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The Zak phase serves as a reliable index for pinpointing topological phase transitions within one-dimensional chains, typically ascertained through numerical methods in complex situations. Nonetheless, in proximity to the transition threshold, the numerical Zak phase may become ambiguous. This ambiguity arises from the discrete nature of numerical approaches and the nonlinear relationship between the wave function's phase and the wave vector (k). In response, this paper presents an innovative method aimed at accurately determining the winding number, thereby facilitating the identification of topological phase transitions. Our approach hinges on analyzing the evolution of the phase difference between the projections of the Bloch wave function onto two distinct sublattices. We demonstrate the efficacy of this method through three illustrative examples: the Su–Schrieffer–Heeger model, a magnetic vortex chain, and a trimer lattice chain. The results indicate that our proposed numerical lagging-phase examination method achieves superior precision in identifying topological phase transitions, particularly at critical junctures, compared to the conventional numerical Zak phase approach. [ABSTRACT FROM AUTHOR]

The localized nature of a flat band is understood by the existence of a compact localized eigenstate. However, the localization properties of a partially flat band, ubiquitous in surface modes of topological semimetals, have been unknown. We show that the partially flat band is characterized by a non-normalizable quasi-compact localized state (Q-CLS), which is compactly localized along several directions but extended in at least one direction. The partially flat band develops at momenta where normalizable Bloch wave functions can be obtained from a linear combination of the non-normalizable Q-CLSs. Outside this momentum region, a ghost flat band, unseen from the band structure, is introduced based on a counting argument. Then, we demonstrate that the Wannier function corresponding to the partially flat band exhibits an algebraic decay behavior. Namely, one can have the Wannier obstruction in a band with a vanishing Chern number if it is partially flat. Finally, we develop the construction scheme of a tight-binding model for a topological semimetal by designing a Q-CLS. Compact localized states constitute an auxiliary state representation for a flat-band lattice system with wave functions non-zero only in a finite portion of the lattice. Here, the authors show that in some flat-band systems, these states can be partially "hidden"; surprisingly, these ghost flat bands present an obstruction to be represented as maximally localized Wannier functions. [ABSTRACT FROM AUTHOR]

A robust density fitting method for calculating Coulomb matrix elements over Bloch functions based on the calculation of two- and three-center matrix elements of the Ewald potential is described and implemented in a Gaussian orbital basis in the Exciton code. The method is tested by comparing Coulomb and exchange energies from density fitting to corresponding energies from SCF Hartree–Fock (HF) calculations for diamond, magnesium oxide, and bulk Ne. Density fitting coefficients from the robust method are compared to coefficients from a variational method applied to wave function orbital products in bulk Ne. Four center Coulomb matrix elements from density fitting are applied to time dependent (TDHF) calculations in diamond, magnesium oxide, and anatase and rutile polytypes of titanium dioxide. Shifting virtual states downwards uniformly relative to occupied states and scaling the electron–hole attraction term in the TDHF Hamiltonian by 0.4 yields good agreement with either experiment or Bethe–Salpeter equation calculations. This approach mirrors similar "scissors" adjustments of occupied and virtual states and introduction of a scaled electron–hole attraction term in some time dependent DFT calculations. [ABSTRACT FROM AUTHOR]

The bulk-boundary correspondence is an integral feature of topological analysis and the existence of boundary or interface modes offers direct insight into the topological structure of the Bloch wave function. While only the topology of the wave function has been considered relevant to boundary modes, we demonstrate that another geometric quantity, the so-called quantum distance, can also host a bulk-interface correspondence. We consider a generic class of two-dimensional flat band systems, where the flat band has a parabolic band-crossing with another dispersive band. While such flat bands are known to be topologically trivial, we show that the nonzero maximum quantum distance between the eigenstates of the flat band around the touching point guarantees the existence of boundary modes at the interfaces between two domains with different chemical potentials or different maximum quantum distance. Moreover, the maximum quantum distance can predict even the explicit form of the dispersion relation and decay length of the interface modes. The bulk-boundary correspondence is a defining feature of non-trivial topological matter and extends from the many topological orders that can exist in these systems. Here, the authors theoretically propose a feature distinct from the conventional bulk-boundary correspondence whereby localised modes exist between two flat-band systems with different geometric characters. [ABSTRACT FROM AUTHOR]

