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Magnetic transition-metal dichalcogenides (TMDs) have been of particular interest due to their unique magnetic properties and layered structure that can be promising for a wide range of spintronic applications. One of the most exciting compounds in this family of magnets is chromium telluride, Cr$_{1+x}$Te$_2$, which has shown rich magnetic phases with varied Cr concentrations. An emergent antiferromagnetic (AFM) ordering has been found in Cr$_{1.25}$Te$_2$ (equivalently, Cr$_{5}$Te$_8$), which is induced by intercalating 0.25 Cr atom per unit cell within the van der Waals (vdW) gaps of CrTe$_2$. In this work, we report an increased N\'eel Temperature ($T_\mathrm{N}$) of the AFM phase in Cr$_{1+x}$Te$_2$ by slightly reducing the concentration of Cr intercalants. Moreover, the intercalated Cr atoms form a metastable 2$\times$2 supercell structure that can be manipulated by electron beam irradiation. This work offers a promising approach to tuning magnetic and structural properties by adjusting the concentration of self-intercalated magnetic atoms., Comment: 5 figures, 1 table

High spin Chern-number insulators (HSCI) have emerged as a novel 2D topological phase of condensed matter that is beyond the classification of topological quantum chemistry. In this work, we report the observation of a semimetallic Sb monolayer carrying the same band topology as HSCI with a spin Chern number equal to 2. Our calculations further indicate a moderate lattice strain can make Sb monolayer an insulator or a semimetal with a tunable spin Chern number from 0 to 3. The results suggest strained Sb monolayers as a promising platform for exploring exotic properties of the HSCI topological matter., Comment: 4 figures

A two-dimensional (2D) Weyl semimetal featuring a spin-polarized linear band dispersion and a nodal Fermi surface is a new topological phase of matter. It is a solid-state realization of Weyl fermions in an intrinsic 2D system. The nontrivial topology of 2D Weyl cones guarantees the existence of a new form of topologically protected boundary states, Fermi string edge states. In this work, we report the realization of a 2D Weyl semimetal in monolayer-thick epitaxial bismuthene grown on SnS(Se) substrate. The intrinsic band gap of bismuthene is eliminated by the space-inversion-symmetry-breaking substrate perturbations, resulting in a gapless spin-polarized Weyl band dispersion. The linear dispersion and spin polarization of the Weyl fermion states are observed in our spin and angle-resolved photoemission measurements. In addition, the scanning tunneling microscopy/spectroscopy reveals a pronounced local density of states at the edge, suggesting the existence of Fermi string edge states. These results open the door for the experimental exploration of the exotic properties of Weyl fermion states in reduced dimensions., Comment: 5 figures

The superconductor PdTe$_2$ is known to host bulk Dirac bands and topological surface states. The coexistence of superconductivity and topological surface states makes PdTe$_2$ a promising platform for exploring topological superconductivity and Majorana bound states. In this work, we report the layer-by-layer molecular beam epitaxy growth and spectroscopic characterization of high quality PdTe$_2$ films with thickness down to 3 monolayers (ML). In the 3 ML PdTe$_2$ film, we observed spin-polarized surface resonance states, which are isolated from the bulk bands due to the quantum size effects. In addition, the hybridization of surface states on opposite faces leads to a thickness-dependent gap in the topological surface Dirac bands. Our photoemission results show clearly that the size of the hybridization gap increases as the film thickness is reduced. The success in growing high quality PdTe$_2$ films by state-of-art molecular beam epitaxy technique and the observation of surface resonances and gaped topological surface states sheds light on the applications of PdTe$_2$ quantum films in spintronics and topological computation., Comment: 4 figures

We report the epitaxial growth of PdTe$_2$ ultrathin films on topological insulator Bi$_2$Se$_3$. A prominent Moir\'e pattern was observed in STM measurements. The Moir\'e periodicity increases as film thickness decreases, indicating a lattice expansion of epitaxial PdTe$_2$ thin films with lower thicknesses. In addition, our simulations based on Moir\'e Metrology reveal uniaxial lattice strains at the edge of PdTe$_2$ domains, and anisotropic strain distributions throughout the Moir\'e supercell with a net change in lattice strain up to ~2.9%. Our DFT calculations show that this strain effect leads to a narrowing of the band gap at $\Gamma$ point near the Fermi level. Under a strain of ~2.8%, the band gap at $\Gamma$ closes completely. Further increasing the lattice strain makes the band gap reopen and the order of conduction band and valence bands inverted in energy. The results offer a proof of concept for constructing quantum grids of topological materials under the modulation of Moir\'e potentials., Comment: 13 pages, 4 figures

