Two-dimensional Brillouin Zone Research Articles

Berry Curvature and Bulk-Boundary Correspondence from Transport Measurement for Photonic Chern Bands.

Berry curvature is a fundamental element to characterize topological quantum physics, while a full measurement of Berry curvature in momentum space was not reported for topological states. Here we achieve two-dimensional Berry curvature reconstruction in a photonic quantum anomalous Hall system via Hall transport measurement of a momentum-resolved wave packet. Integrating measured Berry curvature over the two-dimensional Brillouin zone, we obtain Chern numbers corresponding to -1 and 0. Further, we identify bulk-boundary correspondence by measuring topology-linked chiral edge states at the boundary. The full topological characterization of photonic Chern bands from Berry curvature, Chern number, and edge transport measurements enables our photonic system to serve as a versatile platform for further in-depth study of novel topological physics.

Realization of flat bands by lattice intercalation in kagome metals

Recently there has been intense interest in kagome metals, which are expected to host flat bands (FBs). However, the observed ``FBs'' are not flat over the whole two-dimensional Brillouin zone and overlap strongly with other bands. In fact, the FB does not truly exist in a default $d$-orbital kagome lattice, and the conditions for its existence in kagome metals are unknown. Here, based on tight-binding model analyses of the interplay between orbital and lattice symmetry, we establish such conditions. We show that for a single $d$-orbital kagome lattice assuming large crystal field splitting (CFS), only the ${d}_{{z}^{2}}$ orbital gives rise to a FB, while ${d}_{xy}, {d}_{{x}^{2}\ensuremath{-}{y}^{2}}, {d}_{xz}$, and ${d}_{yz}$ orbitals can only produce a FB with a rotated $d$-orbital basis so that they conform with the underlying kagome lattice symmetry. Most importantly, we demonstrate that both conditions of $d$-orbital rotation and large CFS can be ideally satisfied by intercalating the kagome lattice with a hexagonal sublattice without disrupting the destructive interference of FB wave function. Furthermore, we propose layered metalorganic frameworks as promising candidate kagome metals to realize FBs.

Gapped Collective Charge Excitations and Interlayer Hopping in Cuprate Superconductors.

We use resonant inelastic x-ray scattering to probe the propagation of plasmons in the electron-doped cuprate superconductor Sr_{0.9}La_{0.1}CuO_{2}. We detect a plasmon gap of ∼120 meV at the two-dimensional Brillouin zone center, indicating that low-energy plasmons in Sr_{0.9}La_{0.1}CuO_{2} are not strictly acoustic. The plasmon dispersion, including the gap, is accurately captured by layered t-J-V model calculations. A similar analysis performed on recent resonant inelastic x-ray scattering data from other cuprates suggests that the plasmon gap is generic and its size is related to the magnitude of the interlayer hopping t_{z}. Our work signifies the three dimensionality of the charge dynamics in layered cuprates and provides a new method to determine t_{z}.

Discriminant indicator with generalized rotational symmetry

Discriminant indicators with generalized inversion symmetry are computed only from data at the high-symmetry points. They allow a systematic search for exceptional points. In this paper, we propose discriminant indicators for two- and three-dimensional systems with generalized $n$-fold rotational symmetry ($n=4$, $6$). As is the case for generalized inversion symmetry, the indicator taking a nontrivial value predicts the emergence of exceptional points and loops without ambiguity of the reference energy. A distinct difference from the case of generalized inversion symmetry is that the indicator with generalized $n$-fold rotational symmetry ($n=4$, $6$) can be computed only from data at two of four high-symmetry points in the two-dimensional Brillouin zone. Such a difference is also observed in three-dimensional systems. Furthermore, we also propose how to fabricate a two-dimensional system with generalized four-fold rotational symmetry for an electrical circuit.

Chern number and Berry curvature for Gaussian mixed states of fermions

We generalize the concept of topological invariants for mixed states based on the ensemble geometric phase (EGP) introduced for one-dimensional lattice models to two dimensions. In contrast to the geometric phase for density matrices suggested by Uhlmann, the EGP leads a proper Chern number for Gaussian, finite-temperature or non-equilibrium steady states. The Chern number can be expressed as an integral of the Berry curvature of the so-called fictitious Hamiltonian, constructed from single-particle correlations, over the two-dimensional Brillouin zone. For the Chern number to be non-zero the fictitious Hamiltonian has to break time-reversal symmetry.