A diabatization protocol that combines the recently developed model space diabatization scheme and a mean-field treatment for spin-orbit coupling is presented. The protocol is highly efficient and easy to use. Its robustness is demonstrated in the calculations of the diabatic spin-orbit matrix elements of PH+3, AsH+3, and SbH+3 . It captures subtle geometry-dependence of the spin-orbit couplings and provides wave function information to explain the origin of the dependence. [ABSTRACT FROM AUTHOR]

We study the lattice configuration and electronic structure of a double moiré superlattice, which is composed of a graphene layer encapsulated by two other layers in a way such that the two hexagonal moiré patterns are arranged in a dodecagonal quasicrystalline configuration. We show that there are between 0 and 4 such configurations depending on the lattice mismatch between graphene and the encapsulating layer. We then reveal the resonant interaction, which is distinct from the conventional 2-, 3-, 4-wave mixing of moiré superlattices, that brings together and hybridizes twelve degenerate Bloch states of monolayer graphene. These states do not fully satisfy the dodecagonal quasicrystalline rotational symmetry due to the symmetry of the wave vectors involved. Instead, their wave functions exhibit trigonal quasicrystalline order, which lacks inversion symmetry, at the energies much closer to the charge neutrality point of graphene. [ABSTRACT FROM AUTHOR]

Herein, the original technique for transport measurements at one‐axis deformation is applied for the characterization of In‐doped n‐type Pb1−xSnxTe thin films on a mica substrate. The main advantage of the developed setup is the possibility of obtaining a significantly higher pressure compared with bulk samples. A systematic study of the transport properties as a function of pressure for In‐doped PbTe and PbSnTe films is conducted. A huge deformation of up to ≈6% without sample destruction is obtained within one‐axis deformation and an applied pressure of up to ≈30 kbar for Pb1−xInxTe1−yIy and Pb1−x−ySnxInyTe thin films. This allowed us to determine the position of the In resonance level in the band spectrum of the PbSnTe compound as a function of the pressure. The In impurity level is moving up with the application of pressure. Such a conclusion is confirmed by the Hall effect and electrical conductivity measurements for Pb1−xSnxTe films doped with In and can be explained by the fact that In impurity wave functions are strongly localized and interact weakly with Bloch states of PbSnTe and with each other. [ABSTRACT FROM AUTHOR]

The theory of Bloch electron dynamics for carriers in homogeneous electric fields of arbitrary time dependence is developed in consideration of the electronic transport properties in graphene. The general approach is to use the accelerated Bloch state representation (ABR) as a basis so that the dependence upon the electric field, including Zener tunneling, is treated exactly; also, the electric field is described in the vector potential gauge. Within the ABR, the instantaneous eigenstates for the central Hamiltonian are described and utilized to develop both the time‐dependent wave functions and the single‐particle density matrix pictures with explicit application to intrinsic graphene. The Bloch electron analysis for graphene in a constant electric field reveals the explicit manifestation of electron–hole pair creation, Bloch oscillations, and Wannier–Stark localization. The average velocity and acceleration are established as an explicit function of the electric field, and their behavior is characterized on short and long time scales. Further, it is shown that the average acceleration in a nonvanishing electric field gives rise to an electric field induced dynamical effective mass tensor in the direction of the field. [ABSTRACT FROM AUTHOR]

We report on ground-state properties of a one-dimensional, weakly interacting Bose gas constrained by an infinite multi-rod periodic structure at zero temperature. We solve the stationary Gross–Pitaevskii equation (GPE) to obtain the Bloch wave functions from which we give a semi-analytical solution for the density profile, as well as for the phase of the wave function in terms of the Jacobi elliptic functions, and the incomplete elliptic integrals of the first, second and third kind. Then, we determine numerically the energy of the ground state, the chemical potential and the compressibility of the condensate and show their dependence on the potential height, as well as on the interaction between the bosons. We show the appearance of loops in the energy band spectrum of the system for strong enough interactions, which appear at the edges of the first Brillouin zone for odd bands and at the center for even bands. We apply our model to predict the energy band structure of the Bose gas in an optical lattice with subwavelength spatial structure. To discuss the density range of the validity of the GPE predictions, we calculate the ground-state energies of the free Bose gas using the GPE, which we compare with the Lieb–Liniger exact energies. [ABSTRACT FROM AUTHOR]

