The kinetics of vacancy defect in graphene drives structural modifications leading to disorder, multivacancy complex, and edge reconstruction. Within the first-principles calculations, we study the dynamic Jahn-Teller distortion and diffusion of a vacancy defect. Further, the intricate dependence of carrier doping is systematically investigated. The experimental observation of dynamic Jahn-Teller distortion is argued to be blocked by defect functionalization and charge doping. We demonstrate that lattice relaxation perpendicular to the graphene sheet along with the in-plane strain relaxation play predominant roles in predicting the correct microscopic mechanism for vacancy diffusion. The importance of quantum correction to the classical barrier is discussed. The calculated activation barrier increases upon both electron and hole doping and the observed trends are explained by the differential charge density distribution and hardening of the responsible low-energy phonon modes. Electron doping essentially freezes the vacancy motion, and thus any degradation mediated by it. While tracking and analyzing the vacancy diffusion experimentally in graphene is a difficult task, the present results will motivate new experimental efforts and assist interpretation of the results.

Viewed as an ideal quantum simulator, the optical lattice system with trapped ultracold atoms has been developed into a powerful tool to investigate unusual physical properties due to its considerable tunability. In this paper, we propose to construct a feasible two-dimensional optical lattice which can be continuously transformed from a square lattice to a Lieb lattice by tuning laser intensity. In these transformations, accompanied with the crossover between nonrelativistic particles and relativistic Dirac fermions, the topological and transport properties are varied under artificial gauge potentials. It is found that the fractal energy spectrum in an effective magnetic field splits and reassembles from a traditional Hofstadter butterfly to multiple ones with flat band, characterized by different Chern numbers. In a specific case of half a pseudomagnetic flux quantum, the spectrum has a transition from individual Dirac cones to double-layer Dirac cones, then to the Dirac cones with a flat band, the pseudospin of particles changes from 1/2 to 3/2, then to 1. In the structural evolution, a remarkable birefringent tunneling across a potential barrier is demonstrated. The adjustable lattice structure replenished with synthetic gauge fields for cold atoms offers more opportunities to observe intriguing physical properties.

We investigate the electronic origin of zero Hall conductivity or the axion insulator phase of a Cr-doped $\mathrm{S}{\mathrm{b}}_2\mathrm{T}{\mathrm{e}}_3$ film by analyzing the evolution of its band structure upon a magnetic field sweep. By applying a small electric field or using asymmetric doping, we differentiate the coercivities of the two surfaces, so they can form either ferromagnetic (FM) or antiferromagnetic (AFM) alignments. Through the surface-resolved Berry curvature calculations, we demonstrate that the system manifests robust zero Hall plateaus when the two surfaces remain topologically nontrivial and enter a magnetic state which has intrasurface FM and intersurface AFM orderings. Our results provide insights for the understanding of the uncharted issues of axion insulators, complementary to the exciting experimental progress in this realm.

We show that the edges of quantum spin Hall topological insulators represent a natural platform for the realization of an exotic magnetic phase which has all the properties of a lattice supersolid. On one hand, fermionic edge modes are helical due to the nontrivial topology of the bulk. On the other hand, a disorder at the edge or magnetic adatoms may produce a dense array of localized spins interacting with the helical electrons. The spin subsystem is magnetically frustrated since the indirect exchange favors the formation of helical spin order and the direct one favors (anti)ferromagnetic ordering of the spins. At a moderately strong direct exchange, the competition between these spin interactions results in the spontaneous breaking of parity and in the Ising-type order of the $z$ components at zero temperature. If the total spin is conserved, the spin order does not pin a collective massless helical mode which supports the ideal transport. In this case, the phase transition converts the helical spin order to the order of a chiral lattice supersolid. This represents a promising possibility for experimental studies of the elusive supersolidity.