Fermi surface in La-based cuprate superconductors from Compton scattering imaging

Compton scattering provides invaluable information on the underlying Fermi surface (FS) and is a powerful tool complementary to angle-resolved photoemission spectroscopy and quantum oscillation measurements. Here we perform high-resolution Compton scattering measurements for La2−xSrxCuO4 with x = 0.08 (Tc = 20 K) at 300 K and 150 K, and image the momentum distribution function in the two-dimensional Brillouin zone. We find that the observed images cannot be reconciled with the conventional hole-like FS believed so far. Instead, our data imply that the FS is strongly deformed by the underlying nematicity in each CuO2 plane, but the bulk FSs recover the fourfold symmetry. We also find an unusually strong temperature dependence of the momentum distribution function, which may originate from the pseudogap formation in the presence of the reconstructed FSs due to the underlying nematicity. Additional measurements for x = 0.15 and 0.30 at 300 K suggest similar FS deformation with weaker nematicity, which nearly vanishes at x = 0.30.

Acoustic modes of locally resonant phononic crystals: Comparison with frequency-dependent mass models

Locally resonant acoustic materials were proposed to control sound using structures with feature size orders of magnitude smaller than the acoustic wavelength in air. We analytically derive the effective, frequency-dependent mass densities of resonant oscillators consisting of a heavy mass within a light shell, embedded in foam, in two dimensions, using a rigid core-shell approximation. The effective mass is expressible in closed form by elementary functions, enabling a mapping of the low-frequency physics to a simple one-dimensional model involving a point mass harmonically coupled within a box. The acoustic band structure of a two-dimensional square lattice of these oscillators is evaluated in the effective, frequency-dependent mass density model and compared with the exact solution. For the out-of-plane oscillation within this two-dimensional phononic crystal, the nonlinear eigenvalue equation resulting from the frequency-dependent density is solved with the Cutting Surface Method to yield the acoustic band structure. This agrees with the exact band structure obtained by finite element method calculations within $2%$ numerical error. For the in-plane oscillations involving internal translations or rotations within the core-shell resonators, the one-dimensional mass-in-a-box model recaptures the exact acoustic modes at only certain high-symmetry points of the two-dimensional Brillouin zone. The description of more complex resonances within acoustic metamaterials requires a fundamental generalization of current effective mass models.

Computation of the lattice dynamics for the low-coverage Au/Cu(111) ordered metallic surface alloy system

The lattice dynamics of the ordered Au/Cu(111) surface alloy system are investigated, in particular for the low-coverage ordered Cu(111)-Au 3×3 R30° surface alloy structure. The computations are developed using the phase field matching theory (PFMT) and the Green's functions (GF) formalism. Dispersion branches for surface phonons and resonances along the directions of high symmetry ΓM‾, MK‾, and KΓ‾ of the two-dimensional (2D) surface Brillouin zone (BZ) of the system, are computed. The vibration local densities of states (LDOS) are also determined for representative atomic sites at the surface alloy boundary. The lattice dynamics of the Cu(111)-Au 3×3 R30° surface reveal novel surface phonon and resonance dispersion branches that could be of potential interest for electronic, chemical, and spintronic applications, notably via electron-phonon interactions at this surface.

Electronic structure of graphene under periodic uniaxial tensile strain

Using density functional theory with the generalized gradient approximation, we investigate the electronic structure of graphene with uniaxial tensile strain in terms of its direction and homogeneity. A uniform strain does not affect the conical dispersion band of pristine graphene but simply shifts its position in the two-dimensional Brillouin zone. Graphene with periodic tensile strain along the armchair and chiral directions is a semiconductor with a tiny gap of a few meV, which is slightly dependent on the strain direction.

