The controversy about interference of photons.

In the 1960s, the demonstration of interference effects using two laser-beams raised the question: can two photons interfere? Its plausibility contested Dirac's dictum, "Interference between two different photons never occurs". Disagreements about this conflict led to a controversy. This paper will chart the controversy's contour and show that it evolved over two phases. Subsequently, I investigate the reasons for its perpetuation. The controversy was initiated and fuelled by several misinterpretations of the dictum. I also argue that Dirac's dictum is not applicable to two photon interference as they belong to different contexts of interference. Recognising this resolves the controversy.

Symbolic model checking quantum circuits in Maude.

This article presents a symbolic approach to model checking quantum circuits using a set of laws from quantum mechanics and basic matrix operations with Dirac notation. We use Maude, a high-level specification/programming language based on rewriting logic, to implement our symbolic approach. As case studies, we use the approach to formally specify several quantum communication protocols in the early work of quantum communication and formally verify their correctness: Superdense Coding, Quantum Teleportation, Quantum Secret Sharing, Entanglement Swapping, Quantum Gate Teleportation, Two Mirror-image Teleportation, and Quantum Network Coding. We demonstrate that our approach/implementation can be a first step toward a general framework to formally specify and verify quantum circuits in Maude. The proposed way to formally specify a quantum circuit makes it possible to describe the quantum circuit in Maude such that the formal specification can be regarded as a series of quantum gate/measurement applications. Once a quantum circuit has been formally specified in the proposed way together with an initial state and a desired property expressed in linear temporal logic (LTL), the proposed model checking technique utilizes a built-in Maude LTL model checker to automatically conduct formal verification that the quantum circuit enjoys the property starting from the initial state.

Passively Broadband Tunable Dual Circular Dichroism via Bound States in the Continuum in Topological Chiral Metasurface.

Dynamic control for a strong circular dichroism (CD) response is essential in engineering applications such as polarization manipulation, sensing, and imaging. Here, we propose and experimentally demonstrate a broadband tunable dual CD response via bound states in the continuum (BICs) in two-dimensional topologically protected metasurfaces composed of all-dielectric Si chiral grating structures that generate a pair of mixed and degenerated BIC mode and circular dichroic mode (CDM) as an additional degree of freedom in CD manipulation. It is found that a singular CD peak of nearly 100% at 1.6 µm can be achieved by CDM when BIC is hidden under normal incidence, while the CD peak can be split into two in which peak wavelengths can be precisely and linearly tuned over a bandwidth of 180 nm by the incident angle when the BIC mode is excited under oblique incidence. Additionally, dynamic modulation of output polarization states from linear to circular can be arbitrarily achieved at the split CD peaks by controlling the incident angle when asymmetry perturbations on chiral gratings are introduced due to the decoupling of various polarization states at Γ point by BIC to different positions in K space. The proposed chiral grating metasurface exhibits unique angle-sensitive tunable CD spectral characteristics, making it ideal for hyperspectral and spin-selective wavefront shaping, and holds significant promise in various applications such as optical security, angle sensors, chiral lasers, nonlinear filters, and other active chiral optical devices.

Chiral Honeycomb Lattices of Nonplanar π-Conjugated Supramolecules with Protected Dirac and Flat Bands.

The honeycomb lattice is a fundamental two-dimensional (2D) network that gives rise to surprisingly rich electronic properties. While its expansion to 2D supramolecular assembly is conceptually appealing, its realization is not straightforward because of weak intermolecular coupling and the strong influence of a supporting substrate. Here, we show that the application of a triptycene derivative with phenazine moieties, Trip-Phz, solves this problem due to its strong intermolecular π-π pancake bonding and nonplanar geometry. Our scanning tunneling microscopy (STM) measurements demonstrate that Trip-Phz molecules self-assemble on a Ag(111) surface to form chiral and commensurate honeycomb lattices. Electronically, the network can be viewed as a hybrid of honeycomb and kagome lattices. The Dirac and flat bands predicted by a simple tight-binding model are reproduced by total density functional theory (DFT) calculations, highlighting the protection of the molecular bands from the Ag(111) substrate. The present work offers a rational route for creating chiral 2D supramolecules that can simultaneously accommodate pristine Dirac and flat bands.

