Threefold Rotational Symmetry Research Articles

Kink scattering in a generalized Wess-Zumino model

In this paper, kink scattering in the dimensional reduction of the bosonic sector of a one-parameter family of generalized Wess-Zumino models with three vacuum points is discussed. The value of the model parameter determines the specific location of the vacua. The influence of the vacuum arrangements (evolving from three collinear vacua to three vacua placed at the vertices of an equilateral triangle) on the kink scattering is investigated. Two different regimes can be distinguished: in the first one, two symmetric BPS kinks/antikinks arise whereas in the second one a new different BPS kink/antikink emerges, with the exception of a three-fold rotational symmetry case, where the three topological defects are identical. The scattering between the two symmetric kinks is thoroughly analyzed. Two different scattering channels have been found: kink-kink reflection and kink-kink hybridization. In the last case, the collision between the two symmetric kinks gives rise to the third different kink. Resonance phenomena also appear allowing a vibrating kink to split into two symmetric kinks moving away.

Room-Temperature Chiral Light-Emitting Diode Based on Strained Monolayer Semiconductors.

Room-temperature chiral light sources whose optical helicity can be electrically switched are one of the most important devices for future optical quantum information processing. The emerging valley degree of freedom in monolayer semiconductors allows generation of chiral luminescence via valley polarization. However, relevant valley-polarized light-emitting diodes (LEDs) have only been achieved at low temperatures (typically below 80 K). Here, a room-temperature chiral LED with strained transition metal dichalcogenide monolayers is realized. Spatially resolved polarization spectroscopy reveals that strain effects are crucial to yielding robust valley-polarized electroluminescence. The broken threefold rotational symmetry of strained monolayers induce inequivalent valley drifts at the K/K' valleys, resulting in different amounts of spin recombination driven by electric fields. Based on this scenario, ideally strained conditions are designed for LEDs on flexible substrates, in which the helicity of room-temperature valley-polarized electroluminescence is electrically tuned. The results provide a new pathway for practical chiral light sources based on monolayer semiconductors.

Electric-field-tunable electronic nematic order in twisted double-bilayer graphene

Graphene-based moiré systems have attracted considerable interest in recent years as they display a remarkable variety of correlated phenomena. Besides insulating and superconducting phases in the vicinity of integer fillings of the moiré unit cell, there is growing evidence for electronic nematic order both in twisted bilayer graphene and twisted double-bilayer graphene (tDBG), as signaled by the spontaneous breaking of the threefold rotational symmetry of the moiré superlattices. Here, we combine symmetry-based analysis with a microscopic continuum model to investigate the structure of the nematic phase of tDBG and its experimental manifestations. First, we perform a detailed comparison between the theoretically calculated local density of states and recent scanning tunneling microscopy data (arXiv:2009.11645) to resolve the internal structure of the nematic order parameter in terms of the layer, sublattice, spin, and valley degrees of freedom. We find strong evidence that the dominant contribution to the nematic order parameter comes from states at the moiré scale rather than at the microscopic scale of the individual graphene layers, which demonstrates the key role played by the moiré degrees of freedom and confirms the correlated nature of the nematic phase in tDBG. Secondly, our analysis reveals an unprecedented tunability of the orientation of the nematic director in tDBG by an externally applied electric field, allowing the director to rotate away from high-symmetry crystalline directions. We compute the expected fingerprints of this rotation in both STM and transport experiments, providing feasible ways to probe it. Rooted in the strong sensitivity of the flat bands of tDBG to the displacement field, this effect opens an interesting route to the electrostatic control of electronic nematicity in moiré systems.

Imaging the Néel vector switching in the monolayer antiferromagnet MnPSe3 with strain-controlled Ising order.