A two-dimensional time-reversal symmetric topological superconductor is a fully gapped system possessing a helical Majorana mode on the edges. This helical Majorana edge mode (HMEM), which is a Kramers pair of two chiral Majorana edge modes in the opposite propagating directions, is robust under time-reversal symmetry protection. We propose a feasible setup and accessible measurement to provide the preliminary step of the HMEM realization by studying superconducting antiferromagnetic quantum spin Hall insulators. Since this antiferromagnetic topological insulator hosts a helical electron edge mode and preserves effective time-reversal symmetry, which is the combination of time-reversal symmetry and crystalline symmetry, the proximity effect of the conventional $s$-wave superconducting pairing can directly induce a single HMEM. We further show the HMEM leads to the observation of an $e^2/h$ conductance, and this quantized conductance survives even in the presence of small symmetry-breaking disorders.

The Sachdev-Ye-Kitaev (SYK) model describes a strongly correlated metal with all-to-all random interactions (average strength $J$) between $N$ fermions (complex Dirac fermions or real Majorana fermions). In the large-$N$ limit a conformal symmetry emerges that renders the model exactly soluble. Here we study how the non-Fermi-liquid behavior of the closed system in equilibrium manifests itself in an open system out of equilibrium. We calculate the current-voltage characteristic of a quantum dot, described by the complex-valued SYK model, coupled to a voltage source via a single-channel metallic lead (coupling strength $\mathrm{\Gamma }$). A one-parameter scaling law appears in the large-$N$ conformal regime, where the differential conductance $G=dI/dV$ depends on the applied voltage only through the dimensionless combination $\xi =eVJ/{\mathrm{\Gamma }}^2$. Low and high voltages are related by the duality $G\left(\xi \right)=G(\pi /\xi )$. This provides for an unambiguous signature of the conformal symmetry in tunneling spectroscopy.

Single defect centers in layered hexagonal boron nitride are promising candidates as single-photon sources for quantum optics and nanophotonics applications. However, spectral instability hinders many applications. Here, we perform resonant excitation measurements and observe Fourier transform limited linewidths down to $\approx 50$ MHz. We investigated the optical properties of more than 600 single-photon emitters (SPEs) in hBN. The SPEs exhibit narrow zero-phonon lines distributed over a spectral range from 580 to 800 nm and with dipolelike emission with a high polarization contrast. Finally, the emitters withstand transfer to a foreign photonic platform, namely, a silver mirror, which makes them compatible with photonic devices such as optical resonators and paves the way to quantum photonics applications.

We report on composite-cavity microresonators comprising a vertical cavity and a distributed feedback structure. First-order diffraction on the second-order Bragg grating serves as coupling mechanism between the two resonator elements. Employing the broad emission spectrum of the small molecule system ${\mathrm{Alq}}_3$:DCM, dispersion characteristics of multiple resonances are recorded in continuously tunable devices and related to calculated resonances and their field profiles. In combination with the analysis of input-output measurements, the lasing characteristics of vertical and lateral resonances are controlled by adjusting specific design parameters of the system. For identical resonance energies, the distinct resonators show coherent interaction in the cross-coupled configuration, if the quality of the resonances and the cross-coupling efficiency meet the condition of strong photon-photon interaction. This resonant coupling has a fundamental impact on dispersion and lasing characteristics of the system.

We describe an approach to fix the gauge degrees of freedom in tensor networks, including those with closed loops, which allows a canonical form for arbitrary tensor networks to be realized. Additionally, a measure for the internal correlations present in a tensor network is proposed, which quantifies the extent of resonances around closed loops in the network. Finally we describe an algorithm for the optimal truncation of an internal index from a tensor network, based upon proper removal of the redundant internal correlations. These results, which offer a unified theoretical framework for the manipulation of tensor networks with closed loops, can be applied to improve existing tensor network methods for the study of many-body systems and may also constitute key algorithmic components of sophisticated new tensor methods.