In this paper, we present a torus solution to the generalized Weierstrass-Enneper representation of surfaces in R 4 . The key analytical technique will be Bloch wave functions with complex wave vectors. We will also discuss some possible uses of these solutions which derive from their explicit nature, such as Dehn surgery and the physics of exotic smooth structure. [ABSTRACT FROM AUTHOR]

We investigate the complex band structure of elastic waves in a magnetoelectroelastic phononic crystal, consisting of a polymeric matrix reinforced by BaTiO 3 -CoFe 2 O 4 inclusions in a square lattice. We also study the influence of inclusion geometry - circular, square and rotated square on the band structure. The elastic wave propagation inside periodic structures consists of both propagating and evanescent modes. Accordingly, improved plane wave expansion ( ω ( k ) ) and extended plane wave expansion ( k ( ω ) ) methods are formulated to solve the governing equations of motion of a magnetoelectroelastic solid considering wave propagation in the xy plane in order to obtain the propagating and evanescent modes. Complete band gaps between the XY and Z modes are observed for all types of inclusion. Piezoelectricity and piezomagnetism influence significantly the band gaps and the attenuation. There are many modes of the magnetoelectroelastic phononic crystal that are not identified by the ω ( k ) method. However, they are well identified by the k ( ω ) method proposed. These complex modes are related to the evanescent nature of Bloch waves. The new findings of this study should be useful for vibration control using smart periodic structures with piezoelectricity and piezomagnetism. [ABSTRACT FROM AUTHOR]

The introduction of nonlinearity alters the dispersion of elastic waves in solid media. In this paper, we present an analytical formulation for the treatment of finite-strain Bloch waves in one-dimensional phononic crystals consisting of layers with alternating material properties. Considering longitudinal waves and ignoring lateral effects, the exact nonlinear dispersion relation in each homogeneous layer is first obtained and subsequently used within the transfer matrix method to derive an approximate nonlinear dispersion relation for the overall periodic medium. The result is an amplitude-dependent elastic band structure that upon verification by numerical simulations is accurate for up to an amplitude-to-unitcell length ratio of one-eighth. The derived dispersion relation allows us to interpret the formation of spatial invariance in the wave profile as a balance between hardening and softening effects in the dispersion that emerge due to the nonlinearity and the periodicity, respectively. For example, for a wave amplitude of the order of one-eighth of the unit-cell size in a demonstrative structure, the two effects are practically in balance for wavelengths as small as roughly three times the unit-cell size. [ABSTRACT FROM AUTHOR]

Both Bloch wave (BW) and real space multislice (RS-MS) methods are based on the Schrödinger equation. The BW method is considered to be the most accurate and popular simulation method used in various simulations, and it has solved many crystallographic problems. In this paper, we verified that the aforementioned two methods are similar to each other with minor difference. First, the RS-MS method is implemented in the real space, while the BW method is carried out in the reciprocal space. Furthermore, by quantitatively calculating exit wavefunctions of four crystals, i.e., Cu, MgAl 2 O 4 , Mg 44 Rh 7 and KNbO 3 , we found that the RS-MS method offers certain advantages over the BW method. Firstly, the RS-MS method takes all the structure factors into account to calculate crystal potentials, so accuracy loss caused from the crystal potential is effectively avoided. Secondly, the dimension of the scattering matrix for the RS-MS method is proportional to the area of the 2 dimensional (2D) unit cell which is perpendicular to the incident beam direction, whereas that for the BW method is proportional to the square of the area of the 2D unit cell. To reduce computation time, reduction of dimension of the scattering matrix for the BW method has to be performed, thus accuracy loss is inevitable even if compensated by the Bethe potential method. For crystals with a small 2D unit cell, both simulation methods work well. When the area of the 2D unit cell is larger than square of 2.0 nm, for example KNbO 3 , the size of the scattering matrix of the BW method is too large to be carried out by simulation and one should use the RS-MS method instead. This paper presents a tutorial comparison of the different methods. [ABSTRACT FROM AUTHOR]

