Vortex Crystals Research Articles

Dynamics of Polar Vortex Crystallization.

Vortex crystals are commonly observed in ultrathin ferroelectrics. However, a clear physical picture of origin of this topological state is currently lacking. Here, we show that vortex crystallization in ultrathin Pb(Zr_{0.4},Ti_{0.6})O_{3} films stems from the softening of a phonon mode and can be described as a Z_{2}×SU(1) symmetry-breaking transition. This result sheds light on the topology of the polar vortex patterns and bridges polar vortices with smectic phases, spin spirals, and other modulated states. Finally, we predict a resonant switching of the vortex tube orientation driven by midinfrared laser light which could enable new technologies.

From a vortex gas to a vortex crystal in instability-driven two-dimensional turbulence

We study structure formation in two-dimensional turbulence driven by an external force, interpolating between linear instability forcing and random stirring, subject to nonlinear damping. Using extensive direct numerical simulations, we uncover a rich parameter space featuring four distinct branches of stationary solutions: large-scale vortices, hybrid states with embedded shielded vortices (SVs) of either sign, and two states composed of many similar SVs. Of the latter, the first is a dense vortex gas where all SVs have the same sign and diffuse across the domain. The second is a hexagonal vortex crystal forming from this gas when the linear instability is sufficiently weak. These solutions coexist stably over a wide parameter range. The late-time evolution of the system from small-amplitude initial conditions is nearly self-similar, involving three phases: initial inverse cascade, random nucleation of SVs from turbulence and, once a critical number of vortices is reached, a phase of explosive nucleation of SVs, leading to a statistically stationary state. The vortex gas is continued in the forcing parameter, revealing a sharp transition towards the crystal state as the forcing strength decreases. This transition is analysed in terms of the diffusivity of individual vortices using ideas from statistical physics. The crystal can also decay via an inverse cascade resulting from the breakdown of shielding or insufficient nonlinear damping acting on SVs. Our study highlights the importance of the forcing details in two-dimensional turbulence and reveals the presence of non-trivial SV states in this system, specifically the emergence and melting of a vortex crystal.

Self-trapping of vortex crystals via competing nonlinearities.

We investigate the existence of self-trapped nonlinear waves with multiple phase singularities. Working with the cubic-quintic nonlinear Schrödinger equation, we focus on configurations with an antivortex surrounded by a triangular arrangement of vortices within a hosting soliton. We find stationary patterns that can be interpreted as stable self-trapped vortex crystals, constituting the first example of a configuration of this sort with space-independent potentials. Their stability is linked to their norm, transitioning from unstable to stable as their size increases, with an intermediate region where the structure is marginally unstable, undergoing a remarkable and puzzling self-reconstruction during its evolution.

A 2D ferroelectric vortex pattern in twisted BaTiO3 freestanding layers

The wealth of complex polar topologies1–10 recently found in nanoscale ferroelectrics results from a delicate balance between the intrinsic tendency of the materials to develop a homogeneous polarization and the electric and mechanical boundary conditions imposed on them. Ferroelectric–dielectric interfaces are model systems in which polarization curling originates from open circuit-like electric boundary conditions, to avoid the build-up of polarization charges through the formation of flux-closure11–14 domains that evolve into vortex-like structures at the nanoscale15–17 level. Although ferroelectricity is known to couple strongly with strain (both homogeneous18 and inhomogeneous19,20), the effect of mechanical constraints21 on thin-film nanoscale ferroelectrics has been comparatively less explored because of the relative paucity of strain patterns that can be implemented experimentally. Here we show that the stacking of freestanding ferroelectric perovskite layers with controlled twist angles provides an opportunity to tailor these topological nanostructures in a way determined by the lateral strain modulation associated with the twisting. Furthermore, we find that a peculiar pattern of polarization vortices and antivortices emerges from the flexoelectric coupling of polarization to strain gradients. This finding provides opportunities to create two-dimensional high-density vortex crystals that would enable us to explore previously unknown physical effects and functionalities.

Generation of femtosecond optical vortices with multiple separate phase singularities from a Kerr-lens mode-locked Yb:KGW oscillator.

Femtosecond optical vortices with a phase singular point have diverse applications such as microscopic particles manipulation, special-structure micro-processing and quantum information. Raising the number of singularity points can provide additional dimensions of control. Here we report for what we believe is the first time the generation of femtosecond optical vortices with multiple (two and five) singularities directly from a laser oscillator. The average powers and pulse durations of the resulting vortex pulses are several hundred milliwatts and less than 300 fs, respectively. This work represents an innovate way for obtaining femtosecond multi-vortices, opening the way to the further studies of optical vortex crystals and their applications.