Mirror skin effect and its electric circuit simulation

We analyze impacts of crystalline symmetry on the non-Hermitian skin effects. Focusing on mirror symmetry, we propose a novel type of skin effects, a mirror skin effect, which results in significant dependence of energy spectrum on the boundary condition only for the mirror invariant line in the two-dimensional Brillouin zone. This effect arises from the topological properties characterized by a mirror winding number. We further reveal that the mirror skin effect can be observed for an electric circuit composed of negative impedance converters with current inversion where switching the boundary condition significantly changes the admittance eigenvalues only along the mirror invariant lines. Furthermore, we demonstrate that extensive localization of the eigenstates for each mirror sector result in an anomalous voltage response.

Visualization of localized perturbations on a (001) surface of the ferromagnetic semimetal EuB6

We performed scanning tunneling microscopy (STM) and spectroscopy on a (001) surface of the ferromagnetic semimetal ${\mathrm{EuB}}_{6}$. Large-amplitude oscillations emanating from the elastic scattering of electrons by the surface impurities are observed in topography and in differential conductance maps. Fourier transform of the conductance maps embracing these regions indicate a holelike dispersion centered around the $\overline{\mathrm{\ensuremath{\Gamma}}}$ point of the two-dimensional Brillouin zone. Using density functional theory slab calculations, we identify a spin-split surface state, which stems from the dangling ${p}_{z}$ orbitals of the apical boron atom. Hybridization with bulk electronic states leads to a resonance enhancement in certain regions around the $\overline{\mathrm{\ensuremath{\Gamma}}}$ point, contributing to the remarkably strong real-space response around static point defects, which are observed in STM measurements.

Axion coupling in the hybrid Wannier representation

Many magnetic point-group symmetries induce a topological classification on crystalline insulators, dividing them into those that have a nonzero quantized Chern-Simons magnetoelectric coupling ("axion-odd" or "topological"), and those that do not ("axion-even" or "trivial"). For time-reversal or inversion symmetries, the resulting topological state is usually denoted as a "strong topological insulator" or an "axion insulator" respectively, but many other symmetries can also protect this "axion $Z_2$" index. Topological states are often insightfully characterized by choosing one crystallographic direction of interest, and inspecting the hybrid Wannier (or equivalently, the non-Abelian Wilson-loop) band structure, considered as a function of the two-dimensional Brillouin zone in the orthogonal directions. Here, we systematically classify the axion-quantizing symmetries, and explore the implications of such symmetries on the Wannier band structure. Conversely, we clarify the conditions under which the axion$Z_2$ index can be deduced from the Wannier band structure. In particular, we identify cases in which a counting of Dirac touchings between Wannier bands, or a calculation of the Chern number of certain Wannier bands, or both, allows for a direct determination of the axion $Z_2$ index. We also discuss when such symmetries impose a "flow" on the Wannier bands, such that they are always glued to higher and lower bands by degeneracies somewhere in the projected Brillouin zone, and the related question of when the corresponding surfaces can remain gapped, thus exhibiting a half-quantized surface anomalous Hall conductivity. Our formal arguments are confirmed and illustrated in the context of tight-binding models for several paradigmatic axion-odd symmetries including time reversal, inversion, simple mirror, and glide mirror symmetries.

Regions of existence of surface spin-waves in surface Brillouin zones of hexagonal simple ferromagnetic thin films

Abstract A quantum statistic Green's-function method is used to study the surface in-plane propagating spin-waves (k// ≠ 0) in a symmetric magnetic film consisting of twelve atomic layers with damping. The effects of temperature, external magnetic field, surface exchange coupling, surface anisotropy and thickness on the regions of existence of low- and high-frequency surface spin-waves in the surface two-dimensional Brillouin zone have been investigated. The regions of existence of surface spin-waves increase first and then decrease with increasing temperature or external magnetic field. There is a special temperature or external magnetic field, at which the regions of existence of surface spin-waves exhibit a maximum. Additionally, the spatial distribution of spin-waves in different spin-wave frequency range is also studied. For a film with low-frequency surface spin-waves, there is a special spin-wave frequency range, in which no spin-waves propagate in surfaces, the spin-waves propagate only in internal layers of the film. Namely, in this frequency range, the film is a spin-waves surface insulator . For a film with high-frequency surface spin-waves, there is also a special spin-wave frequency range, in which the spin-waves propagate only in surfaces. Namely, in this frequency range, the film is a spin-waves internal insulator . In the present work, a quantum statistic approach is developed to study the surface spin-wave of magnetic film. The results show the method to adjust the regions of existence of surface spin-waves in the surface two-dimensional Brillouin zone and are beneficial for building future microwave device based upon magnetic film.