In this paper, we have done a comparative theoretical study of electronic and magnetic properties of iron phthalocyanine (FePc) and cobalt phthalocyanine (CoPc) molecules physisorbed on a monolayer of ${\mathrm{MoS}}_2$ and graphene by density functional theory. Various types of physisorption sites have been considered for both surfaces. The lowest energy structure for both metal phthalocyanine ($M\mathrm{Pc}$) molecules physisorbed on ${\mathrm{MoS}}_2$ is a sulfur-top position, i.e., when the metal center of the molecule is on top of a sulfur atom. However, on graphene, the lowest energy structure for the FePc molecule is when a metal atom is on top of a bridge position. In contrast to this, the CoPc molecule prefers a carbon-top position. The adsorption of $M\mathrm{Pc}$ molecules is stronger on the ${\mathrm{MoS}}_2$ surface than on graphene ($\sim 2.5$ eV higher physisorption energy). In these systems, spin dipole moments of the metal centers are antiparallel to the spin moments and hence a huge reduction of effective spin moment can be seen. The calculations of magnetic anisotropy energies using both variational and second-order perturbation approaches indicate no significant changes after physisorption. In case of the FePc and CoPc physisorption, respectively, an out-of-plane and an in-plane easy axis of magnetization can be observed. Our calculations indicate a reduction of ${\mathrm{MoS}}_2$ work function $\sim 1$ eV due to physisorption of $M\mathrm{Pc}$ molecules while it does not change significantly in the case of graphene.

Two-dimensional arrays of periodically driven qubits can host inherently dynamical topological phases with anomalous chiral edge dynamics. These chiral Floquet phases are formally characterized by a dynamical topological invariant, the chiral unitary index. Introducing a quantity called the chiral mutual information, we show that this invariant can be precisely interpreted in terms of a quantized chiral transfer of quantum information along the edge of the system, and devise a physical setup to measure it.

Higher-order topological insulators are distinguished by the existence of topologically protected modes with codimension two or higher. Here, we report the manifestation of a second-order topological insulator in a two dimensional frustrated quantum magnet, which exhibits topological corner modes. Our exactly solvable model is a generalization of the Kitaev honeycomb model to the Shastry-Sutherland lattice that, besides a chiral spin liquid phase, exhibits a gapped spin liquid with Majorana corner modes, which are protected by two mirror symmetries. This second-order Kitaev spin liquid remains stable in the presence of thermal fluctuations and undergoes a finite-temperature phase transition evidenced in large-scale quantum Monte Carlo simulations.

We study the Kitaev model in a [001] magnetic field employing the mean-field theory in the Majorana fermion representation. The mean-field Hamiltonian of the system has the Bogoliubov–de Gennes (BdG) form of a two-dimensional superconductor. We discover a robust gapless regime in the intermediate magnetic field for both gapless and gapped antiferromagnetic Kitaev models with $J_x=J_y$ before the system is polarized in a high magnetic field. A topological phase transition connecting two gapless phases with a nodal line phase takes place at a critical magnetic field $h_{c_1}$ in this regime. While the nodal line phase at $h_{c_1}$ disappears when the mirror symmetry $J_x=J_y$ is broken, the nodal point gapless phase can exist at intermediate fields even without the mirror symmetry. We reveal that the phase evolution of the system in the magnetic field is driven by the competition between the magnetic field and the particle-hole asymmetry of the normal state of the BdG Hamiltonian, which results in the robust intermediate gapless phase for the antiferromagnetic case. For the ferromagnetic case, there is no intermediate phase transition before polarization. The above phase diagrams are confirmed by dynamical mean-field theory results.

