A tropical geometric approach to exceptional points.

Non-Hermitian systems have been widely explored in platforms ranging from photonics to electric circuits. A defining feature of non-Hermitian systems is exceptional points (EPs), where both eigenvalues and eigenvectors coalesce. Tropical geometry is an emerging field of mathematics at the interface between algebraic geometry and polyhedral geometry, with diverse applications to science. Here, we introduce and develop a unified tropical geometric framework to characterize different facets of non-Hermitian systems. We illustrate the versatility of our approach using several examples and demonstrate that it can be used to select from a spectrum of higher-order EPs in gain and loss models, predict the skin effect in the non-Hermitian Su-Schrieffer-Heeger model, and extract universal properties in the presence of disorder in the Hatano-Nelson model. Our work puts forth a framework for studying non-Hermitian physics and unveils a connection of tropical geometry to this field.

Quantum Hall effect based on Weyl orbits in Cd3As2.

Discovered decades ago, the quantum Hall effect remains one of the most studied phenomena in condensed matter physics and is relevant for research areas such as topological phases, strong electron correlations and quantum computing1-5. The quantized electron transport that is characteristic of the quantum Hall effect typically originates from chiral edge states-ballistic conducting channels that emerge when two-dimensional electron systems are subjected to large magnetic fields2. However, whether the quantum Hall effect can be extended to higher dimensions without simply stacking two-dimensional systems is unknown. Here we report evidence of a new type of quantum Hall effect, based on Weyl orbits in nanostructures of the three-dimensional topological semimetal Cd3As2. The Weyl orbits consist of Fermi arcs (open arc-like surface states) on opposite surfaces of the sample connected by one-dimensional chiral Landau levels along the magnetic field through the bulk6,7. This transport through the bulk results in an additional contribution (compared to stacked two-dimensional systems and which depends on the sample thickness) to the quantum phase of the Weyl orbit. Consequently, chiral states can emerge even in the bulk. To measure these quantum phase shifts and search for the associated chiral modes in the bulk, we conduct transport experiments using wedge-shaped Cd3As2 nanostructures with variable thickness. We find that the quantum Hall transport is strongly modulated by the sample thickness. The dependence of the Landau levels on the magnitude and direction of the magnetic field and on the sample thickness agrees with theoretical predictions based on the modified Lifshitz-Onsager relation for the Weyl orbits. Nanostructures of topological semimetals thus provide a way of exploring quantum Hall physics in three-dimensional materials with enhanced tunability.

Thickness-Dependent Magnetic Breakdown in ZrSiSe Nanoplates.

We report a study of thickness-dependent interband and intraband magnetic breakdown by thermoelectric quantum oscillations in ZrSiSe nanoplates. Under high magnetic fields of up to 30 T, quantum oscillations arising from degenerated hole pockets were observed in thick ZrSiSe nanoplates. However, when decreasing the thickness, plentiful multifrequency quantum oscillations originating from hole and electron pockets are captured. These multiple frequencies can be explained by the emergent interband magnetic breakdown enclosing individual hole and electron pockets and intraband magnetic breakdown within spin-orbit coupling (SOC) induced saddle-shaped electron pockets, resulting in the enhanced contribution to thermal transport in thin ZrSiSe nanoplates. These experimental frequencies agree well with theoretical calculations of the intriguing tunneling processes. Our results introduce a new member of magnetic breakdown to the field and open up a dimension for modulating magnetic breakdown, which holds fundamental significance for both low-dimensional topological materials and the physics of magnetic breakdown.

Nondegenerate Integer Quantum Hall Effect from Topological Surface States in Ag2Te Nanoplates.

The quantum Hall effect is one of the exclusive properties displayed by Dirac Fermions in topological insulators, which propagates along the chiral edge state and gives rise to quantized electron transport. However, the quantum Hall effect formed by the nondegenerate Dirac surface states has been elusive so far. Here, we demonstrate the nondegenerate integer quantum Hall effect from the topological surface states in three-dimensional (3D) topological insulator ß-Ag2Te nanostructures. Surface-state dominant conductance renders quantum Hall conductance plateaus with a step of e2/h, along with typical thermopower behaviors of two-dimensional (2D) massless Dirac electrons. The 2D nature of the topological surface states is proven by the electrical and thermal transport responses under tilted magnetic fields. Moreover, the degeneracy of the surface states is removed by structure inversion asymmetry (SIA). The evidenced SIA-induced nondegenerate integer quantum Hall effect in low-symmetry ß-Ag2Te has implications for both fundamental study and the realization of topological magneto-electric effects.