The article focuses on a study that report a spectroscopic search for line degeneracies in the electronic bands of NbAs2, a topological semimetal (TSM). It mentions that topology of a TSM arises from a fundamental feature of propagating Bloch waves in crystals. It also mentions that angle-resolved photoemission spectroscopy (ARPES) uses the photoelectric effect and measurement of the energies and momenta of electrons ejected from a material are analyzed to reconstruct its band structure.

Electron vortex beams are distorted by scattering within a crystal, so that the wavefunction can effectively be decomposed into many vortex components. Using a Bloch wave approach equations are derived for vortex beam decomposition at any given depth and with respect to any frame of reference. In the kinematic limit (small specimen thickness) scattering largely takes place at the neighbouring atom columns with a local phase change of π /2 rad. When viewed along the beam propagation direction only one vortex component is present at the specimen entrance surface (i.e. the ‘free space’ vortex in vacuum), but at larger depths the probe is in a mixed state due to Bragg scattering. Simulations show that there is no direct correlation between vortex components and the 〈 L z 〉 pendellösung, i.e. at a given depth probes with relatively constant 〈 L z 〉 can be in a more mixed state compared to those with more rapidly varying 〈 L z 〉. This suggests that minimising oscillations in the 〈 L z 〉 pendellösung by probe channelling is not the only criterion for generating a strong electron energy loss magnetic circular dichroism (EMCD) signal. [ABSTRACT FROM AUTHOR]

Abstract: A novel Bloch band based level set method is proposed for computing the semiclassical limit of Schrödinger equations in periodic media. For the underlying equation, subject to a highly oscillatory initial data, a hybrid of the WKB approximation and homogenization leads to the Bloch eigenvalue problem and an associated Hamilton–Jacobi system for the phase in each Bloch band, with the Bloch eigenvalue be part of the Hamiltonian. We formulate a level set description to capture multi-valued solutions to the band WKB system, and then evaluate total homogenized density over a sample set of bands. A superposition of band densities is established over all bands and solution branches when away from caustic points. The numerical approach splits the solution process into several parts: (i) initialize the level set function from the band decomposition of the initial data; (ii) solve the Bloch eigenvalue problem to compute Bloch waves; (iii) evolve the band level set equation to compute multi-valued velocity and density on each Bloch band; (iv) evaluate the total position density over a sample set of bands using Bloch waves and band densities obtained in steps (ii) and (iii), respectively. Numerical examples with different number of bands are shown to demonstrate the capacity of the method. [Copyright &y& Elsevier]

Abstract: In a previous study the BCS wave function was generalized to include the motion of the center-of-mass of Cooper pairs. It was found that the Ginsburg–Landau order parameter could be identified with a wave function describing this motion. Here attention is focussed on the effects of replacing free electron orbitals by Bloch waves. In the copper oxides, where the Fermi surface lies contiguous to the Brillouin zone boundary, and with pairs coupled by soft phonon modes, limitations on the wave vectors associated with the center-of-mass coordinates are found. These limitations lead to nodes in the Fourier transform of the Ginsburg–Landau order parameter. Possible relevance of the nodes to the more recently discovered properties of high T C superconductors is discussed. [Copyright &y& Elsevier]

Calculations of quantum trajectories associated to a propagated wave function provide new insight into quantum processes such as particle scattering and diffraction. Here, hydrodynamic calculations of electron beam imaging under conditions comparable to those of a transmission electron microscope display the mechanisms behind different commonly investigated diffraction conditions. The Bloch wave method is used to propagate the electron wave function and associated trajectories are computed to map the wave function as it is transmitted through the material. Simulations at normal incidence and of the two-beam condition are performed and electron diffraction is analyzed through a real space interpretation of the wave function. In future work, this method can be coupled with Monte Carlo modeling in order to create all encompassing simulations of electron imaging. [ABSTRACT FROM AUTHOR]

