Integer Vortices Research Articles

Phase diagram of generalized XY model using the tensor renormalization group

We use the higher-order tensor renormalization group method to study the two-dimensional generalized XY model that admits integer and half-integer vortices. This model is the deformation of the classical XY model and has a rich phase structure consisting of nematic, ferromagnetic, and disordered phases and three transition lines belonging to the Berezinskii-Kosterlitz-Thouless and Ising class. We explore the model for a wide range of temperatures, T, and the deformation parameter, Δ, and compute specific heat along with integer and half-integer magnetic susceptibility, finding both Berezinskii-Kosterlitz-Thouless-like and Ising-like transitions and the region where they meet. Published by the American Physical Society 2024

Pulsar glitches from quantum vortex networks

Neutron stars or pulsars are very rapidly rotating compact stars with extremely high density. One of the unsolved long-standing problems of these enigmatic celestial bodies is the origin of pulsars’ glitches, i.e., the sudden rapid deceleration in the rotation speed of neutron stars. Although many glitch events have been reported, there is no consensus on the microscopic mechanism responsible for them. One of the important characterizations of the glitches is the scaling law P(E)∼E-α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P(E) \\sim E^{-\\alpha }$$\\end{document} of the probability distribution for a glitch with energy E. Here, we reanalyse the accumulated up-to-date observation data to obtain the exponent α≈0.88\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha \\approx 0.88$$\\end{document} for the scaling law, and propose a simple microscopic model that naturally deduces this scaling law without any free parameters. Our model explains the appearance of these glitches in terms of the presence of quantum vortex networks arising at the interface of two different kinds of superfluids in the core of neutron stars; a p-wave neutron superfluid in the inner core which interfaces with the s-wave neutron superfluid in the outer core, where each integer vortex in the s-wave superfluid connects to two half-quantized vortices in the p-wave superfluid through structures called “boojums”.

Observation of spatial self-phase modulation excited by off-axis integer and fractional vortex beams

Spatial self-phase modulation (SSPM) is widely used to characterize the nonlinear refractive index of two-dimensional layered materials and design passive nonlinear photonic devices. In this work, we investigate the SSPM phenomena of black phosphorus dispersed in agarose solid under the excitation of on-axis and off-axis integer and half-integer vortex beams. It is found that with the increase of the off-axis distance of the Gaussian vortex beam, the pattern of the far-field self-diffraction intensity gradually breaks from the initial ring and finally evolves into a tail outside the self-diffraction ring. Moreover, for half-integer vortex beams, the size of the innermost tail of the self-diffraction ring is nearly half smaller than that of the second outer tail. The underlying mechanism of the observed SSPM phenomenon is also analyzed. Interestingly, the trailing number and rotation direction of the self-diffraction ring pattern quantitatively reflect the magnitude and sign of the topological charge (TC) of the off-axis vortex beam, respectively. This work provides a novel method to measure the TC of off-axis vortex beam in the nonlinear optics regime.

Generation of integer and fractional vortex beams based on liquid crystal electronically reconfigurable spiral phase plates.

The manufacturing and characterization of a large-size 72-electrode liquid crystal-based reconfigurable spiral phase plate (SPP) is presented. The SPP is addressed by a custom-made driver with 72 independent channels, which allows for the generation of any arbitrary integer or fractional optical vortex beam with topological charges ranging from -24 to +24. The 25 mm diameter device is fabricated using direct laser writing, leading to a fill factor over 99%. The device performance and flexibility exceed previous transparent reconfigurable SPP in terms of size, tuning range, and fill factor. The device and the light path have been simulated using the angular spectrum propagation method, showing excellent correspondence.