Multiferroic quantum criticality.

The zero-temperature limit of a continuous phase transition is marked by a quantum critical point, which can generate physical effects that extend to elevated temperatures. Magnetic quantum criticality is now well established, and has been explored in systems ranging from heavy fermion metals to quantum Ising materials. Ferroelectric quantum critical behaviour has also been recently demonstrated, motivating a flurry of research investigating its consequences. Here, we introduce the concept of multiferroic quantum criticality, in which both magnetic and ferroelectric quantum criticality occur in the same system. We develop the phenomenology of multiferroic quantum criticality and describe the associated experimental signatures, such as phase stability and modified scaling relations of observables. We propose several material systems that could be tuned to multiferroic quantum criticality utilizing alloying and strain as control parameters. We hope that these results stimulate exploration of the interplay between different kinds of quantum critical behaviours.

Ultrahigh conductivity in Weyl semimetal NbAs nanobelts.

In two-dimensional (2D) systems, high mobility is typically achieved in low-carrier-density semiconductors and semimetals. Here, we discover that the nanobelts of Weyl semimetal NbAs maintain a high mobility even in the presence of a high sheet carrier density. We develop a growth scheme to synthesize single crystalline NbAs nanobelts with tunable Fermi levels. Owing to a large surface-to-bulk ratio, we argue that a 2D surface state gives rise to the high sheet carrier density, even though the bulk Fermi level is located near the Weyl nodes. A surface sheet conductance up to 5-100 S per □ is realized, exceeding that of conventional 2D electron gases, quasi-2D metal films, and topological insulator surface states. Corroborated by theory, we attribute the origin of the ultrahigh conductance to the disorder-tolerant Fermi arcs. The evidenced low-dissipation property of Fermi arcs has implications for both fundamental study and potential electronic applications.

Charge Density Wave and Narrow Energy Gap at Room Temperature in 2D Pb3-xSb1+xS4Te2-δ with Square Te Sheets.

We report a new two-dimensional compound, Pb3-xSb1+xS4Te2-δ, that has a charge density wave (CDW) at room temperature. The CDW is incommensurate with q-vector of 0.248(6)a* + 0.246(8)b* + 0.387(9)c* for x = 0.29(2) and δ = 0.37(3) due to positional and occupational long-range ordering of Te atoms in the sheets. The modulated structure was refined from the single-crystal X-ray diffraction data with a superspace group P1̅(αßγ)0 using (3 + 1)-dimensional crystallography. The resistivity increases with decreasing temperature, suggesting semiconducting behavior. The transition temperature (TCDW) of the CDW is ∼345 K, above which the Te square sheets become disordered with no q-vector. First-principles density functional theory calculations on the undistorted structure and an approximate commensurate supercell reveal that the gap is due to the structure modulation.

Semiconducting Ba3Sn3Sb4 and Metallic Ba7-xSn11Sb15-y (x = 0.4, y = 0.6) Zintl Phases.

We report the discovery of two ternary Zintl phases Ba3Sn3Sb4 and Ba7-xSn11Sb15-y (x = 0.4, y = 0.6). Ba3Sn3Sb4 adopts the monoclinic space group P21/c with a = 14.669(3) Å, b = 6.9649(14) Å, c = 13.629(3) Å, and ß = 104.98(3)°. It features a unique corrugated two-dimensional (2D) structure consisting of [Sn3Sb4]6- layers extending along the ab-plane with Ba2+ atoms sandwiched between them. The nonstoichiometric Ba6.6Sn11Sb14.4 has a complex one-dimensional (1D) structure adopting the orthorhombic space group Pnma, with unit cell parameters a = 37.964(8) Å, b = 4.4090(9) Å, and c = 24.682(5) Å. It consists of large double Sn-Sb ribbons separated by Ba2+ atoms. Ba3Sn3Sb4 is an n-type semiconductor which has a narrow energy gap of ∼0.18 eV and a room temperature carrier concentration of ∼4.2 × 1018 cm-3. Ba6.6Sn11Sb14.4 is determined to be a metal with electrons being the dominant carriers.

Spin-Valve Effect in NiFe/MoS2/NiFe Junctions.

