arXiv subjects

Ivo Souza

Publications and source records attributed to Ivo Souza.

At least 37 records · Page 2Linked to original sources

arXiv2023

Covariant derivatives of Berry-type quantities: Application to nonlinear transport

The derivatives of the Berry curvature $Ω$ and intrinsic orbital magnetic moment m in momentum space are relevant to various problems, including the nonlinear anomalous Hall effect and magneto-transport within the Boltzmann-equation formalism. To investigate these properties using first-principles methods, we have developed a Wannier interpolation scheme that evaluates the ''covariant derivatives'' of the non-Abelian $Ω$ and m matrices for a group of bands within a specific energy range of interest. Unlike the simple derivative, the covariant derivative does not involve couplings within the groups and preserves the gauge covariance of the $Ω$ and m matrices. In the simulation of nonlinear anomalous Hall conductivity, the resulting ''Fermi-sea'' formula for the Berry curvature dipole are more robust and converges faster with the density of the integration k-grid than the ''Fermi-surface'' formula implemented earlier. The developed methodology is made available via the open-source code WannierBerri and we demonstrate the efficiency of this method through first-principles calculations on trigonal Tellurium.

cond-mat.mtrl-sci↗

arXiv2022

On the separation of Hall and Ohmic nonlinear responses

The symmetric and antisymmetric parts of the linear conductivity describe the dissipative (Ohmic) and nondissipative (Hall) parts of the current. The Hall current is always transverse to the applied electric field regardless of its orientation; the Ohmic current is purely longitudinal in cubic crystals, but in lower-symmetry crystals it has a transverse component whenever the field is not aligned with a principal axis. In this work, we extend that analysis beyond the linear regime. We consider all possible ways of partitioning the current at any order in the electric field without taking symmetry into account, and find that the Hall vs Ohmic decomposition is the only one that satisfies certain basic requirements. A general prescription is given for achieving that decomposition, and the case of the quadratic conductivity is analyzed in detail. By performing a symmetry analysis we find that in five of the 122 magnetic point groups the quadratic dc conductivity is purely Ohmic and even under time reversal, a type of response that is entirely disorder mediated.

cond-mat.mtrl-sci↗

arXiv2022

Assessing the role of interatomic position matrix elements in tight-binding calculations of optical properties

We study the role of hopping matrix elements of the position operator $\mathbf{\hat{r}}$ in tight-binding calculations of linear and nonlinear optical properties of solids. Our analysis relies on a Wannier-interpolation scheme based on \textit{ab initio} calculations, which automatically includes matrix elements of $\mathbf{\hat{r}}$ between different Wannier orbitals. A common approximation, both in empirical tight-binding and in Wannier-interpolation calculations, is to discard those matrix elements, in which case the optical response only depends on the on-site energies, Hamiltonian hoppings, and orbital centers. We find that interatomic $\mathbf{\hat{r}}$-hopping terms make a sizeable contribution to the shift photocurrent in monolayer BC$_2$N, a covalent acentric crystal. If a minimal basis of $p_z$ orbitals on the carbon atoms is used to model the band-edge response, even the dielectric function becomes strongly dependent on those terms.

cond-mat.mes-hall↗

arXiv2021

Mirror Chern numbers in the hybrid Wannier representation

The topology of electronic states in band insulators with mirror symmetry can be classified in two different ways. One is in terms of the mirror Chern number, an integer that counts the number of protected Dirac cones in the Brillouin zone of high-symmetry surfaces. The other is via a $\mathbb{Z}_2$ index that distinguishes between systems that have a nonzero quantized orbital magnetoelectric coupling ("axion-odd"), and those that do not ("axion-even"); this classification can also be induced by other symmetries in the magnetic point group, including time reversal and inversion. A systematic characterization of the axion $\mathbb{Z}_2$ topology has previously been obtained by representing the valence states in terms of hybrid Wannier functions localized along one chosen crystallographic direction, and inspecting the associated Wannier band structure. Here we focus on mirror symmetry, and extend that characterization to the mirror Chern number. We choose the direction orthogonal to the mirror plane as the Wannierization direction, and show that the mirror Chern number can be determined from the winding numbers of the touching points between Wannier bands on mirror-invariant planes, and from the Chern numbers of flat bands pinned to those planes. In this representation, the relation between the mirror Chern number and the axion $\mathbb{Z}_2$ index is readily established. The formalism is illustrated by means of $\textit{ab initio}$ calculations for SnTe in the monolayer and bulk forms, complemented by tight-binding calculations for a toy model.

