Berry Connection Research Articles

PYATB: An efficient Python package for electronic structure calculations using ab initio tight-binding model

We present PYATB, a Python package designed for computing band structures and related properties of materials using the ab initio tight-binding Hamiltonian. The Hamiltonian is directly obtained after conducting self-consistent calculations with first-principles packages using numerical atomic orbital bases, such as ABACUS. The package comprises three modules: Bands, Geometric, and Optical. In the Bands module, one can calculate essential properties of band structures, including the partial density of states, fat bands, Fermi surfaces, and Weyl/Dirac points. The band unfolding method is utilized to obtain the energy band spectra of a supercell by projecting the electronic structure of the supercell onto the Brillouin zone of the primitive cell. With the Geometric module, one can compute the Berry phase and Berry curvature-related quantities, such as electric polarization, Wilson loops, Chern numbers, and anomalous Hall conductivities. The Optical module offers a range of optical property calculations, including optical conductivity and nonlinear optical responses, such as shift current and Berry curvature dipole. Program summaryProgram title: PYATBCPC Library link to program files:https://doi.org/10.17632/fsfpdy9t5r.1Developer's repository link:https://github.com/pyatb/pyatbCode Ocean capsule:https://codeocean.com/capsule/9408682Licensing provisions: GPLv3Programming language: C++, PythonNature of problem: This program is to study the electronic structure, electronic polarization, band topological properties, topological classification, linear and nonlinear optical response of solid crystal systems.Solution method: Based on the tight binding method to solve the band structure, the Wilson loop is used to classify the topological phases, and the optical response is calculated by Berry curvature and Berry connection.

Symmetry-conserving maximally projected Wannier functions

Wannier functions are a powerful tool for the determination of topological properties of solids based on band structure calculations. Here, the authors present a simple and efficient scheme to obtain symmetry-conserving Wannier functions especially suitable for local orbital codes. They analyze which approximation of the basis contribution to the Berry connection and curvature matrices preserves symmetry and argue which Bloch function gauge is best suited for model systems, where the basis is not explicitly known.

Quantum geometry induced anapole superconductivity

Anapole superconductivity recently proposed for multiband superconductors [S. Kanasugi and Y. Yanase, Commun. Phys. 5, 39 (2022)] is a key feature of time-reversal ($\mathcal{T}$) symmetry broken polar superconductors. The anapole moment was shown to arise from the asymmetric Bogoliubov spectrum, which induces finite center of mass momenta of Cooper pairs at the zero magnetic field. In this paper, we show an alternative mechanism of anapole superconductivity: the quantum geometry induces the anapole moment when the interband pairing and Berry connection are finite. Thus, the anapole superconductivity is a ubiquitous feature of $\mathcal{T}$-broken multiband polar superconductors. Applying the theory to a minimal model of ${\mathrm{UTe}}_{2}$, we demonstrate the quantum geometry induced anapole superconductivity. Furthermore, we show the Bogoliubov Fermi surfaces (BFS) in an anapole superconducting state and predict an unusual temperature dependence of BFS due to the quantum geometry. Experimental verification of these phenomena may clarify the superconducting state in ${\mathrm{UTe}}_{2}$ and reveal the ubiquitous importance of quantum geometry in exotic superconductors.

Multipole theory of optical spatial dispersion in crystals

Natural optical activity is the paradigmatic example of an effect originating in the weak spatial inhomogeneity of the electromagnetic field on the atomic scale. In molecules, such effects are well described by the multipole theory of electromagnetism, where the coupling to light is treated semiclassically beyond the electric-dipole approximation. That theory has two shortcomings: it is limited to bounded systems, and its building blocks -the multipole transition moments- are origin dependent. In this work, we recast the multipole theory in a translationally-invariant form that remains valid for crystals. Working in the independent-particle approximation, we introduce "intrinsic" multipole transition moments that are origin independent and transform covariantly under gauge transformations of the Bloch eigenstates. Electric-dipole transitions are given by the interband Berry connection, while magnetic-dipole and electric-quadrupole transitions are described by matrix generalizations of the intrinsic magnetic moment and quantum metric. In addition to multipole-like terms, the response of crystals at first order in the wavevector of light contains band-dispersion terms that have no counterpart in molecular theories. The full response is broken down into magnetoelectric and quadrupolar parts, which can be isolated in the static limit where electric and magnetic fields become decoupled. The rotatory-strength sum rule for crystals is found to be equivalent to the topological constraint for a vanishing chiral magnetic effect in equilibrium, and the formalism is validated by numerical tight-binding calculations.

