Catalogue of flat-band stoichiometric materials

Ferromagnetism and correlated insulating states in monolayer Mo33Te56

Although the kagome model is fundamentally two-dimensional, the essential kagome physics, i.e., the kagome-bands-driven emergent electronic states, has yet to be explored in the monolayer limit. Here, we present the experimental realization of kagome physics in monolayer Mo33Te56, showcasing both ferromagnetic ordering and a correlated insulating state with an energy gap of up to 15 meV. Using a combination of scanning tunnelling microscopy and theoretical calculations, we find a structural phase of the monolayer Mo-Te compound, which forms a mirror-twin boundary loop superlattice exhibiting kagome geometry and multiple sets of kagome bands. The partial occupancy of these nearly flat bands results in Fermi surface instability, counteracted by the emergence of ferromagnetic order (with a coercive field ~0.1 T, as observed by spin-polarized STM) and the opening of a correlated hard gap. Our work establishes a robust framework featuring well-defined atomic and band structures, alongside the intrinsic two-dimensional nature, essential for the rigorous examination of kagome physics.

General Method to Construct Flat Bands in Two-Dimensional Lattices

Searching for new materials hosting flat bands is pivotal for exploring strongly correlated effects and designing sensitive quantum devices, but remains challenging. We present a general method for realizing flat bands based on mathematical optimization and symmetry analysis. The method enables the discovery of ∼1000 types of two-dimensional lattices that can host flat bands, in sharp contrast with ∼10 flat-band lattices predicted previously besides the well-known ones. We further verify the method using first-principles calculations. Our approach provides new insights for the design of flat-band lattices, particularly when aiming to create experimentally feasible configurations.

1D Flat Bands in Phosphorene Nanoribbons with Pentagonal Nature

Materials with flat bands can serve as a promising platform to investigate strongly interacting phenomena. However, experimental realization of ideal flat bands is mostly limited to artificial lattices or moiré systems. Here, a general way is reported to construct 1D flat bands in phosphorene nanoribbons (PNRs) with a pentagonal nature: penta-hexa-PNRs and penta-dodeca-PNRs, wherein the corresponding 1D flat bands are directly verified by using angle-resolved photoemission spectroscopy. It is confirmed that the observed 1D flat bands originate from the electronic 1D zigzag and Lieb lattices, respectively, as revealed by the combination of bond-resolved scanning tunneling microscopy, scanning tunneling spectroscopy, tight-binding models, and first-principles calculations. The study demonstrates a general way to construct 1D flat bands in 1D solid materials system, which provides a robust platform to explore strongly interacting phases of matter.

Elf autoencoder for unsupervised exploration of flat-band materials using electronic band structure fingerprints

Two-dimensional materials with flat electronic bands are promising for realising exotic quantum phenomena such as unconventional superconductivity and nontrivial topology. However, exploring their vast chemical space is a significant challenge. Here we introduce elf, an unsupervised convolutional autoencoder that encodes electronic band structure images into fingerprint vectors, enabling the autonomous clustering of materials by electronic properties beyond traditional chemical paradigms. Unsupervised visualisation of the fingerprint space then uncovers hidden chemical trends and identifies promising candidates based on similarities to well-studied exemplars. This approach complements high-throughput ab initio methods by rapidly screening candidates and guiding further investigations into the mechanisms underlying flat-band physics. The elf autoencoder is a powerful tool for autonomous discovery of unexplored flat-band materials, enabling unbiased identification of compounds with desirable electronic properties across the 2D chemical space.

Solution Processable Metal-Organic Frameworks: Synthesis Strategy and Applications

Metal-organic frameworks (MOFs), constructed by inorganic secondary building units with organic linkers via reticular chemistry, inherently suffer from poor solution processability due to their insoluble nature, resulting from their extensive crystalline networks and structural rigidity. The ubiquitous occurrence of precipitation and agglomeration of MOFs upon formation poses a significant obstacle to the scale-up production of MOF-based monolith, aerogels, membranes, and electronic devices, thus restricting their practical applications in various scenarios. To address the previously mentioned challenge, significant strides have been achieved over the past decade in the development of various strategies aimed at preparing solution-processable MOF systems. In this review, the latest advance in the synthetic strategies for the construction of solution-processable MOFs, including direct dispersion in ionic liquids, surface modification, controllable calcination, and bottom-up synthesis, is comprehensively summarized. The respective advantages and disadvantages of each method are discussed. Additionally, the intriguing applications of solution-processable MOF systems in the fields of liquid adsorbent, molecular capture, sensing, and separation are systematically discussed. Finally, the challenges and opportunities about the continued advancement of solution-processable MOFs and their potential applications are outlooked.

