Three-dimensional Momentum Space Research Articles

Differentiating Three-Dimensional Molecular Structures Using Laser-Induced Coulomb Explosion Imaging.

Coulomb explosion imaging (CEI) with x-ray free electron lasers has recently been shown to be a powerful method for obtaining detailed structural information of gas-phase planar ring molecules [R. Boll et al., X-ray multiphoton-induced Coulomb explosion images complex single molecules, Nat. Phys. 18, 423 (2022).NPAHAX1745-247310.1038/s41567-022-01507-0]. In this Letter, we investigate the potential of CEI driven by a tabletop laser and extend this approach to differentiating three-dimensional structures. We study the static CEI patterns of planar and nonplanar organic molecules that resemble the structures of typical products formed in ring-opening reactions. Our results reveal that each molecule exhibits a well-localized and distinctive pattern in three-dimensional fragment-ion momentum space. We find that these patterns yield direct information about the molecular structures and can be qualitatively reproduced using a classical Coulomb explosion simulation. Our findings suggest that laser-induced CEI can serve as a robust method for differentiating molecular structures of organic ring and chain molecules. As such, it holds great promise as a method for following ultrafast structural changes, e.g., during ring-opening reactions, by tracking the motion of individual atoms in pump-probe experiments.

Observation of Higher-Order Nodal-Line Semimetal in Phononic Crystals.

Higher-order topological insulators and semimetals, which generalize the conventional bulk-boundary correspondence, have attracted extensive research interest. Among them, higher-order Weyl semimetals feature twofold linear crossing points in three-dimensional momentum space, 2D Fermi-arc surface states, and 1D hinge states. Higher-order nodal-point semimetals possessing Weyl points or Dirac points have been implemented. However, higher-order nodal-line or nodal-surface semimetals remain to be further explored in experiments in spite of many previous theoretical efforts. In this work, we realize a second-order nodal-line semimetal in 3D phononic crystals. The bulk nodal lines, 2D drumhead surface states guaranteed by Zak phases, and 1D flat hinge states attributed to k_{z}-dependent quadrupole moments are observed in simulations and experiments. Our findings of nondispersive surface and hinge states may promote applications in acoustic sensing and energy harvesting.

Realization and topological properties of third-order exceptional lines embedded in exceptional surfaces

As the counterpart of Hermitian nodal structures, the geometry formed by exceptional points (EPs), such as exceptional lines (ELs), entails intriguing spectral topology. We report the experimental realization of order-3 exceptional lines (EL3) that are entirely embedded in order-2 exceptional surfaces (ES2) in a three-dimensional periodic synthetic momentum space. The EL3 and the concomitant ES2, together with the topology of the underlying space, prohibit the evaluation of their topology in the eigenvalue manifold by prevailing topological characterization methods. We use a winding number associated with the resultants of the Hamiltonian. This resultant winding number can be chosen to detect only the EL3 but ignores the ES2, allowing the diagnosis of the topological currents carried by the EL3, which enables the prediction of their evolution under perturbations. We further reveal the connection between the intersection multiplicity of the resultants and the winding of the resultant field around the EPs and generalize the approach for detecting and topologically characterizing higher-order EPs. Our work exemplifies the unprecedented topology of higher-order exceptional geometries and may inspire new non-Hermitian topological applications.

A conservative Galerkin solver for the quasilinear diffusion model in magnetized plasmas