The Dynamics of Jupiter's and Saturn's Weather Layers: A Synthesis After Cassini and Juno

Until recently, observations of the giant planets of our Solar System were confined to sampling relatively shallow regions of their atmospheres, leaving many uncertainties as to the dynamics of deeper layers. The Cassini and Juno missions to Saturn and Jupiter, however, have begun to address these issues, for example, by measuring their gravity and magnetic fields. The results show that the zonally coherent jets and cloud bands extend to levels where the electrical conductivity of the fluid becomes significant, whereas large-scale vortices, such as the Great Red Spot, are relatively shallow but may have deep-seated roots. The polar regions also exhibit intense cyclonic vortices that, on Jupiter, arrange themselves into remarkably regular “vortex crystals.” Numerical models seem able to capture some of this complexity, but many issues remain unresolved, suggesting a need for models that can represent both deep and shallow processes sufficiently realistic ally to compare with observations.

Theory of Magnetic Vortex Crystals Induced by Electric Dipole Interactions

Theory of Magnetic Vortex Crystals Induced by Electric Dipole Interactions

Antiferromagnetic order and its interplay with superconductivity in CaK(Fe1−x Mn x )4As4

The magnetic order for several compositions of CaK(FeMn x )4As4 has been studied by nuclear magnetic resonance (NMR), Mössbauer spectroscopy, and neutron diffraction. Our observations for the Mn-doped 1144 compound are consistent with the hedgehog spin vortex crystal (hSVC) order which has previously been found for Ni-doped . The hSVC state is characterized by the stripe-type propagation vectors and just as in the doped 122 compounds. The hSVC state preserves tetragonal symmetry at the Fe site, and only this SVC motif with simple antiferromagnetic (AFM) stacking along c is consistent with all our observations using NMR Mössbauer spectroscopy, and neutron diffraction. We find that the hSVC state in the Mn-doped 1144 compound coexists with superconductivity, and by combining the neutron scattering and Mössbauer spectroscopy data we can infer a quantum phase transition, hidden under the superconducting dome, associated with the suppression of the AFM transition temperature (T N) to zero for x ≈ 0.01. In addition, unlike several 122 compounds and Ni-doped 1144, the ordered magnetic moment is not observed to decrease at temperatures below the superconducting transition temperature (T c).

Coupling Navier-Stokes and Gross-Pitaevskii equations for the numerical simulation of two-fluid quantum flows

Numerical methods for solving the Navier-Stokes equations for classical (or normal) viscous fluids are well established. This is also the case for the Gross-Pitaevskii equation, governing quantum inviscid flows (or superfluids) in the zero temperature limit. In quantum flows, like liquid helium II at intermediate temperatures between zero and 2.17 K, a normal fluid and a superfluid coexist with independent velocity fields. The most advanced existing models for such systems use the Navier-Stokes equations for the normal fluid and a simplified description of the superfluid, based on the dynamics of quantized vortex filaments, with ad hoc reconnection rules. There was a single attempt (C. Coste, 1998 [22]) to couple Navier-Stokes and Gross-Pitaevskii equations in a global model intended to describe the compressible two-fluid liquid helium II. We present in this contribution a new numerical model to couple a Navier-Stokes incompressible fluid with a Gross-Pitaevskii superfluid. Coupling terms in the global system of equations involve new definitions of the following concepts: the regularized superfluid vorticity and velocity fields, the friction force exerted by quantized vortices to the normal fluid, the covariant gradient operator in the Gross-Pitaevskii model based on a slip velocity respecting the dynamics of vortex lines in the normal fluid. A numerical algorithm based on pseudo-spectral Fourier methods is presented for solving the coupled system of equations. Finally, we numerically test and validate the new numerical system against well-known benchmarks for the evolution in a normal fluid of different types or arrangements of quantized vortices (vortex crystal, vortex dipole and vortex rings). The new coupling model has the advantage to keep the full Gross-Pitaevskii model for the superfluid, and thus describe quantized vortex dynamics without any phenomenological approximation. This opens new possibilities to revisit and enrich existing numerical results for complex quantum fluids, such as quantum turbulent flows.

Multiple-[formula omitted] instability under fourth-order inplane single-ion anisotropy

Multiple-Q instability under tetragonal single-ion anisotropy is theoretically investigated. By performing the simulated annealing for a spin model with the momentum-resolved interaction and fourth-order inplane single-ion anisotropy on a square lattice, we find that the interplay between the competing exchange interactions in momentum space and single-ion anisotropy leads to various field-induced double-Q states in the ground state. In particular, we show that a double-Q vortex crystal emerges when the interactions at high-harmonic wave vectors are comparable to those at wave vectors that give the dominant contributions. Our results underscore the importance of magnetic anisotropy, which brings about the multiple-Q instability in centrosymmetric magnets.