cond-mat.mes-hall↗

arXiv2021

Quadrupole moments, edge polarizations, and corner charges in the Wannier representation

The modern theory of polarization allows for the determination of the macroscopic end charge of a truncated one-dimensional insulator, modulo the charge quantum $e$, from a knowledge of bulk properties alone. A more subtle problem is the determination of the corner charge of a two-dimensional insulator, modulo $e$, from a knowledge of bulk and edge properties alone. While previous works have tended to focus on the quantization of corner charge in the presence of symmetries, here we focus on the case that the only bulk symmetry is inversion, so that the corner charge can take arbitrary values. We develop a Wannier-based formalism that allows the corner charge to be predicted, modulo $e$, only from calculations on ribbon geometries of two different orientations. We elucidate the dependence of the interior quadrupole and edge dipole contributions upon the gauge used to construct the Wannier functions, finding that while these are individually gauge-dependent, their sum is gauge-independent. From this we conclude that the edge polarization is not by itself a physical observable, and that any Wannier-based method for computing the corner charge requires the use of a common gauge throughout the calculation. We satisfy this constraint using two Wannier construction procedures, one based on projection and another based on a gauge-consistent nested Wannier construction. We validate our theory by demonstrating the correct prediction of corner charge for several tight-binding models. We comment on the relations between our approach and previous ones that have appeared in the literature.

cond-mat.mes-hall↗

arXiv2020

Gapless hinge states from adiabatic pumping of axion coupling

We demonstrate that chiral hinge modes naturally emerge in insulating crystals undergoing a slow cyclic evolution that changes the Chern-Simons axion angle $θ$ by $2π$. This happens when the surface (not just the bulk) returns to its initial state at the end of the cycle, in which case it must pass through a metallic state to dispose of the excess quantum of surface anomalous Hall conductivity pumped from the bulk. If two adjacent surfaces become metallic at different points along the cycle, there is an interval in which they are in topologically distinct insulating states, with chiral modes propagating along the connecting hinge. We illustrate these ideas for a tight-binding model consisting of coupled layers of the Haldane model with alternating parameters. The surface topology is determined in a slab geometry using two different markers, surface anomalous Hall conductivity and surface-localized charge pumping (flow of surface-localized Wannier bands), and we find that both correctly predict the appearance of gapless hinge modes in a rod geometry. When viewing the axion pump as a four-dimensional (4D) crystal with one synthetic dimension, the hinge modes trace Fermi arcs in the Brillouin zone of the 2D hinge connecting a pair of 3D surfaces of the 4D crystal.

cond-mat.mtrl-sci↗

arXiv2020

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.

cond-mat.mes-hall↗

arXiv2019

Wannier90 as a community code: new features and applications

Wannier90 is an open-source computer program for calculating maximally-localised Wannier functions (MLWFs) from a set of Bloch states. It is interfaced to many widely used electronic-structure codes thanks to its independence from the basis sets representing these Bloch states. In the past few years the development of Wannier90 has transitioned to a community-driven model; this has resulted in a number of new developments that have been recently released in Wannier90 v3.0. In this article we describe these new functionalities, that include the implementation of new features for wannierisation and disentanglement (symmetry-adapted Wannier functions, selectively-localised Wannier functions, selected columns of the density matrix) and the ability to calculate new properties (shift currents and Berry-curvature dipole, and a new interface to many-body perturbation theory); performance improvements, including parallelisation of the core code; enhancements in functionality (support for spinor-valued Wannier functions, more accurate methods to interpolate quantities in the Brillouin zone); improved usability (improved plotting routines, integration with high-throughput automation frameworks), as well as the implementation of modern software engineering practices (unit testing, continuous integration, and automatic source-code documentation). These new features, capabilities, and code development model aim to further sustain and expand the community uptake and range of applicability, that nowadays spans complex and accurate dielectric, electronic, magnetic, optical, topological and transport properties of materials.