Terahertz tunable three-dimensional photonic jets.

Highly localized electromagnetic field distributions near the "shadow-side" surface of certain transparent mesoscale bodies illuminated by light waves are called photonic jets. We demonstrated formation of three-dimensional (3D) tunable photonic jets in terahertz regime (terajets, TJs) by dielectric micro-objects -including spheres, cylinders, and cubes-coated with a bulk Dirac semimetal (BDS) layer, under uniform beam illumination. The optical characteristics of the produced TJs can be modulated dynamically through tuning the BDS layer's index of refraction via changing its Fermi energy. It is demonstrated that the Fermi energy of BDS layer has a significant impact on tuning the optical characteristics of the produced photonic jets for both TE and TM polarizations. A notable polarization dependency of the characteristics of the TJs was also observed. The impact of obliquity of the incident beam was studied as well and it was demonstrated that electromagnetic field distributions corresponding to asymmetric photonic jets can be formed in which the intensity at the focal region is preserved in a wide angular range which could find potential application in scanning devices. It was found that the maximum intensity of the TJ occurs at a non-trivial morphology-dependent source-angle.

Electric field modulated electronic, thermoelectric and transport properties of 2D tetragonal silicene and its nanoribbons.

Using both first principles and analytical approaches, we investigate the role of a transverse electric field in tuning the electrical, thermoelectric, optical and transport properties of a buckled tetragonal silicene (TS) structure. The transverse electric field transforms the linear spectrum to parabolic at the Fermi level and opens a band gap. The gap is similar at the two Dirac points present in the irreducible Brillouin zone of the TS structure and increases in proportion to the applied field strength. However, a sufficiently strong electric field converts the system into a metallic one. A comparable band opening is also seen in the TS nanoribbons. Electric field-induced semiconducting nature improves its thermoelectric properties. Estimated Debye temperature reveals its superiority over graphene in terms of thermoelectric performance. The optical response of the structures is very asymmetric. Large values of imaginary and real components of the dielectric function are seen. The absorption frequency lies in the UV region. Plasma frequencies are identified and are red-shifted with the applied field. The current-voltage characteristics of the symmetric type nanoribbons show oscillation in current whereas the voltage-rectifying capability of anti-symmetric type nanoribbons under a transverse electric field is interesting.

Designing edge states from fractional polarization insulators.

We theoretically investigated disconnected dispersive edge states in an anisotropic honeycomb lattice without chiral symmetry. When both mirror and chiral symmetries are present, this system is defined by a topological quantity known as fractional polarization (FP) term and exhibits a bulk band gap, classifying it as an FP insulator. While the FP insulator accommodates robust, flat topological edge states (TES), it also offers the potential to engineer these edge states by deliberately disrupting a critical symmetry that safeguards the underlying topology. These symmetry-breaking terms allow the edge states to become dispersive and generate differing configurations along the open boundaries. Furthermore, disconnected helical-like and chiral-like edge states analogous to TES seen in quantum spin and anomalous hall effect are achieved by the finite size effect, not possible from the symmetry-breaking terms alone. The demonstration of manipulating these edge states from a FP insulator can open up new avenues in constructing devices that utilize topological domain walls.

Pressure-Dependent Superconductivity in Topological Dirac Semimetal SrCuBi.

