Quantum theory of orbital magnetization and its generalization to interacting systems

Theory of topological exciton insulators and condensates in flat Chern bands

Excitons are the neutral quasiparticles that form when Coulomb interactions create bound states between electrons and holes. Due to their bosonic nature, excitons are expected to condense and exhibit superfluidity at sufficiently low temperatures. In interacting Chern insulators, excitons may inherit the nontrivial topology and quantum geometry from the underlying electron wavefunctions. We theoretically investigate the excitonic bound states and superfluidity in flat-band insulators pumped with light. We find that the exciton wavefunctions exhibit vortex structures in momentum space, with the total vorticity being equal to the difference of Chern numbers between the conduction and valence bands. Moreover, both the exciton binding energy and the exciton superfluid density are proportional to the Brillouin-zone average of the quantum metric and the Coulomb potential energy per unit cell. Spontaneous emission of circularly polarized light from radiative decay is a detectable signature of the exciton vorticity. We propose that the vorticity can also be experimentally measured via the nonlinear anomalous Hall effect, whereas the exciton superfluidity can be detected by voltage-drop quantization through a combination of quantum geometry and Aharonov-Casher effect. Topological excitons and their superfluid phase could be realized in flat bands of twisted Van der Waals heterostructures.

Theory of spin and orbital Edelstein effects

In systems with broken spatial inversion symmetry, such as surfaces, interfaces, or bulk systems lacking an inversion center, the application of a charge current can generate finite spin and orbital densities associated with a nonequilibrium magnetization, which is known as spin and orbital Edelstein effect (SEE and OEE), respectively. Early reports on this current-induced magnetization focus on two-dimensional Rashba systems, in which an in-plane nonequilibrium spin density is generated perpendicular to the applied charge current. However, until today, a large variety of materials have been theoretically predicted and experimentally demonstrated to exhibit a sizeable Edelstein effect, which comprises contributions from the spin as well as the orbital degrees of freedom, and whose associated magnetization may be out of plane, nonorthogonal, and even parallel to the applied charge current, depending on the system's particular symmetries. In this review, we give an overview on the most commonly used theoretical approaches for the discussion and prediction of the SEE and OEE. Further, we introduce a selection of the most intensely discussed materials exhibiting a finite Edelstein effect, and give a brief summary of common experimental techniques.

Magnon Orbital Nernst Effect in Honeycomb Antiferromagnets without Spin-Orbit Coupling

Recently, topological responses of magnons have emerged as a central theme in magnetism and spintronics. However, resulting Hall responses are typically weak and infrequent, since, according to present understanding, they arise from effective spin-orbit couplings, which are weaker compared to the exchange energy. Here, by investigating transport properties of magnon orbital moments, we predict that the magnon orbital Nernst effect is an intrinsic characteristic of the honeycomb antiferromagnet and therefore, it manifests even in the absence of spin-orbit coupling. For the electric detection, we propose an experimental scheme based on the magnetoelectric effect. Our results break the conventional wisdom that the Hall transport of magnons requires spin-orbit coupling by predicting the magnon orbital Nernst effect in a system without it, which leads us to envision that our work initiates the intensive search for various magnon Hall effects in generic magnetic systems with no reliance on spin-orbit coupling.

Topological aspects of multi-kantiferromagnetism in cubic rare-earth compounds

We advertise rare-earth intermetallics with high-symmetry crystal structures and competing interactions as a possible materials platform hosting spin structures with non-trivial topological properties. Focusing on the series of cubicRCu compounds, whereR= Ho, Er, Tm, the bulk properties of these systems display exceptionally rich magnetic phase diagrams hosting an abundance of different phase pockets characteristic of antiferromagnetic order in the presence of delicately balanced interactions. The electrical transport properties exhibit large anomalous contributions suggestive of topologically non-trivial winding in the electronic and magnetic structures. Neutron diffraction identifies spontaneous long-range magnetic order in terms of commensurate and incommensurate variations of(ππ0)antiferromagnetism with the possibility for various multi-kconfigurations. Motivated by general trends in these materials, we discuss the possible existence of topologically non-trivial winding in real and reciprocal space in the class ofRCu compounds including antiferromagnetic skyrmion lattices. Putatively bringing together different limits of non-trivial topological winding in the same material, the combination of properties inRCu systems promises access to advanced functionalities.