Flat-band based high-temperature ferromagnetic semiconducting state in the graphitic C4N3 monolayer

Half-filled isolated flat-band paves a new way to realize high-temperature ferromagnetic semiconductor for spintronics applications, but it is extremely rare in lattice models and lacking in realistic materials. Herein, the 2 × 2 super-cell of the honeycomb lattice with a single-hole defect is proposed as a new lattice model (HL-D-1/8) to realize nontrivial isolated flat-bands. We further demonstrate that C4N3 monolayer of the experimentally realized graphitic carbon nitride (Adv. Mater., 22, 1004, 2010; Nat. Commun., 9, 3366, 2018) is a perfect system holding such a lattice to host flat-bands. A new corrugated Pca21 configuration is proposed as the ground state for the free-standing C4N3 monolayer, which is dynamically stable and energetically more favorable than the widely-used flat one without dynamical stability. The Pca21 configuration is found to be an intrinsic ferromagnetic half-semiconductor (T c ≈ 241 K) with one semiconducting spin-channel (1.75 eV) and one insulating spin-channel (3.64 eV), which is quite rare in two-dimensional systems. Its ferromagnetic semiconducting property originates from the isolated pz -state flat-band as the corrugation shifts the flat-band upward to the Fermi level. Interestingly, such a corrugated Pca21 C4N3 monolayer is found to be both piezoelectric and ferroelectric, which makes it an unusual metal-free two-dimensional multiferroic.

Three-Dimensional Multiorbital Flat Band Models and Materials

Flat band (FB) systems are essential for uncovering exotic quantum phenomena associated with strong electron correlations. Here we present a systematic theoretical framework for constructing multiorbital FB models and identifying feasible material candidates. This framework integrates group theory and crystallography into a symmetry-adapted tight-binding model incorporating lattice, site, and orbital degrees of freedom. Using this approach, we unveil a novel three-dimensional (3D) multiorbital FB model in the face-centered-cubic lattice, distinct from well-known single-orbital Lieb and kagome models. Critically, we identify numerous high-quality binary materials with ultraclean 3D FBs near the Fermi level. Furthermore, we explore diverse orbital bases within this model and extend our analysis to other cubic lattices with different space groups, broadening the scope for realizing 3D multiorbital FB systems. Our findings provide a foundational platform for exploring correlated physics in multiorbital FB systems and guiding the discovery of new quantum materials.

Multifold Topological Point with Quadratic Order in Binary Skutterudite Rhodium Triarsenide

The exploration of topological nodal point states has recently evolved, moving beyond traditional linear crossings to include higher-order dispersions and multifold degeneracies. This study utilizes first-principles calculations to uncover an ideal multifold nodal point of quadratic order in the binary skutterudite rhodium triarsenide. The band structures around this nodal point show not only simple configuration but also clean distribution. Notably, a type-III dispersion condition has also been identified. When considering the effects of spin-orbit coupling, the nodal point retains both its multiple degeneracy and quadratic characteristics, although the band degeneracy transitions from 3-fold to quadruple. Detailed symmetry argument and model analysis have been provided, and precise band surface distribution has been obtained. Furthermore, the material is characterized by multiple significant arc surface states, as confirmed by the projected topological surface states. The clear separation of these states from the bulk bands facilitates experimental investigation. In summary, the multifold nodal point state identified in this research, along with the corresponding material candidate, presents an exceptional platform for the further study of higher-order topological point states, potentially catalyzing advancements in this emerging field.

On the engineering of higher-order Van Hove singularities in two dimensions

The properties of correlated electron materials are often intricately linked to Van Hove singularities (VHS) in the vicinity of the Fermi energy. The class of these VHS is of great importance, with higher-order ones-with power-law divergence in the density of states-leaving frequently distinct signatures in physical properties. We use a new theoretical method to detect and analyse higher-order VHS (HOVHS) in two-dimensional materials and apply it to the electronic structure of the surface layer of Sr2RuO4. We then constrain a low energy model of the VHS of the surface layer of Sr2RuO4 against angle-resolved photoemission spectroscopy and quasiparticle interference data to analyse the VHS near the Fermi level. We show how these VHS can be engineered into HOVHS.

Persistent flat band splitting and strong selective band renormalization in a kagome magnet thin film

Magnetic kagome materials provide a fascinating playground for exploring the interplay of magnetism, correlation and topology. Many magnetic kagome systems have been reported including the binary FemXn (X = Sn, Ge; m:n = 3:1, 3:2, 1:1) family and the rare earth RMn6Sn6 (R = rare earth) family, where their kagome flat bands are calculated to be near the Fermi level in the paramagnetic phase. While partially filling a kagome flat band is predicted to give rise to a Stoner-type ferromagnetism, experimental visualization of the magnetic splitting across the ordering temperature has not been reported for any of these systems due to the high ordering temperatures, hence leaving the nature of magnetism in kagome magnets an open question. Here, we probe the electronic structure with angle-resolved photoemission spectroscopy in a kagome magnet thin film FeSn synthesized using molecular beam epitaxy. We identify the exchange-split kagome flat bands, whose splitting persists above the magnetic ordering temperature, indicative of a local moment picture. Such local moments in the presence of the topological flat band are consistent with the compact molecular orbitals predicted in theory. We further observe a large spin-orbital selective band renormalization in the Fed x y + d x 2 - y 2 spin majority channel reminiscent of the orbital selective correlation effects in the iron-based superconductors. Our discovery of the coexistence of local moments with topological flat bands in a kagome system echoes similar findings in magic-angle twisted bilayer graphene, and provides a basis for theoretical effort towards modeling correlation effects in magnetic flat band systems.