The discovery of superconducting states in diverse topological materials generates a burgeoning interest to explore a topological superconductor and to realize a fault-tolerant topological quantum computation. A variety of routes to realize topological superconductors are proposed, and many types of topological materials are developed. However, a pristine topological material with a natural superconducting state is relatively rare as compared to topological materials with artificially induced superconductivity. Here, it is reported that the planar honeycomb structured 3D topological Dirac semimetal (TDS) SrCuBi, which is the Zintl phase, shows a natural surface superconductivity at 2.1 K under ambient pressure. It is clearly identified from theoretical calculations that a topologically nontrivial state exists on the (100) surface. Further, its superconducting transition temperature (Tc) increases by applying pressure, exhibiting a maximal Tc of 4.8 K under 6.2 GPa. It is believed that this discovery opens up a new possibility of exploring exotic Majorana fermions at the surface of 3D TDS superconductors.

Ultralow lattice thermal conductivity in type-I Dirac MBene TiB2.

MBenes, the emergent novel two-dimensional family of transition metal borides have recently attracted remarkable attention. Transport studies of such two-dimensional structures are very rare and are of sparking interest. In this paper Using Boltzmann transport theory with ab-initio inputs from density functional theory, we examined the transport in TiB2MBene system, which is highly dependent on number of layers. We have shown that the addition of an extra layer (as in bilayer BL) destroys the formation of type-I Dirac state by introducing the positional change and tilt to the Dirac cones, thereby imparting the type-II Weyl metallic character in contrast to Dirac-semimetallic character in monolayer ML. Such non-trivial electronic ordering significantly impacts the transport behavior. We further show that the anisotropic room temperature lattice thermal conductivityκLfor ML (BL) is observed to be 0.41 (0.52) and 2.00 (2.04) W m-1 K-1forxandydirections, respectively, while the high temperatureκL(ML 0.13 W m-1 K-1and BL 0.21 W m-1 K-1at 900 K inxdirection) achieves ultralow values. Our analysis reveals that such values are attributed to enhanced anharmonic phonon scattering, enhanced weighted phase space and co-existence of electronic and phononic Dirac states. We have further calculated the electronic transport coefficients for TiB2MBene, where the layer dependent competing behavior is observed at lower temperatures. Our results further unravels the layer dependent thermoelectric performance, where ML is shown to have promising room-temperature thermoelectric figure of merit (ZT) as 1.71 compared to 0.38 for BL.

Field-Effect Thermoelectric Hotspot in Monolayer Graphene Transistor.

Graphene is a promising candidate for the thermal management of downscaled microelectronic devices owing to its exceptional electrical and thermal properties. Nevertheless, a comprehensive understanding of the intricate electrical and thermal interconversions at a nanoscale, particularly in field-effect transistors with prevalent gate operations, remains elusive. In this study, nanothermometric imaging is used to examine a current-carrying monolayer graphene channel sandwiched between hexagonal boron nitride dielectrics. It is revealed for the first time that beyond the expected Joule heating, the thermoelectric Peltier effect actively plays a significant role in generating hotspots beneath the gated region. With gate-controlled charge redistribution and a shift in the Dirac point position, an unprecedented systematic evolution of thermoelectric hotspots, underscoring their remarkable tenability is demonstrated. This study reveals the field-effect Peltier contribution in a single graphene-material channel of transistors, offering valuable insights into field-effect thermoelectrics and future on-chip energy management.

A Topological Parametric Phonon Oscillator.

Topological bosonic systems have recently aroused intense interests in exploring exotic phenomena that have no counterparts in electronic systems. The squeezed bosonic interaction in these systems is particularly interesting, because it can modify the vacuum fluctuations of topological states, drive them into instabilities, and lead to topological parametric oscillators. However, these phenomena remain experimentally elusive because of limited nonlinearities in most existing topological bosonic systems. Here, a topological parametric phonon oscillator is experimentally realized based on a nonlinear nanoelectromechanical Dirac-vortex cavity with strong squeezed interaction. Specifically, the Dirac-vortex cavity is parametrically driven to provide phase-sensitive amplification for topological phonons, leading to the observation of coherent parametric phonon oscillation above the threshold. Additionally, it is confirmed that the random frequency variation caused by fabrication disorders can be suppressed effectively by increasing the cavity size, while the free spectral range reduces at a much slower rate, which benefit the realization of large-area single-mode lasers. Our results represent an important advance in experimental investigations of topological physics with large bosonic nonlinearities and parametric gain. This article is protected by copyright. All rights reserved.