When two-dimensional van der Waals materials are stacked to build heterostructures, moir\'e patterns emerge from twisted interfaces or from mismatch in lattice constant of individual layers. Relaxation of the atomic positions is a direct, generic consequence of the moir\'e pattern, with many implications for the physical properties. Moir\'e driven atomic relaxation may be naively thought to be restricted to the interfacial layers and thus irrelevant for multi-layered heterostructures. However, we provide experimental evidence for the importance of the three dimensional nature of the relaxation in two types of van der Waals heterostructures: First, in multi-layer graphene twisted on graphite at small twist angles ($\theta\approx0.14^\circ$) we observe propagation of relaxation domains even beyond 18 graphene layers. Second, we show how for multi-layer PdTe$_2$ on Bi$_2$Se$_3$ the moir\'e lattice constant depends on the number of PdTe$_2$ layers. Motivated by the experimental findings, we developed a continuum approach to model multi-layered relaxation processes based on the generalized stacking fault energy functional given by ab-initio simulations. Leveraging the continuum property of the approach enables us to access large scale regimes and achieve agreement with our experimental data for both systems. Furthermore it is well known that the electronic structure of graphene sensitively depends on local lattice deformations. Therefore we study the impact of multi-layered relaxation on the local density of states of the twisted graphitic system. We identify measurable implications for the system, experimentally accessible by scanning tunneling microscopy. Our multi-layered relaxation approach is not restricted to the discussed systems, and can be used to uncover the impact of an interfacial defect on various layered systems of interest., Comment: 17 pages, 7 figures

Non-equilibrium thermodynamics has long been an area of substantial interest to ecologists because most fundamental biological processes, such as protein synthesis and respiration, are inherently energy-consuming. Microbial communities are a natural system to decipher this mechanistic basis because their interactions in the form of substrate consumption, metabolite production, and cross-feeding can be described explicitly in thermodynamic terms. Previous work has considered how thermodynamic constraints impact competition between pairs of species, but restrained from analysing how this manifests in complex dynamical systems. To address this gap, we develop a thermodynamic microbial community model with fully reversible reaction kinetics, which allows direct consideration of free-energy dissipation. This also allows species to interact via products rather than just substrates, increasing the dynamical complexity, and allowing a more nuanced classification of interaction types to emerge. Using this model, we find that community diversity increases with substrate lability, because greater free-energy availability allows for faster generation of niches. Thus, more niches are generated in the time frame of community establishment, leading to higher final species diversity. We also find that allowing species to make use of near-to-equilibrium reactions increases diversity in a low free-energy regime. In such a regime, two new thermodynamic interaction types that we identify here reach comparable strengths to the conventional (competition and facilitation) types, emphasising the key role that thermodynamics plays in community dynamics. Our results suggest that accounting for realistic thermodynamic constraints is vital for understanding the dynamics of real-world microbial communities., Comment: 23 pages, 5 figures

The discovery of graphene has stimulated enormous interest in two-dimensional (2D) electron gas with linear band structure. 2D Dirac materials possess many intriguing physical properties such as high carrier mobility and zero-energy Landau level thanks to the relativistic dispersion and chiral spin/pseudospin texture. 2D Dirac states discovered so far are exclusively pinned at high-symmetry points of the Brillouin zone, for example, surface Dirac states at $\overline{\Gamma}$ in topological insulators Bi$_2$Se(Te)$_3$ and Dirac cones at $K$ and $K'$ in graphene. In this work, we report the realization of 2D Dirac states at generic $k$-points in antimony atomic layers with phosphorene structure ($i.e.$ $\alpha$-antimonene). The unpinned nature enables versatile ways to control the locations of the Dirac points in momentum space. In addition, dispersions around the unpinned Dirac points exhibit intrinsically anisotropic behaviors due to the reduced symmetry of generic momentum points. These properties make the $\alpha$-antimonene films a promising platform for exploring interesting physics in unpinned 2D Dirac fermions that are distinct from the conventional Dirac states in graphene., Comment: 5 figures