Mapping phonon modes from reduced-dimensional to bulk systems

A lattice-dynamics-based algorithm for mapping the true phonon modes of a film, defined by a two-dimensional (2D) Brillouin zone, to the modes of the corresponding bulk material, defined by a three-dimensional (3D) Brillouin zone, is proposed. The mapping allows for an assessment of the common assumption that film phonon modes have the same harmonic properties as the corresponding bulk phonon modes. The algorithm is based on normal mode decomposition and is inspired by the observation that the atomic trajectories generated by the 2D eigenvectors lead to standing-wave-like behaviors in the cross-plane direction. This behavior enables an unfolding scheme that does not require an assumption of cross-plane periodicity, as used in previous approaches for mapping phonon modes. The algorithm is applied to films between two and ten unit cells thick built from Lennard-Jones (LJ) argon, whose bulk is isotropic, and graphene, whose bulk (graphite) is anisotropic. For LJ argon, the density of states deviates from that of the bulk as the film gets thinner due to phonon frequencies that shift to lower values. This shift is a result of transverse branch splitting due to the film’s anisotropy and the emergence of a quadratic acoustic branch. As such, while the mapping algorithm works well for the thicker LJ argon films, it does not perform as well for the thinner films as there is a weaker correspondence between the 2D and 3D modes. For graphene, the density of states of even the thinnest films closely matches that of graphite due to the inherent anisotropy, except for a small shift at low frequencies. As a result, the mapping algorithm works well for all thicknesses of the graphene films, indicating a strong correspondence between the 2D and 3D modes.

Realistic estimates of superconducting properties for the cuprates: reciprocal-space diagrammatic expansion combined with variational approach

We propose a systematic approach to the systems of correlated electrons, the so-called -DE-GWF method, based on reciprocal-space (-resolved) diagrammatic expansion of the variational Gutzwiller-type wave function for parametrized models of correlated fermions. The present approach, in contrast to either variational Monte-Carlo (VMC), or the recently developed real-space diagrammatic expansion of the Gutzwiller-type wave function (direct-space DE-GWF technique), is applicable directly in the thermodynamic limit and thus is suitable for describing selected singular features of the wave-vector-dependent quantities. We employ the -DE-GWF method to extract the non-analytic part of the two leading moments of the fermion spectral-density function across the (two-dimensional) Brillouin zone for the Hubbard model and away from the half-filling. Those moments are used to evaluate the nodal quasiparticle velocities and their spectral weights in the correlated superconducting state. The two velocities determined in that manner exhibit scaling with the electron concentration qualitatively different from that obtained earlier for the excited states of the high-Tc cuprates within the projected quasi-particle ansatz, and the results are in a very good quantitative agreement with experimental data if interpreted as those characterizing the spectrum below and above the observed kink. We provide a detailed discussion of the two gaps and two excitation branches (two velocities) appearing naturally within our DE-GWF approach. The two separate sets of characteristics distinguish the renormalized quasiparticle states very close to the Fermi surface from the deeper correlated-state properties. Also, an enhancement of the -dependent magnetic susceptibility is shown to contain a spin-fluctuation contribution within our language. Finally, the -DE-GWF approach is compared to both the VMC and real-space DE-GWF results for the cases of Hubbard and t-J-U models.

External electric field driven modification of the anomalous and spin Hall conductivities in Fe thin films on MgO(001)

The effects of applying external electric fields to the anomalous and spin Hall conductivities in Fe thin-film models with different layer thicknesses on MgO(001) are investigated by using first-principles calculations. We observe that, for the considered systems, the application of positive electric field associated with the accumulation of negative charges on the Fe side generally decreases (increases) the anomalous (spin) Hall conductivities. The mapping of the Hall conductivities within the two-dimensional Brillouin zone shows that the electric-field-induced modifications are related to the modification of the band structures of the atoms at the interface with the MgO substrate. In particular, the external electric field affects the Hall conductivities via the modifications of the ${d}_{xz},{d}_{yz}$ orbitals, in which the application of positive electric field pushes the minority-spin states of the ${d}_{xz},{d}_{yz}$ bands closer to the Fermi level. Better agreement with the anomalous Hall conductivity for bulk Fe and a more realistic scenario for the electric field modification of Hall conductivities are obtained by using the thicker layers of Fe on MgO (${\mathrm{Fe}}_{3}$/MgO and ${\mathrm{Fe}}_{5}$/MgO).