Surface terminations control charge transfer from bulk to surface states in topological insulators.

Topological insulators (TI) hold significant potential for various electronic and optoelectronic devices that rely on the Dirac surface state (DSS), including spintronic and thermoelectric devices, as well as terahertz detectors. The behavior of electrons within the DSS plays a pivotal role in the performance of such devices. It is expected that DSS appear on a surface of three dimensional(3D) TI by mechanical exfoliation. However, it is not always the case that the surface terminating atomic configuration and corresponding band structures are homogeneous. In order to investigate the impact of surface terminating atomic configurations on electron dynamics, we meticulously examined the electron dynamics at the exfoliated surface of a crystalline 3D TI (Bi 2 Se 3 ) with time, space, and energy resolutions. Based on our comprehensive band structure calculations, we found that on one of the Se-terminated surfaces, DSS is located within the bulk band gap, with no other surface states manifesting within this region. On this particular surface, photoexcited electrons within the conduction band effectively relax towards DSS and tend to linger at the Dirac point for extended periods of time. It is worth emphasizing that these distinct characteristics of DSS are exclusively observed on this particular surface.

Klein Tunneling in ß12 Borophene.

Motivated by the recent observation of Klein tunneling in 8-Pmmn borophene, we delve into the phenomenon in ß12 borophene by employing tight-binding approximation theory to establish a theoretical mode. The tight-binding model is a semi-empirical method for establishing the Hamiltonian based on atomic orbitals. A single cell of ß12 borophene contains five atoms and multiple central bonds, so it creates the complexity of the tight-binding model Hamiltonian of ß12 borophene. We investigate transmission across one potential barrier and two potential barriers by changing the width and height of barriers and the distance between two potential barriers. Regardless of the change in the barrier heights and widths, we find the interface to be perfectly transparent for normal incidence. For other angles of incidence, perfect transmission at certain angles can also be observed. Furthermore, perfect and all-angle transmission across a potential barrier takes place when the incident energy approaches the Dirac point. This is analogous to the "super", all-angle transmission reported for the dice lattice for Klein tunneling across a potential barrier. These findings highlight the significance of our theoretical model in understanding the complex dynamics of Klein tunneling in borophene structures.

Asymmetric Tilt-Induced Quantum Beating of Conductance Oscillation in Magnetically Modulated Dirac Matter Systems.

Herein, we investigate the effect of tilt mismatch on the quantum oscillations of spin transport properties in two-dimensional asymmetrically tilted Dirac cone systems. This study involves the examination of conductance oscillation in two distinct junction types: transverse- and longitudinal-tilted Dirac cones (TTDCs and LTDCs). Our findings reveal an unusual quantum oscillation of spin-polarized conductance within the TTDC system, characterized by two distinct anomaly patterns within a single period, labeled as the linear conductance phase and the oscillatory conductance phase. Interestingly, these phases emerge in association with tilt-induced orbital pseudo-magnetization and exchange interaction. Our study also demonstrates that the structure of the LTDC can modify the frequency of spin conductance oscillation, and the asymmetric effect within this structure results in a quantum beating pattern in oscillatory spin conductance. We note that an enhancement in the asymmetric longitudinal tilt velocity ratio within the structure correspondingly amplifies the beating frequency. Our research potentially contributes valuable insights for detecting the asymmetry of tilted Dirac fermions in type-I Dirac semimetal-based spintronics and quantum devices.

Semimetallic state transition in Two-Dimensional carbon nitride covalent networks and enhanced electrocatalytic activity for nitrate to ammonia conversion.

