Existing theories of superconductivity emphasize either electron-electron or electron-phonon interaction, each of which tends to cancel the effect of the other. We present direct evidence from unbiased numerical calculations of cooperative, as opposed to competing, effects of electron-electron and electron-phonon interactions within the frustrated Hubbard Hamiltonian with bond-coupled phonons, uniquely at the band filling of one-quarter. The two interactions cooperatively reinforce $d$-wave superconducting pair-pair correlations at this filling while competing with one another at all other densities. Our work gives insight into how intertwined charge order and superconductivity appear in real materials.

We investigate the conditions under which periodically driven quantum systems subject to dissipation exhibit a stable subharmonic response. Noting that coupling to a bath introduces not only cooling but also noise, we point out that a system subject to the latter for the entire cycle tends to lose coherence of the subharmonic oscillations, and thereby the long-time temporal symmetry breaking. We provide an example of a short-ranged two-dimensional system which does not suffer from this and therefore displays persistent subharmonic oscillations stabilized by the dissipation. We also show that this is fundamentally different from the disordered discrete time crystal previously found in closed systems, both conceptually and in its phenomenology. The framework we develop here clarifies how fully connected models constitute a special case where subharmonic oscillations are stable in the thermodynamic limit.

In this Rapid Communication, we have examined the superconducting ground state of the ${\mathrm{HfV}}_2{\mathrm{Ga}}_4$ compound using resistivity, magnetization, zero-field (ZF), and transverse-field (TF) muon-spin relaxation and rotation ($\mu \mathrm{SR}$) measurements. Resistivity and magnetization unveil the onset of bulk superconductivity with $T_{\mathrm{C}}\sim 3.9$ K. TF-$\mu \mathrm{SR}$ measurements show the temperature dependence of the superfluid density, indicating, surprisingly, a nodal two-gap $\left(s+d\right)$-wave superconducting order parameter. In addition, the ZF muon relaxation rate increases with decreasing temperature below 4.6 K, suggesting the presence of weak spin fluctuations. These observations pointed to an unconventional multiband nature of the superconducting ground state. To better understand these findings, we carry out first-principles electronic-structure calculations, further highlighting multiple disconnected sheets with very different orbital weights and spin-orbit coupling composing the Fermi surface, bridging the way for a nodal multiband superconductivity scenario. In this vein, therefore, the ${\mathrm{HfV}}_2{\mathrm{Ga}}_4$ family stands out as an open avenue to novel unexplored unconventional superconducting compounds and an ideal playground to investigate the mechanisms behind such phenomena.

Topological phenomena in non-Hermitian systems have recently become a subject of great interest in the photonics and condensed-matter communities. In particular, the possibility of observing topologically protected edge states in non-Hermitian lattices has sparked an intensive search for systems where this kind of states are sustained. Here we present a study of the emergence of topological edge states in two-dimensional Haldane lattices exhibiting balanced gain and loss. In line with recent studies on other Chern insulator models, we show that edge states can be observed in the so-called broken $\mathcal{P}\mathcal{T}$-symmetric phase, that is, when the spectrum of the gain-loss-balanced system's Hamiltonian is not entirely real. More importantly, we find that such topologically protected edge states emerge irrespective of the lattice boundaries, namely, zigzag, bearded, or armchair.

One-dimensional (1D) Anderson localization phenomena are strongly affected by a bias force or equivalently a voltage in electronic systems. We experimentally study this case, launching a noninteracting ${}^{39}\mathrm{K}$ Bose-Einstein condensate in a 1D disordered potential induced by a far-off-resonance laser speckle, while controlling a bias force. In agreement with theoretical predictions, we observe a transition between algebraic localization and delocalization as a function of our control parameter that is the relative strength of the disorder against the bias force. We also demonstrate that the transition is intrinsically energy independent and that the initial velocity of the wave packet only plays a role through an effective disorder strength due to the correlation of the disorder.