The quasilinear theory describes the resonant interaction between particles and waves with two coupled equations: one for the evolution of the particle probability density function (pdf), the other for the wave spectral energy density (sed). In this paper, we propose a conservative Galerkin scheme for the quasilinear model in three-dimensional momentum space and three-dimensional spectral space, with cylindrical symmetry.We construct an unconditionally conservative weak form, and propose a discretization that preserves the unconditional conservation property, by “unconditional” we mean that conservation is independent of the singular transition probability. The discrete operators, combined with a consistent quadrature rule, will preserve all the conservation laws rigorously. The technique we propose is quite general: it works for both relativistic and non-relativistic systems, for both magnetized and unmagnetized plasmas, and even for problems with time-dependent dispersion relations.We represent the particle pdf by continuous basis functions, and use discontinuous basis functions for the wave sed, thus enabling the application of a positivity-preserving technique. The marching simplex algorithm, which was initially designed for computer graphics, is adopted for numerical integration on the resonance manifold. We introduce a semi-implicit time discretization, and discuss the stability condition. In addition, we present numerical examples with a “bump on tail” initial configuration, showing that the particle-wave interaction results in a strong anisotropic diffusion effect on the particle pdf.

The “most classical” states of Euclidean invariant elementary quantum mechanical systems

Complex techniques of general relativity are used to determine all the states in two- and three-dimensional momentum spaces in which the equality holds in uncertainty relations for non-commuting basic observables of Euclidean invariant elementary quantum mechanical systems, even with non-zero intrinsic spin. It is shown that while there is a 1-parameter family of such states for any two components of the angular momentum vector operator with any angle between them, such states exist for a component of the linear and angular momenta only if these components are orthogonal to each other, and hence, the problem is reduced to the two-dimensional Euclidean invariant case. We also show that analogous states exist for a component of the linear momentum and of the center-of-mass vector only if the angle between them is zero or an acute angle. No such state (represented by a square integrable and differentiable wave function) can exist for any pair of components of the center-of-mass vector operator. Therefore, the existence of such states depends not only on the Lie algebra but on the choice of its generators as well.

Higher-order nodal ring photonic semimetal.

The intriguing discovery of higher-order topology has tremendously promoted the development of topological physics. Three-dimensional topological semimetals have emerged as an ideal platform for investigating novel topological phases. Consequently, new proposals have been theoretically revealed and experimentally realized. However, most existing schemes are implemented on the acoustic system, while similar concepts are rarely launched in photonic crystals due to the complicated optical manipulation and geometrical design. In this Letter, we propose a higher-order nodal ring semimetal protected by C2 symmetry originating from C6 symmetry. The higher-order nodal ring is predicted in three-dimensional momentum space with desired hinge arcs connected by two nodal rings. Fermi arcs and topological hinge modes generate significant marks in higher-order topological semimetals. Our work successfully proves the presence of a novel higher-order topological phase in photonic systems that we will strive to apply practically in high-performance photonic devices.

Straight Photonic Nodal Lines with Quadrupole Berry Curvature Distribution and Superimaging "Fermi Arcs".

In periodic systems, nodal lines are loops in the three-dimensional momentum space with each point on them representing a band degeneracy. Nodal lines exhibit rich topological features, as they can take various configurations such as rings, links, chains, and knots. These line nodes are generally protected by mirror or PT symmetry and frequently accompanied by drumhead surface states. Here, we propose and demonstrate a novel type of photonic straight nodal lines in a D_{2D} metacrystal, which are protected by an unusual rotoinversion time (roto-PT) symmetry. These nodal lines are located at the central axis and hinges of the Brillouin zone. They appear as quadrupole sources of Berry curvature flux in contrast to the Weyl points, which are monopoles. Interestingly, topological surface states exist at all three cutting surfaces, as guaranteed by π-quantized Zak phases along all three directions. As frequency changes, the surface state equifrequency contours evolve from closed to open and become straight lines at a critical transition frequency, at which diffractionless surface wave propagations are experimentally demonstrated, paving the way toward development of superimaging topological devices.

Tracking Valley Topology with Synthetic Weyl Paths.