Drastic magnetic-field-induced chiral current order and emergent current-bond-field interplay in kagome metals

In kagome metals, the chiral current order parameter [Formula: see text] with time-reversal-symmetry-breaking is the source of various exotic electronic states, while the method of controlling the current order and its interplay with the star-of-David bond order [Formula: see text] are still unsolved. Here, we reveal that tiny uniform orbital magnetization [Formula: see text] is induced by the chiral current order, and its magnitude is prominently enlarged under the presence of the bond order. Importantly, we derive the magnetic-field ([Formula: see text])-induced Ginzburg-Landau (GL) free energy expression [Formula: see text], which enables us to elucidate the field-induced current-bond phase transitions in kagome metals. The emergent current-bond-[Formula: see text] trilinear coupling term in the free energy, [Formula: see text], naturally explains the characteristic magnetic-field sensitive electronic states in kagome metals, such as the field-induced current order and the strong interplay between the bond and current orders. The GL coefficients of [Formula: see text] derived from the realistic multiorbital model are appropriate to explain various experiments. Furthermore, we discuss the field-induced loop current orders in the square lattice models that have been studied in cuprate superconductors.

A quantum ruler for orbital magnetism in moiré quantum matter

For almost a century, magnetic oscillations have been a powerful "quantum ruler" for measuring Fermi surface topology. In this study, we used Landau-level spectroscopy to unravel the energy-resolved valley-contrasting orbital magnetism and large orbital magnetic susceptibility that contribute to the energies of Landau levels of twisted double-bilayer graphene. These orbital magnetism effects led to substantial deviations from the standard Onsager relation, which manifested as a breakdown in scaling of Landau-level orbits. These substantial magnetic responses emerged from the nontrivial quantum geometry of the electronic structure and the large length scale of the moiré lattice potential. Going beyond traditional measurements, Landau-level spectroscopy performed with a scanning tunneling microscope offers a complete quantum ruler that resolves the full energy dependence of orbital magnetic properties in moiré quantum matter.

Transport of Spin Magnetic Multipole Moments Carried by Bloch Quasiparticles

Magnetic ordering beyond the standard dipolar order has attracted significant attention in recent years, but it remains an open question how to effectively manipulate such nontrivial order parameters using external perturbations such as electric currents or fields. In particular, it is desirable to have a conceptual tool similar to nonequilibrium spin currents in spintronics to describe the creation and transport of multipole moments. In this context, we present a theory for Cartesian spin magnetic multipole moments of Bloch quasiparticles and their transport based on a general gauge-invariant formula obtained using the wave packet approach. As a concrete example, we point out that the low-energy Hamiltonian of phosphorene subject to a perpendicular electric field has a valley structure that hosts magnetic octupole moments. The magnetic octupole moments can be exhibited by an in-plane electric current and lead to accumulation of staggered spin densities at the corners of a rectangular sample. Our Letter paves the way for systematically seeking and utilizing quasiparticles with higher-order magnetic multipole moments in crystal materials towards the emergence of multipoletronics.

General nonlinear Hall current in magnetic insulators beyond the quantum anomalous Hall effect

Can a generic magnetic insulator exhibit a Hall current? The quantum anomalous Hall effect (QAHE) is one example of an insulating bulk carrying a quantized Hall conductivity while insulators with zero Chern number present zero Hall conductance in the linear response regime. Here, we find that a general magnetic insulator possesses a nonlinear Hall conductivity quadratic to the electric field if the system breaks inversion symmetry, which can be identified as a new type of multiferroic coupling. This conductivity originates from an induced orbital magnetization due to virtual interband transitions. We identify three contributions to the wavepacket motion, a velocity shift, a positional shift, and a Berry curvature renormalization. In contrast to the crystalline solid, we find that this nonlinear Hall conductivity vanishes for Landau levels of a 2D electron gas, indicating a fundamental difference between the QAHE and the integer quantum Hall effect.

Time-Reversal-Even Nonlinear Current Induced Spin Polarization

We propose a time-reversal-even spin generation in second order of electric fields, which dominates the current induced spin polarization in a wide class of centrosymmetric nonmagnetic materials, and leads to a novel nonlinear spin-orbit torque in magnets. We reveal a quantum origin of this effect from the momentum space dipole of the anomalous spin polarizability. First-principles calculations predict sizable spin generations in several nonmagnetic hcp metals, in monolayer TiTe_{2}, and in ferromagnetic monolayer MnSe_{2}, which can be detected in experiment. Our work opens up the broad vista of nonlinear spintronics in both nonmagnetic and magnetic systems.