Diradicals as Topological Charge Carriers in Metal-Organic Toy Model Pt3(HIB)2

We explore the eclipsed stacking of a metal-organic Kagome lattice containing heavy-metal nodes. Our model is Pt3(HIB)2, a hypothetical but viable member of a well-known family of hexaaminobenzene based metal-organic frameworks (MOFs). Applying space group theory, it is shown how molecular diradicals, brought into play by a noninnocent ligand, become topologically nontrivial bands when moving in a periodic potential. Three factors are required to enable this: (1) eclipsed stacking, which shifts the Fermi level near a symmetry-protected band crossing (2) the emergence of an electride-like band that renders the topological Z 2 invariant equal to 1, thus nontrivial, and (3) Pt-induced spin-orbit coupling, to turn the crossing into a bulk band gap. The electride band, with its unforeseen role, bears kinship to the interlayer band in hexagonal superconductors. It places its charge density in the voids of the crystal, rather than around the atomic nuclei, and we name it a "pore band". While the synthesis of truly conductive MOFs has proven challenging, the analysis shows that intrinsically nonlocal physics may emerge from tunable molecular building blocks. With the richness of redox-active MOF chemistry, this offers a pathway to tailored topological electronics.

Prediction of metal-free Stoner and Mott-Hubbard magnetism in triangulene-based two-dimensional polymers

Ferromagnetism and antiferromagnetism require robust long-range magnetic ordering, which typically involves strongly interacting spins localized at transition metal atoms. However, in metal-free systems, the spin orbitals are largely delocalized, and weak coupling between the spins in the lattice hampers long-range ordering. Metal-free magnetism is of fundamental interest to physical sciences, unlocking unprecedented dimensions for strongly correlated materials and biocompatible magnets. Here, we present a strategy to achieve strong coupling between spin centers of planar radical monomers in π-conjugated two-dimensional (2D) polymers and rationally control the orderings. If the π-states in these triangulene-based 2D polymers are half-occupied, then we predict that they are antiferromagnetic Mott-Hubbard insulators. Incorporating a boron or nitrogen heteroatom per monomer results in Stoner ferromagnetism and half-metallicity, with the Fermi level located at spin-polarized Dirac points. An unprecedented antiferromagnetic half-semiconductor is observed in a binary boron-nitrogen-centered 2D polymer. Our findings pioneer Stoner and Mott-Hubbard magnetism emerging in the electronic π-system of crystalline-conjugated 2D polymers.

Realization of a 2D Lieb Lattice in a Metal-Inorganic Framework with Partial Flat Bands and Topological Edge States

Flat bands and Dirac cones in materials are the source of the exotic electronic and topological properties. The Lieb lattice is expected to host these electronic structures, arising from quantum destructive interference. Nevertheless, the experimental realization of a 2D Lieb lattice remained challenging to date due to its intrinsic structural instability. After computationally designing a Platinum-Phosphorus (Pt-P) Lieb lattice, it has successfully overcome its structural instability and synthesized on a gold substrate via molecular beam epitaxy. Low-temperature scanning tunneling microscopy and spectroscopy verify the Lieb lattice's morphology and electronic flat bands. Furthermore, topological Dirac edge states stemming from pronounced spin-orbit coupling induced by heavy Pt atoms are predicted. These findings convincingly open perspectives for creating metal-inorganic framework-based atomic lattices, offering prospects for strongly correlated phases interplayed with topology.

Accessing bands with extended quantum metric in kagome Cs2Ni3S4 through soft chemical processing

Flat bands that do not merely arise from weak interactions can produce exotic physical properties, such as superconductivity or correlated many-body effects. The quantum metric can differentiate whether flat bands will result in correlated physics or are merely dangling bonds. A potential avenue for achieving correlated flat bands involves leveraging geometrical constraints within specific lattice structures, such as the kagome lattice; however, materials are often more complex. In these cases, quantum geometry becomes a powerful indicator of the nature of bands with small dispersions. We present a simple, soft-chemical processing route to access a flat band with an extended quantum metric below the Fermi level. By oxidizing Ni-kagome material Cs2Ni3S4 to CsNi3S4, we see a two orders of magnitude drop in the room temperature resistance. However, CsNi3S4 is still insulating, with no evidence of a phase transition. Using experimental data, density functional theory calculations, and symmetry analysis, our results suggest the emergence of a correlated insulating state of unknown origin.

Decoding the Drive-Bath Interplay: A Guideline to Enhance Superconductivity

Driven-dissipative physics lie at the core of quantum optics. However, the full interplay between a driven quantum many-body system and its environment remains relatively unexplored in the solid state realm. In this Letter, we inspect this interplay beyond the commonly employed stroboscopic Hamiltonian picture based on the specific example of a driven superconductor. Using the Shirley-Floquet and Keldysh formalisms as well as a generalization of the notion of superconducting fitness to the driven case, we show how a drive which anticommutes with the superconducting gap operator generically induces an unusual particle-hole structure in the spectral functions from the perspective of the thermal bath. Concomitant with a driving frequency which is near resonant with the intrinsic cutoff frequency of the underlying interaction, this spectral structure can be harnessed to enhance the superconducting transition temperature. Our work paves the way for further studies for driven-dissipative engineering of exotic phases of matter in solid-state systems.