Antiferromagnets are interesting materials for spintronics because of their faster dynamics and robustness against perturbations from magnetic fields. Control of the antiferromagnetic order constitutes an important step towards applications, but has been limited to bulk materials so far. Here, using spatially resolved second-harmonic generation, we show direct evidence of long-range antiferromagnetic order and Ising-type Néel vector switching in monolayer MnPSe3 with large XY anisotropy. In additional to thermally induced switching, uniaxial strain can rotate the Néel vector, aligning it to a general in-plane direction irrespective of the crystal axes. A change of the universality class of the phase transition in the XY model under uniaxial strain causes this emergence of strain-controlled Ising order in the XY magnet MnPSe3. Our discovery is a further ingredient for compact antiferromagnetic spintronic devices in the two-dimensional limit.

Symmetry analysis of tensors in the honeycomb lattice of edge-sharing octahedra

We obtain the most general forms of rank-2 and rank-3 tensors allowed by the crystal symmetries of the honeycomb lattice of edge-sharing octahedra for crystals belonging to different crystallographic point groups, including the monoclinic point group $2/m$ and the trigonal (or rhombohedral) point group $\bar{3}$. Our results are relevant for two-dimensional materials, such as $\alpha$-RuCl$_3$, CrI$_3$, and the honeycomb iridates. We focus on the magnetic-field-dependent thermal conductivity tensor $\kappa_{ij}(\mathbf{H})$, which describes a system's longitudinal and thermal Hall responses, for the cases when the magnetic field is applied along high-symmetry directions, perpendicular to the plane and in the plane. We highlight some unexpected results, such as the equality of fully-longitudinal components to partially-transverse components in rank-3 tensors for systems with three-fold rotational symmetry, and make testable predictions for the thermal conductivity tensor.

Engineering interface polarization

Interface Physics Many properties can emerge at the interface of van der Waals materials created by rotating the layers of a single material or by creating heterointerfaces between different materials. Akamatsu et al. formed an interface that intentionally broke in-plane inversion symmetry by combining crystals of tungsten diselenide with threefold rotational symmetry and black phosphorus with twofold rotational symmetry. This interface creates in-plane electronic polarization that results in a spontaneous photovoltaic effect only along the polarization direction. This effect was explained in terms of a shift current mechanism. Science , this issue p. [68][1] [1]: /lookup/doi/10.1126/science.aaz9146

A van der Waals interface that creates in-plane polarization and a spontaneous photovoltaic effect.

Van der Waals interfaces can be formed by layer stacking without regard to lattice constants or symmetries of individual building blocks. We engineered the symmetry of a van der Waals interface of tungsten selenide and black phosphorus and realized in-plane electronic polarization that led to the emergence of a spontaneous photovoltaic effect. Spontaneous photocurrent was observed along the polar direction and was absent in the direction perpendicular to it. The observed spontaneous photocurrent was explained by a quantum-mechanical shift current that reflects the geometrical and topological electronic nature of this emergent interface. The present results offer a simple guideline for symmetry engineering that is applicable to a variety of van der Waals interfaces.

Cooperative hydrophobic core interactions in the β-trefoil architecture.

Symmetric protein architectures have a compelling aesthetic that suggests a plausible evolutionary process (i.e., gene duplication/fusion) yielding complex architecture from a simpler structural motif. Furthermore, symmetry inspires a practical approach to computational protein design that substantially reduces the combinatorial explosion problem, and may provide practical solutions for structure optimization. Despite such broad relevance, the role of structural symmetry in the key area of hydrophobic core-packing cooperativity has not been adequately studied. In the present report, the threefold rotational symmetry intrinsic to the β-trefoil architecture is shown to form a geometric basis for highly-cooperative core-packing interactions that both stabilize the local repeating motif and promote oligomerization/long-range contacts in the folding process. Symmetry in the β-trefoil structure also permits tolerance towards mutational drift that involves a structural quasi-equivalence at several key core positions.

