconducting Bi2Se3

The doped topological insulator ${A}_{x}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, with $A={\mathrm{Cu},\mathrm{Sr},\mathrm{Nb}}$, becomes a nematic superconductor below ${T}_{c}\ensuremath{\approx}3$--4 K. The associated electronic nematic director is described by an angle $\ensuremath{\alpha}$ and is experimentally manifested in the elliptical shape of the in-plane critical magnetic field ${H}_{c2}$. Because of the threefold rotational symmetry of the lattice, $\ensuremath{\alpha}$ is expected to align with one of three high-symmetry directions corresponding to the in-plane nearest-neighbor bonds, consistent with a ${Z}_{3}$-Potts nematic transition. Here, we show that the nematic coupling to the acoustic phonons, which makes the nematic correlation length tend to diverge along certain directions only, can fundamentally alter this phenomenology in trigonal lattices. Compared to hexagonal lattices, the former possesses a sixth independent elastic constant ${c}_{14}$ due to the fact that the in-plane shear strain doublet $({\ensuremath{\epsilon}}_{xx}\ensuremath{-}{\ensuremath{\epsilon}}_{yy},\ensuremath{-}2{\ensuremath{\epsilon}}_{xy})$ and the out-of-plane shear strain doublet $(2{\ensuremath{\epsilon}}_{yz},\ensuremath{-}2{\ensuremath{\epsilon}}_{zx})$ transform as the same irreducible representation. We find that, when ${c}_{14}$ overcomes a threshold value, which is expected to be the case in doped ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, the nematic director $\ensuremath{\alpha}$ unlocks from the high-symmetry directions due to the competition between the quadratic phonon-mediated interaction and the cubic nematic anharmonicity. This implies the breaking of the residual in-plane twofold rotational symmetry (${C}_{2x}$), resulting in a triclinic phase. We discuss the implications of these findings for the structure of nematic domains, for the shape of the in-plane ${H}_{c2}$ in ${A}_{x}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, and for the presence of nodes inside the superconducting state.

Spirals in Galaxies

Spirals in galaxies have long been thought to be caused by gravitational instability in the stellar component of the disk, but discerning the precise mechanism had proved elusive. Tidal interactions, and perhaps bars, may provoke some spiral responses, but spirals in many galaxies must be self-excited. We survey the relevant observational data and aspects of disk dynamical theory. The origin of the recurring spiral patterns in simulations of isolated disk galaxies has recently become clear, and it is likely that the mechanism is the same in real galaxies, although evidence to confirm this supposition is hard to obtain. As transient spiral activity increases random motion, the patterns must fade over time unless the disk also contains a dissipative gas component. Continuing spiral activity alters the structure of the disks in other ways: reducing metallicity gradients and flattening rotation curves are two of the most significant. The overwhelming majority of spirals in galaxies have two- or three-fold rotational symmetry, indicating that the cool, thin disk component is massive. Spirals in simulations of halo-dominated disks instead manifest many arms and, consequently, do not capture the expected full spiral-driven evolution. We conclude by identifying areas where further work is needed.

Chiral phonons in lattices with C4 symmetry

Chiral phonons were initially proposed and further verified experimentally in two-dimensional (2D) hexagonal crystal lattices. Many intriguing features brought about by chiral phonons are attributed to the pseudoangular momenta which are associated with the threefold rotational symmetry of hexagonal lattices. Here, we go beyond the hexagonal crystals and investigate the chiral phonons in systems with fourfold rotational symmetry. We clarify the symmetry requirements for the emergence of chiral phonons in both 2D square lattices and three-dimensional tetragonal lattices. For two dimensions, the realization of ${C}_{4}$ chiral phonons requires the breaking of time-reversal symmetry; while for three dimensions, they can exist on the ${C}_{4}$-invariant path in a chiral tetragonal lattice. These phonons have the advantage that they can be more readily coupled with optical transitions, which facilitates their experimental detection. We demonstrate our idea via model analysis and first-principles calculations of concrete materials, including the MnAs monolayer and the $\ensuremath{\alpha}$-cristobalite. Our work reveals chiral phonons beyond the hexagonal lattices and paves the way for further exploration of chiral phonon physics in square/tetragonal materials and metamaterials.

Synchronization of a genetic oscillator with the cell division cycle

Genetic circuits that control specific cellular functions are never fully insulated against influences of other parts of the cell. For example, they are subject to periodic modulation by the cell cycle through volume growth and gene doubling. To investigate possible effects of the cell cycle on oscillatory gene circuits dynamics, we modelled a simple synthetic genetic oscillator, the repressilator, and studied hallmarks of the resulting nonlinear dynamics. We found that the repressilator coupled to the cell cycle shows typical quasiperiodic motion with discrete Fourier spectra and windows in parameter space with synchronization of the two oscillators, with a devil’s stair case indicating the Arnold tongues of synchronization. In the case of identical parameters for the three genes of the repressilator and simultaneous gene duplication, we identify two classes of synchronization windows, symmetric and asymmetric, depending on whether the trajectories satisfy a discrete three-fold rotation symmetry, corresponding to cyclic permutation of the three genes. Unexpectedly changing the gene doubling time revealed that the width of the Arnold tongues is connected to that three-fold symmetry of the synchronization trajectories: non-simultaneous gene duplication increases the width of asymmetric synchronization regions, for some of them by an order of magnitude. By contrast, there is only a small or even a negative effect on the window size for symmetric synchronization. This observation points to a control mechanism of synchronization via the location of the genes on the chromosome.

Evidence for a single-layer van der Waals multiferroic.

