Topological Field Research Articles

Topological symmetry in quantum field theory

We introduce a definition and framework for internal topological symmetries in quantum field theory, including “noninvertible symmetries” and “categorical symmetries”. We outline a calculus of topological defects which takes advantage of well-developed theorems and techniques in topological field theory. Our discussion focuses on finite symmetries, and we give indications for a generalization to other symmetries. We treat quotients and quotient defects (often called “gauging” and “condensation defects”), finite electromagnetic duality, and duality defects, among other topics. We include an appendix on finite homotopy theories, which are often used to encode finite symmetries and for which computations can be carried out using methods of algebraic topology. Throughout we emphasize exposition and examples over a detailed technical treatment.

The Spin-Statistics Theorem for Topological Quantum Field Theories

The Spin-Statistics Theorem for Topological Quantum Field Theories

Customizable Generation of Arbitrary‐Order Polarization Vortices by Spin‐Decoupled Geometric Phases

AbstractDue to nontrivial field topologies, polarization vortices carrying polarization singularities have attracted increasing interest in the fields of optics and photonics. However, the conventional methods for generation of polarization vortcies still suffer from the complexity and the heavy bulk. In this work, taking advantage of spin‐decoupled geometric phase in metasurfaces, a novel method is proposed to generate polarization vortices with customizable topological charges. By tailoring spin‐decoupled geometric phases generated by the metasurface, distinct helical phase profiles can be imparted to left‐ and right‐handed circularly polarized components. Consequently, two scalar vortices with different topological charges can be simultaneously generated in the two orthogonal circular polarization components, leading to the generation of the polarization vortices with the needed topological charges and the diverse morphologies. Both simulation and experimental results well demonstrate the method. Owing to the nondispersive feature of geometric phases, the designed metasurface works well in a wide frequency band. The proposed method contributes a reliable and feasible solution for customizing the generation of polarization vortices, which can be further transposed to other frequencies and applied to information encoding and polarization detection etc.

Abelian TQFTS and Schrödinger local systems

In this paper, we construct an action of 3-cobordisms on the finite dimensional Schrödinger representations of the Heisenberg group by Lagrangian correspondences. In addition, we review the construction of the abelian Topological Quantum Field Theory (TQFT) associated with a q -deformation of U(1) for any root of unity q . We prove that, for 3-cobordisms compatible with Lagrangian correspondences, there is a normalization of the associated Schrödinger bimodule action that reproduces the abelian TQFT.The full abelian TQFT provides a projective representation of the mapping class group \mathrm{Mod}(\Sigma) on the Schrödinger representation, which is linearizable at odd root of 1. Motivated by homology of surface configurations with Schrödinger representation as local coefficients, we define another projective action of \mathrm{Mod}(\Sigma) on Schrödinger representations. We show that the latter is not linearizable by identifying the associated 2-cocycle.

Dualities of Paired Quantum Hall Bilayer States at ν_{T}=1/2+1/2.

Density-balanced, widely separated quantum Hall bilayers at ν_{T}=1 can be described as two copies of composite Fermi liquids (CFLs). The two CFLs have interlayer weak-coupling Bardeen-Cooper-Schrieffer instabilities mediated by gauge fluctuations, the resulting pairing symmetry of which depends on the CFL hypothesis used. If both layers are described by the conventional Halperin-Lee-Read (HLR) theory-based composite electron liquid (CEL), the dominant pairing instability is in the p+ip channel; whereas if one layer is described by CEL and the other by a composite hole liquid (CHL, in the sense of anti-HLR), the dominant pairing instability occurs in the s-wave channel. Using the Dirac composite fermion (CF) picture, we show that these two pairing channels can be mapped onto each other by particle-hole (PH) transformation. Furthermore, we derive the CHL theory as the nonrelativistic limit of the PH-transformed massive Dirac CF theory. Finally, we prove that an effective topological field theory for the paired CEL-CHL in the weak-coupling limit is equivalent to the exciton condensate phase in the strong-coupling limit.