Experiments reveal that the superconductors ${\text{UPt}}_3$, ${\text{PrOs}}_4{\text{Sb}}_{12}$, and ${\text{U}}_{1-x}{\text{Th}}_x{\text{Be}}_{13}$ undergo two superconducting transitions in the absence of an applied magnetic field. The prevalence of these multiple transitions suggests a common underlying mechanism. A natural candidate theory which accounts for these two transitions is the existence of a small symmetry-breaking field; however, such a field has not been observed in ${\text{PrOs}}_4{\text{Sb}}_{12}$ or ${\text{U}}_{1-x}{\text{Th}}_x{\text{Be}}_{13}$ and has been called into question for ${\text{UPt}}_3$. Motivated by arguments originally developed for superfluid ${}^3\mathrm{He}$, we propose that a generalized spin fluctuation feedback effect is responsible for these two transitions. We first develop a phenomenological theory for ${}^3\mathrm{He}$ that couples spin fluctuations to superfluidity, which correctly predicts that a high-temperature broken time-reversal superfluid ${}^3\mathrm{He}$ phase can emerge as a consequence. The transition at lower temperatures into a time-reversal invariant superfluid phase must then be first order by symmetry arguments. We then apply this phenomenological approach to the three superconductors ${\text{UPt}}_3$, ${\text{PrOs}}_4{\text{Sb}}_{12}$, and ${\text{U}}_{1-x}{\text{Th}}_x{\text{Be}}_{13}$, revealing that this naturally leads to a high-temperature time-reversal invariant nematic superconducting phase, which can be followed by a second-order phase transition into a broken time-reversal symmetry phase, as observed.

The frictional dynamics of interacting surfaces under forced translation are critically dependent on lattice commensurability. The highly nonlinear system of an elastic atomic chain sliding on an incommensurate periodic potential exhibits topological defects, known as kinks, that govern the frictional and translational dynamics. Performing experiments in a trapped-ion friction emulator, we observe two distinct structural and frictional phases: a commensurate high-friction phase where the ions stick-slip simultaneously over the lattice, and an incommensurate low-friction phase where the propagation of a kink breaks that simultaneity. We experimentally track the kink's propagation with atom-by-atom and sublattice site resolution and show that its velocity increases with commensurability. Our results elucidate the commensurate-incommensurate transition and the connection between the appearance of kinks and the reduction of friction in a finite system, with important consequences for controlling friction at nanocontacts.

Motivated by the need to understand and simulate the ubiquitous experimentally observed zero-bias conductance peaks in superconductor-semiconductor hybrid structures, we theoretically investigate the tunneling conductance spectra in one-dimensional nanowires in proximity to superconductors in a systematic manner taking into account several different physical mechanisms producing zero-bias conductance peaks. The mechanisms we consider are the presence of quantum dots, inhomogeneous potential, random disorder in the chemical potential, random fluctuations in the superconducting gap, and in the effective $g$ factor with the self-energy renormalization induced by the parent superconductor in both short ($L\sim 1\mu \mathrm{m}$) and long nanowires ($L\sim 3\mu \mathrm{m}$). We classify all foregoing theoretical results for zero-bias conductance peaks into three types: the good, the bad, and the ugly, according to the physical mechanisms producing the zero-bias peaks and their topological properties. We find that, although the topological Majorana zero modes are immune to weak disorder, strong disorder (“ugly”) completely suppresses topological superconductivity and generically leads to trivial zero-bias peaks. Compared qualitatively with the extensive existing experimental results in the superconductor-semiconductor nanowire structures, we conclude that most current experiments are likely exploring trivial zero-bias peaks in the “ugly” situation dominated by strong disorder. We also study the nonlocal end-to-end correlation measurement in both the short and long wires, and point out the limitation of the nonlocal correlation in ascertaining topological properties particularly when applied to short wires. Although we present results for “good” and “bad” zero-bias peaks, arising respectively from topological Majorana bound states and trivial Andreev bound states, strictly for the sake of direct comparison with the “ugly” zero-bias conductance peaks arising from strong disorder, the main goal of the current work is to establish with a very high confidence level the real physical possibility that essentially all experimentally observed zero-bias peaks in Majorana nanowires are most likely ugly, i.e., purely induced by strong disorder, and are as such utterly nontopological. Our work clearly suggests that an essential prerequisite for any future observation of topological Majorana zero modes in nanowires is a substantial materials improvement of the semiconductor-superconductor hybrid systems leading to much cleaner wires.