Exploring flat-band properties in two-dimensional M3QX7 compounds

We present a computational study of the M3QX7 family of two-dimensional compounds, focusing specifically on their flat-band properties. We use a high-throughput search methodology, accelerated by machine learning, to explore the vast chemical space spawned by this family. In this way, we identify numerous stable 2D compounds within the M3QX7 family. We investigate how the chemical composition can be manipulated to modulate the position and dispersion of the flat bands. By employing a tight-binding model we explain the formation of flat bands as a result of a relatively loose connection between triangular M3QX3 clusters via bridges of X atoms. The model provides an understanding of the residual interactions that can impact the band dispersion. The same loose connection between clusters not only leads to strongly localized electronic states and thus flat electronic bands but also leads to localized phonon modes and flat bands in the phonon dispersion.

Emergent flat band and topological Kondo semimetal driven by orbital-selective correlations

Flat electronic bands are expected to show proportionally enhanced electron correlations, which may generate a plethora of novel quantum phases and unusual low-energy excitations. They are increasingly being pursued in d-electron-based systems with crystalline lattices that feature destructive electronic interference, where they are often topological. Such flat bands, though, are generically located far away from the Fermi energy, which limits their capacity to partake in the low-energy physics. Here we show that electron correlations produce emergent flat bands that are pinned to the Fermi energy. We demonstrate this effect within a Hubbard model, in the regime described by Wannier orbitals where an effective Kondo description arises through orbital-selective Mott correlations. Moreover, the correlation effect cooperates with symmetry constraints to produce a topological Kondo semimetal. Our results motivate a novel design principle for Weyl Kondo semimetals in a new setting, viz. d-electron-based materials on suitable crystal lattices, and uncover interconnections among seemingly disparate systems that may inspire fresh understandings and realizations of correlated topological effects in quantum materials and beyond.

Rashba-splitting-induced topological flat band detected by anomalous resistance oscillations beyond the quantum limit in ZrTe5

Topological flat bands - where the kinetic energy of electrons is quenched - provide a platform for investigating the topological properties of correlated systems. Here, we report the observation of a topological flat band formed by polar-distortion-assisted Rashba splitting in the three-dimensional Dirac material ZrTe5. The polar distortion and resulting Rashba splitting on the band are directly detected by torque magnetometry and the anomalous Hall effect, respectively. The local symmetry breaking further flattens the band, on which we observe resistance oscillations beyond the quantum limit. These oscillations follow the temperature dependence of the Lifshitz-Kosevich formula but are evenly distributed in B instead of 1/B at high magnetic fields. Furthermore, the cyclotron mass gets anomalously enhanced about 102 times at fields ~ 20 T. Our results provide an intrinsic platform without invoking moiré or order-stacking engineering, which opens the door for studying topologically correlated phenomena beyond two dimensions.

Anomalous Hall effect and magnetic transition in the kagome material YbMn6Sn6

We investigate the magnetic and transport properties of a kagome magnet YbMn6Sn6. We have grown YbMn6Sn6single crystals having a HfFe6Ge6type structure with ordered Yb and Sn atoms, which is different from the distorted structure previously reported. The single crystal undergoes a ferromagnetic phase transition around 300 K and a ferrimagnetic transition at approximately 30 K, and the magnetic transition at low temperature may be correlated to the ordered Yb sublattice. Negative magnetoresistance has been observed at high temperatures, while the positive magnetoresistance appears at 5 K when the current is oriented out of kagome plane. We observe a large anisotropic anomalous Hall effect with the intrinsic Hall contribution of 141 Ω-1cm-1forσzxintand 32 Ω-1cm-1forσxyint, respectively. These values are similar to those in YMn6Sn6with the same anisotropy. The magnetic transition and anomalous Hall behavior in YbMn6Sn6highlights the influence of the ordered Yb atoms and rare earth elements on its magnetic and transport properties.

Reversible non-volatile electronic switching in a near-room-temperature van der Waals ferromagnet

Non-volatile phase-change memory devices utilize local heating to toggle between crystalline and amorphous states with distinct electrical properties. Expanding on this kind of switching to two topologically distinct phases requires controlled non-volatile switching between two crystalline phases with distinct symmetries. Here, we report the observation of reversible and non-volatile switching between two stable and closely related crystal structures, with remarkably distinct electronic structures, in the near-room-temperature van der Waals ferromagnet Fe5-δGeTe2. We show that the switching is enabled by the ordering and disordering of Fe site vacancies that results in distinct crystalline symmetries of the two phases, which can be controlled by a thermal annealing and quenching method. The two phases are distinguished by the presence of topological nodal lines due to the preserved global inversion symmetry in the site-disordered phase, flat bands resulting from quantum destructive interference on a bipartite lattice, and broken inversion symmetry in the site-ordered phase.