Quantum frequency doubling in the topological insulator Bi2Se3

The nonlinear Hall effect due to Berry curvature dipole (BCD) induces frequency doubling, which was recently observed in time-reversal-invariant materials. Here we report novel electric frequency doubling in the absence of BCD on a surface of the topological insulator Bi2Se3 under zero magnetic field. We observe that the frequency-doubling voltage transverse to the applied ac current shows a threefold rotational symmetry, whereas it forbids BCD. One of the mechanisms compatible with the symmetry is skew scattering, arising from the inherent chirality of the topological surface state. We introduce the Berry curvature triple, a high-order moment of the Berry curvature, to explain skew scattering under the threefold rotational symmetry. Our work paves the way to obtain a giant second-order nonlinear electric effect in high mobility quantum materials, as the skew scattering surpasses other mechanisms in the clean limit.

Evidence of Potts-Nematic Superfluidity in a Hexagonal sp^{2} Optical Lattice.

As in between liquid and crystal phases lies a nematic liquid crystal, which breaks rotation with preservation of translation symmetry, there is a nematic superfluid phase bridging a superfluid and a supersolid. The nematic order also emerges in interacting electrons and has been found to largely intertwine with multiorbital correlation in high-temperature superconductivity, where Ising nematicity arises from a four-fold rotation symmetry C_{4} broken down to C_{2}. Here, we report an observation of a three-state (Z_{3}) quantum nematic order, dubbed "Potts-nematicity", in a system of cold atoms loaded in an excited band of a hexagonal optical lattice described by an sp^{2}-orbital hybridized model. This Potts-nematic quantum state spontaneously breaks a three-fold rotation symmetry of the lattice, qualitatively distinct from the Ising nematicity. Our field theory analysis shows that the Potts-nematic order is stabilized by intricate renormalization effects enabled by strong interorbital mixing present in the hexagonal lattice. This discovery paves a way to investigate quantum vestigial orders in multiorbital atomic superfluids.

A chiral uranyl-Kemp's tricarboxylate cubic framework: structure-directing effect of counterions with three-fold rotational symmetry.

In the presence of PPh3Me+ cations, Kemp's tricarboxylate (kta3-) complexes the uranyl cation to give [PPh3Me][UO2(kta)] (1), a triperiodic framework with cubic symmetry and srs topology. The PPh3Me+ cation is held by weak interactions into cavities with matching three-fold rotational symmetry. Comparison with the diperiodic hemi-hydrate polymorph previously reported points to the disrupting role of OHO hydrogen bonds in the latter.

Double-Layered Supramolecular Prisms Self-Assembled by Geometrically Non-equivalent Tetratopic Subunits.

Supramolecular cages/vesicles in biology display sophisticated structures and functions by utilizing a few types of protein subunit quasi-equivalently at distinct geometrical locations. However, synthetic supramolecular cages still lack comparable complexity to reach the high levels of functionality found in natural systems. Herein we report the self-assembly of giant pentagonal supramolecular prisms (molecular weight >50 kDa) with tetratopic pyridinyl subunits serving different geometrical roles within the structures, and their packing into a novel superstructure with unexpected three-fold rotational symmetry in a single two-dimensional layer of crystalline state. The formation of these complicated structures is controlled by both the predetermined angles of the ligands and the mismatched structural tensions created from the multi-layered geometry of the building blocks. Such a self-assembly strategy is extensively used by viruses to increase the volume and complexity of capsids and would provide a new approach to construct highly sophisticated supramolecular architectures.

Localized spin-orbit polaron in magnetic Weyl semimetal Co3Sn2S2

The kagome lattice Co3Sn2S2 exhibits the quintessential topological phenomena of a magnetic Weyl semimetal such as the chiral anomaly and Fermi-arc surface states. Probing its magnetic properties is crucial for understanding this correlated topological state. Here, using spin-polarized scanning tunneling microscopy/spectroscopy (STM/S) and non-contact atomic force microscopy (nc-AFM) combined with first-principle calculations, we report the discovery of localized spin-orbit polarons (SOPs) with three-fold rotation symmetry nucleated around single S-vacancies in Co3Sn2S2. The SOPs carry a magnetic moment and a large diamagnetic orbital magnetization of a possible topological origin associated relating to the diamagnetic circulating current around the S-vacancy. Appreciable magneto-elastic coupling of the SOP is detected by nc-AFM and STM. Our findings suggest that the SOPs can enhance magnetism and more robust time-reversal-symmetry-breaking topological phenomena. Controlled engineering of the SOPs may pave the way toward practical applications in functional quantum devices.