Helicoids and vortices

We point out an interesting connection between fluid dynamics and minimal surface theory: When gluing helicoids into a minimal surface, the limit positions of the helicoids correspond to a ‘vortex crystal’, an equilibrium of point vortices in two-dimensional fluid that move together as a rigid body. While vortex crystals have been studied for almost 150 years, the gluing construction of minimal surfaces is relatively new. As a consequence of the connection, we obtain many new minimal surfaces and some new vortex crystals by simply comparing notes.

Skyrmion and vortex crystals in the Hubbard model

A mutual interplay between the charge and spin degrees of freedom in itinerant magnets leads to a plethora of topological spin textures, such as magnetic skyrmion and vortex crystals, in both centrosymmetric and noncentrosymmetric hosts. Meanwhile, their stabilization has been extensively studied in the system including the classical localized spins. We here study a realization of the skyrmion crystal in the centrosymmetric triangular-lattice Hubbard model, where the itinerant nature of electrons plays a more significant role. By performing the self-consistent mean-field calculations, we find that two types of skyrmion crystals with spatially nonuniform charge modulations appear in the ground state at a zero magnetic field. Moreover, we obtain another noncoplanar vortex crystal phase without a net scalar chirality in the vicinity of the skyrmion crystal phase. We show that the latter vortex crystal exhibits the topological phase transition to a different skyrmion phase in an applied magnetic field. Our results provide a possibility of the skyrmion and vortex crystals in itinerant magnets without the localized moments.

Zero-field skyrmion, meron, and vortex crystals in centrosymmetric hexagonal magnets

We investigate the interaction conditions to stabilize topological spin textures in centrosymmetric magnets without an external magnetic field. By performing the simulated annealing for an effective spin model with the momentum-resolved interaction on a two-dimensional triangular lattice, we find that the competing interactions between the finite-q channel and the uniform (q = 0) channel lead to a zero-field skyrmion crystal with the skyrmion number of ±1. Moreover, we obtain a variety of topological spin textures, such as a meron crystal, vortex crystal, and other multiple-Q states with and without a scalar chirality. We especially show that the bond-dependent anisotropic exchange interaction originating from the relativistic spin–orbit coupling plays an important role in inducing such topological spin textures, while the single-ion anisotropy does not.

Surface waves and bulk Ruderman mode of a bosonic superfluid vortex crystal in the lowest Landau level

We determine and analyze collective normal modes of a finite disk-shaped two-dimensional vortex crystal formed in a compressible bosonic superfluid in an artificial magnetic field. Using the microscopic Gross-Pitaevskii theory in the lowest Landau level approximation, we generate vortex crystal ground states and solve the Bogoliubov-de Gennes equations for small amplitude collective oscillations. We find chiral surface waves that propagate at frequencies larger than those of the bulk Tkachenko modes. Furthermore, we study low frequency bulk excitations and identify a torsional Ruderman mode, which we find is well-described by a previously developed low-energy effective field theory.

Hall anomaly by vacancies in a pinned lattice of vortices: A quantitative analysis based on the thin-film data from Bi2.1Sr1.9CaCu2.0O8+δ

Hall anomaly, as appears in the mixed-state Hall resistivity of type-II superconductors, has had numerous theories but yet a consensus on its origin. In this work, we conducted a quantitative analysis of the magnetotransport measurements on BSCCO thin films by Zhao et al. [Phys. Rev. Lett. 122, 247001 (2019)] and validate a previously proposed vacancy mechanism [cf. J. Phys. Condens. Matter. 10, L677 (1998)] with many-body vortex correlations for the phenomenon. The model attributes the Hall anomaly to the motion of vacancies in pinned fragments of vortex lattice. Its validity is first examined by an exploration on the vortex states near the Kosterlitz-Thouless transition on the vortex crystal. Comparisons are then carried out between the measured activation energies with the calculated creation energy of the vortex-anti-vortex pair and the vacancy energy on the flux-line lattice, with no adjustable parameter. Our analysis elucidates the theoretical basis and prerequisites of the vacancy model. In particular, the vacancy activation energies are an order of magnitude smaller than that of a sole vortex line. The proposed mechanism may provide a macro-theoretical framework for other studies.