Two-dimensional (2D) layered transition metal dichalcogenides (TMDs) have been recently proposed as appealing candidate materials for spintronic applications owing to their distinctive atomic crystal structure and exotic physical properties arising from the large bonding anisotropy. Here we introduce the first MoS2-based spin-valves that employ monolayer MoS2 as the nonmagnetic spacer. In contrast with what is expected from the semiconducting band-structure of MoS2, the vertically sandwiched-MoS2 layers exhibit metallic behavior. This originates from their strong hybridization with the Ni and Fe atoms of the Permalloy (Py) electrode. The spin-valve effect is observed up to 240 K, with the highest magnetoresistance (MR) up to 0.73% at low temperatures. The experimental work is accompanied by the first principle electron transport calculations, which reveal an MR of ∼9% for an ideal Py/MoS2/Py junction. Our results clearly identify TMDs as a promising spacer compound in magnetic tunnel junctions and may open a new avenue for the TMDs-based spintronic applications.

Controlling the Spin Texture of Topological Insulators by Rational Design of Organic Molecules.

We present a rational design approach to customize the spin texture of surface states of a topological insulator. This approach relies on the extreme multifunctionality of organic molecules that are used to functionalize the surface of the prototypical topological insulator (TI) Bi2Se3. For the rational design we use theoretical calculations to guide the choice and chemical synthesis of appropriate molecules that customize the spin texture of Bi2Se3. The theoretical predictions are then verified in angular-resolved photoemission experiments. We show that, by tuning the strength of molecule-TI interaction, the surface of the TI can be passivated, the Dirac point can energetically be shifted at will, and Rashba-split quantum-well interface states can be created. These tailored interface properties-passivation, spin-texture tuning, and creation of hybrid interface states-lay a solid foundation for interface-assisted molecular spintronics in spin-textured materials.

Topological tuning in three-dimensional dirac semimetals.

We study with first-principles methods the interplay between bulk and surface Dirac fermions in three dimensional Dirac semimetals. By combining density functional theory with the coherent potential approximation, we reveal a topological phase transition in Na_{3}Bi_{1-x}Sb_{x} and Cd_{3}[As_{1-x}P_{x}]_{2} alloys, where the material goes from a Dirac semimetal to a trivial insulator upon changing Sb or P concentrations. Tuning the composition allows us to engineer the position of the bulk Dirac points in reciprocal space. Interestingly, the phase transition coincides with the reversal of the band ordering between the conduction and valence bands.

Observation of Type-II Topological Nodal-Line Fermions in ZrSiSe.

Recently, there has been significant interest in topological nodal-line semimetals due to their linear energy dispersion with one-dimensional nodal lines or loops. These materials exhibit fascinating physical properties, such as drumhead surface states and 3D anisotropic nodal-line structures. Similar to Weyl semimetals, type-II nodal-line semimetals have two crossing bands that are both electron-like or hole-like along a certain direction. However, the direct observation of type-II nodal-line Fermions has been challenging due to the lack of suitable material platforms and the low density of states. Here we present experimental evidence for the coexistence of both type-I and type-II nodal-line Fermions in ZrSiSe, which was obtained through magneto-optical and angle-resolved photoemission spectroscopy (ARPES) measurements. Our density functional theory calculations predict that the type-II nodal-line structure can be developed in the Z-R line of the first Brillouin zone based on the lattice constants of the grown single crystal. Indeed, ARPES measurements reveal the type-II nodal-line band structure. The extracted type-II Landau level transitions from magneto-optical measurements exhibit good agreement with the calculated type-II energy dispersion model based on the band structure. Our experimental results demonstrate that ZrSiSe possesses two types of nodal-line Fermions, distinguishing it from other ZrSiX (X = S, Te) materials and positioning it as an ideal platform for investigating type-II nodal-line semimetals.

Nonlinear Hall effect in Rashba systems with hexagonal warping.

Rashba spin-orbit coupled systems are an important class of materials noted for diverse fundamental and applied phenomena. Recently, the emergence of non-linear Hall effect under conditions of time-reversal symmetry has been discovered in materials with broken inversion symmetry. In this work, we study the second- and third-order Hall response in Rashba systems with hexagonal warping. Starting with a low-energy model, we obtain the analytic expressions and discover the unique dipole profile in Rashba systems with hexagonal warping. Furthermore, we extend the analysis using a realistic tight-binding model. Next, we predict the existence of a third-order Hall effect in these systems, and calculate the Berry connection polarizability tensor analytically. We also show how the model parameters affect the third-order conductivity. Our predictions can help in the experimental realization of Berry curvature multipole physics in Rashba materials with hexagonal warping, and provide a new platform for engineering the non-linear Hall effects.