Inspired by the newly emergent valleytronics, great interest has been attracted to the topological valley transport in classical metacrystals. The presence of nontrivial domain-wall states is interpreted with a concept of valley Chern number, which is well defined only in the limit of small band gap. Here, we propose a new visual angle to track the intricate valley topology in classical systems. Benefiting from the controllability of our acoustic metacrystals, we construct Weyl points in synthetic three-dimensional momentum space through introducing an extra structural parameter (rotation angle here). As such, the two-dimensional valley-projected band topology can be tracked with the strictly quantized topological charge in three-dimensional Weyl crystal, which features open surface arcs connecting the synthetic Weyl points and gapless chiral surface states along specific Weyl paths. All theoretical predictions are conclusively identified by our acoustic experiments. Our findings may promote the development of topological valley physics, which is less well defined yet under hot debate in multiple physical disciplines.

Learning the crystal structure genome for property classification

Materials property predictions have improved from advances in machine learning algorithms, delivering materials discoveries and novel insights through data-driven models of structure-property relationships. Nearly all available models rely on featurization of materials composition, however, whether the exclusive use of structural knowledge in such models has the capacity to make comparable predictions remains unknown. Here we employ a deep neural network model to decode structure-property relationships in crystalline materials without explicitly considering chemical compositions. The focus is on classification of crystal systems, mechanical elasticity, electronic band gap, and phase stability. Our model utilizes a three-dimensional (3D) momentum space representation of structure from elastic x-ray scattering theory that exhibits rotation and permutation invariance. We perform novel ablation studies to help interpret the model performance by perturbing the physically meaningful input features (i.e., the diffraction patterns) instead of tuning the architecture of the learning model as in conventional ablation methods. We find that the spatial symmetry of the 3D diffraction patterns, which reflects crystalline symmetry operations, is more important than the diffraction intensities contained within for the model to make a successful classification. Our work showcases the potential of using statistical learning models to help understand materials physics, rather than performing predictive and generative tasks as in most materials informatics research. We also argue that learning the crystal structure genome in a chemistry-agnostic manner demonstrates that some crystal structures inherently host high propensities for optimal materials properties, which enables the decoupling of structure and composition for future codesign of multifunctionality.

The 3-dimensional distribution of quarks in momentum space

We present the distribution of unpolarized quarks in a transversely polarized proton in three-dimensional momentum space. Our results are based on the extractions of the unpolarized and Sivers transverse momentum dependent parton distributions (TMDs) in a fully consistent TMD framework.

Structured photoelectron distributions in photodetachment induced by trains of laser pulses: Vortices versus spirals

The formation of quantum vortices in photodetachment by a sequence of left- and right-handed circularly polarized laser pulses in various configurations is analyzed using either the strong-field approximation or numerical solution of the time-dependent Schr\"odinger equation. Two types of pulse sequences are considered: reducible and irreducible. While the former can be decomposed into series of consecutive and identical (irreducible) subtrains of pulses, the latter cannot be decomposed in that way. As we show, the vortex pattern in the probability amplitude of photodetachment is fully determined by the irreducible pulse configuration. Additional repetitions of an irreducible train create, in the three-dimensional momentum space, nonvortex nodal surfaces, the position of which is estimated. The conditions for the experimental observation of quantum vortices are also determined.

Angle-resolved photoemission spectroscopy study of the electronic structure evolution in Sn4X3 (X = P, As, Sb)

The binary compounds Sn4X3 (X = P, As, Sb) have recently surfaced as a unique platform in which to study the interplay between nontrivial band topology and superconductivity with both strengthening as X varies from P to Sb. Here we perform the first systematic comparison of the electronic structure between the three member compounds. Results of our angle-resolved photoemission spectroscopy experiments on both Sn4Sb3 and Sn4P3 are compared with first-principles bulk and surface electronic structure calculations as well as with those recently reported for Sn4As3. Based on a thorough characterization of the electronic structure in three-dimensional momentum space, three salient aspects have been identified, including a monotonic chemical potential shift in a non-rigid-band fashion, a monotonic variation in the magnitude of a band splitting of surface origin, and the emergence of additional surface-related bands. We show that these observations can be reasonably captured by the evolution of the initial-state band structure given by first-principles calculations, and argue that the covalent bonding between the surface Sn monolayer and the underlying X monolayer, rather than the spin–orbit coupling, likely plays an important role in accounting for the observed differences in the electronic structure across the Sn4X3 family. Our finding depicts a comprehensive picture for the electronic structure evolution in the Sn4X3 family, and paves the way to deciphering the relationship between the nontrivial band topology and superconductivity therein.

