Energy-momentum Space Research Articles

Diffeomorphisms of the energy-momentum space: Perturbative QED

A perturbative formulation of quantum electrodynamics is given in terms of geometrical invariants of the energy-momentum space, whose geometry is taken to be one of a constant curvature. The construction is relevant for different classes of noncommutativity: the Snyder model and the so called GUP models. For the Snyder model it is shown that all the amplitudes are finite at every order of the perturbation expansion.

Constructing Berry-Maxwell equations with Lorentz invariance and Gauss's law of Weyl monopoles in four-dimensional energy-momentum space

Constructing Berry-Maxwell equations with Lorentz invariance and Gauss's law of Weyl monopoles in four-dimensional energy-momentum space

Repelling Planet Pairs by Ping-pong Scattering

The Kepler mission reveals a peculiar trough-peak feature in the orbital spacing of close-in planets near mean-motion resonances: a deficit and an excess that are, respectively, a couple of percent interior to and wide of the resonances. This feature has received two main classes of explanations: one involving eccentricity damping and the other scattering with small bodies. Here, we point out a few issues with the damping scenario and study the scattering scenario in more detail. We elucidate why scattering small bodies tends to repel two planets. As the small bodies random-walk in energy and angular momentum space, they tend to absorb fractionally more energy than angular momentum. This, which we call “ping-pong repulsion,” transports angular momentum from the inner to the outer planet and pushes the two planets apart. Such a process, even if ubiquitous, leaves identifiable marks only near first-order resonances: diverging pairs jump across the resonance quickly and produce the mean-motion resonance asymmetry. To explain the observed positions of the trough-peaks, a total scattering mass of order a few percent of the planet masses is required. Moreover, if this mass is dominated by a handful of Mercury-sized bodies, one can also explain the planet eccentricities as inferred from transit-time variations. Last, we suggest how these conditions may have naturally arisen during the late stages of planet formation.

Time- and angle-resolved photoemission spectroscopy with wavelength-tunable pump and extreme ultraviolet probe enabled by twin synchronized amplifiers.

We describe a setup for time- and angle-resolved photoemission spectroscopy with wavelength-tunable excitation and an extreme ultraviolet probe. It is enabled by using the 10kHz twin Ti:sapphire amplifiers seeded by the common Ti:sapphire oscillator. The typical probe energy is 21.7eV, and the wavelength of the pump excitation is tuned between 2400 and 1200nm by using the optical parametric amplifier. The spectral width of the extreme ultraviolet probe is 53meV, and the time resolution is dependent on the wavelength for the pump, better than 60fs for the pump energy >0.7eV. This system enables the pump energy to be matched with a specific interband transition and to probe a wider energy-momentum space. We present the results for the prototypical materials of highly oriented pyrolytic graphite and Bi2Se3 to show the performance of our system.

Revealing the spatial nature of sublattice symmetry

The sublattice symmetry on a bipartite lattice is commonly regarded as the chiral symmetry in the AIII class of the tenfold Altland–Zirnbauer classification. Here, we reveal the spatial nature of sublattice symmetry and show that this assertion holds only if the periodicity of primitive unit cells agrees with that of the sublattice labeling. In cases where the periodicity does not agree, sublattice symmetry is represented as a glide reflection in energy–momentum space, which inverts energy and simultaneously translates some k by π, leading to substantially different physics. Particularly, it introduces novel constraints on zero modes in semimetals and completely alters the classification table of topological insulators compared to class AIII. Notably, the dimensions corresponding to trivial and nontrivial classifications are switched, and the nontrivial classification becomes Z2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathbb{Z}}}_{2}$$\\end{document} instead of Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb{Z}}$$\\end{document}. We have applied these results to several models, including the Hofstadter model both with and without dimerization.

Autonomous micro-focus angle-resolved photoemission spectroscopy.