Third-order charge transport in a magnetic topological semimetal

Magnetic topological materials and their physical signatures are a focus of current research. By first-principles calculations and symmetry analysis, we reveal exotic topological semimetal states in an existing antiferromagnet $\mathrm{Th}\mathrm{Mn}{}_{2}{\mathrm{Si}}_{2}$. Depending on the N\'eel vector orientation, the topological band crossings near the Fermi level form either a double-nodal loop or two pairs of Dirac points, which are all fourfold degenerate and robust under spin-orbit coupling. These topological features produce large Berry connection polarizability, leading to pronounced nonlinear transport effects. Particularly, we evaluate the third-order current response, which dominates the transverse current in $\mathrm{Th}\mathrm{Mn}{}_{2}{\mathrm{Si}}_{2}$. We show that the third-order response can be much more sensitive to topological phase transitions than a linear response, which offers a powerful tool for characterizing topological states of matter.

Symmetry dictionary on charge and spin nonlinear responses for all magnetic point groups with nontrivial topological nature.

Recently, charge or spin nonlinear transport with nontrivial topological properties in crystal materials has attracted much attention. In this paper, we perform a comprehensive symmetry analysis for all 122 magnetic point groups (MPGs) and provide a useful dictionary for charge and spin nonlinear transport from the Berry curvature dipole, Berry connection polarizability and Drude term with nontrivial topological nature. The results are obtained by conducting a full symmetry investigation of the matrix representations of six nonlinear response tensors. We further identify every MPG that can accommodate two or three of the nonlinear tensors. The present work gives a solid theoretical basis for an overall understanding of the second-order nonlinear responses in realistic materials.

Long-range electrostatic contribution to electron-phonon couplings and mobilities of two-dimensional and bulk materials

Charge transport plays a crucial role in manifold potential applications of two-dimensional materials, including field-effect transistors, solar cells, and transparent conductors. At most operating temperatures, charge transport is hindered by scattering of carriers by lattice vibrations. Assessing the intrinsic phonon-limited carrier mobility is thus of paramount importance to identify promising candidates for next-generation devices. Here we provide a framework to efficiently compute the drift and Hall carrier mobility of two-dimensional materials through the Boltzmann transport equation by relying on a Fourier-Wannier interpolation. Building on a recent formulation of long-range contributions to dynamical matrices and phonon dispersions [Phys. Rev. X 11, 041027 (2021)], we extend the approach to electron-phonon coupling including the effect of dynamical dipoles and quadrupoles. We identify an unprecedented contribution associated with the Berry connection that is crucial to preserve the Wannier-gauge covariance of the theory. This contribution is not specific to two-dimensional crystals, but also concerns the three-dimensional case, as we demonstrate via an application to bulk SrO. We showcase our method on a wide selection of relevant monolayers ranging from ${\mathrm{SnS}}_{2}$ to ${\mathrm{MoS}}_{2}$, graphene, BN, InSe, and phosphorene. We also discover a nontrivial temperature evolution of the Hall hole mobility in InSe whereby the mobility increases with temperature above 150 K due to the Mexican-hat electronic structure of the InSe valence bands. Overall, we find that dynamical quadrupoles are essential and can impact the carrier mobility in excess of 75%.

Accurate Prediction of Hall Mobilities in Two-Dimensional Materials through Gauge-Covariant Quadrupolar Contributions.

Despite considerable efforts, accurate computations of electron-phonon and carrier transport properties of low-dimensional materials from first principles have remained elusive. By building on recent advances in the description of long-range electrostatics, we develop a general approach to the calculation of electron-phonon couplings in two-dimensional materials. We show that the nonanalytic behavior of the electron-phonon matrix elements depends on the Wannier gauge, but that a missing Berry connection restores invariance to quadrupolar order. We showcase these contributions in a MoS_{2} monolayer, calculating intrinsic drift and Hall mobilities with precise Wannier interpolations. We also find that the contributions of dynamical quadrupoles to the scattering potential are essential, and that their neglect leads to errors of 23% and 76% in the room-temperature electron and hole Hall mobilities, respectively.