cond-mat.mtrl-sci↗

arXiv2019

Computation of intrinsic spin Hall conductivities from first principles using maximally-localized Wannier functions

We present a method to compute the intrinsic spin Hall conductivity from first principles using an interpolation scheme based on maximally-localized Wannier functions. After obtaining the relevant matrix elements among the ab initio Bloch states calculated on a coarse k-point mesh, we Fourier transform them to find the corresponding matrix elements between Wannier states. We then perform an inverse Fourier transform to interpolate the velocity and spin-current matrix elements onto a dense k-point mesh, and use them to evaluate the spin Hall conductivity as a Brillouin-zone integral. This strategy has a much lower computational cost than a direct ab initio calculation, without sacrificing the accuracy. We demonstrate that the spin Hall conductivities of platinum and doped gallium arsenide, computed with our interpolation scheme as a function of the Fermi energy, are in good agreement with those obtained in previous first-principles studies. We also discuss certain approximations that can be made, in the spirit of the tight-binding method, to simplify the calculation of the velocity and spin-current matrix elements in the Wannier representation.

cond-mat.mes-hall↗

arXiv2018

Geometric and nongeometric contributions to the surface anomalous Hall conductivity

A static electric field generates circulating currents at the surfaces of a magnetoelectric insulator. The anomalous Hall part of the surface conductivity tensor describing such bound currents can change by multiples of $e^2/h$ depending on the insulating surface preparation, and a bulk calculation does not fix its quantized part. To resolve this ambiguity, we develop a formalism for calculating the full surface anomalous Hall conductivity in a slab geometry. We identify a Berry-curvature term, closely related to the expression for the bulk anomalous Hall conductivity, whose value can change by quantized amounts by adjusting the surface Hamiltonian. In addition, the surface anomalous Hall conductivity contains a nongeometric part that does not depend on the surface preparation.

cond-mat.mes-hall↗

arXiv2018

Ab initio calculation of the shift photocurrent by Wannier interpolation

We describe and implement a first-principles algorithm based on maximally-localized Wannier functions for calculating the shift-current response of piezoelectric crystals in the independent-particle approximation. The proposed algorithm presents several advantages over existing ones, including full gauge invariance, low computational cost, and a correct treatment of the optical matrix elements with nonlocal pseudopotentials. Band-truncation errors are avoided by a careful formulation of $k\cdot p$ perturbation theory within the subspace of wannierized bands. The needed ingredients are the matrix elements of the Hamiltonian and of the position operator in the Wannier basis, which are readily available at the end of the wannierization step. If the off-diagonal matrix elements of the position operator are discarded, our expressions reduce to the ones that have been used in recent tight-binding calculations of the shift current. We find that this `diagonal' approximation can introduce sizeable errors, highlighting the importance of carefully embedding the tight-binding model in real space for an accurate description of the charge transfer that gives rise to the shift current.

cond-mat.mtrl-sci↗

arXiv2018

Gyrotropic effects in trigonal tellurium studied from first principles

We present a combined ab initio study of several gyrotropic effects in p-doped trigonal tellurium (effects that reverse direction with the handedness of the spiral chains in the atomic structure). The key ingredients in our study are the k-space Berry curvature and intrinsic orbital magnetic moment imparted on the Bloch states by the chirality of the crystal structure. We show that the observed sign reversal with temperature of the circular photogalvanic effect can be explained by the presence of Weyl points near the bottom of the conduction band acting as sources and sinks of Berry curvature. The passage of a current along the trigonal axis induces a rather small parallel magnetization, which can nevertheless be detected by optical means (Faraday rotation of transmitted light) due to the high transparency of the sample. In agreement with experiment, we find that when infrared light propagates antiparallel to the current at low doping the current-induced optical rotation enhances the natural optical rotation. According to our calculations the plane of polarization rotates in the opposite sense to the bonded atoms in the spiral chains, in agreement with a recent experiment that contradicts earlier reports.