Non-Hermitian topological phases: principles and prospects.

The synergy between non-Hermitian concepts and topological ideas have led to very fruitful activity in the recent years. Their interplay has resulted in a wide variety of new non-Hermitian topological phenomena being discovered. In this review, we present the key principles underpinning the topological features of non-Hermitian phases. Using paradigmatic models-Hatano-Nelson, non-Hermitian Su-Schrieffer-Heeger and non-Hermitian Chern insulator-we illustrate the central features of non-Hermitian topological systems, including exceptional points, complex energy gaps and non-Hermitian symmetry classification. We discuss the non-Hermitian skin effect and the notion of the generalized Brillouin zone, which allows restoring the bulk-boundary correspondence. Using concrete examples, we examine the role of disorder, describe the Floquet engineering, present the linear response framework, and analyze the Hall transport properties of non-Hermitian topological systems. We also survey the rapidly growing experimental advances in this field. Finally, we end by highlighting possible directions which, in our view, may be promising for explorations in the near future.

Spin-state switching: chemical modulation and the impact of intermolecular interactions in manganese(III) complexes.

A series of mononuclear manganese(III) complexes [Mn(X-sal2-323)](ReO4) (X = 5 Cl, 1; X = 5 Br, 2; X = 3,5 Cl, 3; X = 3,5 Br, 4; and X = 5 NO2, 5), containing hexadentate ligands prepared using the condensation of N,N'-bis(3-aminopropyl)ethylenediamine and 5- or 3,5-substituted salicylaldehyde, has been synthesized. Variable temperature single-crystal X-ray diffraction, magnetic, spectroscopic, electrochemical, and spectroelectrochemical analyses, and theoretical calculations have been used to explore the role of various ligand substituents in the spin-state switching behavior of the prepared manganese(III) complexes. All five complexes consist of an analogous distorted octahedral monocationic MnN4O2 surrounding offered by the flexible hexadentate ligand and ReO4- as the counter anion. However, a disordered water molecule was detected in complex 4. Complexes 1 (X = 5 Cl) and 5 (X = 5 NO2) show gradual and complete spin-state switching between the high-spin (HS) (S = 2) and the low-spin (LS) (S = 1) state with T1/2 values of 146 and 115 K respectively, while an abrupt and complete transition at 95 K was observed for complex 2 (X = 5 Br). Alternatively, complex 3 (X = 3, 5 Cl) exhibits an incomplete and sharp transition between the HS and LS states at 104 K, while complex 4 (X = 3, 5 Br) (desolvated) remains almost LS up to 300 K and then displays gradual and incomplete SCO at a higher temperature. The nature of the spin-state switch and transition temperature suggest that the structural effect (cooperativity) plays a more significant role in comparison with the electronic effect coming from various substituents (Cl, Br, and NO2), which is further supported by the detailed structural, electrochemical, and theoretical studies.

Weak electronic correlations observed in magnetic Weyl Semimetal Mn3Ge.

Using angle-resolved photoemission spectroscopy (ARPES) and density functional theory (DFT) calculations, we systematically studied the electronic band structure of Mn3Ge in the vicinity of the Fermi level. We observe several bands crossing the Fermi level, confirming the metallic nature of the studied system. We further observe several flat bands along various high symmetry directions, consistent with the DFT calculations. The calculated partial density of states suggests a dominant Mn 3dorbital contribution to the total valence band DOS. With the help of orbital-resolved band structure calculations, we qualitatively identify the orbital information of the experimentally obtained band dispersions. Out-of-plane electronic band dispersions are explored by measuring the ARPES data at various photon energies. Importantly, our study suggests relatively weaker electronic correlations in Mn3Ge compared to Mn3Sn.

Giant Superlinear Power Dependence of Photocurrent Based on Layered Ta2 NiS5 Photodetector.