Angle-resolved photoemission spectroscopy (ARPES) is a technique used to map the occupied electronic structure of solids. Recent progress in x-ray focusing optics has led to the development of ARPES into a microscopic tool, permitting the electronic structure to be spatially mapped across the surface of a sample. This comes at the expense of a time-consuming scanning process to cover not only a three-dimensional energy-momentum (E, kx, ky) space but also the two-dimensional surface area. Here, we implement a protocol to autonomously search both k- and real-space in order to find positions of particular interest, either because of their high photoemission intensity or because of sharp spectral features. The search is based on the use of Gaussian process regression and can easily be expanded to include additional parameters or optimization criteria. This autonomous experimental control is implemented on the SGM4 micro-focus beamline of the synchrotron radiation source ASTRID2.

Surface spectroscopy and surface–bulk hybridization of Weyl semimetals

Weyl semimetal showing open-arc surface states is a prominent example of topological quantum matter in three dimensions. With the bulk-boundary correspondence present, nontrivial surface-bulk hybridization is inevitable but less understood. Spectroscopies have been often limited to verifying the existence of surface Fermi arcs, whereas its spectral shape related to the hybridization profile in energy-momentum space is not well studied. We present an exactly solvable formalism at the surface for a wide range of prototypical Weyl semimetals. The resonant surface state and the bulk influence coexist as a surface-bulk hybrid and are treated in a unified manner. Directly accessible to angle-resolved photoemission spectroscopy, we analytically reveal universal information about the system obtained from the spectroscopy of resonant topological states. We systematically find inhomogeneous and anisotropic singular responses around the surface-bulk merging borderline crossing Weyl points, highlighting its critical role in the Weyl topology. The response in scanning tunneling spectroscopy is also discussed. The results will provide much-needed insight into the surface-bulk-coupled physical properties and guide in-depth spectroscopic investigation of the nontrivial hybrid in many topological semimetal materials.

Monte Carlo Stellar Dynamics near Massive Black Holes: Two-dimensional Fokker–Planck Solutions of Multiple Mass Components

In this study we present a novel Monte Carlo code, referred to as GNC, which enables the investigation of dynamical relaxation in clusters comprising multiple mass components in the vicinity of supermassive black holes at the centers of galaxies. Our method is based on two-dimensional Fokker–Planck equations in the energy and angular momentum space, and allows the evolution of multiple mass components, including stars and compact objects. The code demonstrates remarkable flexibility in incorporating additional complex dynamics. By employing a weighting method, we effectively enhance the statistical accuracy of rare particle results. In this initial publication, we present the fundamental version of our method, focusing on two-body relaxations and loss cone effects. Through comparisons with previous studies, we establish consistent outcomes in terms of relaxation processes, energy and angular momentum distributions, density profiles, and loss cone consumption rates. We consistently observe the development of tangential anisotropy within the cluster, while the outer regions tend to retain near-isotropic characteristics. GNC holds great promise for exploring a wide range of intriguing phenomena within galactic nuclei, including relativistic stellar dynamics, providing detailed and insightful outcomes.

Field control of quasiparticle decay in a quantum antiferromagnet

Dynamics in a quantum material is described by quantized collective motion: a quasiparticle. The single-quasiparticle description is useful for a basic understanding of the system, whereas a phenomenon beyond the simple description such as quasiparticle decay which affects the current carried by the quasiparticle is an intriguing topic. The instability of the quasiparticle is phenomenologically determined by the magnitude of the repulsive interaction between a single quasiparticle and the two-quasiparticle continuum. Although the phenomenon has been studied in several materials, thermodynamic tuning of the quasiparticle decay in a single material has not yet been investigated. Here we show, by using neutron scattering, magnetic field control of the magnon decay in a quantum antiferromagnet RbFeCl3, where the interaction between the magnon and continuum is tuned by the field. At low fields where the interaction is small, the single magnon decay process is observed. In contrast, at high fields where the interaction exceeds a critical magnitude, the magnon is pushed downwards in energy and its lifetime increases. Our study demonstrates that field control of quasiparticle decay is possible in the system where the two-quasiparticle continuum covers wide momentum-energy space, and the phenomenon of the magnon avoiding decay is ubiquitous.