Supercurrent and electromotive force generations by the Berry connection from many-body wave functions

The velocity field composed of the electromagnetic field vector potential and the Berry connection from many-body wave functions explains supercurrent generation, Faraday’s law for the electromotive force (EMF) generation, and other EMF generations whose origins are not electromagnetism. An example calculation for the EMF from the Berry connection is performed using a model for the cuprate superconductivity.

Semiclassical analysis of Dirac fields on curved spacetime

We present a semiclassical analysis for Dirac fields on an arbitrary spacetime background and in the presence of a fixed electromagnetic field. Our approach is based on a Wentzel-Kramers-Brillouin approximation, and the results are analyzed at leading and next-to-leading order in the small expansion parameter $\ensuremath{\hbar}$. Taking into account the spin-orbit coupling between the internal and external degrees of freedom of wave packets, we derive effective ray equations with spin-dependent terms. These equations describe the gravitational spin Hall effect of localized Dirac wave packets. We treat both massive and massless Dirac fields and show how a covariantly defined Berry connection and the associated Berry curvature govern the semiclassical dynamics. The gravitational spin Hall equations are shown to be particular cases of the Mathisson-Papapetrou equations for spinning objects.

Nonlinear spin current of photoexcited magnons in collinear antiferromagnets

We study the nonlinear magnon spin current induced by an ac electric field under light irradiation in collinear antiferromagnets with broken inversion symmetry. For linearly polarized light, we find that a dc spin current appears through ``the magnon spin shift current" mechanism, which is driven by a spin polarization generation in the two magnon creation process and has a close relationship to the geometry of magnon bands through Berry connection. For circularly polarized light, a dc spin current appears through ``the spin injection current" mechanism, which is proportional to the relaxation time of magnons and can be large when the magnon lifetime is long. We demonstrate generation of the magnon spin shift and injection currents, based on a few toy models and a realistic model for a multiferroic material M$_2$Mo$_3$O$_8$.

A comprehensive study of the velocity, momentum and position matrix elements for Bloch states: Application to a local orbital basis

We present a comprehensive study of the velocity operator, \hat{\boldsymbol{v}}=\frac{i}{\hbar} [\hat{H},\hat{\boldsymbol{r}}]\,𝐯̂=iℏ[Ĥ,𝐫̂], when used in crystalline solids calculations. The velocity operator is key to the evaluation of a number of physical properties and its computation, both from a practical and fundamental perspective, has been a long-standing debate for decades. Our work summarizes the different approaches found in the literature, but never connected before in a comprehensive manner. In particular we show how one can compute the velocity matrix elements following two different routes. One where the commutator is explicitly used and another one where the commutator is avoided by relying on the Berry connection. We work out an expression in the latter scheme to compute velocity matrix elements, generalizing previous results. In addition, we show how this procedure avoids ambiguous mathematical steps and how to properly deal with the two popular gauge choices that coexist in the literature. As an illustration of all this, we present several examples using tight-binding models and local density functional theory calculations, in particular using Gaussian-type localized orbitals as basis sets. We show how the the velocity operator cannot be approximated, in general, by the k-gradient of the Bloch Hamiltonian matrix when a non-orthonormal basis set is used. Finally, we also compare with its real-space evaluation through the identification with the canonical momentum operator when possible. This comparison offers us, in addition, a glimpse of the importance of non-local corrections, which may invalidate the naive momentum-velocity correspondence.