Tuning of the flat band and its impact on superconductivity in Mo5Si3-xPx

The superconductivity in systems containing dispersionless (flat) bands is seemingly paradoxical, as traditional Bardeen-Cooper-Schrieffer theory requires an infinite enhancement of the carrier masses. However, the combination of flat and steep (dispersive) bands within the multiple band scenario might boost superconducting responses, potentially explaining high-temperature superconductivity in cuprates and metal hydrides. Here, we report on the magnetic penetration depths, the upper critical field, and the specific heat measurements, together with the first-principles calculations for the Mo5Si3-xPx superconducting family. The band structure features a flat band that gradually approaches the Fermi level as a function of phosphorus doping x, reaching the Fermi level at x ≃ 1.3. This leads to an abrupt change in nearly all superconducting quantities. The superfluid density data placed on the 'Uemura plot' results in two separated branches, thus indicating that the emergence of a flat band enhances correlations between conducting electrons.

Decoding flat bands from compact localized states

The flat band system is an ideal quantum platform to investigate the kaleidoscope created by the electron-electron correlation effects. The central ingredient of realizing a flat band is to find its compact localized states. In this work, we develop a systematic way to generate compact localized states by designing destructive interference patterns from 1-dimensional chains. A variety of 2-dimensional new flat band systems are constructed with this method. Furthermore, we show that the method can be extended to generate the compact localized states in multi-orbital systems by carefully designing the block hopping scheme, as well as in quasicrystal and disorder systems.

Three-dimensional flat bands in pyrochlore metal CaNi2

Electronic flat-band materials host quantum states characterized by a quenched kinetic energy. These flat bands are often conducive to enhanced electron correlation effects and emergent quantum phases of matter1. Long studied in theoretical models2-4, these systems have received renewed interest after their experimental realization in van der Waals heterostructures5,6 and quasi-two-dimensional (2D) crystalline materials7,8. An outstanding experimental question is if such flat bands can be realized in three-dimensional (3D) networks, potentially enabling new materials platforms9,10 and phenomena11-13. Here we investigate the C15 Laves phase metal CaNi2, which contains a nickel pyrochlore lattice predicted at a model network level to host a doubly-degenerate, topological flat band arising from 3D destructive interference of electronic hopping14,15. Using angle-resolved photoemission spectroscopy, we observe a band with vanishing dispersion across the full 3D Brillouin zone that we identify with the pyrochlore flat band as well as two additional flat bands that we show arise from multi-orbital interference of Ni d-electrons. Furthermore, we demonstrate chemical tuning of the flat-band manifold to the Fermi level that coincides with enhanced electronic correlations and the appearance of superconductivity. Extending the notion of intrinsic band flatness from 2D to 3D, this provides a potential pathway to correlated behaviour predicted for higher-dimensional flat-band systems ranging from tunable topological15 to fractionalized phases16.

Reveal Ultrafast Electron Relaxation across Sub-bands of Tellurium by Time- and Energy-Resolved Photoemission Microscopy

Exploring ultrafast carrier dynamics is crucial for the materials' fundamental properties and device design. In this work, we employ time- and energy-resolved photoemission electron microscopy with tunable pump wavelengths from visible to near-infrared to reveal the ultrafast carrier dynamics of the elemental semiconductor tellurium. We find that two discrete sub-bands around the Γ point of the conduction band are involved in excited-state electron ultrafast relaxation and reveal that hot electrons first go through ultrafast intra sub-band cooling on a time scale of about 0.3 ps and then transfer from the higher sub-band to the lower one on a time scale of approximately 1 ps. Additionally, theoretical calculations reveal that the lower one has flat-band characteristics, possessing a large density of states and a long electron lifetime. Our work demonstrates that TR- and ER-PEEM with broad tunable pump wavelengths are powerful techniques in revealing the details of ultrafast carrier dynamics in time and energy domains.

Inducing Single Spin-Polarized Flat Bands in Monolayer Graphene

Due to the fundamental and technological implications in driving the appearance of non-trivial, exotic topological spin textures and emerging symmetry-broken phases, flat electronic bands in 2D materials, including graphene, are nowadays a relevant topic in the field of spintronics. Here, via europium doping, single spin-polarized bands are generated in monolayer graphene supported by the Co(0001) surface. The doping is controlled by Eu positioning, allowing for the formation of aK ¯ $\bar{\mathrm{K}}$ -valley localized single spin-polarized low-dispersive parabolic band close to the Fermi energy when Eu is on top, and of a π* flat band with single spin character when Eu is intercalated underneath graphene. In the latter case, Eu also induces a bandgap opening at the Dirac point while the Eu 4f states act as a spin filter, splitting the π band into two spin-polarized branches. The generation of flat bands with single spin character, as revealed by the spin- and angle-resolved photoemission spectroscopy (ARPES) experiments, complemented by density functional theory (DFT) calculations, opens up new pathways toward the realization of spintronic devices exploiting such novel exotic electronic and magnetic states.