Unveiling complex magnetic field configurations in red giant stars

The recent measurement of magnetic field strength inside the radiative interior of red giant stars has opened the way toward full 3D characterization of the geometry of stable large-scale magnetic fields. However, current measurements, which are limited to dipolar (ℓ = 1) mixed modes, do not properly constrain the topology of magnetic fields due to degeneracies on the observed magnetic field signature on such ℓ = 1 mode frequencies. Efforts focused toward unambiguous detections of magnetic field configurations are now key to better understand angular momentum transport in stars. We investigated the detectability of complex magnetic field topologies (such as the ones observed at the surface of stars with a radiative envelope with spectropolarimetry) inside the radiative interior of red giants. We focused on a field composed of a combination of a dipole and a quadrupole (quadrudipole) and on an offset field. We explored the potential of probing such magnetic field topologies from a combined measurement of magnetic signatures on ℓ = 1 and quadrupolar (ℓ = 2) mixed mode oscillation frequencies. We first derived the asymptotic theoretical formalism for computing the asymmetric signature in the frequency pattern for ℓ = 2 modes due to a quadrudipole magnetic field. To access asymmetry parameters for more complex magnetic field topologies, we numerically performed a grid search over the parameter space to map the degeneracy of the signatures of given topologies. We demonstrate the crucial role played by ℓ = 2 mixed modes in accessing internal magnetic fields with a quadrupolar component. The degeneracy of the quadrudipole compared to pure dipolar fields is lifted when considering magnetic asymmetries in both ℓ = 1 and ℓ = 2 mode frequencies. In addition to the analytical derivation for the quadrudipole, we present the prospect for complex magnetic field inversions using magnetic sensitivity kernels from standard perturbation analysis for forward modeling. Using this method, we explored the detectability of offset magnetic fields from ℓ = 1 and ℓ = 2 frequencies and demonstrate that offset fields may be mistaken for weak and centered magnetic fields, resulting in underestimating the magnetic field strength in stellar cores. We emphasize the need to characterize ℓ = 2 mixed-mode frequencies, (along with the currently characterized ℓ = 1 mixed modes), to unveil the higher-order components of the geometry of buried magnetic fields and to better constrain angular momentum transport inside stars.

CFT Correlators and Mapping Class Group Averages

Mapping class group averages appear in the study of 3D gravity partition functions. In this paper, we work with 3D topological field theories to establish a bulk-boundary correspondence between such averages and correlators of 2D rational CFTs whose chiral mapping class group representations are irreducible and satisfy a finiteness property. We show that Ising-type modular fusion categories satisfy these properties on surfaces with or without field insertions, extending results in [1], and we comment on the absence of invertible global symmetries in the examples we consider.

Impact of the Out‐Of‐Plane Flow Shear on Magnetic Reconnection at the Flanks of Earth's Magnetopause

AbstractMagnetic reconnection changes the magnetic field topology and facilitates the energy and particle exchange at magnetospheric boundaries such as the Earth's magnetopause. The flow shear perpendicular to the reconnecting plane prevails at the flank magnetopause under southward interplanetary magnetic field conditions. However, the effect of the out‐of‐plane flow shear on asymmetric reconnection is an open question. In this study, we utilize kinetic simulations to investigate the impact of the out‐of‐plane flow shear on asymmetric reconnection. By systematically varying the flow shear strength, we analyze the flow shear effects on the reconnection rate, the diffusion region structure, and the energy conversion rate. We find that the reconnection rate increases with the upstream out‐of‐plane flow shear, and for the same upstream conditions, it is higher at the dusk side than at the dawn side. The diffusion region is squeezed in the outflow direction due to magnetic pressure which is proportional to the square of the Alfvén Mach number of the shear flow. The out‐of‐plane flow shear increases the energy conversion rate , and for the same upstream conditions, the magnitude of is larger at the dusk side than at the dawn side. This study reveals that out‐of‐plane flow shear not only enhances the reconnection rate but also significantly boosts energy conversion, with more pronounced effects on the dusk‐side flank than on the dawn‐side flank. These insights pave the way for better understanding the solar wind‐magnetosphere interactions.

Development of a compact bolometer camera concept for investigation of radiation asymmetries at Wendelstein 7-X.

Power exhaust is one of the central challenges in magnetically confined fusion plasmas. Radiative detachment can be employed to reduce particle and heat fluxes to the divertor target, mitigating divertor damage and erosion. However, accomplishing this for a non-axisymmetric machine such as Wendelstein 7-X is a non-trivial task because of the complex role of transport and plasma-wall interaction in a three-dimensional magnetic field topology. We introduce a new bolometer camera design that can be easily installed in multiple toroidal locations and adapted to the required geometry, providing additional spatial coverage. This can be used to locally enhance tomographic capabilities or to resolve spatial variations of the plasma emissivity. By including these non-uniformities in the total radiated power estimate, global power balance measurements can be improved. We model each bolometer camera using ray tracing. We then analyze the forward-modeled detector response to several physically motivated synthetic emission phantoms with respect to its capability to quantify the local average emissivity. The results prove this concept as a promising asset for the investigation of poloidal and toroidal radiated power asymmetries in Wendelstein 7-X. The first CBC prototypes have undergone development and installation for the next experimental campaign.