Full-Brillouin-zone calculation of high-order harmonic generation from solid-state media

Solid-state high-order harmonic generation (HHG) continues to attract a lot of interest. From the theory and simulation standpoint, two issues are still open; The first is the so-called transition-dipole phase problem. It has been recognized that the dipoles must be treated as complex-valued quantities and that their corresponding Berry connections must be included to ensure phase-gauge invariance. However, while this has been successfully implemented for lower-dimensional systems, fully vectorial and three-dimensional simulations remain challenging. The second issue concerns the symmetry of the high-order harmonic response, when simulations sometimes fail to honor the symmetry of the crystalline material. This work addresses both of these problems with the help of a HHG-simulation approach which (a) is manifestly free of the transition-dipole phase problem, (b) does not require calculation of dipole moments, (c) can account for the contributions from the entire Brillouin zone, and (d) faithfully preserves the symmetry of the simulated crystalline material. We use the method to show that high-order harmonic sources are distributed throughout the Brillouin zone with various phase shifts giving rise to significant cancellations. As a consequence, for the simulated response to correctly capture the material symmetry, contributions from the entire Brillouin zone must be included. Our results have important implications for a number of HHG applications, including all-optical band and dipole reconstruction.

Singular-connection approach to topological phases and resonant optical responses

We introduce a class of singular connections as an alternative to the Berry connection for any family of quantum states defined over a parameter space. We find a natural application of the singular connection in the context of transition dipoles between two bands. We find that the shift vector is nothing but the difference between the singular connection and the connection induced from the Berry connections of involved bands; the gauge invariance of the shift vector is transparent from this expression. We show, using singular connections, that the topological invariant in two dimensions associated with optical transitions between the two bands can be computed, by means of this connection, by algebraically counting the points in the zero locus of a transition dipole matrix element of the two bands involved. It follows that this invariant provides a natural topological lower bound on the number of momenta in the Brillouin zone for which an electron cannot be excited from one Bloch band to the other by absorbing a photon.

Diverse lateral shifts of beams in non-Hermitian waveguide arrays.

Non-Hermitian systems have attracted considerable attention in optics due to the rich physics introduced by the existence of real spectra and exceptional points (EPs), which is exploited in lasers, optical sensors and on-chip manipulations of light. Here, focusing on the dynamics of beams in non-Hermitian waveguide arrays supporting a ring of EPs (exceptional ring) and 3rd-order EPs, we theoretically demonstrate that the center of energy of a beam prepared around an eigenstate of the waveguide array near EPs could exhibit non-zero shifts in the lateral direction during its propagation. When the initial state of the beam prepared around an eigenstate inside (outside) the exceptional ring with the imaginary (real) eigenvalue, the lateral shifts of the beams are manifested by the non-oscillating (Zitterbewegung-like) motions, which are robust to the perturbations of coupling coefficients between waveguides. Remarkably, the amplitude of the non-oscillating shift is dependent on a non-Hermitian Berry connection (U(1) gauge invariance). It contradicts the conventional wisdom that the Berry connection cannot induce the dynamic effect. Furthermore, near the high-order EPs, the initial-state-dependent lateral shifts of the beams present diversity, such as multifrequencies and destructive interferences. The counterintuitive lateral shifts of the beams stem from the non-orthogonal nature of eigenstate of the non-Hermitian systems, which may open a gateway towards the non-Hermitian beam dynamics and manipulations of beams.

Complex Berry curvature and complex energy band structures in non-Hermitian graphene model

AbstractNon-Hermitian quantum systems exhibit many novel physical properties of quantum states. We consider a non-Hermtian graphene model based on the tight-binding approximation with the coupling of the graphene and the substrate. We analyze the complex energy structure of this model and its exceptional points as well as relevant topological invariants. We give the analytic complex Berry connection and Berry curvature in the Brillouin zone and investigate numerically the relationships between the complex Berry curvature and the complex energy band structures. We find that the behaviors of the complex Berry curvature depend on the complex energy band structures. The occurrence of the peaks of both real and imaginary parts of the complex Berry curvature corresponds to the exceptional (gapless) points in the Brillouin zone. In particular, the Dirac cone of the imaginary part of the Berry curvature occurs and corresponding to the occurrence of the flat real energy band for the non-Hermitian parameter $$\eta =3$$ η = 3 . These results provide some novel insights to the relationship between the non-Hermitian graphene, geometry, and topological invariants.