Second-Harmonic Generation of the Vortex Beams with Integer and Fractional Topological Charges

The single-pass second-harmonic generation (SHG) of a vortex beam under low fundamental wave depletion is systematically studied. Vortex modes at 1064 nm with integer topological charges from ±1 to ±9 and fractional ones at ±0.75 are generated by modulating the fundamental Gaussian beam with different spiral phase plates. The frequency doubling of these fundamental vortex modes is realized via single-pass SHG through the KTP. A detailed theoretical model is set up in the single-pass SHG of the vortex beams. Theoretical analysis indicates that the higher the order of the vortex beams, the lower the SHG efficiency, when the beam waists and fundamental power are given. The experimentally measured SHG output characteristics verify those obtained via theoretical analysis. Conservation of the orbital angular momentum during the SHG process is also verified, regardless of the fractional or integer vortex beams. SH LG0,2l vortex beams with high mode purity are obtained. The beam waists of fundamental/SH in KTP measured using a 4f system demonstrate that the Rayleigh ranges of the fundamental wave and SH wave are the same. The paper comprehensively presents some basic laws in the single-pass SHG of a vortex beam. In addition, it also indicates that SHG is an effective method to improve the mode purity of vortex beam.

Free-space creation of a perfect vortex beam with fractional topological charge.

Perfect vortex beams can only propagate stably with integer topological charges. Thus, creating perfect fractional vortex beams capable of stable propagation in free space, as perfect integer vortex beams, is crucial. This study proposed perfect vortex beams carrying fractional topological charge of l + 0.5, which are special solutions of the wave equation, and can maintain stable propagation with physical laws same as integer topological charge. Perfect fractional vortex beams were created in free space, which can break the cognition of traditional fractional perfect vortex beams and promote the development of scientific fields such as optical communication, quantum sensing, and optical imaging.

Phase transitions driven by topological excitations and their tensor network approach

The fundamental concepts of phases of matter and thermal phase transitions constitute the cornerstone of our understanding of the physical universe. The historical development of the phase transition theory from Landau's spontaneous symmetry breaking paradigm to modern topological phase transition theories represents a major milestone in the evolution of numerous scientific disciplines. From the perspective of emergent philosophy, the interplay of topological excitations leads to enriched physical phenomena. One prominent prototype is the Berezinskii-Kosterlitz-Thouless (BKT) phase transition, where unbinding of integer vortices occurs in the absence of spontaneous breaking of continuous <i>U</i>(1) symmetry. Using the state-of-the-art tensor network methods, we express the partition function of the two-dimensional <i>XY</i>-related system in terms of a product of one-dimensional transfer operators. From the singularities of the entanglement entropy of the one-dimensional transfer operator, we accurately determine the complete phase diagram of the partition function. This method provides new insights into the emergent phenomenon driven by topological excitations, and sheds new light on future studies of 2D systems with continuous symmetries.

Fractional vortices, Z2 gauge theory, and the confinement-deconfinement transition

In this Letter we discuss the classical three-dimensional XY model whose nearest-neighbor interaction is a mixture of $cos({\ensuremath{\theta}}_{i}\ensuremath{-}{\ensuremath{\theta}}_{j})$ (ferromagnetic) and $cos2({\ensuremath{\theta}}_{i}\ensuremath{-}{\ensuremath{\theta}}_{j})$ (nematic). This model is dual to a theory with integer and half-integer vortices. While both types of vortices interact with a noncompact $U(1)$ gauge field, the half-integer vortices interact with an extra interaction mediated by a ${\mathbb{Z}}_{2}$ gauge field. We shall discuss the confinement-deconfinement transition of the half-integer vortices, the Wilson and the `t Hooft loops, and their mutual statistics in path integral language. In addition, we shall present a quantum version of the classical model which exhibits these physics.

