Bloch Wave Functions Research Articles

Shift current from electromagnon excitations in multiferroics

Electromagnon is the spin wave in multiferroic materials and is known to accompany electric polarization due to the cross correlation between the charge and spin. Here, we theoretically show that the electromagnons also induce dc current upon their photoexcitations. The proposed dc current response originates from the shift current mechanism which is characterized by the so called shift vector, a geometric quantity of the Bloch wavefunctions.

Periodic Ultranarrow Rods as 1D Subwavelength Optical Lattices

We report on ground state properties of a one-dimensional, weakly-interacting Bose gas constrained by an infinite multi-rods 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.

Fractional corner charges in spin-orbit coupled crystals

We study two-dimensional spinful insulating phases of matter that are protected by time-reversal and crystalline symmetries. To characterize these phases we employ the concept of corner charge fractionalization: Corners can carry charges that are fractions of even multiples of the electric charge. The charges are quantized and topologically stable as long as all symmetries are preserved. We classify the different corner charge configurations for all point groups, and match them with the corresponding bulk topology. For this we employ symmetry indicators and (nested) Wilson loop invariants. We provide formulas that allow for a convenient calculation of the corner charge from Bloch wavefunctions and illustrate our results using the example of arsenic and antimony monolayers. Depending on the degree of structural buckling, these materials can exhibit two distinct obstructed atomic limits. We present density functional theory calculations for open flakes to support our findings.

Topological current for transverse electrical and thermal conductivity in thermoelectric effect

Thermoelectric efficiency of the traditional thermoelectric material is low, which restricts the large scale applications. Recently, the developing of the topological insulator provides a new opportunity to get high thermoelectric efficiency material. There are two effects in topological insulator: anomalous Hall and Nernst effect, which have contribution to the transport properties. Because of anomalous Hall and Nernst effect the electrical thermal conductivity have transverse parts, which affect the Seebeck coefficient. However, the transverse parts can be expressed by Berry curvature. By using of φ-mapping topological theory, the Berry curvature is studied and we find there is topological vortex in the momentum space. The Bloch wave function is zero at the topological vortex. Finally, the relationships between the topological vortex and the transverse electrical and thermal conductivity is given and how the topology affects the Seebeck coefficient is researched in detail.

Unifying topological phase transitions in non-interacting, interacting, and periodically driven systems

Topological phase transitions occur in real materials as well as quantum engineered systems, all of which differ greatly in terms of dimensionality, symmetries, interactions, and driving, and hence require a variety of techniques and concepts to describe their topological properties. For instance, topology may be accessed from single-particle Bloch wave functions, Green's functions, or many-body wave functions. We demonstrate that despite this diversity, all topological phase transitions display a universal feature: namely, a divergence of the curvature function that composes the topological invariant at the critical point. This feature can be exploited via a renormalization-group–like methodology to describe topological phase transitions. This approach serves to extend notions of correlation function, critical exponents, scaling laws and universality classes used in Landau theory to characterize topological phase transitions in a unified manner.

Non-Bloch Band Theory of Non-Hermitian Systems.

In spatially periodic Hermitian systems, such as electronic systems in crystals, the band structure is described by the band theory in terms of the Bloch wave functions, which reproduce energy levels for large systems with open boundaries. In this paper, we establish a generalized Bloch band theory in one-dimensional spatially periodic tight-binding models. We show how to define the Brillouin zone in non-Hermitian systems. From this Brillouin zone, one can calculate continuum bands, which reproduce the band structure in an open chain. As an example, we apply our theory to the non-Hermitian Su-Schrieffer-Heeger model. We also show the bulk-edge correspondence between the winding number and existence of the topological edge states.

Torus solutions to the Weierstrass-Enneper representation of surfaces

In this paper, we present a torus solution to the generalized Weierstrass-Enneper representation of surfaces in R4. 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.

Symmetry representation approach to topological invariants in C2zT -symmetric systems