Relation between spin current and spin torque in Rashba ferromagnets

We establish a brief relation between spin current and spin torque, including spin-orbit torque and spin transfer torque in 2D Rashba ferromagnets with nonuniform magnetic texture. Both electrically and thermally induced charge, heat, and spin current are investigated by the Luttinger's mechanical method, and we derive the contributions of magnetization corresponding to the thermal spin current and the thermal spin torque. The novel transport currents are also found in this paper when the interplay between spin-orbit coupling and nonuniform magnetic texture is taken into account.

Size and shape effects in the orbital magnetization of TMDs monolayers

The intrinsic orbital magnetization of a TMD monolayer is usually calculated for periodic crystals without mentioning the geometrical shape of samples and boundary conditions (BCs) for the electron wave functions. Such approaches, based on Bloch's theorem, involves a contribution from the Berry curvature, also in the case when the system is described by the two-band minimal model (Xiaoet al2012Phys. Rev. Lett.108196802). In the present paper, we show that the geometrical and topological properties of the specimen, as well as the BCs, play an important role in the problem of valley orbital magnetization even for a macroscopic sample.

Axial Magnetoelectric Effect in Dirac Semimetals

We propose a mechanism to generate a static magnetization via the "axial magnetoelectric effect" (AMEE). Magnetization M∼E_{5}(ω)×E_{5}^{*}(ω) appears as a result of the transfer of the angular momentum of the axial electric field E_{5}(t) into the magnetic moment in Dirac and Weyl semimetals. We point out similarities and differences between the proposed AMEE and a conventional inverse Faraday effect. As an example, we estimated the AMEE generated by circularly polarized acoustic waves and find it to be on the scale of microgauss for gigahertz frequency sound. In contrast to a conventional inverse Faraday effect, magnetization rises linearly at small frequencies and fixed sound intensity as well as demonstrates a nonmonotonic peak behavior for the AMEE. The effect provides a way to investigate unusual axial electromagnetic fields via conventional magnetometry techniques.

Universal Behavior of the Thermal Hall Conductivity

We report theoretical and experimental analyses of the thermal Hall conductivity in correlated systems. For both fermionic and bosonic excitations with nontrivial topology, we show that at "intermediate" temperatures, the thermal Hall conductivity exhibits an unexpected universal scaling with a simple exponential form. At low temperatures, it behaves differently and reflects the spectral properties of underlying excitations. Our predictions are examined as examples in two prototype compounds, the quantum paraelectric SrTiO_{3} and the spin-liquid compound RuCl_{3}. The experimental data can be largely covered by our proposed minimal phenomenological model independent of microscopic details, revealing dominant bosonic contributions in SrTiO_{3} and gapped fermionic excitations in RuCl_{3}. Our work establishes a phenomenological link between microscopic models and experimental data and provides a unified basis for analyzing the thermal Hall conductivity in correlated systems over a wide temperature region.

Berry Phase Effects in Dipole Density and the Mott Relation

We provide a unified semiclassical theory for thermoelectric responses of any observable represented by an operator θ[over ^] that is well defined in periodic crystals. The Einstein and Mott relations are established generally in the presence of Berry phase effects for various physical realizations of θ[over ^] in electronic systems, including the familiar case of the electric current as well as the currently controversial cases of the spin polarization and spin current. The magnetization current, which has been proven indispensable in the thermoelectric response of electric current, is generalized to the cases of various θ[over ^]. In our theory the dipole density of a physical quantity emerges and plays a vital role, which contains not only the statistical sum of the dipole moment of θ[over ^] but also a Berry phase correction.

Theory of Current-Induced Angular Momentum Transfer Dynamics in Spin-Orbit Coupled Systems