Fully spin-polarized double-Weyl fermions with type-III dispersion in the quasi-one-dimensional materials X2RhF6 ( X=K , Rb, Cs)

Double-Weyl fermions, as novel topological states of matter, have been mostly discussed in nonmagnetic materials. Here, based on density-functional theory and symmetry analysis, we propose the realization of fully spin-polarized double-Weyl fermions in a family ferromagnetic materials ${\mathrm{X}}_{2}{\mathrm{RhF}}_{6}$ ($X=\text{K}$, Rb, Cs). These materials have the half-metal ground states, where only the bands from the spin-down channel present near the Fermi energy. The spin-down bands form a pair of triply degenerate nodal points (TDNPs) if spin-orbit coupling (SOC) is not included. Under SOC, one TDNP splits into two double-Weyl points featuring quadratic dispersion along two momentum directions with the chiral charge of $\ifmmode\pm\else\textpm\fi{}2$, and they are protected by the three-fold rotation (${C}_{3}$) symmetry. Unlike most double-Weyl semimetals, the Weyl points proposed here have the type-III dispersion with one of the crossing bands being saddle-shaped. An effective model is constructed, which describes well the nature of the Weyl points. These Weyl points are fully spin-polarized, and are characterized with double Fermi arcs on the surface spectrum. Breaking ${C}_{3}$ symmetry by lattice strain could shift one double-Weyl point into a pair of type-II single-Weyl points. The ${\mathrm{X}}_{2}{\mathrm{RhF}}_{6}$ materials proposed here are excellent candidates to investigate the novel properties of type-III double-Weyl fermions in ferromagnetic system, as well as generate potential applications in spintronics.

Harmonic spin–orbit angular momentum cascade in nonlinear optical crystals

Optical angular momentum-based photonic technologies demonstrate the key role of the optical spin–orbit interaction that usually refers to linear optical processes in spatially engineered optical materials1. Re-examining the basics of nonlinear optics of homogeneous crystals under circularly polarized light2,3, we report experiments on the enrichment of the spin–orbit angular momentum spectrum of paraxial light. The demonstration is made within the framework of second-harmonic generation using a crystal with three-fold rotational symmetry. Four spin–orbit optical states for the second harmonic field are predicted from a single fundamental state owing to the interplay between linear spin–orbit coupling and nonlinear wave mixing; three of these states are experimentally verified. Besides representing a spin-controlled nonlinear route to orbital angular multiplexing4, modal vortex light sources5,6, high-dimensional parametric processes7 and multi-state optical magnetization8, our findings suggest that the fundamentals of nonlinear optics are worth revisiting through the prism of the spin–orbit interaction of light.

On the topological immunity of corner states in two-dimensional crystalline insulators

A higher-order topological insulator (HOTI) in two dimensions is an insulator without metallic edge states but with robust zero-dimensional topological boundary modes localized at its corners. Yet, these corner modes do not carry a clear signature of their topology as they lack the anomalous nature of helical or chiral boundary states. Here, we demonstrate using immunity tests that the corner modes found in the breathing kagome lattice represent a prime example of a mistaken identity. Contrary to previous theoretical and experimental claims, we show that these corner modes are inherently fragile: the kagome lattice does not realize a higher-order topological insulator. We support this finding by introducing a criterion based on a corner charge-mode correspondence for the presence of topological midgap corner modes in n-fold rotational symmetric chiral insulators that explicitly precludes the existence of a HOTI protected by a threefold rotational symmetry.

Topological Modes Protected by Chiral and Two-Fold Rotational Symmetry in a Spring-Mass Model with a Lieb Lattice Structure

We propose how to realize the topological modes, which correspond to topological zero modes for a quantum system, protected by chiral and rotation symmetry for a mechanical system. Specifically, we show the emergence of topological modes protected by chiral and two-fold rotational symmetry by a spring-mass system with a Lieb lattice structure and dents on the floor. Moreover, comparing the results of a tight-binding model, we have found the additional topological modes for our spring-mass model due to the extra degrees of freedoms. Our approach to realize the topological modes can be applied to other cases with rotation symmetry, e.g., a system of a honeycomb lattice with three-fold rotational symmetry.