Topological Superconductors from a Materials Perspective

Topological superconductors (TSCs) have garnered significant research and industry attention in the past two decades. By hosting Majorana bound states which can be used as qubits that are robust against local perturbations, TSCs offer a promising platform toward (nonuniversal) topological quantum computation. However, there has been a scarcity of TSC candidates, and the experimental signatures that identify a TSC are often elusive. In this Perspective, after a short review of the TSC basics and theories, we provide an overview of the TSC materials candidates, including natural compounds and synthetic material systems. We further introduce various experimental techniques to probe TSCs, focusing on how a system is identified as a TSC candidate and why a conclusive answer is often challenging to draw. We conclude by calling for new experimental signatures and stronger computational support to accelerate the search for new TSC candidates.

1D Electronic Flat Bands in Untwisted Moiré Superlattices

After the preparation of 2D electronic flat band (EFB) in van der Waals (vdW) superlattices, recent measurements suggest the existence of 1D electronic flat bands (1D-EFBs) in twisted vdW bilayers. However, the realization of 1D-EFBs is experimentally elusive in untwisted 2D layers, which is desired considering their fabrication and scalability. Herein, the discovery of 1D-EFBs is reported in an untwisted in situ-grown two atomic-layer Bi(110) superlattice self-aligned on an SnSe(001) substrate using scanning probe microscopy measurements and density functional theory calculations. While the Bi-Bi dimers of Bi zigzag (ZZ) chains are buckled, the epitaxial lattice mismatch between the Bi and SnSe layers induces two 1D buckling reversal regions (BRRs) extending along the ZZ direction in each Bi(110)-11 × 11 supercell. A series of 1D-EFBs arises spatially following BRRs that isolate electronic states along the armchair (AC) direction and localize electrons in 1D extended states along ZZ due to quantum interference at a topological node. This work provides a generalized strategy for engineering 1D-EFBs in utilizing lattice mismatch between untwisted rectangular vdW layers.

First-Principles Insights into the Relative Stability, Physical Properties, and Chemical Properties of MoSe2

A fascinating transition-metal dichalcogenide (TMDC) compound, MoSe₂, has attracted a lot of interest in electrochemical, photocatalytic, and optoelectronic systems. However, detailed studies on the structural stability of the various MoSe₂ polymorphs are still lacking. For the first time, the relative stability of 11 different MoSe₂ polymorphs (1H, 2H, 3H_a, 3H_b, 2T, 4T, 2R₁, 1T₁, 1T₂, 3T, and 2R₂) is proposed, and a detailed analysis of these polymorphs is carried out by employing the first-principles calculations based on density functional theory (DFT). We computed the physical properties of the polymorphs such as band structure, phonon, and elastic constants to examine the viability for real-world applications. The electronic properties of the involved polymorphs were calculated by employing the hybrid functional of Heyd, Scuseria, and Ernzerhof (HSE06). The energy band gap of the polymorphs (1H, 2H, 3H_a, 3H_b, 2T, 4T, and 2R₁) is in the range of 1.6-1.8 eV, coinciding with the experimental value for the polymorph 2H. The covalent bonding nature of MoSe₂ is analyzed from the charge density, charge transfer, and electron localization function. Among the 11 polymorphs, 1H, 2H, 2T, and 3H_b polymorphs are predicted as stable polymorphs based on the calculation of the mechanical and dynamical properties. Even though the 4T and 3H_a polymorphs' phonons are stable, they are mechanically unstable; hence, they are considered to be under a metastable condition. Additionally, we computed the direction-dependent elastic moduli and isotropic factors for both mechanically and dynamically stable polymorphs. Stable polymorphs are analyzed spectroscopically using IR and Raman spectra. The thermal stability of the polymorphs is also studied.

Flat band ferromagnetism in Pb[Formula: see text]Sb[Formula: see text]O[Formula: see text] via a self-doped mechanism

Electron systems with strong geometrical frustrations have flat bands, and their unusual band dispersions are expected to induce a wide variety of physical properties. However, for the emergence of such properties, the Fermi level must be pinned within the flat band. In this study, we performed first-principles calculations on pyrochlore oxide Pb[Formula: see text]Sb[Formula: see text]O[Formula: see text] and theoretically clarified that the self-doping mechanism induces pinning of the Fermi level in the flat band in this system. Therefore, a very high density of states is realized at the Fermi level, and the ferromagnetic state transforms into the ground state via a flat band mechanism, although the system does not contain any magnetic elements. This compound has the potential to serve as a new platform for projecting the properties of flat band systems in the real world.

A carbon allotrope with twisted Dirac cones induced by grain boundaries composed of pentagons and octagons

The grain boundaries (GBs) composed of pentagons and octagons (558 GBs) have been demonstrated to induce attractive transport properties such as Van Hove singularity (VHS) and quasi-one-dimensional metallic wires. Here, we propose a monolayer carbon allotrope which is formed from the introduction of periodic 558 GBs to decorate intact graphene, termed as PHO-graphene. The calculated electronic properties indicate that PHO-graphene not only inherits the previously superior characteristics such as Van Hove singularity and quasi-one-dimensional metallic wires, but also possesses two twisted Dirac cones near the Fermi level. Further calculation finds that the Berry phase is quantized to ± π at the two Dirac points, which is consistent with the distribution of the corresponding Berry curvature. The parity argument uncovers that PHO-graphene hosts a nontrivial band topology and the edge states connecting the two Dirac points are clearly visible. Our findings not only provide a reliable avenue to realize the abundant and extraordinary properties of carbon allotropes, but also offer an attractive approach for designing all carbon-based devices.