Spin-Wave Excitations Evidencing the Kitaev Interaction in Single Crystalline α-RuCl_{3}.

Kitaev interactions underlying a quantum spin liquid have long been sought, but experimental data from which their strengths can be determined directly, are still lacking. Here, by carrying out inelastic neutron scattering measurements on high-quality single crystals of α-RuCl_{3}, we observe spin-wave spectra with a gap of ∼2 meV around the M point of the two-dimensional Brillouin zone. We derive an effective-spin model in the strong-coupling limit based on energy bands obtained from first-principles calculations, and find that the anisotropic Kitaev interaction K term and the isotropic antiferromagnetic off-diagonal exchange interaction Γ term are significantly larger than the Heisenberg exchange coupling J term. Our experimental data can be well fit using an effective-spin model with K=-6.8 meV and Γ=9.5 meV. These results demonstrate explicitly that Kitaev physics is realized in real materials.

Edge states of mechanical diamond and its topological origin

A mechanical diamond, with the classical mechanics of a spring-mass model arrayed on a diamond lattice, is discussed topologically. Its frequency dispersion possesses an intrinsic nodal structure in the three-dimensional Brillouin zone (BZ) protected by the chiral symmetry. Topological changes of the line nodes are demonstrated, associated with the modification of the tension. The line nodes projected into two-dimensional BZ, form loops, which are characterized by the quantized Berry phases with 0 or π. With boundaries, the edge states are discussed in relation to the Berry phases and winding numbers, and the bulk-edge correspondence of the mechanical diamond is established.

Line nodes and surface Majorana flat bands in static and kicked p -wave superconducting Harper model

We investigate the effect of introducing nearest-neighbor $p$-wave superconducting pairing to both the static and kicked extended Harper model with two periodic phase parameters acting as artificial dimensions to simulate three-dimensional systems. It is found that in both the static model and the kicked model, by varying the $p$-wave pairing order parameter, the system can switch between a fully gapped phase and a gapless phase with point nodes or line nodes. The topological property of both the static and kicked model is revealed by calculating corresponding topological invariants defined in the one-dimensional lattice dimension. Under open boundary conditions along the physical dimension, Majorana flat bands at energy zero (quasienergy zero and $\pi$) emerge in the static (kicked) model at the two-dimensional surface Brillouin zone. For certain values of pairing order parameter, (Floquet) Su-Schrieffer-Heeger-like edge modes appear in the form of arcs connecting different (Floquet) Majorana flat bands. Finally, we find that in the kicked model, it is possible to generate two controllable Floquet Majorana modes, one at quasienergy zero and the other at quasienergy $\pi$, at the same parameter values.

Shubnikov-de Haas quantum oscillations reveal a reconstructed Fermi surface near optimal doping in a thin film of the cuprate superconductorPr1.86Ce0.14CuO4±δ

We study magnetotransport properties of the electron-doped superconductor Pr$_{2-x}$Ce$_x$CuO$_{4\pm\delta}$ with $x$ = 0.14 in magnetic fields up to 92~T, and observe Shubnikov de-Haas magnetic quantum oscillations. The oscillations display a single frequency $F$=255$\pm$10~T, indicating a small Fermi pocket that is $\sim$~1\% of the two-dimensional Brillouin zone and consistent with a Fermi surface reconstructed from the large hole-like cylinder predicted for these layered materials. Despite the low nominal doping, all electronic properties including the effective mass and Hall effect are consistent with overdoped compounds. Our study demonstrates that the exceptional chemical control afforded by high quality thin films will enable Fermi surface studies deep into the overdoped cuprate phase diagram.