Motivated by the importance of understanding various competing mechanisms to the current-induced spin-orbit torque on magnetization in complex magnets, we develop a theory of current-induced spin-orbital coupled dynamics in magnetic heterostructures. The theory describes angular momentum transfer between different degrees of freedom in solids, e.g., the electron orbital and spin, the crystal lattice, and the magnetic order parameter. Based on the continuity equations for the spin and orbital angular momenta, we derive equations of motion that relate spin and orbital current fluxes and torques describing the transfer of angular momentum between different degrees of freedom, achieved in a steady state under an applied external electric field. We then propose a classification scheme for the mechanisms of the current-induced torque in magnetic bilayers. We evaluate the sources of torque using density functional theory, effectively capturing the impact of the electronic structure on these quantities. We apply our formalism to two different magnetic bilayers, Fe/W(110) and Ni/W(110), which are chosen such that the orbital and spin Hall effects in W have opposite sign and the resulting spin- and orbital-mediated torques can compete with each other. We find that while the spin torque arising from the spin Hall effect of W is the dominant mechanism of the current-induced torque in Fe/W(110), the dominant mechanism in Ni/W(110) is the orbital torque originating in the orbital Hall effect of the non-magnetic substrate. Thus the effective spin Hall angles for the total torque are negative and positive in the two systems. Our prediction can be experimentally identified in moderately clean samples, where intrinsic contributions dominate. This clearly demonstrates that our formalism is ideal for studying the angular momentum transfer dynamics in spin-orbit coupled systems as it goes beyond the "spin current picture" by naturally incorporating the spin and orbital degrees of freedom on an equal footing. Our calculations reveal that, in addition to the spin and orbital torque, other contributions such as the interfacial torque and self-induced anomalous torque within the ferromagnet are not negligible in both material systems.

Mixed topological semimetals driven by orbital complexity in two-dimensional ferromagnets

The concepts of Weyl fermions and topological semimetals emerging in three-dimensional momentum space are extensively explored owing to the vast variety of exotic properties that they give rise to. On the other hand, very little is known about semimetallic states emerging in two-dimensional magnetic materials, which present the foundation for both present and future information technology. Here, we demonstrate that including the magnetization direction into the topological analysis allows for a natural classification of topological semimetallic states that manifest in two-dimensional ferromagnets as a result of the interplay between spin-orbit and exchange interactions. We explore the emergence and stability of such mixed topological semimetals in realistic materials, and point out the perspectives of mixed topological states for current-induced orbital magnetism and current-induced domain wall motion. Our findings pave the way to understanding, engineering and utilizing topological semimetallic states in two-dimensional spin-orbit ferromagnets.

Accurate Formula for the Macroscopic Polarization of Strongly Correlated Materials

The many-body Berry phase formula for macroscopic polarization is approximated by a sum of natural orbital geometric phases with fractional occupation numbers accounting for the dominant correlation effects. This formula accurately reproduces the exact polarization in the Rice-Mele-Hubbard model across the band insulator-Mott insulator transition. A similar formula based on a reduced Berry curvature accurately predicts the interaction-induced quenching of Thouless topological charge pumping.

Orbital Edelstein Effect as a Condensed-Matter Analog of Solenoids

We theoretically study current-induced orbital magnetization in a chiral crystal. This phenomenon is an orbital version of the Edelstein effect. We propose an analogy between the current-induced orbital magnetization and an Ampère field in a solenoid in classical electrodynamics. To quantify this effect, we define a dimensionless parameter from the response coefficients relating a current density with an orbital magnetization. This dimensionless parameter can be regarded as a number of turns within a unit cell when the crystal is regarded as a solenoid, and it represents how "chiral" the crystal is. By focusing on the dimensionless parameter, one can design a band structure that realizes the induction of large orbital magnetization. In particular, a Weyl semimetal with all of the Weyl nodes close to the Fermi energy can have a large value for this dimensionless parameter, which can exceed that of a classical solenoid.

Zero-field magnetic response functions in Landau levels

We present a fresh perspective on the Landau level quantization rule; that is, by successively including zero-field magnetic response functions at zero temperature, such as zero-field magnetization and susceptibility, the Onsager's rule can be corrected order by order. Such a perspective is further reinterpreted as a quantization of the semiclassical electron density in solids. Our theory not only reproduces Onsager's rule at zeroth order and the Berry phase and magnetic moment correction at first order but also explains the nature of higher-order corrections in a universal way. In applications, those higher-order corrections are expected to curve the linear relation between the level index and the inverse of the magnetic field, as already observed in experiments. Our theory then provides a way to extract the correct value of Berry phase as well as the magnetic susceptibility at zero temperature from Landau level fan diagrams in experiments. Moreover, it can be used theoretically to calculate Landau levels up to second-order accuracy for realistic models.