Topological Flat Bands in Self-Complementary Plasmonic Metasurfaces

Photonics can be confined in real space with dispersion vanishing in the momentum space due to destructive interference. In this Letter, we report the experimental realization of flat bands with nontrivial topology in a self-complementary plasmonic metasurface. The band diagram and compact localized states are measured. In these nontrivial band gaps, we observe the topological edge states by near-field measurements. Furthermore, we propose a digitalized metasurface by loading controllable diodes with C_{3} symmetry in every unit cell. By pumping a digital signal into the metasurface, we investigate the interaction between incident waves and the dynamic metasurface. Experimental results indicate that compact localized states in the nontrivial flat band could enhance the wave-matter interactions to convert more incident waves to time-modulated harmonic photonics. Although our experiments are conducted in the microwave regime, extending the related concepts into the optical plasmonic systems is feasible. Our findings pave an avenue toward planar integrated photonic devices with nontrivial flat bands and exotic transmission phenomena.

Topological kagome magnets and superconductors

A kagome lattice naturally features Dirac fermions, flat bands and van Hove singularities in its electronic structure. The Dirac fermions encode topology, flat bands favour correlated phenomena such as magnetism, and van Hove singularities can lead to instabilities towards long-range many-body orders, altogether allowing for the realization and discovery of a series of topological kagome magnets and superconductors with exotic properties. Recent progress in exploring kagome materials has revealed rich emergent phenomena resulting from the quantum interactions between geometry, topology, spin and correlation. Here we review these key developments in this field, starting from the fundamental concepts of a kagome lattice, to the realizations of Chern and Weyl topological magnetism, to various flat-band many-body correlations, and then to the puzzles of unconventional charge-density waves and superconductivity. We highlight the connection between theoretical ideas and experimental observations, and the bond between quantum interactions within kagome magnets and kagome superconductors, as well as their relation to the concepts in topological insulators, topological superconductors, Weyl semimetals and high-temperature superconductors. These developments broadly bridge topological quantum physics and correlated many-body physics in a wide range of bulk materials and substantially advance the frontier of topological quantum matter.

Upper limit of the transition temperature of superconducting materials

Why are the transition temperatures (T c) of superconducting materials so different? The answer to this question is not only of great significance in revealing the mechanism of high-T c superconductivity but also can be used as a guide for the design of new superconductors. However, so far, it is still challenging to identify the governing factors affecting the T c. In this work, with the aid of machine learning and first-principles calculations, we found a close relevance between the upper limit of the T c and the energy-level distribution of valence electrons. It implies that some additional inter-orbital electron-electron interaction should be considered in the interpretation of high-T c superconductivity.

High-Throughput Investigations of Topological and Nodal Superconductors

The theory of symmetry indicators has enabled database searches for topological materials in normal conducting phases, which has led to several encyclopedic topological material databases. To date, such a database for topological superconductors is yet to be achieved because of the lack of information about pairing symmetries of realistic materials. In this Letter, sidestepping this issue, we tackle an alternative problem: the predictions of topological and nodal superconductivity in materials for each single-valued representation of point groups. Based on recently developed symmetry indicators for superconductors, we provide comprehensive mappings from pairing symmetries to the topological or nodal superconducting nature for nonmagnetic materials listed in the Inorganic Crystal Structure Database. We quantitatively show that around 90% of computed materials are topological or nodal superconductors when a pairing that belongs to a one-dimensional nontrivial representation of point groups is assumed. When materials are representation-enforced nodal superconductors, positions and shapes of the nodes are also identified. When combined with experiments, our results will help us understand the pairing mechanism and facilitate realizations of the long-sought Majorana fermions promising for topological quantum computations.

Peculiar Physics of Heavy-Fermion Metals: Theory versus Experiment

This review considers the topological fermion condensation quantum phase transition (FCQPT) that leads to flat bands and allows the elucidation of the special behavior of heavy-fermion (HF) metals that is not exhibited by common metals described within the framework of the Landau Fermi liquid (LFL) theory. We bring together theoretical consideration within the framework of the fermion condensation theory based on the FCQPT with experimental data collected on HF metals. We show that very different HF metals demonstrate universal behavior induced by the FCQPT and demonstrate that Fermi systems near the FCQPT are controlled by the Fermi quasiparticles with the effective mass M* strongly depending on temperature T, magnetic field B, pressure P, etc. Within the framework of our analysis, the experimental data regarding the thermodynamic, transport and relaxation properties of HF metal are naturally described. Based on the theory, we explain a number of experimental data and show that the considered HF metals exhibit peculiar properties such as: (1) the universal T/B scaling behavior; (2) the linear dependence of the resistivity on T, ρ(T)∝A1T (with A1 is a temperature-independent coefficient), and the negative magnetoresistance; (3) asymmetrical dependence of the tunneling differential conductivity (resistivity) on the bias voltage; (4) in the case of a flat band, the superconducting critical temperature Tc∝g with g being the coupling constant, while the M* becomes finite; (5) we show that the so called Planckian limit exhibited by HF metals with ρ(T)∝T is defined by the presence of flat bands.