Fermiology of Topological Metals

The modern scope of fermiology encompasses not just the classical geometry of Fermi surfaces but also the geometry of quantum wave functions over the Fermi surface. This enlarged scope is motivated by the advent of topological metals—metals whose Fermi surfaces are characterized by a robustly nontrivial Berry phase. We review the extent to which topological metals can be diagnosed from magnetic-field-induced quantum oscillations of transport and thermodynamic quantities. A holistic analysis of the oscillatory wave form is proposed, in which different characteristics of the wave form (e.g., phase offset, high-harmonic amplitudes, temperature-dependent frequency) encode different aspects of a topologically nontrivial Fermi surface. Which characteristic to focus on depends on ( a) the orientation of the magnetic field relative to certain crystallographic axes, ( b) the symmetry class of the topological metal, and ( c) the separation of Fermi-surface pockets in quasimomentum [Formula: see text] space. Closely proximate pockets arise when (1) spin–split pockets are nearly overlapping due to a weak spin–orbit force or when (2) two pockets touch at an isolated [Formula: see text] point, which can be a topological band-touching point or a saddlepoint in the energy-momentum dispersion. The emergence of a pseudospin degree of freedom (in case 1) and the implications of magnetic breakdown (in case 2) are reviewed, with emphasis on new aspects originating from the (nonabelian) Berry connection of the Fermi surface. Future extensions of topofermiology are suggested in the directions of interaction-induced Fermi-liquid instabilities and two-dimensional electron liquids.

Tunable interband and intraband plasmons in twisted double bilayer graphene

Flat bands in twisted moir\'e superlattices support a variety of topological and strongly correlated phenomena along with easily tunable electrical and optical properties. Here, we demonstrate the existence of long lived, flat intraband and interband plasmons in twisted double bilayer graphene. We show that the interband plasmons originate from the presence of a Van Hove singularity in the joint density of states and a finite Berry connection between the pair of bands involved. We find that the gapped interband plasmon mode has a universal dispersion, and the plasmon gap is specified by the location of the Van Hove singularity in the joint density of states. Metallic moir\'e systems support an additional intraband plasmon mode which becomes flat in the large momentum limit because of the influence of higher interband transitions. We demonstrate that the undamped and flat plasmon modes in moir\'e systems are highly tunable and can be controlled by varying the vertical electric field and electron doping, and they persist over a wide range of twist angles.

On supersymmetric interface defects, brane parallel transport, order-disorder transition and homological mirror symmetry

We concentrate on a treatment of a Higgs-Coulomb duality as an absence of manifest phase transition between ordered and disordered phases of 2d mathcal{N} = (2, 2) theories. We consider these examples of QFTs in the Schrödinger picture and identify Hilbert spaces of BPS states with morphisms in triangulated categories of D-brane boundary conditions. As a result of Higgs-Coulomb duality D-brane categories on IR vacuum moduli spaces are equivalent, this resembles an analog of homological mirror symmetry. Following construction ideas behind the Gaiotto-Moore-Witten algebra of the infrared one is able to introduce interface defects in these theories and associate them to D-brane parallel transport functors. We concentrate on surveying simple examples, analytic when possible calculations, numerical estimates and simple physical picture behind curtains of geometric objects. Categorification of hypergeometric series analytic continuation is derived as an Atiyah flop of the conifold. Finally we arrive to an interpretation of the braid group action on the derived category of coherent sheaves on cotangent bundles to flag varieties as a categorification of Berry connection on the Fayet-Illiopolous parameter space of a sigma-model with a quiver variety target space.

Magnetic-translational sum rule and approximate models of the molecular Berry curvature.

The Berry connection and curvature are key components of electronic structure calculations for atoms and molecules in magnetic fields. They ensure the correct translational behavior of the effective nuclear Hamiltonian and the correct center-of-mass motion during molecular dynamics in these environments. In this work, we demonstrate how these properties of the Berry connection and curvature arise from the translational symmetry of the electronic wave function and how they are fully captured by a finite basis set of London orbitals but not by standard Gaussian basis sets. This is illustrated by a series of Hartree-Fock calculations on small molecules in different basis sets. Based on the resulting physical interpretation of the Berry curvature as the shielding of the nuclei by the electrons, we introduce and test a series of approximations using the Mulliken fragmentation scheme of the electron density. These approximations will be particularly useful in ab initio molecular dynamics calculations in a magnetic field since they reduce the computational cost, while recovering the correct physics and up to 95% of the exact Berry curvature.