Toward surface orbitronics: giant orbital magnetism from the orbital Rashba effect at the surface of sp-metals

As the inversion symmetry is broken at a surface, spin-orbit interaction gives rise to spin-dependent energy shifts - a phenomenon which is known as the spin Rashba effect. Recently, it has been recognized that an orbital counterpart of the spin Rashba effect - the orbital Rashba effect - can be realized at surfaces even without spin-orbit coupling. Here, we propose a mechanism for the orbital Rashba effect based on sp orbital hybridization, which ultimately leads to the electric polarization of surface states. For the experimentally well-studied system of a BiAg₂ monolayer, as a proof of principle, we show from first principles that this effect leads to chiral orbital textures in k-space. In predicting the magnitude of the orbital moment arising from the orbital Rashba effect, we demonstrate the crucial role played by the Berry phase theory for the magnitude and variation of the orbital textures. As a result, we predict a pronounced manifestation of various orbital effects at surfaces, and proclaim the orbital Rashba effect to be a key platform for surface orbitronics.

Prototypical topological orbital ferromagnet γ-FeMn

We predict from first principles an entirely topological orbital magnetization in the noncoplanar bulk antiferromagnet γ-FeMn originating in the nontrivial topology of the underlying spin structure, without any reference to spin-orbit interaction. Studying the influence of strain, composition ratio, and spin texture on the topological orbital magnetization and the accompanying topological Hall effect, we promote the scalar spin chirality as key mechanism lifting the orbital degeneracy. The system is thus a prototypical topological orbital ferromagnet, the macroscopic orbital magnetization of which is prominent even without spin-orbit coupling. One of the remarkable features of γ-FeMn is the possibility for pronounced orbital magnetostriction mediated by the complex spin topology in real space.

The inverse thermal spin-orbit torque and the relation of the Dzyaloshinskii-Moriya interaction to ground-state energy currents

Using the Kubo linear-response formalism we derive expressions to calculate the electronic contribution to the heat current generated by magnetization dynamics in ferromagnetic metals with broken inversion symmetry and spin-orbit interaction (SOI). The effect of producing heat currents by magnetization dynamics constitutes the Onsager reciprocal of the thermal spin-orbit torque (TSOT), i.e. the generation of torques on the magnetization due to temperature gradients. We find that the energy current driven by magnetization dynamics contains a contribution from the Dzyaloshinskii-Moriya interaction (DMI), which needs to be subtracted from the Kubo linear response of the energy current in order to extract the heat current. We show that the expressions of the DMI coefficient can be derived elegantly from the DMI energy current. Guided by formal analogies between the Berry phase theory of DMI on the one hand and the modern theory of orbital magnetization on the other hand we are led to an interpretation of the latter in terms of energy currents as well. Based on ab initio calculations we investigate the electronic contribution to the heat current driven by magnetization dynamics in Mn/W(0 0 1) magnetic bilayers. We predict that fast domain walls drive strong heat currents.

Irrelevance of the Boundary on the Magnetization of Metals

The macroscopic current density responsible for the mean magnetization M of a uniformly magnetized bounded sample is localized near its surface. In order to evaluate M one needs the current distribution in the whole sample: bulk and boundary. In recent years it has been shown that the boundary has no effect on M in insulators: therein, M admits an alternative expression not based on currents. M can be expressed in terms of the bulk electron distribution only, which is "nearsighted" (exponentially localized); this virtue is not shared by metals, having a qualitatively different electron distribution. We show, by means of simulations on paradigmatic model systems, that even in metals the M value can be retrieved in terms of the bulk electron distribution only.

The Calculation of NMR Chemical Shifts in Periodic Systems Based on Gauge Including Atomic Orbitals and Density Functional Theory

We present here a method that can calculate NMR shielding tensors from first principles for systems with translational invariance. Our approach is based on Kohn-Sham density functional theory and gauge-including atomic orbitals. Our scheme determines the shielding tensor as the second derivative of the total electronic energy with respect to an external magnetic field and a nuclear magnetic moment. The induced current density due to a periodic perturbation from nuclear magnetic moments is obtained through numerical differentiation, whereas the influence of the responding perturbation in terms of the external magnetic field is evaluated analytically. The method is implemented into the periodic program BAND. It employs a Bloch basis set made up of Slater-type or numeric atomic orbitals and represents the Kohn-Sham potential fully without the use of effective core potentials. Results from calculations of NMR shielding constants based on the present approach are presented for isolated molecules as well as systems with one-, two- and three-dimensional periodicity. The reported values are compared to experiment and results from calculations on cluster models.

Current-induced Orbital and Spin Magnetizations in Crystals with Helical Structure

We theoretically show that in a crystal with a helical lattice structure, orbital and spin magnetizations along a helical axis are induced by an electric current along the helical axis. We propose a simple tight-binding model for calculations, and the results can be generalized to any helical crystals. The induced magnetizations are opposite for right-handed and left-handed helices. The current-induced spin magnetization along the helical axis comes from a radial spin texture on the Fermi surface. This is in sharp contrast to Rashba systems where the induced spin magnetization is perpendicular to the applied current.