Time, momentum, and energy resolved pump-probe tunneling spectroscopy of two-dimensional electron systems

Real-time probing of electrons can uncover intricate relaxation mechanisms and many-body interactions in strongly correlated materials. Here, we introduce time, momentum, and energy resolved pump-probe tunneling spectroscopy (Tr-MERTS). The method allows the injection of electrons at a particular energy and observation of their subsequent decay in energy-momentum space. Using Tr-MERTS, we visualize electronic decay processes, with lifetimes from tens of nanoseconds to tens of microseconds, in Landau levels formed in a GaAs quantum well. Although most observed features agree with simple energy-relaxation, we discovered a splitting in the nonequilibrium energy spectrum in the vicinity of a ferromagnetic state. An exact diagonalization study suggests that the splitting arises from a maximally spin-polarized state with higher energy than a conventional equilibrium skyrmion. Furthermore, we observe time-dependent relaxation of the splitting, which we attribute to single-flipped spins forming skyrmions. These results establish Tr-MERTS as a powerful tool for studying the properties of a 2DES beyond equilibrium.

Angle-resolved photoelectron spectroscopy in a low-energy electron microscope.

Spectroscopic photoemission microscopy is a well-established method to investigate the electronic structure of surfaces. In modern photoemission microscopes, the electron optics allow imaging of the image plane, momentum plane, or dispersive plane, depending on the lens setting. Furthermore, apertures allow filtering of energy-, real-, and momentum space. Here, we describe how a standard spectroscopic and low-energy electron microscope can be equipped with an additional slit at the entrance of the already present hemispherical analyzer to enable an angle- and energy-resolved photoemission mode with micrometer spatial selectivity. We apply a photogrammetric calibration to correct for image distortions of the projective system behind the analyzer and present spectra recorded on Au(111) as a benchmark. Our approach makes data acquisition in energy-momentum space more efficient, which is a necessity for laser-based pump-probe photoemission microscopy with femtosecond time resolution.

Direct Observation of Topological Phonons in Graphene.

Phonons, as the most fundamental emergent bosons in condensed matter systems, play an essential role in the thermal, mechanical, and electronic properties of crystalline materials. Recently, the concept of topology has been introduced to phonon systems, and the nontrivial topological states also exist in phonons due to the constraint by the crystal symmetry of the space group. Although the classification of various topological phonons has been enriched theoretically, experimental studies were limited to several three-dimensional (3D) single crystals with inelastic x-ray or neutron scatterings. The experimental evidence of topological phonons in two-dimensional (2D) materials is absent. Here, using high-resolution electron energy loss spectroscopy following our theoretical predictions, we directly map out the phonon spectra of the atomically thin graphene in the entire 2D Brillouin zone, and observe two nodal-ring phonons and four Dirac phonons. The closed loops of nodal-ring phonons and the conical structure of Dirac phonons in 2D momentum space are clearly revealed by our measurements, in nice agreement with our theoretical calculations. The ability of 3D mapping (2D momentum space and energy space) of phonon spectra opens up a new avenue to the systematic identification of the topological phononic states. Our work lays a solid foundation for potential applications of topological phonons in superconductivity, dynamic instability, and phonon diode.

High‐Contrast Switching of Light Enabled by Zero Diffraction

Diffraction allows to change the direction of light. Therefore, controlling the diffraction efficiency with high contrast enables controlling the pathway of light within optical systems. However, a high contrast requires that the diffraction efficiency is tunable close to zero. Probably the most prominent example for zero diffraction in a waveguide grating is a bound state in the continuum (BIC). Herein, zero diffraction of two plane waves under symmetric incidence to a leaky symmetric waveguide grating is found. The phenomenon not only occurs at singular spectral positions but on continuous curves in the energy–momentum space. The relative phase of the two waves enables large contrast control over diffraction in a wide spectral range. The practical meaning of this finding for local switching is demonstrated. Light is trapped into a nonlinear optical waveguide and detrapped at a desired position with electric control. A switching contrast exceeding 1000 is experimentally shown.