Multiferroic materials have attracted wide interest because of their exceptional static1-3 and dynamical4-6 magnetoelectric properties. In particular, type-II multiferroics exhibit an inversion-symmetry-breaking magnetic order that directly induces ferroelectric polarization through various mechanisms, such as the spin-current or the inverse Dzyaloshinskii-Moriya effect3,7. This intrinsic coupling between the magnetic and dipolar order parameters results in high-strength magnetoelectric effects3,8. Two-dimensional materials possessing such intrinsic multiferroic properties have been long sought for to enable the harnessing of magnetoelectric coupling in nanoelectronic devices1,9,10. Here we report the discovery of type-II multiferroic order in a single atomic layer of the transition-metal-based van der Waals material NiI2. The multiferroic state of NiI2 is characterized by a proper-screw spin helix with given handedness, which couples to the charge degrees of freedom to produce a chirality-controlled electrical polarization. We use circular dichroic Raman measurements to directly probe the magneto-chiral ground state and its electromagnon modes originating from dynamic magnetoelectric coupling. Combining birefringence and second-harmonic-generation measurementswith theoretical modelling and simulations, we detect a highly anisotropic electronic state that simultaneously breaks three-fold rotational and inversion symmetry, and supports polar order. The evolution of the optical signatures as a function of temperature and layer number surprisingly reveals an ordered magnetic polar state that persists down to the ultrathin limit of monolayer NiI2. These observations establish NiI2 and transition metal dihalides as a new platform for studying emergent multiferroic phenomena, chiral magnetic textures and ferroelectricity in the two-dimensional limit.

Second-harmonic generation in atomically thin 1T−TiSe2 and its possible origin from charge density wave transitions

Optical second-harmonic generation (SHG) can only occur in noncentrosymmetric crystals in the leading electric-dipole approximation. Transition metal dichalco genides with the $1T$ octahedral coordination is centrosymmetric, hence precluding SHG. Here we report the surprising observation of SHG in atomically thin $1T\text{\ensuremath{-}}\mathrm{Ti}{\mathrm{Se}}_{2}$, a prototypical charge density wave (CDW) material. Its intensity peaks in the trilayer, reaching 2% of that in monolayer $\mathrm{Mo}{\mathrm{S}}_{2}$, a two-dimensional crystal featuring pronounced nonlinear optical effect. The SHG signal exhibits a sixfold polarization dependence characteristic of a lattice with threefold rotational symmetry. It monotonically decreases with increasing temperature and persists at room temperature. Raman spectroscopy demonstrates that the CDW order is robust in atomically thin samples, with the transition temperature slightly lower than in the bulk. The SHG can be explained by the lattice distortion associated with the CDW as well as its fluctuation above the transition temperature. These results challenge the exciton insulator scenario and the chiral nature of the CDW, but support the band Jahn-Teller mechanism. Our work demonstrates SHG as a sensitive probe of the stacking order of the CDW in $1T\text{\ensuremath{-}}\mathrm{Ti}{\mathrm{Se}}_{2}$ and enriches the material base for nonlinear optical effects.

Design of continuously deformable topological phononic structures and their application to elastic waveguides

We design a topological phononic crystal in a thin plate, aiming at the development of an efficient elastic waveguide based on the concept of band topology in phonon dispersion. We adopt a snowflake-like structure for the crystal unit cell and search for the optimal structure that exhibits the topological phase transition of the three-dimensional phononic crystal by changing the unit cell structure. The bandgap width can be adjusted by varying the length of the branch of the snow side, and a topological phase transition occurs at the unit cell structure with three-fold rotational symmetry. Elastic waveguides based on edge modes appearing at interfaces between crystals with different band topologies are designed, and their transmission efficiency is evaluated numerically and experimentally. The results demonstrate robustness of the elastic wave propagation in thin plates. Moreover, we experimentally estimate a robustness of the topologically protected propagating states against structural inhomogeneities. The results obtained in this study pave the way for practical development of new surface acoustic wave technologies for high-frequency communication devices.

Moiré nematic phase in twisted double bilayer graphene

Graphene moir\'e superlattices display electronic flat bands. At integer fillings of these flat bands, energy gaps due to strong electron-electron interactions are generally observed. However, the presence of other correlation-driven phases in twisted graphitic systems at non-integer fillings is unclear. Here, we report scanning tunneling microscopy (STM) measurements that reveal the existence of threefold rotational (C3) symmetry breaking in twisted double bilayer graphene (tDBG). Using spectroscopic imaging over large and uniform areas to characterize the direction and degree of C3 symmetry breaking, we find it to be prominent only at energies corresponding to the flat bands and nearly absent in the remote bands. We demonstrate that the C3 symmetry breaking cannot be explained by heterostrain or the displacement field, and is instead a manifestation of an interaction-driven electronic nematic phase, which emerges even away from integer fillings. Comparing our experimental data with a combination of microscopic and phenomenological modeling, we show that the nematic instability is not associated with the local scale of the graphene lattice, but is an emergent phenomenon at the scale of the moir\'e lattice, pointing to the universal character of this ordered state in flat band moir\'e materials.

Topological invariants of rotationally symmetric crystals

Recent formal classifications of crystalline topological insulators predict that the combination of time-reversal and rotational symmetry gives rise to topological invariants beyond the ones known for other lattice symmetries. Although the classification proves their existence, it does not indicate a way of calculating the values of those invariants. Here, we show that a specific set of concentric Wilson loops and line invariants yields the values of all topological invariants in two-dimensional systems with pure rotation symmetry in class AII. Examples of this analysis are given for specific models with two-fold and three-fold rotational symmetry. We find new invariants that relate to the presence of higher-order topology and corner charges.