Berry phase theory of Dzyaloshinskii-Moriya interaction and spin-orbit torques

Recent experiments on current-induced domain-wall motion in chiral domain walls reveal important contributions both from spin-orbit torques (SOTs) and from the Dzyaloshinskii-Moriya interaction (DMI). We derive a Berry phase expression for the DMI and show that within this Berry phase theory DMI and SOTs are intimately related, in a way formally analogous to the relation between orbital magnetization (OM) and anomalous Hall effect (AHE). We introduce the concept of the twist torque moment, which probes the internal twist of wavepackets in chiral magnets in a similar way as the orbital moment probes the wavepacket's internal self-rotation. We propose to interpret the Berry phase theory of DMI as a theory of spiralization in analogy to the modern theory of OM. We show that the twist torque moment and the spiralization together give rise to a Berry phase governing the response of the SOT to thermal gradients, in analogy to the intrinsic anomalous Nernst effect. The Berry phase theory of DMI is computationally very efficient because it only needs the electronic structure of the collinear magnetic system as input. As an application of the formalism we compute the DMI in Co/Pt(111), O/Co/Pt(111) and Al/Co/Pt(111) magnetic bi- and trilayers and show that the DMI is highly anisotropic in these systems.

Energy magnetization and the thermal Hall effect

We obtain a set of general formulas for determining magnetizations, including the usual electromagnetic magnetization as well as the gravitomagnetic energy magnetization. The magnetization corrections to the thermal transport coefficients are explicitly demonstrated. Our theory provides a systematic approach for properly evaluating the thermal transport coefficients of magnetic systems, eliminating the unphysical divergence from the direct application of the Kubo formula. For a noninteracting anomalous Hall system, the corrected thermal Hall conductivity obeys the Wiedemann-Franz law.

Electrical polarization and orbital magnetization: the modern theories

Macroscopic polarization P and magnetization M are the most fundamental concepts in any phenomenological description of condensed media. They are intensive vector quantities that intuitively carry the meaning of dipole per unit volume. But for many years both P and the orbital term in M evaded even a precise microscopic definition, and severely challenged quantum-mechanical calculations. If one reasons in terms of a finite sample, the electric (magnetic) dipole is affected in an extensive way by charges (currents) at the sample boundary, due to the presence of the unbounded position operator in the dipole definitions. Therefore P and the orbital term in M--phenomenologically known as bulk properties--apparently behave as surface properties; only spin magnetization is problemless. The field has undergone a genuine revolution since the early 1990s. Contrary to a widespread incorrect belief, P has nothing to do with the periodic charge distribution of the polarized crystal: the former is essentially a property of the phase of the electronic wavefunction, while the latter is a property of its modulus. Analogously, the orbital term in M has nothing to do with the periodic current distribution in the magnetized crystal. The modern theory of polarization, based on a Berry phase, started in the early 1990s and is now implemented in most first-principle electronic structure codes. The analogous theory for orbital magnetization started in 2005 and is partly work in progress. In the electrical case, calculations have concerned various phenomena (ferroelectricity, piezoelectricity, and lattice dynamics) in several materials, and are in spectacular agreement with experiments; they have provided thorough understanding of the behaviour of ferroelectric and piezoelectric materials. In the magnetic case the very first calculations are appearing at the time of writing (2010). Here I review both theories on a uniform ground in a density functional theory (DFT) framework, pointing out analogies and differences. Both theories are deeply rooted in geometrical concepts, elucidated in this work. The main formulae for crystalline systems express P and M in terms of Brillouin-zone integrals, discretized for numerical implementation. I also provide the corresponding formulae for disordered systems in a single k-point supercell framework. In the case of P the single-point formula has been widely used in the Car-Parrinello community to evaluate IR spectra.

Polarization and adiabatic pumping in inhomogeneous crystals

We develop a general theory of electric polarization in crystals with inhomogeneous order. We show that the inhomogeneity-induced polarization can be classified into two parts: a perturbative contribution stemming from a correction to the basis functions and a topological contribution described in terms of the Chern-Simons form of the Berry gauge fields. The latter is determined up to an uncertainty quantum, which is the second Chern number in appropriate units. Our theory provides an exhaustive link between microscopic models and the macroscopic polarization.