We study the homotopy classification of symmetry representations to describe the bulk topological invariants protected by the combined operation of a twofold rotation ${C}_{2z}$ and time-reversal $T$ symmetries. We define topological invariants as obstructions to having smooth Bloch wave functions compatible with a momentum-independent symmetry representation. When the Bloch wave functions are required to be smooth, the information on the band topology is contained in the symmetry representation. This implies that the $d$-dimensional homotopy class of the unitary matrix representation of the symmetry operator corresponds to the $d$-dimensional topological invariants. Here, we prove that the second Stiefel-Whitney number, a two-dimensional (2D) topological invariant protected by ${C}_{2z}T$, is the homotopy invariant that characterizes the second homotopy class of the matrix representation of ${C}_{2z}T$. As an application of our result, we show that the three-dimensional (3D) bulk topological invariant for the ${C}_{2z}T$-protected topological crystalline insulator proposed by C. Fang and L. Fu in Phys. Rev. B 91, 161105 (2015), which we call the 3D strong Stiefel-Whitney insulator, is identical to the quantized magnetoelectric polarizability. The bulk-boundary correspondence associated with the quantized magnetoelectric polarizability shows that the 3D strong Stiefel-Whitney insulator has chiral hinge states as well as 2D massless surface Dirac fermions. This shows that the 3D strong Stiefel-Whitney insulator has the characteristics of both the first- and the second-order topological insulators, simultaneously, which is in consistence with the recent classification of higher-order topological insulators protected by an order-two symmetry.

Non-Wigner-Dyson level statistics and fractal wave function of disordered Weyl semimetals

Finding fingerprints of disordered Weyl semimetals (WSMs) is an unsolved task. Here we report such findings in the level statistics and the fractal nature of electron wave function around Weyl nodes (WNs) of disordered WSMs. The nearest-neighbor level spacing follows a new universal distribution ${P}_{c}(s)={C}_{1}{s}^{2}exp[\ensuremath{-}{C}_{2}{s}^{2\ensuremath{-}{\ensuremath{\gamma}}_{0}}]$ originally proposed for the level statistics of critical states in the integer quantum Hall systems or normal dirty metals (diffusive metals) at metal-to-insulator transitions, instead of the Wigner-Dyson distribution for diffusive metals. Numerically we find ${\ensuremath{\gamma}}_{0}=0.62\ifmmode\pm\else\textpm\fi{}0.07$. In contrast to the Bloch wave functions of clean WSMs that uniformly distribute over the whole space of $(D=3)$ at large length scale, the wave function of disordered WSMs at a WN occupies a fractal space of dimension $D=2.18\ifmmode\pm\else\textpm\fi{}0.05$. Away from the WN, wave function is a fractal at a length scale below a correlation length diverging at the WN as $\ensuremath{\xi}\ensuremath{\propto}{|E|}^{\ensuremath{-}\ensuremath{\nu}}$ with $\ensuremath{\nu}=0.89\ifmmode\pm\else\textpm\fi{}0.05$. Beyond the length scale, the wave function is homogeneous. In the ergodic limit, the level number variance ${\mathrm{\ensuremath{\Sigma}}}_{2}$ around Weyl nodes increases linearly with the average level number $N,{\mathrm{\ensuremath{\Sigma}}}_{2}=\ensuremath{\chi}N$, where $\ensuremath{\chi}=0.2\ifmmode\pm\else\textpm\fi{}0.1$ is independent of system sizes and disorder strengths.

A rational framework for dynamic homogenization at finite wavelengths and frequencies.

In this study, we establish an inclusive paradigm for the homogenization of scalar wave motion in periodic media (including the source term) at finite frequencies and wavenumbers spanning the first Brillouin zone. We take the eigenvalue problem for the unit cell of periodicity as a point of departure, and we consider the projection of germane Bloch wave function onto a suitable eigenfunction as descriptor of effective wave motion. For generality the finite wavenumber, finite frequency homogenization is pursued in via second-order asymptotic expansion about the apexes of 'wavenumber quadrants' comprising the first Brillouin zone, at frequencies near given (acoustic or optical) dispersion branch. We also consider the junctures of dispersion branches and 'dense' clusters thereof, where the asymptotic analysis reveals several distinct regimes driven by the parity and symmetries of the germane eigenfunction basis. In the case of junctures, one of these asymptotic regimes is shown to describe the so-called Dirac points that are relevant to the phenomenon of topological insulation. On the other hand, the effective model for nearby solution branches is found to invariably entail a Dirac-like system of equations that describes the interacting dispersion surfaces as 'blunted cones'. For all cases considered, the effective description turns out to admit the same general framework, with differences largely being limited to (i) the eigenfunction basis, (ii) the reference cell of medium periodicity, and (iii) the wavenumber-frequency scaling law underpinning the asymptotic expansion. We illustrate the analytical developments by several examples, including Green's function near the edge of a band gap and clusters of nearby dispersion surfaces.