We present femtosecond pump-probe measurements of the nondegenerate ($1960\text{−}\mathrm{nm}$ excitation and 1176–$1326\text{−}\mathrm{nm}$ probe) two-photon absorption spectra of $8\text{−}\mathrm{nm}$ GaAs/$12\mathrm{nm}{\mathrm{Al}}_{0.32}{\mathrm{Ga}}_{0.68}\mathrm{As}$ quantum well waveguides. Experiments were performed with light pulses copolarized normal and tangential to the quantum well plane. The results are compared to perturbative calculations of transition rates between states determined by the $\mathbf{k}·\mathbf{p}$ method with an 8- or 14-band basis. We find excellent agreement between theory and experiment for normal polarization, then use the model to support predictions of orders-of-magnitude enhancement of nondegenerate two-photon absorption as one constituent photon energy nears an intersubband resonance.

We report on the preparation and the physical properties of superconducting ${\left(\mathrm{TaNb}\right)}_{1-x}{\left(\mathrm{ZrHfTi}\right)}_x$ high-entropy alloy films. The films were prepared by means of magnetron sputtering at room temperature, with $x$ ranging from 0 to 1 with an average thickness of 600–950 nm. All films crystallize in a pseudo body-centered cubic (BCC) structure. For samples with $x<0.65$, the normal-state properties are metallic, while for $x\ge 0.65$ the films are weakly insulating. The transition from metallic to weakly insulating occurs right at the near-equimolar stoichiometry. We find all films, except for $x=0$ or 1, to become superconducting at low temperatures, and we interpret their superconducting properties within the Bardeen-Cooper-Schrieffer (BCS) framework. The highest transition temperature $T_c=6.9\mathrm{K}$ of the solid solution is observed for $x\sim 0.43$. The highest upper-critical field $B_{c2}\left(0\right)=11.05$ T is found for the near-equimolar ratio $x\sim 0.65$, where the mixing entropy is the largest. The superconducting parameters derived for all the films from transport measurements are found to be close to those that are reported for amorphous superconductors. Our results indicate that these films of high-entropy alloys are promising candidates for superconducting device fabrication.

Twisting moiré heterostructures to the flatband regime allows for the formation of strongly correlated quantum states, since the dramatic reduction of the bandwidth can cause the residual electronic interactions to set the principal energy scale. An effective description for such correlated moiré heterostructures, derived in the strong-coupling limit at integer filling, generically leads to spin-valley Heisenberg models. Here we explore the emergence and stability of spin-liquid behavior in an ${\mathrm{SU}\left(2\right)}^{\mathrm{spin}}\otimes {\mathrm{SU}\left(2\right)}^{\mathrm{valley}}$ Heisenberg model upon inclusion of Hund's-induced and longer-ranged exchange couplings, employing a pseudofermion functional renormalization group approach. We consider two lattice geometries, triangular and honeycomb (relevant to different moiré heterostructures), and find, for both cases, an extended parameter regime surrounding the SU(4) symmetric point where no long-range order occurs, indicating a stable realm of quantum spin-liquid behavior. For large Hund's coupling, we identify the adjacent magnetic orders, with both antiferromagnetic and ferromagnetic ground states emerging in the separate spin and valley degrees of freedom. For both lattice geometries the inclusion of longer-ranged exchange couplings is found to have both stabilizing and destabilizing effects on the spin-liquid regime depending on the sign of the additional couplings.