Giant and Tunable Out-of-Plane Spin Polarization of Topological Antimonene.

Topological insulators are bulk insulators with metallic and fully spin-polarized surface states displaying Dirac-like band dispersion. Due to spin-momentum locking, these topological surface states (TSSs) have a predominant in-plane spin polarization in the bulk fundamental gap. Here, we show by spin-resolved photoemission spectroscopy that the TSS of a topological insulator interfaced with an antimonene bilayer exhibits nearly full out-of-plane spin polarization within the substrate gap. We connect this phenomenon to a symmetry-protected band crossing of the spin-polarized surface states. The nearly full out-of-plane spin polarization of the TSS occurs along a continuous path in the energy-momentum space, and the spin polarization within the gap can be reversibly tuned from nearly full out-of-plane to nearly full in-plane by electron doping. These findings pave the way to advanced spintronics applications that exploit the giant out-of-plane spin polarization of TSSs.

Time dispersion in quantum electrodynamics

If we use the path integral approach, we can write quantum electrodynamics (QED) in a way that is manifestly relativistic. However the path integrals are confined to paths that are on mass-shell. What happens if we extend QED by computing the path integrals over all paths in energy momentum space, not only those on mass-shell? We use the requirement of covariance to do this in an unambiguous way. This gives a QED where the time/energy components appear in a way that is manifestly parallel to the space/momentum components: we have dispersion in time, entanglement in time, full equivalence of the Heisenberg uncertainty principle (HUP) in time to the HUP in space, and so on. Entanglement in time has the welcome side effect of eliminating the ultraviolet divergences. We recover standard QED in the long time limit. We predict effects at scales of attoseconds. With recent developments in attosecond physics and in quantum computing, these effects should be detectable. Since the predictions are unambiguous and testable the approach is falsifiable. Falsification would sharpen our understanding of the role of time in QED. Confirmation would have significant implications for attosecond physics, quantum computing and communications, and quantum gravity.

Effect of magnetic order on longitudinal Tomonaga-Luttinger liquid spin dynamics in weakly coupled spin- 12 chains

The quantum many-body interactions in one-dimensional spin-$\frac{1}{2}$ systems are subject to Tomonaga-Luttinger liquid (TLL) physics, which predict an array of multiparticle excitations that form continua in momentum-energy space. Here we use inelastic neutron spectroscopy to study the TLL spin dynamics in ${\mathrm{SrCo}}_{2}{\mathrm{V}}_{2}{\mathrm{O}}_{8}$, a compound which contains weakly coupled spin-$\frac{1}{2}$ chains of Co atoms, at 0.05 K and in a longitudinal magnetic field up to 9.0 T. The measurements were performed above 3.9 T, where the ground-state N\'eel antiferromagnetic (AFM) order is completely suppressed and the multiparticle excitations are exclusively of the TLL type. In this region and below 7.0 T, the longitudinal TLL mode---psinon/antipsinon (P/AP)---is unexpectedly well described by a damped harmonic oscillator (DHO) while approaching the zone center defining the static spin-spin correlations. A non-DHO-type, continuumlike signal is seen at higher fields, but deviations from the ideal one-dimensional TLL still remain. This change in the P/AP mode coincides with the phase transition between the longitudinal spin density wave (LSDW) and transverse AFM order. Inside the LSDW state, the DHO-type P/AP spectral weight increases and the linewidth broadens as the magnetic order parameter decreases. These results reveal the impact of three-dimensional magnetic order on the TLL spin dynamics; they call for beyond-mean-field treatment for the interchain exchange interactions.