Photodetector based on two-dimensional (2D) materials is an ongoing quest in optoelectronics. 2D photodetectors are generally efficient at low illuminating power but suffer severe recombination processes at high power, which results in the sublinear power-dependent photoresponse and lower optoelectronic efficiency. The desirable superlinear photocurrent is mostly achieved by sophisticated 2D heterostructures or device arrays, while 2D materials rarely show intrinsic superlinear photoresponse. This work reports the giant superlinear power dependence of photocurrent based on multilayer Ta2 NiS5 . While the fabricated photodetector exhibits good sensitivity (3.1 mS W-1 per □) and fast photoresponse (31 µs), the bias-, polarization-, and spatial-resolved measurements point to an intrinsic photoconductive mechanism. By increasing the incident power density from 1.5 to 200 µW µm-2 , the photocurrent power dependence varies from sublinear to superlinear. At higher illuminating conditions, prominent superlinearity is observed with a giant power exponent of γ = 1.5. The unusual photoresponse can be explained by a two-recombination-center model where density of states of the recombination centers (RC) effectively closes all recombination channels. The photodetector is integrated into camera for taking photos with enhanced contrast due to superlinearity. This work provides an effective route to enable higher optoelectronic efficiency at extreme conditions.

Modulation-doping a correlated electron insulator.

Correlated electron materials (CEMs) host a rich variety of condensed matter phases. Vanadium dioxide (VO2) is a prototypical CEM with a temperature-dependent metal-to-insulator (MIT) transition with a concomitant crystal symmetry change. External control of MIT in VO2-especially without inducing structural changes-has been a long-standing challenge. In this work, we design and synthesize modulation-doped VO2-based thin film heterostructures that closely emulate a textbook example of filling control in a correlated electron insulator. Using a combination of charge transport, hard X-ray photoelectron spectroscopy, and structural characterization, we show that the insulating state can be doped to achieve carrier densities greater than 5 × 1021 cm-3 without inducing any measurable structural changes. We find that the MIT temperature (TMIT) continuously decreases with increasing carrier concentration. Remarkably, the insulating state is robust even at doping concentrations as high as ~0.2 e-/vanadium. Finally, our work reveals modulation-doping as a viable method for electronic control of phase transitions in correlated electron oxides with the potential for use in future devices based on electric-field controlled phase transitions.

Non-linear Hall effect in multi-Weyl semimetals.

In the presence of time reversal symmetry, a non-linear Hall effect can occur in systems without an inversion symmetry. One of the prominent candidates for detection of such Hall signals are Weyl semimetals. In this article, we investigate the Berry curvature induced second and third order Hall effect in multi-Weyl semimetals with topological chargesn=1,2,3. We use low energy effective models to obtain general analytical expressions and discover the presence of a large Berry curvature dipole (BCD) in multi-Weyl semimetals, compared to usual (n = 1) Weyl semimetals. We also study the BCD in a realistic tight-binding lattice model and observe two different kinds of variation with increasing topological charge-these can be attributed to different underlying Berry curvature components. We provide estimates of the signatures of second harmonic of Hall signal in multi-Weyl semimetals, which can be detected experimentally. Furthermore, we predict the existence of a third order Hall signal in multi-Weyl semimetals. We derive the analytical expressions of Berry connection polarizability tensor, which is responsible for third order effects, using a low energy model and estimate the measurable conductivity. Our work can help guide experimental discovery of Berry curvature multipole physics in multi-Weyl semimetals.

Tuning 2D magnetism in Fe3+XGeTe2 films by element doping.

Two-dimensional (2D) ferromagnetic materials have been discovered with tunable magnetism and orbital-driven nodal-line features. Controlling the 2D magnetism in exfoliated nanoflakes via electric/magnetic fields enables a boosted Curie temperature (T C) or phase transitions. One of the challenges, however, is the realization of high T C 2D magnets that are tunable, robust and suitable for large scale fabrication. Here, we report molecular-beam epitaxy growth of wafer-scale Fe3+XGeTe2 films with T C above room temperature. By controlling the Fe composition in Fe3+XGeTe2, a continuously modulated T C in a broad range of 185-320 K has been achieved. This widely tunable T C is attributed to the doped interlayer Fe that provides a 40% enhancement around the optimal composition X = 2. We further fabricated magnetic tunneling junction device arrays that exhibit clear tunneling signals. Our results show an effective and reliable approach, i.e. element doping, to producing robust and tunable ferromagnetism beyond room temperature in a large-scale 2D Fe3+XGeTe2 fashion.