Fermi arc states are features of Weyl semimetal (WSM) surfaces which are robust due to the topological character of the bulk band structure. We demonstrate that Fermi arcs may undergo profound restructurings when surfaces of different systems with a well-defined twist angle are tunnel coupled. The twisted WSM interface supports a moiré pattern which may be approximated as a periodic system with a large real-space unit cell. States bound to the interface emerge, with interesting consequences for the magneto-oscillations expected when a magnetic field is applied perpendicular to the system surfaces. As the twist angle passes through special “arcless angles,” for which open Fermi arc states are absent at the interface, Fermi loops of states with no connection to bulk states appear. Such states have interesting resonance signatures in the optical conductivity of the system in a magnetic field perpendicular to the interface. These signatures are independent of the bulk thickness or disorder.

We study a class of topological charge density wave states exhibiting monopole harmonic symmetries. The density wave ordering is equivalent to pairing in the particle-hole channel due to Fermi surface nesting under interactions. When electron and hole Fermi surfaces carry different Chern numbers, the particle-hole pairing exhibits a nontrivial Berry phase inherited from band-structure topology independent of the concrete density wave ordering mechanism. The associated density wave gap functions become nodal, and the net nodal vorticity is determined by the monopole charge of the pairing Berry phase. The gap function nodes become zero-energy Weyl nodes of the bulk spectra of quasiparticle excitations. These states can occur in doped Weyl semimetals with nested electron and hole Fermi surfaces enclosing Weyl nodes of the same chirality in the weak-coupling regime. Topologically nontrivial low-energy Fermi arc surface states appear in the density wave ordering state as a consequence of the emergent zero-energy Weyl nodes.

Quantum matter with exotic topological order has potential applications in quantum computation. However, in present experiments, the manipulations on topological states are still challenging. We here propose an architecture for optical control of topological matter. We consider a topological superconducting qubit array with a Su-Schrieffer-Heeger (SSH) Hamiltonian which couples to a microwave cavity. Based on parity properties of the topological qubit array, we propose an optical spectroscopy method to observe the topological phase transition, i.e., edge-to-bulk transition. This new method can be achieved by designing cavity-qubit couplings. A main purpose of this work is to understand how the topological phase transition affects light-matter interaction. We find that the topological bandgap plays an essential role on this issue. In the topological phase, the resonant vacuum Rabi splitting of degenerate edge states coupling to the cavity field is protected from those of bulk states by the bandgap. In the dispersive regime, the cavity induced coupling between edge states is dominant over couplings between edge and bulk states, due to the topological bandgap. As a result, quantum interference between topological edge states occurs and enables single-photon transport through boundaries of the topological qubit array. Our work may pave a way for topological quantum state engineering.

We report a detailed theoretical model of a recently demonstrated magnetic trap system based on a pair of magnetic tips. The model takes into account key parameters such as tip diameter, facet angle, and gap separation. It yields quantitative descriptions consistent with experiments such as the vertical and radial frequency, equilibrium position, and the optimum facet angle that produces the strongest confinement. We arrive at the striking conclusion that a maximum confinement enhancement can be achieved at an optimum facet angle ${\theta }_{\text{max}}=arccos\sqrt{2/3}$. We also discuss a critical gap which is the maximum gap between the tips for the confinement enhancement effect to occur. This magnetic trap and its theoretical model serve as an interesting example of a simple and elementary magnetic trap in physics.

Calculations of electronic and optical properties of solids at finite temperature including electron-phonon interactions and quantum zero-point renormalization have enjoyed considerable progress during the past few years. Among the emerging methodologies in this area, we recently proposed an approach to compute optical spectra at finite temperature including phonon-assisted quantum processes via a single supercell calculation [Zacharias and Giustino, Phys. Rev. B 94, 075125 (2016)]. In the present work we considerably expand the scope of our previous theory starting from a compact reciprocal space formulation, and we demonstrate that this improved approach provides accurate temperature-dependent band structures in three-dimensional and two-dimensional materials, using a special set of atomic displacements in a single supercell calculation. We also demonstrate that our special displacement reproduces the thermal ellipsoids obtained from x-ray crystallography and yields accurate thermal averages of the mean-square atomic displacements. This enhanced version of the special displacement method enables nonperturbative, robust, and straightforward ab initio calculations of the electronic and optical properties of solids at finite temperature, and can easily be used as a postprocessing step to any electronic structure code. To illustrate the capabilities of this method, we investigate the temperature-dependent band structures and atomic displacement parameters of prototypical nonpolar and polar semiconductors and of a prototypical two-dimensional semiconductor, namely Si, GaAs, and monolayer ${\mathrm{MoS}}_2$, and we obtain excellent agreement with previous calculations and experiments. Given its simplicity and numerical stability, the present development is suited for high-throughput calculations of band structures, quasiparticle corrections, optical spectra, and transport coefficients at finite temperature.