Sunward Oxygen Ion Fluxes and the Magnetic Field Topology at Mars From Hybrid Simulations

AbstractIt is commonly believed that because of the direct solar wind interaction with the Martian atmosphere/ionosphere, the planet could have lost a significant part of its atmosphere. Closed field lines of the crustal magnetic field can weaken a transport of the ionospheric ions to the tail. Reconnection of the interplanetary magnetic field lines draping around Mars and the crustal magnetic field can also lead to a presense of sunward fluxes of planetary ions that might affect the total ion loss. The LatHyS (LATMOS Hybrid Simulation) three‐dimensional multispecies hybrid model is used here to characterize sunward fluxes of O+ ions and the magnetic field topology at Mars. It is shown that although reconnection between the interplanetary magnetic field (IMF) and the crustal magnetic fields strongly modifies the field topology, then sunward ion fluxes are rather small and do not significantly change the total ion loss.

Global and local dynamics of X-flare-producing active regions during solar cycle 25 peak phase

The configuration of the longitudinally elongated region that active regions (ARs) cluster around, known as a toroid belt, has been shown to be an indicator of intense activity. In particular, complex ARs at locations in the north and/or south toroids tend to appear "tipped-away" with respect to each other. On the other hand, magnetic helicity has been used as an indicator of flare activity in ARs. As solar cycle (SC) 25 approaches its peak, a number of significant (X-class) flares have been produced. Here, we investigate the circumstances surrounding two of the most flare-prolific ARs of solar cycle 25, namely, ARs 13590 and 13514. Two aspects of the evolution of these ARs are investigated in this work: the global-scale magnetic toroid configuration and small-scale magnetic field morphology and topology -- before, during, and after the onset of major flares. We studied the global morphology of the solar magnetic fields near intense flares in terms of the spatial distribution of ARs on magnetic fields synoptic maps. On AR scales, we analyzed the magnetic helicity accumulation, as well as its current-carrying and potential components. Our results are consistent with major flare-prolific ARs from solar cycles 23 and 24. In particular, we observe a consistent dominance of current-carrying magnetic helicity at the time of major flares. The evolution of global magnetic toroids, indicating the occurrence of flare-prolific ARs in the tipped-away portion of the toroid, together with the local dynamics of complex ARs, could offer a few weeks of lead time to prepare for upcoming space weather hazards.

On fusing matrices associated with conformal boundary conditions

In the context of rational conformal field theories (RCFT) we look at the fusing matrices that arise when a topological defect is attached to a conformal boundary condition. We call such junctions open topological defects. One type of fusing matrices arises when two open defects fuse while another arises when an open defect passes through a boundary operator. We use the topological field theory approach to RCFTs based on Frobenius algebra objects in modular tensor categories to describe the general structure associated with such matrices and how to compute them from a given Frobenius algebra object and its representation theory. We illustrate the computational process on the rational free boson theories. Applications to boundary renormalisation group flows are briefly discussed.

Generalized cluster states from Hopf algebras: non-invertible symmetry and Hopf tensor network representation

Cluster states are crucial resources for measurement-based quantum computation (MBQC). It exhibits symmetry-protected topological (SPT) order, thus also playing a crucial role in studying topological phases. We present the construction of cluster states based on Hopf algebras. By generalizing the finite group valued qudit to a Hopf algebra valued qudit and introducing the generalized Pauli-X operator based on the regular action of the Hopf algebra, as well as the generalized Pauli-Z operator based on the irreducible representation action on the Hopf algebra, we develop a comprehensive theory of Hopf qudits. We demonstrate that non-invertible symmetry naturally emerges for Hopf qudits. Subsequently, for a bipartite graph termed the cluster graph, we assign the identity state and trivial representation state to even and odd vertices, respectively. Introducing the edge entangler as controlled regular action, we provide a general construction of Hopf cluster states. To ensure the commutativity of the edge entangler, we propose a method to construct a cluster lattice for any triangulable manifold. We use the 1d cluster state as an example to illustrate our construction. As this serves as a promising candidate for SPT phases, we construct the gapped Hamiltonian for this scenario and provide a detailed discussion of its non-invertible symmetries. We demonstrate that the 1d cluster state model is equivalent to the quasi-1d Hopf quantum double model with one rough boundary and one smooth boundary. We also discuss the generalization of the Hopf cluster state model to the Hopf ladder model through symmetry topological field theory. Furthermore, we introduce the Hopf tensor network representation of Hopf cluster states by integrating the tensor representation of structure constants with the string diagrams of the Hopf algebra, which can be used to solve the Hopf cluster state model.