Multiple magnetic orders in LaFeAs1-xPxO uncover universality of iron-pnictide superconductors

The iron-pnictide superconductors have generated tremendous excitement as the competition between magnetism and superconductivity has allowed unique in-roads towards elucidating a microscopic theory of unconventional high-temperature superconductivity. In addition to the stripe spin density wave ({C}_{2M}^{a}) phase observed in the parent compounds of all iron-pnictide superconductors, two novel magnetic orders have recently been discovered in different parent structures: an out-of-plane collinear double-Q ({C}_{4M}^{c}) structure in the hole-doped (Ca, Sr, Ba)1-x(Na)xFe2As2 and Ba1-xKxFe2As2 families, and a spin vortex crystal “hedgehog” ({C}_{4M}^{{ab}}) structure in the CaKFe4As4 family. Using neutron diffraction, we demonstrate that LaFeAs1-xPxO contains all three magnetic orders within a single-phase diagram as a function of substitution, all of which compete strongly with superconductivity. Our experimental observations combined with theoretical modeling demonstrate how the reduction in electronic correlations by chemical substitution results in larger Fermi surfaces and the sequential stabilization of multiple magnetic anisotropies. Our work presents a unified narrative for the competing magnetic and superconducting phases observed in various iron-pnictide systems with different crystal structures and chemistry.

Effective field theories for gapless phases with fractons via a coset construction

Fractons are particles with restricted mobility. We give a symmetry-based derivation of effective field theories of gapless phases with fractonic topological defects, such as solids and supersolids, using a coset construction. The resulting theory is identified as the Cosserat elasticity theory, which reproduces the conventional symmetric elasticity theory at low energies. The construction can be viewed as a dynamical realization of the inverse Higgs mechanism. We incorporate topological defects such as dislocations and disclinations, which are nontrivially related by the Bianchi identities of defect gauge fields. The origin of the fractonic nature of defects in those systems can be traced back to the semidirect product structure of translational and rotational groups. The construction is immediately extendable to higher dimensions and systems with broken translational symmetries, such as solids, supersolids, and vortex crystals. We identify Wess-Zumino terms in supersolids, which induce quasiparticle scatterings on topological defects.

Polar vortex crystals: Emergence and structure

Vortex crystals are quasiregular arrays of like-signed vortices in solid-body rotation embedded within a uniform background of weaker vorticity. Vortex crystals are observed at the poles of Jupiter and in laboratory experiments with magnetized electron plasmas in axisymmetric geometries. We show that vortex crystals form from the free evolution of randomly excited two-dimensional turbulence on an idealized polar cap. Once formed, the crystals are long lived and survive until the end of the simulations (300 crystal-rotation periods). We identify a fundamental length scale, Lγ=(U/γ)1/3, characterizing the size of the crystal in terms of the mean-square velocity U of the fluid and the polar parameter γ=fp/a2p, with fp the Coriolis parameter at the pole and ap the polar radius of the planet.

Vortex laser arrays with topological charge control and self-healing of defects

AbstractGeometric arrays of vortices found in various systems owe their regular structure to mutual interactions within a confined system. In optics, such vortex crystals may form spontaneously within a resonator. Their crystallization is relevant in many areas of physics, although their usefulness is limited by the lack of control over their topology. On the other hand, programmable devices like spatial light modulators allow the design of nearly arbitrary vortex distributions but without any intrinsic evolution. By combining non-Hermitian optics with on-demand topological transformations enabled by metasurfaces, we report a solid-state laser that generates 10 × 10 vortex laser arrays with actively tunable topologies and non-local coupling dictated by the array’s topology. The vortex arrays exhibit sharp Bragg diffraction peaks, witnessing their coherence and topological charge purity, which we spatially resolve over the whole lattice by introducing a parallelized analysis technique. By structuring light at the source, we enable complex transformations that allow to arbitrarily partition orbital angular momentum within the cavity and to heal topological charge defects, thus realizing robust and versatile resonators for applications in topological optics.

Volume-preserving diffeomorphism as nonabelian higher-rank gauge symmetry

We propose nonabelian higher-rank gauge theories in 2+1D and 3+1D. The gauge group is constructed from the volume-preserving diffeomorphisms of space. We show that the intriguing physics of the lowest Landau level (LLL) limit can be interpreted as the consequences of the symmetry. We derive the renowned Girvin-MacDonald-Platzman (GMP) algebra as well as the topological Wen-Zee term within our formalism. Using the gauge symmetry in 2+1D, we derive the LLL effective action of vortex crystal in rotating Bose gas as well as Wigner crystal of electron in an applied magnetic field. We show that the nonlinear sigma models of ferromagnets in 2+1D and 3+1D exhibit the higher-rank gauge symmetries that we introduce in this paper. We interpret the fractonic behavior of the excitations on the lowest Landau level and of skyrmions in ferromagnets as the consequence of the higher-rank gauge symmetry.