Discoveries of interfacial topological Hall effect (THE) provide an ideal platform for exploring physics arising from the interplay between topology and magnetism. The interfacial topological Hall effect is closely related to the Dzyaloshinskii-Moriya interaction (DMI) at interface and topological spin textures. However, it is difficult to achieve a sizable THE in heterostructures due to the stringent constraints on the constituents of THE heterostructures such as strong spin-orbit coupling (SOC). Here we report the observation of a giant THE signal of 1.39 $\mu\Omega\cdot$cm in the van der Waals heterostructures of CrTe2/Bi2Te3 fabricated by molecular beam epitaxy, a prototype of two-dimensional (2D) ferromagnet (FM)/topological insulator (TI). This large magnitude of THE is attributed to an optimized combination of 2D ferromagnetism in CrTe2, strong SOC in Bi2Te3, and an atomically sharp interface. Our work reveals CrTe2/Bi2Te3 as a convenient platform for achieving large interfacial THE in hybrid systems, which could be utilized to develop quantum science and high-density information storage., Comment: 5 figures and 1 table

Tuning interactions between Dirac states in graphene has attracted enormous interest because it can modify the electronic spectrum of the two-dimensional material, enhance electron correlations, and give rise to novel condensed-matter phases such as superconductors, Mott insulators, Wigner crystals and quantum anomalous Hall insulators. Previous works predominantly focus on the flat band dispersion of coupled Dirac states from different graphene layers. In this work, we propose a new route to realizing flat band physics in monolayer graphene under a periodic modulation from substrates. We take gaphene/SiC heterostructure as a role model and demonstrate experimentally the substrate modulation leads to Dirac fermion cloning and consequently, the proximity of the two Dirac cones of monolayer graphene in momentum space. Our theoretical modeling captures the cloning mechanism of Dirac states and indicates that flat bands can emerge at certain magic lattice constants of substrate when the period of modulation becomes nearly commensurate with the $(\sqrt{3}\times\sqrt{3})R30^{\circ}$ supercell of graphene. The results show that the epitaxial monolayer graphene is a promising platform for exploring exotic many-body quantum phases arising from interactions between Dirac electrons.

Two-dimensional (2D) Dirac states with linear band dispersion have attracted enormous interest since the discovery of graphene. However, to date, 2D Dirac semimetals are still very rare due to the fact that 2D Dirac states are generally fragile against perturbations such as spin-orbit couplings. Nonsymmorphic crystal symmetries can enforce the formation of Dirac nodes, providing a new route to establishing symmetry-protected Dirac states in 2D materials. Here we report the symmetry-protected Dirac states in nonsymmorphic alpha-antimonene. The antimonene was synthesized by the method of molecular beam epitaxy. Two Dirac cones with large anisotropy were observed by angle-resolved photoemission spectroscopy. The Dirac state in alpha-antimonene is of spin-orbit type in contrast to the spinless Dirac states in graphene. The result extends the 'graphene' physics into a new family of 2D materials where spin-orbit coupling is present., Comment: 4 figures

Multistable non-equilibrium systems are abundant outcomes of nonlinear dynamics with feedback but still relatively little is known about what determines the stability of the steady states and their switching rates in terms of entropy and entropy production. Here, we will link fluctuation theorems for the entropy production along trajectories with the action obtainable from the Freidlin--Wentzell theorem to elucidate the thermodynamics of switching between states in the large volume limit of multistable systems. We find that the entropy production at steady state plays no role, but the entropy production during switching is key. Steady-state entropy and diffusive noise strength can be neglected in this limit. The relevance to biology, ecological, and climate models is apparent., Comment: 8 pages, 3 figures