Supersymmetry dictated topology in periodic gauge fields and realization in strained and twisted 2D materials

Supersymmetry (SUSY) of a Hamiltonian dictates double degeneracy between a pair of superpartners (SPs) transformed by supercharge, except at zero energy where modes remain unpaired in many cases. Here we explore a SUSY of complete isospectrum between SPs-with paired zero modes-realized by 2D electrons in zero-flux periodic gauge fields, which can describe twisted or periodically strained 2D materials. We find their low-energy sector containing zero (or threshold) modes must be topologically non-trivial, by proving that Chern numbers of the two SPs have a finite difference dictated by the number of zero modes and energy dispersion in their vicinity. In 30° twisted bilayer (double bilayer) transition metal dichalcogenides subject to periodic strain, we find one SP is topologically trivial in its lowest miniband, while the twin SP of identical dispersion has a Chern number of 1 (2), in stark contrast to time-reversal partners that have to be simultaneously trivial or nontrivial. For systems whose physical Hamiltonian corresponds to the square root of a SUSY Hamiltonian, such as twisted or strained bilayer graphene, we reveal that topological properties of the two SUSY SPs are transferred respectively to the conduction and valence bands, including the contrasted topology in the low-energy sector and identical topology in the high-energy sector. This offers a unified perspective for understanding topological properties in many flat-band systems described by such square-root models. Both types of SUSY systems provide unique opportunities for exploring correlated and topological phases of matter.

A real-time temperature field prediction method for steel rolling heating furnaces based on graph neural networks

In the billet reheating process during steel rolling, the real-time and accurate prediction of the temperature field is a prerequisite for the dynamic regulation of the heating process, which is crucial for ensuring the quality of billet reheating and reducing the energy consumption of the reheating furnace. The most commonly used finite element thermal field simulation and analysis methods are unable to meet the demand for real-time prediction under dynamic working conditions. Furthermore, traditional machine learning methods struggle to handle irregular data that includes spatial location information, resulting in inaccurate temperature field predictions, which in turn affect the furnace control efficiency and the quality of the product. In light of these limitations, this study proposes a Graph Neural Network (GNN)-based temperature field prediction method for steel rolling reheating furnaces. This method considers the interactions between the nodes within the reheating furnace and constructs a temperature field topology graph to effectively capture these interactions, including heat conduction and convection. Additionally, in the feature fusion and state updating mechanism of the model, we introduce process parameters, such as the air-fuel ratio, as additional inputs to enhance the ability of the GNN to handle complex variables. This enables real-time accurate simulation of the internal temperature distribution of the reheating furnace. The experimental results demonstrate that the model prediction error is controlled within 2.9 % and the response time is maintained within 20 ms, thereby validating the reliability and efficacy of the method in practical applications.

The four-components link invariant in the framework of topological field theories

In this work, we undertake a perturbative analysis of the topological non-Abelian Chern–Simons-Wong model with the aim to explicitly construct the second-order on-shell action. The resulting action is a topological quantity depending solely on closed curves, so it correspond to an analytical expression of a link invariant. Additionally, we construct an Abelian model that reproduces the same second-order on-shell action as its non-Abelian Chern–Simons-Wong counterpart so it functions as an intermediate model, featuring Abelian fields generated by currents supported on closed paths. By geometrically analyzing each term, we demonstrate that this topological invariant effectively detects the knotting of a four-component link.

Intercalating Architecture for the Design of Charge Density Wave in Metallic MA2Z4 Materials.