We review the recent progress in the study of topological phases in systems with space–time inversion symmetry IST. IST is an anti-unitary symmetry which is local in momentum space and satisfies such as PT in two dimensions (2D) and three dimensions (3D) without spin–orbit coupling and C2T in 2D with or without spin–orbit coupling, where P, T, C2 indicate the inversion, time-reversal, and two-fold rotation symmetries, respectively. Under IST, the Hamiltonian and the periodic part of the Bloch wave function can be constrained to be real-valued, which makes the Berry curvature and the Chern number vanish. In this class of systems, gapped band structures of real wave functions can be topologically distinguished by the Stiefel–Whitney numbers instead. The first and second Stiefel–Whitney numbers w1 and w2, respectively, are the corresponding invariants in 1D and 2D, which are equivalent to the quantized Berry phase and the Z2 monopole charge, respectively. We first describe the topological phases characterized by the first Stiefel–Whitney number, including 1D topological insulators with quantized charge polarization, 2D Dirac semimetals, and 3D nodal line semimetals. Next we review how the second Stiefel–Whitney class characterizes the 3D nodal line semimetals carrying a Z2 monopole charge. In particular, we explain how the second Stiefel–Whitney number w2, the Z2 monopole charge, and the linking number between nodal lines are related. Finally, we review the properties of 2D and 3D topological insulators characterized by the nontrivial second Stiefel Whitney class. [ABSTRACT FROM AUTHOR]

We present a method to compute optical spectra and exciton binding energies of molecules and solids based on the solution of the Bethe-Salpeter equation and the calculation of the screened Coulomb interaction in a finite field. The method does not require either the explicit evaluation of dielectric matrices or of virtual electronic states, and can be easily applied without resorting to the random phase approximation. In addition, it utilizes localized orbitals obtained from Bloch states using bisection techniques, thus greatly reducing the complexity of the calculation and enabling the efficient use of hybrid functionals to obtain single particle wave functions. We report exciton binding energies of several molecules and absorption spectra of condensed systems of unprecedented size, including water and ice samples with hundreds of atoms. [ABSTRACT FROM AUTHOR]

This work addresses the problem of elastic scattering on a localized impurity in a one-dimensional crystal with sublattice degrees of freedom. The impurity yields long-range interferences in the local density of states known as Friedel oscillations. Here we show that the internal degrees of freedom of Bloch waves are responsible for a geometrical phase shift in Friedel oscillations. The Fourier transform of the energy-resolved interference pattern reveals a topological property of this phase shift, which is intrinsically related to a wave-function topological property (Zak phase) in the absence of impurity. As a result, Friedel oscillations in the local density of states can be regarded as a probe of wave topological properties in a broad class of classical and quantum systems, such as acoustic and photonic crystals, ultracold atomic gases in optical lattices, and electronic compounds. [ABSTRACT FROM AUTHOR]

In periodic systems, electronic wave functions of the eigenstates exhibit the periodically modulated Bloch phases and are characterized by their wave numbers k. We theoretically address the effects of the Bloch phase in general layered materials with a stacking shift. When the interlayer shift and the Bloch wave vector k satisfy certain conditions, interlayer transitions of electrons are prohibited by the interference of the Bloch phase. We specify the manifolds in the k space where the hybridization of the Bloch states between the layers is suppressed in accord with the stacking shift. These manifolds, named stacking-adapted interference manifolds (SAIM), are obviously applicable to general layered materials regardless of a detailed atomic configuration within the unit cell. We demonstrate the robustness and usefulness of the SAIM with first-principles calculations for layered boron nitride, transition-metal dichalcogenide, graphite, and black phosphorus. We also apply the SAIM to general three-dimensional crystals to derive special k-point paths for the respective Bravais lattices, along which the Bloch-phase interference strongly suppresses the band dispersion. Our theory provides a general view on the anisotropic electronic motion intrinsic to the periodic-lattice structure. [ABSTRACT FROM AUTHOR]