Classification of flat bands according to the band-crossing singularity of Bloch wave functions

We show that flat bands can be categorized into two distinct classes, that is, singular and nonsingular flat bands, by exploiting the singular behavior of their Bloch wave functions in momentum space. In the case of a singular flat band, its Bloch wave function possesses immovable discontinuities generated by the band-crossing with other bands, and thus the vector bundle associated with the flat band cannot be defined. This singularity precludes the compact localized states from forming a complete set spanning the flat band. Once the degeneracy at the band crossing point is lifted, the singular flat band becomes dispersive and can acquire a finite Chern number in general, suggesting a new route for obtaining a nearly flat Chern band. On the other hand, the Bloch wave function of a nonsingular flat band has no singularity, and thus forms a vector bundle. A nonsingular flat band can be completely isolated from other bands while preserving the perfect flatness. All one-dimensional flat bands belong to the nonsingular class. We show that a singular flat band displays a novel bulk-boundary correspondence such that the presence of the robust boundary mode is guaranteed by the singularity of the Bloch wave function. Moreover, we develop a general scheme to construct a flat band model Hamiltonian in which one can freely design its singular or nonsingular nature. Finally, we propose a general formula for the compact localized state spanning the flat band, which can be easily implemented in numerics and offer a basis set useful in analyzing correlation effects in flat bands.

Current-Voltage Characteristic and Shot Noise of Shift Current Photovoltaics.

We theoretically study the current-voltage relation, the I-V characteristic, of the photovoltaics due to the shift current, i.e., the photocurrent generated without the external dc electric field in noncentrosymmetric crystals through the Berry connection of the Bloch wave functions. We find that the I-V characteristic and shot noise are controlled by the difference of group velocities between conduction and valence bands, i.e., v_{11}-v_{22}, and the relaxation time τ. Since the shift current itself is independent of these quantities, there are a wide variety of possibilities to design it to maximize the energy conversion rate and also to suppress the noise. We propose that the Landau levels in noncentrosymmetric two-dimensional systems are a promising candidate for energy conversion.

Unconventional Flatband Line States in Photonic Lieb Lattices

Flatband systems typically host "compact localized states" (CLS) due to destructive interference and macroscopic degeneracy of Bloch wave functions associated with a dispersionless energy band. Using a photonic Lieb lattice (LL), such conventional localized flatband states are found to be inherently incomplete, with the missing modes manifested as extended line states that form noncontractible loops winding around the entire lattice. Experimentally, we develop a continuous-wave laser writing technique to establish a finite-sized photonic LL with specially tailored boundaries and, thereby, directly observe the unusually extended flatband line states. Such unconventional line states cannot be expressed as a linear combination of the previously observed boundary-independent bulk CLS but rather arise from the nontrivial real-space topology. The robustness of the line states to imperfect excitation conditions is discussed, and their potential applications are illustrated.

Two-Particle Collisional Coordinate Shifts and Hydrodynamic Anomalous Hall Effect in Systems without Lorentz Invariance.

We show that electrons undergoing a two-particle collision in a crystal experience a coordinate shift that depends on their single-particle Bloch wave functions and derive a gauge-invariant expression for such a shift, valid for arbitrary band structures and arbitrary two-particle interaction potentials. As an application of the theory, we consider two-particle coordinate shifts for Weyl fermions in space of three spatial dimensions. We demonstrate that such shifts in general contribute to the anomalous Hall conductivity of a clean electron liquid.

Berry Phase and Model Wave Function in the Half-Filled Landau Level.

We construct model wave functions for the half-filled Landau level parametrized by "composite fermion occupation-number configurations" in a two-dimensional momentum space, which correspond to a Fermi sea with particle-hole excitations. When these correspond to a weakly excited Fermi sea, they have a large overlap with wave functions obtained by the exact diagonalization of lowest-Landau-level electrons interacting with a Coulomb interaction, allowing exact states to be identified with quasiparticle configurations. We then formulate a many-body version of the single-particle Berry phase for adiabatic transport of a single quasiparticle around a path in momentum space, and evaluate it using a sequence of exact eigenstates in which a single quasiparticle moves incrementally. In this formulation the standard free-particle construction in terms of the overlap between "periodic parts of successive Bloch wave functions" is reinterpreted as the matrix element of a "momentum boost" operator between the full Bloch states, which becomes the matrix elements of a Girvin-MacDonald-Platzman density operator in the many-body context. This allows the computation of the Berry phase for the transport of a single composite fermion around the Fermi surface. In addition to a phase contributed by the density operator, we find a phase of exactly π for this process.

Geometry of projected connections, Zak phase, and electric polarization

The concept of the Zak phase lies at the core of the modern theory of electric polarization. It is defined using the components of the Bloch wave functions in a certain basis, which is not captured by the standard expression for Berry potential. We provide a consistent geometric interpretation of the Zak phase in terms of projected connections. In the context of Bloch states, we relate the transformation law of projected Berry potential with classical currents that contribute to the time derivative of the electric polarization. This gives a new argument for the Zak phase formula for the electronic contribution to the polarization. We demonstrate that the Wannier functions play a key role in the description of an adiabatic current in a periodic system.