Proximity-induced superconductivity in three dimensional (3D) topological insulators forms a new quantum phase of matter and accommodates exotic quasiparticles such as Majorana bound states. One of the biggest drawbacks of the commonly studied 3D topological insulators is the presence of conducting bulk that obscures both surface states and low energy bound states. Introducing superconductivity in topological Kondo insulators such as SmB$_6$, however, is promising due to their true insulating bulk at low temperatures. In this work, we develop an unconventional Josephson junction by coupling superconducting Nb leads to the surface states of a SmB$_6$ crystal. We observe a robust critical current at low temperatures that responds to the application of an out-of-plane magnetic field with significant deviations from usual Fraunhofer patterns. The appearance of Shaphiro steps under microwave radiation gives further evidence of a Josephson effect. Moreover, we explore the effects of Kondo breakdown in our devices, such as ferromagnetism at the surface and anomalous temperature dependence of supercurrent. Particularly, the magnetic diffraction patterns show an anomalous hysteresis with the field sweep direction suggesting the coexistence of magnetism with superconductivity at the SmB$_6$ surface. The experimental work will advance the current understanding of topologically nontrivial superconductors and emergent states associated with such unconventional superconducting phases., Comment: accepted to Phys. Rev. B

This article is concerned with classifying the provably total set-functions of Kripke-Platek set theory, KP, and Power Kripke-Platek set theory, KP(P), as well as proving several (partial) conservativity results. The main technical tool used in this paper is a relativisation technique where ordinal analysis is carried out relative to an arbitrary but fixed set x. A classic result from ordinal analysis is the characterisation of the provably recursive functions of Peano Arithmetic, PA, by means of the fast growing hierarchy [10]. Whilst it is possible to formulate the natural numbers within KP, the theory speaks primarily about sets. For this reason it is desirable to obtain a characterisation of its provably total set functions. We will show that KP proves the totality of a set function precisely when it falls within a hierarchy of set functions based upon a relativised constructible hierarchy stretching up in length to any ordinal below the Bachmann-Howard ordinal. As a consequence of this result we obtain that IKP + forall x,y (x in y or x not in y) is conservative over KP for forall-exists-formulae, where IKP stands for intuitionistic Kripke-Platek set theory. In a similar vein, utilising [56], it is shown that KP(P) proves the totality of a set function precisely when it falls within a hierarchy of set functions based upon a relativised von Neumann hierarchy of the same length. The relativisation technique applied to KP(P) with the global axiom of choice, AC-global, also yields a parameterised extension of a result in [58], showing that KP(P) + AC-global is conservative over KP(P) + AC and CZF + AC for Pi_2 in powerset statements. Here AC stands for the ordinary axiom of choice and CZF refers to constructive Zermelo-Fraenkel set theory., Comment: 60 pages

Until the 1970s, proof theoretic investigations were mainly concerned with theories of inductive definitions, subsystems of analysis and finite type systems. With the pioneering work of Gerhard Jaeger in the late 1970s and early 1980s, the focus switched to set theories, furnishing ordinal-theoretic proof theory with a uniform and elegant framework. More recently it was shown that these tools can even sometimes be adapted to the context of strong axioms such as the powerset axiom, where one does not attain complete cut elimination but can nevertheless extract witnessing information and characterize the strength of the theory in terms of provable heights of the cumulative hierarchy. Here this technology is applied to intuitionistic Kripke-Platek set theories IKP(P) and IKP(E), where the operation of powerset and exponentiation, respectively, is allowed as a primitive in the separation and collection schemata. In particular, IKP(P) proves the powerset axiom whereas IKP(E) proves the exponentiation axiom. The latter expresses that given any sets A and B, the collection of all functions from A to B is a set, too. While IKP(P) can be dealt with in a similar vein as its classical cousin, the treatment of IKP(E) posed considerable obstacles. One of them was that in the infinitary system the levels of terms become a moving target as they cannot be assigned a fixed level in the formal cumulative hierarchy solely based on their syntactic structure. As adumbrated in an earlier paper, the results of this paper are an important tool in showing that several intuitionistic set theories with the collection axiom possess the existence property, i.e., if they prove an existential theorem then a witness can be provably described in the theory, one example being intuitionistic Zermelo-Fraenkel set theory with bounded separation.