Simple Chemical Rules for Predicting Band Structures of Kagome Materials

Compounds featuring a kagome lattice are studied for a wide range of properties, from localized magnetism to massless and massive Dirac Fermions. These properties come from the symmetry of the kagome lattice, which gives rise to Dirac cones and flat bands. However, not all compounds with a kagome sublattice show properties related to it. We derive chemical rules predicting if the low-energy physics of a material is determined by the kagome sublattice and bands arising from it. After sorting out all known crystals with the kagome lattice into four groups, we use chemical heuristics and local symmetry to explain additional conditions that need to be met to have kagome bands near the Fermi level.

Structural Diversity in Oxoiridates with 1D IrnO3(n+1) Chain Fragments and Flat Bands

A previously unreported series of hexagonal-perovskite-based Rb-oxoiridates, Rb₅Ir₂O₉, Rb₇Ir₃O₁₂, and Rb₁₂Ir₇O₂₄, have been synthesized and structurally analyzed via N₂-protected single-crystal X-ray diffraction (SC-XRD). These materials exhibit different 1D Ir_nO_3(n+1) chain fragments along their c axes. IrO₆ octahedra and RbO_x (x = 6, 8, and 10) polyhedra are their basic building blocks. The IrO₆ octahedra are linked via face-sharing, forming Ir₂O₉ dimers, Ir₃O₁₂ trimers, and Ir₇O₂₄ heptamers. The nonmagnetic RbO_x (x = 6, 8, and 10) polyhedra serve as both bridging units and spacers. Temperature-dependent SC-XRD shows all three to display positive thermal expansion and rules out structural transitions from their triangular symmetries down to 100 K. Density functional theory results suggest semiconducting-like behavior for the title compounds. The flatness of the electronic bands and our structural analysis are of potential interest for understanding and designing 1D quantum materials.

Strongly Correlated Quantum Spin Liquids versus Heavy Fermion Metals: A Review

This review considers the topological fermion condensation quantum phase transition (FCQPT) that explains the complex behavior of strongly correlated Fermi systems, such as frustrated insulators with quantum spin liquid and heavy fermion metals. The review contrasts theoretical consideration with recent experimental data collected on both heavy fermion metals (HF) and frustrated insulators. Such a method allows to understand experimental data. We also consider experimental data collected on quantum spin liquid in Lu3Cu2Sb3O14 and quasi-one dimensional (1D) quantum spin liquid in both YbAlO3 and Cu(C4H4N2)(NO3)2 with the aim to establish a sound theoretical explanation for the observed scaling laws, Landau Fermi liquid (LFL) and non-Fermi-liquid (NFL) behavior exhibited by these frustrated insulators. The recent experimental data on the heavy-fermion metal α−YbAl1−xFexB4, with x=0.014, and on its sister compounds β−YbAlB4 and YbCo2Ge4, carried out under the application of magnetic field as a control parameter are analyzed. We show that the thermodynamic and transport properties as well as the empirical scaling laws follow from the fermion condensation theory. We explain how both the similarity and the difference in the thermodynamic and transport properties of α−YbAl1−xFexB4 and in its sister compounds β−YbAlB4 and YbCo2Ge4 emerge, as well as establish connection of these (HF) metals with insulators Lu3Cu2Sb3O14, Cu(C4H4N2)(NO3)2 and YbAlO3. We demonstrate that the universal LFL and NFL behavior emerge because the HF compounds and the frustrated insulators are located near the topological FCQPT or are driven by the application of magnetic fields.

A Class of Magnetic Topological Material Candidates with Hypervalent Bi Chains

The link between crystal and electronic structure is crucial for understanding structure-property relations in solid-state chemistry. In particular, it has been instrumental in understanding topological materials, where electrons behave differently than they would in conventional solids. Herein, we identify 1D Bi chains as a structural motif of interest for topological materials. We focus on Sm₃ZrBi₅, a new quasi-one-dimensional (1D) compound in the Ln₃MPn₅ (Ln = lanthanide; M = metal; Pn = pnictide) family that crystallizes in the P6₃/mcm space group. Density functional theory calculations indicate a complex, topologically nontrivial electronic structure that changes significantly in the presence of spin-orbit coupling. Magnetic measurements show a quasi-1D antiferromagnetic structure with two magnetic transitions at 11.7 and 10.7 K that are invariant to applied field up to 9 T, indicating magnetically frustrated spins. Heat capacity, electrical, and thermoelectric measurements support this claim and suggest complex scattering behavior in Sm₃ZrBi₅. This work highlights 1D chains as an unexplored structural motif for identifying topological materials, as well as the potential for rich physical phenomena in the Ln₃MPn₅ family.