Single-atom catalysts (SACs), due to their maximum atomic utilization rate, show tremendous potential for application in the electrocatalytic synthesis of ammonia from nitrate. Yet, the development of superior supports that preserve the high selectivity, activity, and stability of SACs remains an imperative challenge. In this work, based on first-principles calculations and tight-binding (TB) model analysis, a new two-dimensional (2D) carbon nitride monolayer, C7N6, is proposed. The C7N6 structure exhibits a strong covalent network, with dynamical, thermal, and mechanical stability. Surprisingly, the structural transition from C9N4 to C7N6 corresponds to a semimetallic state transition. Further symmetry analysis unveils that the Dirac states in C7N6 are protected by space-time inversion symmetry, and the physical origin of the Dirac cone was confirmed using the TB model. Additionally, a non-zero Z2 invariant and significant topological edge states demonstrate its topologically nontrivial nature. Considering the excellent structural and topological properties of C7N6, a three-step screening strategy is designed to identify eligible SACs for electrochemical nitrate reduction reaction (NO3RR), and Ti@C7N6 is identified as possessing the best activity, with the last proton-electron coupling step *NH2→*NH3 being the potential-determining step (PDS), for which the limiting potential is 0.48 V. Moreover, a free energy diagram shows that the *NOH reaction pathway is energetically preferred on Ti@C7N6, and ab initio molecular dynamics (AIMD) calculations at 500 K confirm its good thermal stability. Our study not only provides excellent CN-based support material but also offers theoretical guidance for constructing highly active and selective SACs for nitrate reduction.

Ultrafast Carrier Relaxation Dynamics in a Nodal-Line Semimetal PtSn4.

Topological Dirac nodal-line semimetals host topologically nontrivial electronic structure with nodal-line crossings around the Fermi level, which could affect the photocarrier dynamics and lead to novel relaxation mechanisms. Herein, by using time- and angle-resolved photoemission spectroscopy, we reveal the previously inaccessible linear dispersions of the bulk conduction bands above the Fermi level in a Dirac nodal-line semimetal PtSn4, as well as the momentum and temporal evolution of the gapless nodal lines. A surprisingly ultrafast relaxation dynamics within a few hundred femtoseconds is revealed for photoexcited carriers in the nodal line. Theoretical calculations suggest that such ultrafast carrier relaxation is attributed to the multichannel scatterings among the complex metallic bands of PtSn4 via electron-phonon coupling. In addition, a unique dynamic relaxation mechanism contributed by the highly anisotropic Dirac nodal-line electronic structure is also identified. Our work provides a comprehensive understanding of the ultrafast carrier dynamics in a Dirac nodal-line semimetal.

Extraordinary Thermoelectric Properties of Topological Surface States in Quantum-Confined Cd3As2 Thin Films.

Topological insulators and semimetals have been shown to possess intriguing thermoelectric properties promising for energy harvesting and cooling applications. However, thermoelectric transport associated with the Fermi arc topological surface states on topological Dirac semimetals remains less explored. This work systematically examines thermoelectric transport in a series of topological Dirac semimetal Cd3As2 thin films grown by molecular beam epitaxy. Surprisingly, significantly enhanced Seebeck effect and anomalous Nernst effect are found at cryogenic temperatures when the Cd3As2 layer is thin. In particular, a peak Seebeck coefficient of nearly 500 µV K-1 and a corresponding thermoelectric power factor over 30 mW K-2 m-1 are observed at 5 K in a 25-nm-thick sample. Combining angle-dependent quantum oscillation analysis, magnetothermoelectric measurement, transport modeling, and first-principles simulation, the contributions from bulk and surface conducting channels are isolated and the unusual thermoelectric properties are attributed to the topological surface states. The analysis showcases the rich thermoelectric transport physics in quantum-confined topological Dirac semimetal thin films and suggests new routes to achieving high thermoelectric performance at cryogenic temperatures.