We classify all possible singularities in the electronic dispersion of two-dimensional systems that occur when the Fermi surface changes topology, using catastrophe theory. For systems with up to seven control parameters (i.e., pressure, strain, bias voltage, etc.), the theory guarantees that the singularity belongs to one of seventeen standard types known as catastrophes. We show that at each of these singularities the density of states diverges as a power law, with a universal exponent characteristic of the particular catastrophe, and we provide its universal ratio of amplitudes of the prefactors for energies above and below the singularity. We further show that crystal symmetry restricts which types of catastrophes can occur at the points of high symmetry in the Brillouin zone. For each of the seventeen wallpaper groups in two dimensions, we list which catastrophes are possible at each high-symmetry point.

Using spin-polarized scanning tunneling microscopy (STM) we observe a two-dimensional $c(2\times 2)$ antiferromagnetic state in a pseudomorphic Mn double layer on the W(001) surface. First-principles calculations confirm the antiferromagnetic ground state. Surprisingly, only the surface Mn layer possesses a magnetic moment while the moments of interface Mn atoms vanish. Calculated STM and spin-polarized STM images as well as local vacuum density of states are in good agreement with experiments and provide strong evidence of a dead magnetic interface layer.

Unsupervised machine learning is a cornerstone of artificial intelligence as it provides algorithms capable of learning tasks, such as classification of data, without explicit human assistance. We present an unsupervised deep learning protocol for finding topological indices of quantum systems. The core of the proposed scheme is a “topological data augmentation” procedure that uses seed objects to generate ensembles of topologically equivalent data. Such data, assigned with dummy labels, can then be used to train a neural network classifier for sorting arbitrary objects into topological equivalence classes. Importantly, we also show how to retrieve the local quantities corresponding to the learned topological indices from the intermediate outputs of the trained network. Our protocol is explicitly illustrated on two-band insulators in one and two dimensions, characterized by a winding number and a Chern number respectively. Using the augmentation technique also in the classification step, to classify a family of topologically equivalent objects instead of a single object, we can achieve accuracy arbitrarily close to $100\%$ even for indices outside the training regime. Apart from the method's applicability to topological classification, it also provides a new perspective on data augmentation in supervised machine learning, where given sufficient mathematical structure the set of category-preserving deformations can be rigorously defined.

The concept of the entanglement between spin and orbital degrees of freedom plays a crucial role in our understanding of various phases and exotic ground states in a broad class of materials, including orbitally ordered materials and spin liquids. We investigate how the spin-orbital entanglement in a Mott insulator depends on the value of the spin-orbit coupling of the relativistic origin. To this end, we numerically diagonalize a one-dimensional spin-orbital model with Kugel-Khomskii exchange interactions between spins and orbitals on different sites supplemented by the on-site spin-orbit coupling. In the regime of small spin-orbit coupling with regard to the spin-orbital exchange, the ground state to a large extent resembles the one obtained in the limit of vanishing spin-orbit coupling. On the other hand, for large spin-orbit coupling the ground state can, depending on the model parameters, either still show negligible spin-orbital entanglement or evolve to a highly spin-orbitally-entangled phase with completely distinct properties that are described by an effective XXZ model. The presented results suggest that (i) the spin-orbital entanglement may be induced by large on-site spin-orbit coupling, as found in the $5d$ transition metal oxides, such as the iridates; (ii) for Mott insulators with weak spin-orbit coupling of Ising type, such as, e.g., the alkali hyperoxides, the effects of the spin-orbit coupling on the ground state can, in the first order of perturbation theory, be neglected.