Superconducting Phases in Lithium Decorated Graphene LiC6

A study of possible superconducting phases of graphene has been constructed in detail. A realistic tight binding model, fit to ab initio calculations, accounts for the Li-decoration of graphene with broken lattice symmetry, and includes s and d symmetry Bloch character that influences the gap symmetries that can arise. The resulting seven hybridized Li-C orbitals that support nine possible bond pairing amplitudes. The gap equation is solved for all possible gap symmetries. One band is weakly dispersive near the Fermi energy along Γ → M where its Bloch wave function has linear combination of {d}_{{x}^{2}-{y}^{2}} and dxy character, and is responsible for {d}_{{x}^{2}-{y}^{2}} and dxy pairing with lowest pairing energy in our model. These symmetries almost preserve properties from a two band model of pristine graphene. Another part of this band, along K → Γ, is nearly degenerate with upper s band that favors extended s wave pairing which is not found in two band model. Upon electron doping to a critical chemical potential μ1 = 0.22 eV the pairing potential decreases, then increases until a second critical value μ2 = 1.3 eV at which a phase transition to a distorted s-wave occurs. The distortion of d- or s-wave phases are a consequence of decoration which is not appear in two band pristine model. In the pristine graphene these phases convert to usual d-wave or extended s-wave pairing.

Flatbands and Emergent Ferromagnetic Ordering in Fe_{3}Sn_{2} Kagome Lattices.

A flatband representing a highly degenerate and dispersionless manifold state of electrons may offer unique opportunities for the emergence of exotic quantum phases. To date, definitive experimental demonstrations of flatbands remain to be accomplished in realistic materials. Here, we present the first experimental observation of a striking flatband near the Fermi level in the layered Fe_{3}Sn_{2} crystal consisting of two Fe kagome lattices separated by a Sn spacing layer. The band flatness is attributed to the local destructive interferences of Bloch wave functions within the kagome lattices, as confirmed through theoretical calculations and modelings. We also establish high-temperature ferromagnetic ordering in the system and interpret the observed collective phenomenon as a consequence of the synergetic effect of electron correlation and the peculiar lattice geometry. Specifically, local spin moments formed by intramolecular exchange interaction are ferromagnetically coupled through a unique network of the hexagonal units in the kagome lattice. Our findings have important implications to exploit emergent flat-band physics in special lattice geometries.

Many-body topological invariants for fermionic short-range entangled topological phases protected by antiunitary symmetries

We present a fully many-body formulation of topological invariants for various topological phases of fermions protected by antiunitary symmetry, which does not refer to single particle wave functions. For example, we construct the many-body $\mathbb{Z}_2$ topological invariant for time-reversal symmetric topological insulators in two spatial dimensions, which is a many-body counterpart of the Kane-Mele $\mathbb{Z}_2$ invariant written in terms of single-particle Bloch wave functions. We show that an important ingredient for the construction of the many-body topological invariants is a fermionic partial transpose which is basically the standard partial transpose equipped with a sign structure to account for anti-commuting property of fermion operators. We also report some basic results on various kinds of pin structures -- a key concept behind our strategy for constructing many-body topological invariants -- such as the obstructions, isomorphism classes, and Dirac quantization conditions.

Armchair α-graphyne nanoribbons as negative differential resistance devices: Induced by nitrogen doping

Abstract Using non-equilibrium Green's functions (NEGF) in combination with tight-binding (TB) model, the electronic transport properties of pristine and nitrogen (N) doped armchair α-graphyne nanoribbons (A-α-GYNRs) are studied under finite bias. Initially, we have calculated the total energy in order to find the most stable place for N atom. Then we have investigated the effect of width (W) and length (L) of the ribbon and also the position (edge and center of the ribbon) and concentration of doping on the electronic transport properties. Our results reveal that, doping changes the semiconducting behavior of 3n and 3n+1 A-α-GYNRs to semi metallic. Moreover, it is observed that the electronic transport properties are more affected by central doping rather than the edge doping. Interestingly, both edge and central doped ribbons show negative differential resistance (NDR) in all widths. Our results show that doping concentration and the NDR are inversely proportional to each other. We have also found that, as the length of the central region of the device gets longer, the NDR reaches up to 159. Transmission spectrum, bandstructure of the electrodes, Bloch wave functions and density of states (DOS) are analyzed subsequently to more elucidate the electronic transport properties. Our findings could be used to develop the nano-scale NDR devices.