Approximate solutions of the spin and pseudospin symmetries under coshine Yukawa tensor interaction.

The approximate solutions of the Dirac equation for spin symmetry and pseudospin symmetry are studied with a coshine Yukawa potential model via the traditional supersymmetric approach (SUSY). To remove the degeneracies in both the spin and pseudospin symmetries, a coshine Yukawa tensor potential is proposed and applied to both the spin symmetry and the pseudospin symmetry. The proposed coshine tensor potential removes the energy degenerate doublets in both the spin symmetry and pseudospin symmetry for a very small value of the tensor strength (H = 0.05). This shows that the coshine Yukawa tensor is more effective than the real Yukawa tensor. The non-relativistic limit of the spin symmetry is obtained by using certain transformations. The results obtained showed that the coshine Yukawa potential and the real Yukawa potential has the same variation with the angular momentum number but the variation of the screening parameter with the energy for the two potential models differs. However, the energy eigenvalues of the coshine Yukawa potential model, are more bounded compared to the energies of the real Yukawa potential model.

Recent progress on topological semimetal IrO2: electronic structures, synthesis, and transport properties.

5dtransition metal oxides, such as iridates, have attracted significant interest in condensed matter physics throughout the past decade owing to their fascinating physical properties that arise from intrinsically strong spin-orbit coupling (SOC) and its interplay with other interactions of comparable energy scales. Among the rich family of iridates, iridium dioxide (IrO2), a simple binary compound long known as a promising catalyst for water splitting, has recently been demonstrated to possess novel topological states and exotic transport properties. The strong SOC and the nonsymmorphic symmetry that IrO2possesses introduce symmetry-protected Dirac nodal lines (DNLs) within its band structure as well as a large spin Hall effect in the transport. Here, we review recent advances pertaining to the study of this unique SOC oxide, with an emphasis on the understanding of the topological electronic structures, syntheses of high crystalline quality nanostructures, and experimental measurements of its fundamental transport properties. In particular, the theoretical origin of the presence of the fourfold degenerate DNLs in band structure and its implications in the angle-resolved photoemission spectroscopy measurement and in the spin Hall effect are discussed. We further introduce a variety of synthesis techniques to achieve IrO2nanostructures, such as epitaxial thin films and single crystalline nanowires, with the goal of understanding the roles that each key parameter plays in the growth process. Finally, we review the electrical, spin, and thermal transport studies. The transport properties under variable temperatures and magnetic fields reveal themselves to be uniquely sensitive and modifiable by strain, dimensionality (bulk, thin film, nanowire), quantum confinement, film texture, and disorder. The sensitivity, stemming from the competing energy scales of SOC, disorder, and other interactions, enables the creation of a variety of intriguing quantum states of matter.

Dirac equation for photons in a fibre: Origin of polarisation.

Spin is a fundamental degree of freedom, which was discovered by Dirac for an electron in his relativistic quantum mechanics, known as the Dirac equation. The origin of spin for a photon is unclear because Maxwell's equations in a vacuum are Lorentz invariant without introducing the concept of spin. Here, the propagation of coherent rays of photons in a graded-index optical fibre is considered to discuss the origin of polarisation for photons using exact solutions of the Laguerre-Gauss and Hermite-Gauss modes. The energy spectrum is massive, and the effective mass is a function of the confinement and orbital angular momentum. The propagation is described by the one-dimensional (1D) non-relativistic Schrödinger equation, which is equivalent to the 2D space-time Klein-Gordon equation by a unitary transformation. The probabilistic interpretation and the conservation law require the factorisation of the Klein-Gordon equation, leading to the 2D Dirac equation with spin. The spin expectation values of photons correspond to the polarisation state on the Poincaré sphere. As an application of the theory, a polarisation interferometer is proposed, whose energy spectrum shows a Dirac cone in the Stokes parameter space.