cond-mat.mtrl-sci↗

arXiv2017

Composite Weyl nodes stabilized by screw symmetry with and without time reversal

We classify the band degeneracies in 3D crystals with screw symmetry $n_m$ and broken $\mathcal P*\mathcal T$ symmetry, where $\mathcal P$ stands for spatial inversion and $\mathcal T$ for time reversal. The generic degeneracies along symmetry lines are Weyl nodes: Chiral contact points between pairs of bands. They can be single nodes with a chiral charge of magnitude $|χ|=1$ or composite nodes with $|χ|=2$ or $3$, and the possible $χ$ values only depend on the order $n$ of the axis, not on the pitch $m/n$ of the screw. Double Weyl nodes require $n=4$ or 6, and triple nodes require $n=6$. In all cases the bands split linearly along the axis, and for composite nodes the splitting is quadratic on the orthogonal plane. This is true for triple as well as double nodes, due to the presence in the effective two-band Hamiltonian of a nonchiral quadratic term that masks the chiral cubic dispersion. If $\mathcal T$ symmetry is present and $\mathcal P$ is broken there may exist on some symmetry lines Weyl nodes pinned to $\mathcal T$-invariant momenta, which in some cases are unavoidable. In the absence of other symmetries their classification depends on $n$, $m$, and the type of $\mathcal T$ symmetry. With spinless $\mathcal T$ such $\mathcal T$-invariant Weyl nodes are always double nodes, while with spinful $\mathcal T$ they can be single or triple nodes. $\mathcal T$-invariant triples nodes can occur not only on 6-fold axes but also on 3-fold ones, and their in-plane band splitting is cubic, not quadratic as in the case of generic triple nodes. These rules are illustrated by means of first-principles calculations for hcp cobalt, a $\mathcal T$-broken, $\mathcal P$-invariant crystal with $6_3$ symmetry, and for trigonal tellurium and hexagonal NbSi$_2$, which are $\mathcal T$-invariant, $\mathcal P$-broken crystals with 3-fold and 6-fold screw symmetry respectively.

cond-mat.mtrl-sci↗

arXiv2016

Surface theorem for the Chern-Simons axion coupling

The Chern-Simons axion coupling of a bulk insulator is only defined modulo a quantum of e^2/h. The quantized part of the coupling is uniquely defined for a bounded insulating sample, but it depends on the specific surface termination. Working in a slab geometry and representing the valence bands in terms of hybrid Wannier functions, we show how to determine that quantized part from the excess Chern number of the hybrid Wannier sheets located near the surface of the slab. The procedure is illustrated for a tight-binding model consisting of coupled quantum anomalous Hall layers. By slowly modulating the model parameters, it is possible to transfer one unit of Chern number from the bottom to the top surface over the course of a cyclic evolution of the bulk Hamiltonian. When the evolution of the surface Hamiltonian is also cyclic, the Chern pumping is obstructed by chiral touchings between valence and conduction surface bands.

cond-mat.mtrl-sci↗

arXiv2016

Metal-insulator transition in disordered systems from the one-body density matrix

The insulating state of matter can be probed by means of a ground state geometrical marker, which is closely related to the modern theory of polarization (based on a Berry phase). In the present work we show that this marker can be applied to determine the metal-insulator transition in disordered systems. In particular, for non-interacting systems the geometrical marker can be obtained from the configurational average of the norm-squared one-body density matrix, which can be calculated within open as well as periodic boundary conditions. This is in sharp contrast to a classification based on the static conductivity, which is only sensible within periodic boundary conditions. We exemplify the method by considering a simple lattice model, known to have a metal-insulator transition as a function of the disorder strength and demonstrate that the transition point can be obtained accurately from the one-body density matrix. The approach has a general {\it ab-initio} formulation and can be applied to realistic disordered materials by standard electronic structure methods.