Properties of two $\mathrm{ThC}{\mathrm{r}}_2\mathrm{S}{\mathrm{i}}_2$-type materials are discussed within the context of their established structural and magnetic symmetries—the structure type accounts for more than 400 compounds of the $\mathrm{A}{\mathrm{B}}_2{\mathrm{X}}_2$ composition. Both materials develop collinear, G-type antiferromagnetic order above room temperature, and magnetic ions occupy acentric sites in centrosymmetric structures (magnetic crystal class $4^{\prime }/m^{\prime }{m}^{\prime }m$). We refute a previous conjecture that $\mathrm{BaM}{\mathrm{n}}_2\mathrm{A}{\mathrm{s}}_2$ is an example of a magnetoelectric material with hexadecapole order using Dirac (magnetoelectric) multipoles by exposing flaws in supporting arguments: principally, an omission of discrete symmetries enforced by the symmetry ($\overline{4}{m}^{\prime }{2}^{\prime }$) of sites used by Mn ions and, also, improper classifications of the primary and secondary order parameters. G-type antiferromagnetism using conventional magnetic dipoles provides the primary order parameter. Dirac quadrupoles are secondary, and possibly visible in Bragg diffraction patterns. Patterns caused by conventional magnetic dipoles and Dirac multipoles are predicted to be fundamentally different, which raises the intriguing possibility of a unique and comprehensive examination of the magnetoelectric state by diffraction; Dirac multipoles create Bragg spots with Miller index $l$ even (and $h+k$ even) while magnetic dipoles appear for $l$ odd (and $h+k$ odd). A rotoinversion operation ($\overline{4}$) in Mn site symmetry is the ultimate source of distinguishing features in multipole types.

We investigate the charge and spin transport in half-metallic ferromagnet ($F$) and superconductor ($S$) nanojunctions. We utilize a self-consistent microscopic method that can accommodate the broad range of energy scales present, and ensures that proximity effects that account for the interactions at the interfaces are accurately determined. Two experimentally relevant half-metallic junction types are considered: The first is an $F_1{F}_{2}S$ structure, where a half-metallic ferromagnet $F_1$ adjoins a weaker conventional ferromagnet $F_2$. The current is injected through the $F_1$ layer by means of an applied bias voltage. The second configuration involves an $S{F}_1{F}_2{F}_{3}S$ Josephson junction whereby a phase difference $\mathrm{\Delta }\phi$ between the two superconducting electrodes generates the supercurrent flow. In this case, the central half-metallic $F_2$ layer is surrounded by two weak ferromagnets $F_1$ and $F_3$. By placing a ferromagnet with a weak exchange field adjacent to an $S$ layer, we are able to optimize the conversion process in which opposite-spin triplet pairs are converted into equal-spin triplet pairs that propagate deep into the half-metallic regions in both junction types. For the tunnel junctions, we study the bias-induced local magnetization, spin currents, and spin-transfer torques for various orientations of the relative magnetization angle $\theta$ in the $F$ layers. We find that the bias-induced equal-spin triplet pairs are maximized in the half metal for $\theta \approx {90}^{\circ }$ and, as part of the conversion process, are anticorrelated with the opposite-spin pairs. We show that the charge current density is maximized, corresponding to the occurrence of a large amplitude of equal-spin triplet pairs, when the exchange interaction of the weak ferromagnet is about $0.1E_F.$ For the half-metallic Josephson junctions we often find that the spin current flowing in the half metal is equivalent to the charge supercurrent flowing throughout the junction. This is indicative that the current consists of spin-polarized triplet pairs. The conversion process of the opposite-spin triplet pairs to the equal-spin triplet pairs in the weaker magnets is clearly demonstrated. This is exemplified by the fact that the supercurrent in the half metal was found to be relatively insensitive to its thickness.

During the evolution of discrete nonlinear systems with dynamics dictated by the discrete nonlinear Schrödinger equation, two quantities are conserved: system energy (Hamiltonian) and system density (number of particles). It is then possible to analyze system evolution in relation to an energy-density phase diagram. Previous works have identified a “thermalization zone” on the phase diagram where regular statistical mechanics methods apply. Based on these statistical mechanics methods we have now assigned a specific equilibrium temperature to every point of the thermalization zone. Temperatures were derived in the grand canonical picture through an entropy-temperature relation, modified to suit the nonlinear lattice systems. Generally, everywhere in the thermalization zone of the phase diagram, temperatures along a fixed system-density line, grow monotonously from zero to infinity. Isotherms on the phase diagram are concave.