Machine Learning-Based Fast Integer and Fractional Vortex Modes Recognition of Partially Occluded Vortex Beams

In this work, a machine learning method is proposed to precisely classify partially occluded integer and fractional vortex modes for the first time in radio frequency (RF). Consequently, we introduce three training schemes, i.e., the direct recognition scheme with the phase data or the amplitude data (PD-DRS and AD-DRS), the phase data or amplitude data interpolated by nearest-neighbor interpolation algorithm (PD-NNI and AD-NNI), and the full data (FD) of the electric field with the NNI algorithm (FD-NNI), to recognize the topological charges. Based on the designed deep convolutional neural network (DCNN) models, the relationship between the test accuracy and the number of sampling points of the three schemes is presented. It is shown that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$3\times3$ </tex-math></inline-formula> sampling points are enough for FD-NNI to achieve the classification accuracy of 98.2%. To validate the robustness of the proposed models, we evaluate them on the sample carrying up to 50% Gaussian noise, separately. Besides, the effects of propagation distance and the occlusion angle are also investigated. The numerical results present that the interpolated data performs better in terms of accuracy compared with the pure sampled data, among which FD-NNI possesses better generalization ability, suggesting great potential in the practical application of radio vorticity communication.

Generation of terahertz radiation with fractional or integer OAMs from a fractional-order vortex two-color field

This paper investigates the generation of the ultra-broadwidth (0.1–30 THz) terahertz (THz) radiation carrying fractional/integer orbital angular momentums (OAMs) via the interaction of the two-color (ω 0 and 2ω 0) laser field carrying initial fractional topological charges (TCs) with air in a moderate pump intensity regime (20 TW cm−2 < I pump < 50 TW cm−2). The two four-wave mixing (FWM) processes (i.e., ω 0 + ω′0 − 2ω 0 → ω THz and 2ω 0 − ω 0 − ω′0 → ω THz) are responsible for THz generation. The two processes can produce two THz pulses. They interfere with each other and THz interference vortex beams are obtained. More importantly, the generation probability from the first FWM process grows while that of the second process declines in the positive frequency region over distance. This is largely due to the combined action of phase mismatch and the blue shift of the THz center frequency. For a longer distance, THz fractional vortex beams (FVBs)/integer vortex beams (IVBs) are produced by the dominant FWM process (ω 0 + ω′0 − 2ω 0 → ω THz). Therefore, via employing different combinations of the initial TCs of the ω 0 and 2ω 0 pulses, one can manipulate the generation of the THz vortex beams with arbitrary fractional-order or integer-order TCs at some specific propagation distances. What is even more interesting is that, when employing half-integer TCs, THz FVBs with varying TC over distance can be produced, companied with birth and annihilation of the alternative vortex pair. This is principally due to diffraction-related effects and the unstable nature of the fractional vortex structures. This simple manipulation for THz waves carrying arbitrary fractional or integer TCs in this scheme encourages the applications for optically rotation, manipulation of molecular or cell assays and image edge enhancement in the field of biomedicine.

Integer and fractionalized vortex lattices and off-diagonal long-range order

We analyze the implication of off-diagonal long-range order (ODLRO) for inhomogeneous periodic field configurations and multi-component order parameters. For single component order parameters we show that the only static, periodic field configuration consistent with ODLRO is a vortex lattice with integer flux in units of the flux quantum in each unit cell. For a superconductor with g degenerate components, fractional vortices are allowed. Depending on the precise order-parameter manifold, they tend to occur in units of 1/g of the flux quantum. These results are well known to emerge from the Ginzburg-Landau or BCS theories of superconductivity. Our results imply that they are valid even if these theories no-longer apply. Integer and fractional vortex lattices are transparently seen to emerge as a consequence of the macroscopic coherence and single valuedness of the condensate.

Review on fractional vortex beam

Abstract As an indispensable complement to an integer vortex beam, the fractional vortex beam has unique physical properties such as radially notched intensity distribution, complex phase structure consisting of alternating charge vortex chains, and more sophisticated orbital angular momentum modulation dimension. In recent years, we have noticed that the fractional vortex beam was widely used for complex micro-particle manipulation in optical tweezers, improving communication capacity, controllable edge enhancement of image and quantum entanglement. Moreover, this has stimulated extensive research interest, including the deep digging of the phenomenon and physics based on different advanced beam sources and has led to a new research boom in micro/nano-optical devices. Here, we review the recent advances leading to theoretical models, propagation, generation, measurement, and applications of fractional vortex beams and consider the possible directions and challenges in the future.