cond-mat.dis-nn↗

arXiv2016

Gyrotropic magnetic effect and the magnetic moment on the Fermi surface

The current density ${\bf j}^{\rm{\bf B}}$ induced in a clean metal by a slowly-varying magnetic field ${\bf B}$ is formulated as the low-frequency limit of natural optical activity, or natural gyrotropy. Working with a multiband Pauli Hamiltonian, we obtain from the Kubo formula a simple expression for $α^{\rm gme}_{ij}=j^{\rm{\bf B}}_i/B_j$ in terms of the intrinsic magnetic moment (orbital plus spin) of the Bloch electrons on the Fermi surface. An alternate semiclassical derivation provides an intuitive picture of the effect, and takes into account the influence of scattering processes in dirty metals. This "gyrotropic magnetic effect" is fundamentally different from the chiral magnetic effect driven by the chiral anomaly and governed by the Berry curvature on the Fermi surface, and the two effects are compared for a minimal model of a Weyl semimetal. Like the Berry curvature, the intrinsic magnetic moment should be regarded as a basic ingredient in the Fermi-liquid description of transport in broken-symmetry metals.

cond-mat.mes-hall↗

arXiv2015

Chiral degeneracies and Fermi-surface Chern numbers in bcc Fe

The degeneracies in the spinor bandstructure of bcc Fe are studied from first principles. We find numerous isolated band touchings carrying chiral charges of magnitude one (Weyl points) or two (double-Weyl nodes), as well as nonchiral degeneracy loops (nodal rings). Some degeneracies are located on symmetry lines or planes in the Brillouin zone and others at generic low-symmetry points, realizing all possible scenarios consistent with the magnetic point group. We clarify the general theory relating the chiral band touchings to the Chern numbers of the Fermi sheets enclosing them, and use this approach to determine the Chern numbers on the Fermi surface of bcc Fe. Although most Fermi sheets enclose Weyl nodes, in almost all cases the net enclosed charge vanishes for symmetry reasons, resulting in a vanishing Chern number. The exceptions are two inversion-symmetric electron pockets along the symmetry line $Δ$ parallel to the magnetization. Each of them surrounds a single Weyl point, leading to Chern numbers of $\pm 1$. These small topological pockets contribute a sizable amount to the nonquantized part of the intrinsic anomalous Hall conductivity, proportional to their reciprocal-space separation. Variation of the Fermi level (or other system parameters) may lead to a touching event between Fermi sheets, accompanied by a transfer of Chern number between them.

cond-mat.mtrl-sci↗

arXiv2015

Valley Hall effect in disordered monolayer MoS2 from first principles

Electrons in certain two-dimensional crystals possess a pseudospin degree of freedom associated with the existence of two inequivalent valleys in the Brillouin zone. If, as in monolayer MoS2, inversion symmetry is broken and time-reversal symmetry is present, equal and opposite amounts of k-space Berry curvature accumulate in each of the two valleys. This is conveniently quantified by the integral of the Berry curvature over a single valley - the valley Hall conductivity. We generalize this definition to include contributions from disorder described with the supercell approach, by mapping ("unfolding") the Berry curvature from the folded Brillouin zone of the disordered supercell onto the normal Brillouin zone of the pristine crystal, and then averaging over several realizations of disorder. We use this scheme to study from first-principles the effect of sulfur vacancies on the valley Hall conductivity of monolayer MoS2. In dirty samples the intrinsic valley Hall conductivity receives gating-dependent corrections that are only weakly dependent on the impurity concentration, consistent with side-jump scattering and the unfolded Berry curvature can be interpreted as a k-space resolved side-jump. At low impurity concentrations skew scattering dominates, leading to a divergent valley Hall conductivity in the clean limit. The implications for the recently-observed photoinduced anomalous Hall effect are discussed.

cond-mat.dis-nn↗