We present a novel approach to induce charge density waves (CDWs) in metallic MA2Z4 materials, resembling the behavior observed in transition metal dichalcogenides (TMDCs). This method leverages the intercalating architecture to maintain the same crystal field and Fermi surface topologies. Our investigation reveals that CDW instability in these materials arises from electron-phonon coupling (EPC) between the d band and longitudinal acoustic (LA) phonons, mirroring TMDC's behavior. By combining α-MA2Z4 with 1H-MX2 materials in a predictive CDW phase diagram using critical EPC constants, we demonstrate the feasibility of extending CDW across material families with comparable crystal fields and reveal the crucial role in CDW instability of the competition between ionic charge transfer and electron correlation. We further uncover a strain-induced Mott transition in β2-NbGe2N4 monolayer featuring star-of-David patterns. This work highlights the potential of intercalating architecture to engineer CDW materials, expanding our understanding of CDW instability and correlation physics.

High-resolution observations of recurrent jets from an arch filament system

Solar jets are collimated plasma ejections along magnetic field lines observed in hot (extreme-ultraviolet (EUV) jets) and cool (chromospheric surges) temperature diagnostics. Their trigger mechanisms and the relationship between hot and cool jets are still not completely understood. We aim to investigate the generation of a sequence of active-region solar jets and their evolution from the photospheric to the coronal heights using multithermal observations from ground-based and space-borne instruments. Using the synergy of high-spatial-resolution and high-temporal-resolution observations by the Swedish 1-m Solar Telescope (SST), along with the Solar Dynamics Observatory (SDO), we analyzed a sequence of solar jets originating in a mixed-polarity region between the leading and following sunspots of an active region. We investigated the kinematics of these jets using the spectra from the SST observations. We used a non-force-free field (NFFF) extrapolation technique to derive the magnetic field topology of the active region. A mixed-polarity region is formed over a long period (24 hours) with persistent magnetic flux emergence. This region has been observed as an arch filament system (AFS) in chromospheric SST observations. In this region, negative polarities surrounded by positive polarities create a fan surface with a null point at a height of 6 Mm detected in the NFFF extrapolation. SST observations in the Hbeta spectral line reveal a large flux rope over the AFS moving from north to south, causing successive EUV and cool jets to move in the east--west direction and later towards the south along the long open loops. The high-resolution SST observations (0 per pixel) resolve the dark area observed at the jet base and reveal the existence of an AFS with an extended cool jet, which may be the result of a peeling-like mechanism of the AFS. Based on the combined analysis of SST and AIA observations along with extrapolated magnetic topology, it is suggested that the magnetic reconnection site may move southward by approximately 20 Mm until it reaches a region where the open magnetic field lines are oriented north--south.

Noninvertible symmetries in 2D from type IIB string theory

We propose a top-down approach to noninvertible symmetries in two-dimensional quantum field theories and their three-dimensional (3D) associated symmetry topological field theories. We focus on the gauge theory engineered on D1-branes probing a particular Calabi-Yau 4-fold singularity. We show how to derive the symmetry topological field theory, a 3D Dijkgraaf-Witten theory, from the IIB supergravity under dimensional reduction. We also identify branes behind the noninvertible topological lines by dimensionally reducing their world volume actions. The action of noninvertible lines on charged local operators is then realized as the Hanany-Witten transition. Published by the American Physical Society 2024

Photospheric Pore Rotation Associated with a C-class Flare from Spectropolarimetric Observations with DKIST

We present high-resolution observations of a C4.1-class solar flare (SOL2023-05-03T20:53) in AR 13293 from the Visible Spectro-Polarimeter (ViSP) and Visible Broadband Imager (VBI) instruments at the DKIST. The fast cadence, good resolution, and high polarimetric sensitivity of ViSP data provide a unique opportunity to explore the photospheric magnetic fields before and during the flare. We infer the magnetic field vector in the photosphere from the Fe i 6302 Å line using Milne–Eddington inversions. Combined analysis of the inverted data and VBI images reveals the presence of two opposite polarity pores exhibiting rotational motion both prior to and throughout the flare event. Data-driven simulations further reveal a complex magnetic field topology above the rotating pores, including a null-point-like configuration. We observed a 30% relative change in the horizontal component (δ F h ) of Lorentz force at the flare peak time and roughly no change in the radial component. We find that the changes in δ F h are the most likely driver of the observed pore rotation. These findings collectively suggest that the back reaction of magnetic field line reconfiguration in the corona may influence the magnetic morphology and rotation of pores in the photosphere on a significantly smaller scale.