Intensity interference characteristics of fractional Laguerre–Gaussian optical vortex

The paper reports the intensity characteristics of fractional vortex beam, and controls the coupling between photon orbital angular momentum and spin angular momentum by coherently manipulating Laguerre–Gaussian vortex beam with linearly polarized plane light and its conjugate beam. By comparing theory with experiment, fractional vortex beams with different topological charges are generated by using pure phase liquid crystal spatial light modulator, and their intensity characteristics and interference characteristics are observed. The results show that the radial notch size of the spot is related to the fractional part of the topological charge value, and the closer the fractional part of the topological charge value is to the half integer, the more obvious the radial notch is. With the increase of integer part of topological charge, the spot radius also increases. In addition, interference with linearly polarized plane light reveals that the number of branches is equal to the number of topological charges. Interfere with its conjugate beam, and judge the topological charge number of vortex beam by the relationship between petal number and topological charge number. The petal number of integer vortex beam is twice the topological charge number, and fractional order vortex beam destroys the ring structure of Laguerre–Gaussian beam.

Measurement of multiplexed fractional vortices with integer mode interval

In recent years, light’s orbital angular momentum (OAM) has been used as a new degree of freedom in both classical and quantum communications. Fractional vortex (FV) with non-integer topological charge (TC) has been used in optical communication to increase transmission capacity and improve security capability. However, it is difficult to detect multiplexed FVs accurately, which restricts the application of FV in optical communications. In this work, we reported a scheme to detect TCs of multiplexing FVs with integer OAM mode intervals. It started by raising an algorithm that can calculate the TC of a single FV with the intensities of two integer vortices and a correction function. Then, based on this algorithm, multiplexing FVs with discrete TCs were regarded as four (three) integer vortices and the corresponding TCs could also be measured by the intensities of two integer OAM modes. The experiments were employed to measure multiplexed FVs in the range from −6.9 to +6.9 for proving the strategy, and the results showed that the TC measurement error is less than 0.023. This multiplexed FVs detection scheme can increase the number of available vortex modes in a certain range, which might inspire wide optical communication applications of FV beams.

Hybrid Berezinskii-Kosterlitz-Thouless and Ising topological phase transition in the generalized two-dimensional XY model using tensor networks

In tensor network representation, the partition function of a generalized two-dimensional XY spin model with topological integer and half-integer vortex excitations is mapped to a tensor product of one-dimensional quantum transfer operator, whose eigenequation can be solved by an algorithm of variational uniform matrix product states. Using the singularities of the entanglement entropy, we accurately determine the complete phase diagram of this model. Both the integer vortex-antivortex binding and half-integer vortex-antivortex binding phases are separated from the disordered phase by the usual Berezinskii-Kosterlitz-Thouless (BKT) transitions, while a continuous topological phase transition exists between two different vortex binding phases, exhibiting a logarithmic divergence of the specific heat and exponential divergence of the spin correlation length. A hybrid BKT and Ising universality class of topological phase transition is thus established. We further prove that three phase transition lines meet at a multicritical point from which a deconfinement crossover line extends into the disordered phase.

Fractional and integer vortex dynamics in strongly coupled two-component Bose-Einstein condensates from AdS/CFT correspondence

In order to study the rotating strongly coupled Bose-Einstein condensations(BEC), a holographic model defined in an AdS black hole that duals to a coupled two-component condensations in global $U(1)$ symmetry broken phase with intercomponent coupling $\eta$ and internal coherent coupling $\epsilon$ is proposed. By solving the dynamics of the model, we study the process of formation and also the crossover from fractional to integer vortex phases. With changing only $\eta$ from zero to a finite value, fractional vortex lattices undergo a transition from hexagon to square lattice and finally to vortex sheets. By continuing to turn on $\epsilon$, we find that two fractional vortices in different components constitute dimers, and when $\eta$ transcend a critical value, multi-dimer like hexamer or tetramer made up of two and three dimers appear. As $\epsilon$ keeps increasing, some dimers rotate to adjust themselves and then constitute the lattice of integer vortices. Under an initial conditions similar to an spinor BEC vortices dynamics experiment, the appearance of disordered turbulence is found in the process of fractional vortex generation, which matches the experimental observation. While in the formation process of integer vortices, the appearance of grooves is predicted.