Causal structure of interacting Weyl fermions in condensed matter systems

The spacetime light cone is central to the definition of causality in the theory of relativity. Recently, links between relativistic and condensed matter physics have been uncovered, where relativistic particles can emerge as quasiparticles in the energy-momentum space of matter. Here, we unveil an energy-momentum analogue of the spacetime light cone by mapping time to energy, space to momentum, and the light cone to the Weyl cone. We show that two Weyl quasiparticles can only interact to open a global energy gap if they lie in each other’s energy-momentum dispersion cones–analogous to two events that can only have a causal connection if they lie in each other’s light cones. Moreover, we demonstrate that the causality of surface chiral modes in quantum matter is entangled with the causality of bulk Weyl fermions. Furthermore, we identify a unique quantum horizon region and an associated ‘thick horizon’ in the emergent causal structure.

Fate of the upper polariton: Breakdown of the quasiparticle picture in the continuum

Organic polaritons resulting from the strong hybridisation between photons and matter excitations have arisen as a suitable platform to device light-matter technological interfaces at room temperature. Despite their inherent complexity, organic polaritons are commonly regarded as coherent light-matter excitations that can be described in terms of Landau's quasiparticle approach. Here, we experimentally unveil the role of incoherent matter excitations on the polaritons by exploring the relevant energy-momentum parameter space. We demonstrate the emergence of a well-defined lower polariton branch and an intriguing fading of the upper polariton at its entrance to a continuum of matter excitations. This marks the breakdown of the simplistic quasiparticle picture for this branch and the formation of a more complex quantum state. Our experimental results are sustained by a general theoretical framework that allows understanding the light-matter hybridisation beyond the quasiparticle approach. Our work expands the understanding of organic polaritons in all their complexity and increases their technological significance.

Topological Electromagnetic Effects and Higher Second Chern Numbers in Four-Dimensional Gapped Phases

Higher-dimensional topological phases play a key role in understanding the lower-dimensional topological phases and the related topological responses through a dimensional reduction procedure. In this work, we present a Dirac-type model of four-dimensional Z_{2} topological insulator (TI) protected by CP symmetry, whose 3D boundary supports an odd number of Dirac cones. A specific perturbation splits each bulk massive Dirac cone into two valleys separated in energy-momentum space with opposite second Chern numbers, in which the 3D boundary modes become a nodal sphere or a Weyl semimetallic phase. By introducing the electromagnetic (EM) and pseudo-EM fields, exotic topological responses of our 4D system are revealed, which are found to be described by the (4+1)D mixed Chern-Simons theories in the low-energy regime. Notably, several topological phase transitions occur from a CP-broken Z_{2} TI to a Z TI when the bulk gap closes by giving rise to exotic double-nodal-line or nodal-hyper-torus gapless phases. Finally, we propose to probe experimentally these topological effects in cold atoms.

K-Space Hyperspectral Imaging by a Birefringent Common-Path Interferometer.

Fourier-plane microscopy is a powerful tool for measuring the angular optical response of a plethora of materials and photonic devices. Among them, optical microcavities feature distinctive energy-momentum dispersions, crucial for a broad range of fundamental studies and applications. However, measuring the whole momentum space (k-space) with sufficient spectral resolution using standard spectroscopic techniques is challenging, requiring long and alignment-sensitive scans. Here, we introduce a k-space hyperspectral microscope, which uses a common-path birefringent interferometer to image photoluminescent organic microcavities, obtaining an angle- and wavelength-resolved view of the samples in only one measurement. The exceptional combination of angular and spectral resolution of our technique allows us to reconstruct a three-dimensional (3D) map of the cavity dispersion in the energy-momentum space, revealing the polarization-dependent behavior of the resonant cavity modes. Furthermore, we apply our technique for the characterization of a dielectric nanodisk metasurface, evidencing the angular and spectral behavior of its anapole mode. This approach is able to provide a complete optical characterization for materials and devices with nontrivial angle-/wavelength-dependent properties, fundamental for future developments in the fields of topological photonics and optical metamaterials.