We study quantum phase transitions in the honeycomb Kitaev model under a magnetic field, focusing on the topological nature of Majorana fermion excitations. We find a gapless phase between the low-field gapless quantum spin liquid and the high-field gapped forced-ferromagnetic state for the antiferromagnetic Kitaev model in the [001] field by using the Majorana mean-field theory, in conjunction with the exact diagonalization and the spin-wave theory supporting the validity of this approach. The transition between the two gapless phases is driven by a topological change of the Majorana spectrum—line node formation interconnecting two Majorana cones. The peculiar change of the Majorana band topology is rationalized by a sign change of the effective Kitaev coupling by the magnetic field, which does not occur in the ferromagnetic Kitaev case. Upon tilting the magnetic field away from [001], the two gapless phases become gapped and topologically nontrivial, characterized by nonzero Chern numbers with different signs. The sign change of the Chern number leads to a reversal of the thermal edge current in the half-quantized thermal Hall effect.

We use inelastic neutron scattering to study the energy and wave-vector dependence of the superconductivity-induced resonance in hole-doped ${\mathrm{Ba}}_{0.67}{\mathrm{K}}_{0.33}{\left({\mathrm{Fe}}_{1-x}{\mathrm{Co}}_x\right)}_2{\mathrm{As}}_2$ ($x=0$ and 0.08 with $T_c\approx 37$ and 28 K, respectively). In previous work on electron-doped $\mathrm{Ba}{\left({\mathrm{Fe}}_{0.963}{\mathrm{Ni}}_{0.037}\right)}_2{\mathrm{As}}_2$ ($T_N=26$ K and $T_c=17$ K), the resonance is found to peak sharply at the antiferromagnetic (AF) ordering wave vector ${\mathbf{Q}}_{\mathrm{AF}}$ along the longitudinal direction, but disperses upwards away from ${\mathbf{Q}}_{\mathrm{AF}}$ along the transverse direction [Kim et al., Phys. Rev. Lett. 110, 177002 (2013)]. For hole-doped $x=0$ and 0.08 without AF order, we find that the resonance displays a ringlike upward dispersion away from ${\mathbf{Q}}_{\mathrm{AF}}$ along both the longitudinal and transverse directions. By comparing these results with calculations using the random phase approximation, we conclude that the dispersive resonance is a direct signature of isotropic superconducting gaps arising from nested hole-electron Fermi surfaces.

The first-principles calculation of many material properties, in particular related to defects and disorder, starts with the relaxation of the atomic positions of the system under investigation. This procedure is routine for nonmagnetic and magnetically ordered materials. However, when it comes to magnetically disordered systems, in particular the paramagnetic phase of magnetic materials, it is not clear how the relaxation procedure should be performed or which geometry should be used. Here we propose a method for the structural relaxation of magnetic materials in the paramagnetic regime, in an adiabatic fast-magnetism approximation within the disordered local moment (DLM) picture in the framework of density functional theory. The method is straightforward to implement using any $abinitio$ code that allows for structural relaxations. We illustrate the importance of considering the disordered magnetic state during lattice relaxations by calculating formation energies and geometries for an Fe vacancy and C insterstitial atom in body-centered cubic (bcc) Fe as well as bcc ${\mathrm{Fe}}_{1-x}{\mathrm{Cr}}_x$ random alloys in the paramagnetic state. In the vacancy case, the nearest neighbors to the vacancy relax toward the vacancy of 0.14 Å ($-5\%$ of the ideal bcc nearest-neighbor distance), which is twice as large as the relaxation in the ferromagnetic case. The vacancy formation energy calculated in the DLM state on these positions is 1.60 eV, which corresponds to a reduction of about 0.1 eV compared to the formation energy calculated using DLM but on ferromagnetic-relaxed positions. The carbon interstitial formation energy is found to be 0.41 eV when the DLM relaxed positions are used, as compared to 0.59 eV when the FM-relaxed positions are employed. For bcc ${\mathrm{Fe}}_{0.5}{\mathrm{Cr}}_{0.5}$ alloys, the mixing enthalpy is reduced by 5 meV/atom, or about 10%, when the DLM state relaxation is considered, as compared to positions relaxed in the ferromagnetic state.