Intrinsic in-plane nodal chain and generalized quaternion charge protected nodal link in photonics

Nodal lines are degeneracies formed by crossing bands in three-dimensional momentum space. Interestingly, these degenerate lines can chain together via touching points and manifest as nodal chains. These nodal chains are usually embedded in two orthogonal planes and protected by the corresponding mirror symmetries. Here, we propose and demonstrate an in-plane nodal chain in photonics, where all chained nodal lines coexist in a single mirror plane instead of two orthogonal ones. The chain point is stabilized by the intrinsic symmetry that is specific to electromagnetic waves at the Г point of zero frequency. By adding another mirror plane, we find a nodal ring that is constructed by two higher bands and links with the in-plane nodal chain. The nodal link in momentum space exhibits non-Abelian characteristics on a C2T - invariant plane, where admissible transitions of the nodal link structure are determined by generalized quaternion charges. Through near-field scanning measurements of bi-anisotropic metamaterials, we experimentally mapped out the in-plane nodal chain and nodal link in such systems.

Photonic topological Weyl degeneracies and ideal type-I Weyl points in the gyromagnetic metamaterials

As a new topological phase in three-dimensional momentum space, Weyl semimetals extend the repertoire of exotic topological states. In this paper, we find the existence of photonic Weyl points in electromagnetic continuous gyromagnetic metamaterials. The dispersion of the longitudinal mode of one of the two bands forming the Weyl point is flat; thus, the Weyl degeneracy point of the system is exactly at the critical transition between the type-I and type-II Weyl points. The Fermi arc connects the projections of the Weyl points with opposite chirality at the interface between the gyromagnetic metamaterial and vacuum. Full-wave simulation results show that surface waves can bypass sharp defects and achieve robust, unidirectional optical transmission without reflection. Furthermore, taking the spatial nonlocal effects into account, the ideal type-I Weyl points can exist in this class of metamaterials. This paper sheds light on fundamental understanding of topological physics, and the study of Weyl points in metamaterials will greatly enrich the field of topological photonics.

Ideal type-II Weyl points in topological circuits.

Weyl points (WPs), nodal degenerate points in three-dimensional (3D) momentum space, are said to be ‘ideal’ if they are symmetry-related and well-separated, and reside at the same energy and far from nontopological bands. Although type-II WPs have unique spectral characteristics compared with type-I counterparts, ideal type-II WPs have not yet been reported because of a lack of an experimental platform with enough flexibility to produce strongly tilted dispersion bands. Here, we experimentally realize a topological circuit that hosts only topological bands with a minimal number of four ideal type-II WPs. By stacking two-dimensional (2D) layers of inductor-capacitor (LC) resonator dimers with the broken parity inversion symmetry (P), we achieve a strongly tilted band structure with two group velocities in the same direction, and topological surface states in an incomplete bandgap. Our results establish an ideal system for the further study of Weyl physics and other exotic topological phenomena.

Observation of Non-Abelian Nodal Links in Photonics.

In crystals, two bands may cross each other and form degeneracies along a closed loop in the three-dimensional momentum space, which is called nodal line. Nodal line degeneracy can be designed to exhibit various configurations such as nodal rings, chains, links, and knots. Very recently, non-Abelian band topology was proposed in nodal link systems, where the nodal lines formed by consecutive pairs of bands exhibit interesting braiding structures and the underlying topological charges are described by quaternions. Here, we experimentally demonstrate non-Abelian nodal links in a biaxial hyperbolic metamaterial. The linked nodal lines threading through each other are formed by the crossings between three adjacent bands. Based on the non-Abelian charges, we further analyze various admissible nodal link configurations for the three-band system. On the interface between the metamaterial and air, surface bound states in the continuum are observed, which serves as the symmetry-enforced derivative of drumhead surface states from the linked nodal lines. Our work serves as a direct observation of the global topological structures of nodal links, and provides a platform for studying non-Abelian topological charge in the momentum space.