Zn modified XY and Goldstone models and vortex confinement transition

The modified XY model is a modification of the XY model by addition of a half-periodic term. The modified Goldstone model is a regular and continuum version of the modified XY model. The former admits a vortex molecule, that is, two half-quantized vortices connected by a domain wall, as a regular topological soliton solution to the equation of motion while the latter admits it as a singular configuration. Here we define the ${\mathbb Z}_n$ modified XY and Goldstone models as the $n=2$ case to be the modified XY and Goldstone models, respectively. We exhaust all stable and metastalble vortex solutions for $n=2,3$ and find a vortex confinement transition from an integer vortex to a vortex molecule of $n$ $1/n$-quantized vortices, depending on the ratio between the term of the XY model and the modified term. We find for the case of $n=3$, a rod-shaped molecule is the most stable while a Y-shaped molecule is metastable. We also construct some solutions for the case of $n=4$.The vortex confinement transition can be understood in terms of the ${\mathbb C}/{\mathbb Z}_n$ orbifold geometry.

Partially coherent radially polarized fractional vortex beam.

A new kind of partially coherent vector beam, named a partially coherent radially polarized fractional vortex (PCRPFV) beam, is introduced as a natural extension of the recently introduced scalar partially coherent fractional vortex beams [Zeng et al., Opt. Express26, 26830 (2018)10.1364/OE.26.026830]. Realizability conditions and propagation formulas for a PCRPFV beam are derived. Statistical properties of a focused PCRPFV beam, such as average intensity, degree of polarization, state of polarization and cross-spectral density matrix, are illustrated in detail and compared with that of a partially coherent radially polarized integer vortex beam and a scalar partially coherent fractional vortex beam. It is found that the statistical properties of a PCRPFV beam are qualitatively different from these simpler beam classes and are strongly determined by the vortex phase (i.e., fractional topological charge) and initial coherence width. We demonstrate experimental generation of PCRPFV beams and confirm their behavior. Our results will be useful for the rotating and trapping of particles, the detection of phase objects, and polarization lidar systems.

Half-quantized non-Abelian vortices in neutron 3P2 superfluids inside magnetars

Abstract We point out that half-quantized non-Abelian vortices exist as the minimum energy states in rotating neutron $^3P_2$ superfluids in the inner cores of magnetars with magnetic fields greater than $3 \times 10^{15}$ Gauss, while they do not in ordinary neutron stars with smaller magnetic fields. One integer vortex is split into two half-quantized vortices. The number of vortices is about $10^{19}$ and they are separated at about $\mu$m in a vortex lattice for typical parameters, while the vortex core size is about 10–100 fm. They are non-Abelian vortices characterized by a non-Abelian first homotopy group, and consequently when two vortices corresponding to non-commutative elements collide, a rung vortex must be created between them, implying the formation of an entangled vortex network inside the cores of magnetars. We find spontaneous magnetization in the vortex core showing diamagnetism whose typical magnitude is about $10^{8-9}$ Gauss, which is ten times larger than that of integer vortices, when external magnetic fields are present along the vortex line.

Observing vortex polarization singularities at optical band degeneracies

In the momentum space of two-dimensional (2D) periodic photonic systems, the polarization states of radiating Bloch modes on a single band can form vortex polarization singularities. Here, we study vortex polarization singularities at the band degeneracies. We design a quadratic degeneracy in a plasmonic crystal with ${C}_{4v}$ symmetry which, by symmetry breaking, splits into a pair of Dirac points with half vortices. The integer vortex singularity at the quadratic degeneracy and the half one at the Dirac degeneracy are measured using polarization-resolved momentum-space imaging spectroscopy. The complete evolution of the splitting from the integer vortex to a pair of half vortices is observed in experiments.