Direct observation of two-dimensional magnons in atomically thin CrI3

Exfoliated chromium triiodide (CrI$_3$) is a layered van der Waals (vdW) magnetic insulator that consists of ferromagnetic layers coupled through antiferromagnetic interlayer exchange. The resulting permutations of magnetic configurations combined with the underlying crystal symmetry produces tunable magneto-optical phenomena that is unique to the two-dimensional (2D) limit. Here, we report the direct observation of 2D magnons through magneto-Raman spectroscopy with optical selection rules that are strictly determined by the honeycomb lattice and magnetic states of atomically thin CrI$_3$. In monolayers, we observe an acoustic magnon mode of ~0.3 meV with cross-circularly polarized selection rules locked to the magnetization direction. These unique selection rules arise from the discrete conservation of angular momentum of photons and magnons dictated by threefold rotational symmetry in a rotational analogue to Umklapp scattering. In bilayers, by tuning between the layered antiferromagnetic and ferromagnetic-like states, we observe the switching of two magnon modes. The bilayer structure also enables Raman activity from the optical magnon mode at ~17 meV (~4.2 THz) that is otherwise Raman-silent in the monolayer. From these measurements, we quantitatively extract the spin wave gap, magnetic anisotropy, intralayer and interlayer exchange constants, and establish 2D magnets as a new system for exploring magnon physics.

Nematic superconductivity in twisted bilayer graphene

Twisted bilayer graphene displays insulating and superconducting phases caused by exceptional flattening of its lowest energy bands. Superconductivity with highest $T_c$ appears at hole and electron dopings, near half-filling for valence or conduction bands. In the hole-doped case, the data show that three-fold lattice rotation symmetry is broken in the superconducting phase, i.e., a superconductor is also a nematic. We present the mechanism for nematic superconductivity. We take as an input the fact that at relevant dopings the Fermi energy lies in the vicinity of twist-induced Van Hove singularities in the density of states and argue that the low-energy physics can be properly described by patch models with six Van Hove points for electron doping and twelve Van Hove points for hole doping. We obtain pairing interactions for the patch models in terms of parameters of the microscopic model for the flat bands, which contains both local and twist-induced non-local interactions and show that the latter gives rise to attraction in different superconducting channels. For hole-doping, we find two attractive channels, $g$ and $i$-waves, with almost equal coupling constants. We show that in the co-existence state, where both order parameters are non-zero, the three-fold lattice rotation symmetry is broken, i.e., a superconductor is also a nematic. We find two possible nematic states, one is time-reversal symmetric, the other additionally breaks time-reversal symmetry. Our scenario for nematic superconductivity is based on generic symmetry considerations, and we expect it to be applicable also to other systems with two (or more) attractive channels with similar couplings.

Quantization and interference of a quantum billiard with fourfold rotational symmetry.

Systems with discrete symmetries are highly important in quantum mechanics. We consider a two-dimensional quantum billiard with fourfold rotational symmetry, where the eigenenergies and eigenstates can be grouped into four symmetry subspaces. Unlike the threefold rotational symmetry case, here the interference of the scarring states on the fundamental domain orbits (FDO) is clean, that they either interfere constructively or annihilate completely. We shall show the complex behavior of the interference revealed in the length spectra for eigenenergies that belong to a particular symmetry subspace and combinations of different symmetry subspaces. We then provide detailed analysis of phase accumulation along the FDOs, which are the keys to determine the interference and could explain the enhancement or annihilation of the peaks well. The quantization condition for the scarring states belonging to different symmetry subspaces is discussed and used to reveal the time-reversal symmetry broken for a particular subspace. An experimental scheme to observe such complex behaviors is also proposed.