Predictive power of transverse-momentum-dependent distributions

We investigate the predictive power of transverse-momentum-dependent (TMD) distributions as a function of the light-cone momentum fraction $x$ and the hard scale $Q$ defined by the process. We apply the saddle point approximation to the unpolarized quark and gluon transverse momentum distributions and evaluate the position of the saddle point as a function of the kinematics. We determine quantitatively that the predictive power for an unpolarized transverse momentum distribution is maximal in the large-$Q$ and small-$x$ region. For cross sections the predictive power of the TMD factorization formalism is generally enhanced by considering the convolution of two distributions, and we explicitly consider the case of $Z$ and $H^0$ boson production. In the kinematic regions where the predictive power is not maximal, the distributions are sensitive to the non-perturbative hadron structure. Thus, these regions are critical for investigating hadron tomography in a three-dimensional momentum space.

Photoelectron Momentum Microscope at BL6U of UVSOR-III synchrotron

Photoelectron spectroscopy resolved in three-dimensional momentum space with a microscopic field of view is realized by combining a so-called Momentum Microscope with a soft X-ray synchrotron radiation source. A new experimental station for momentum-resolved photoelectron micro-spectroscopy and spectro-microscopy has been built at BL6U, an undulator-based soft X-ray beamline of the UVSOR synchrotron facility. This experimental station specializes in characterizing the electronic structure of surface atomic sites, thin films, molecular adsorbates, and bulk crystals. The instrument details are described along with possible measurement techniques.

Quench dynamics of Hopf insulators

Hopf insulators are exotic topological states of matter outside the standard ten-fold way classification based on discrete symmetries. Its topology is captured by an integer invariant that describes the linking structures of the Hamiltonian in the three-dimensional momentum space. In this paper, we investigate the quantum dynamics of Hopf insulators across a sudden quench and show that the quench dynamics is characterized by a $\mathbb{Z}_2$ invariant $\nu$ which reveals a rich interplay between quantum quench and static band topology. We construct the $\mathbb{Z}_2$ topological invariant using the loop unitary operator, and prove that $\nu$ relates the pre- and post-quench Hopf invariants through $\nu=(\mathcal{L}-\mathcal{L}_0)\bmod 2$. The $\mathbb{Z}_2$ nature of the dynamical invariant is in sharp contrast to the $\mathbb{Z}$ invariant for the quench dynamics of Chern insulators in two dimensions. The non-trivial dynamical topology is further attributed to the emergence of $\pi$-defects in the phase band of the loop unitary. These $\pi$-defects are generally closed curves in the momentum-time space, for example, as nodal rings carrying Hopf charge.

Experimental demonstration of acoustic semimetal with topologically charged nodal surface.

Weyl points are zero-dimensional band degeneracy in three-dimensional momentum space that has nonzero topological charges. The presence of the topological charges protects the degeneracy points against perturbations and enables a variety of fascinating phenomena. It is so far unclear whether such charged objects can occur in higher dimensions. Here, we introduce the concept of charged nodal surface, a two-dimensional band degeneracy surface in momentum space that is topologically charged. We provide an effective Hamiltonian for this charged nodal surface and show that such a Hamiltonian can be implemented in a tight-binding model. This is followed by an experimental realization in a phononic crystal. The measured topologically protected surface arc state of such an acoustic semimetal reproduces excellently the full-wave simulations. Creating high-dimensional charged geometric objects in momentum space promises a broad range of unexplored topological physics.