The Berry connection encodes the momentum-space geometry of occupied Bloch states in gapped insulators and plays a central role in topological materials. While gauge-invariant quantities can be evaluated from overlap matrices between neighboring k points, accessing the Berry connection itself as a smooth field requires specifying a continuous gauge over the Brillouin zone. Wannier-based workflows achieve this through projection onto localized orbitals, enabling stable evaluation of geometric quantities and response functions. In this setting, the Berry connection enters directly in Wannier-interpolated calculations of polarization, Berry curvature, and related response functions. In practical implementations, however, the projection-gauge Berry connection is typically constructed from finite-difference overlaps between neighboring k points, discretizing momentum derivatives and introducing errors tied to k-mesh spacing and gauge alignment. These effects can become numerically delicate in systems with small band gaps or when evaluating higher-order responses such as the Chern-Simons axion angle. Here, we derive an exact expression for the non-Abelian Berry connection in the projection gauge that is local in crystal momentum. Starting from projected and orthonormalized Bloch-like states, we obtain a closed-form equation expressed entirely in terms of k-local quantities. We validate the formulation in one and three dimensions by computing the Berry phase and Chern-Simons axion angle in tight-binding models. The resulting framework provides a stable route to evaluating geometric properties within Wannier-interpolation schemes and future first-principles implementations.

Loop-current states arise when interacting electronic degrees of freedom collectively generate interatomic currents, producing a rare form of magnetic order in which spin does not play the primary role. The recent proposal of loop-current states in kagome superconductors has stimulated renewed interest in this exotic type of magnetism. We provide an overview of the phenomenological and symmetry properties of loop currents, as well as relevant microscopic models and ab initio methods in kagome materials. We then discuss how loop-current order generates a spin density wave through spin–orbit coupling and a charge density wave through anharmonic couplings present in systems with three-fold rotational symmetry. We briefly discuss the current status of loop-current order in kagome metals and open challenges, including their experimental detection and interplay with other orders such as superconductivity. Loop-current states represent a rare magnetic order driven by orbital motion, rather than by spin. This Perspective examines their symmetry, microscopic origins, detection challenges and interplay with other orders in kagome superconductors.

Breaking spatial-inversion or time-reversal symmetry in solids leads to transverse electromagnetic effects such as the anomalous Hall effect, Faraday rotation, non-reciprocal directional dichroism and off-diagonal linear magnetoelectricity. These are all tied to the framework of magnetic toroidal invariance. Here we introduce a distinct class of transverse electromagnetic responses that arise from electric toroidal invariance in ferro-rotational systems that preserve both inversion and time-reversal symmetries. It is different from that governed by magnetic toroidal invariance. We demonstrate a high-order off-diagonal magnetic susceptibility of ferro-rotational domains and a reduced linear diagonal magnetic susceptibility at these domain walls in doped ilmenite FeTiO3. Our results reveal the presence of anomalous transverse susceptibilities in ferro-rotational materials with spontaneous electric toroidal moments. Therefore, our findings illustrate emergent functionalities of ferro-rotational materials.

The exploration of transverse electromagnetic responses in solids with broken spatial-inversion (I) and/or time-reversal (T) symmetries has unveiled numerous captivating phenomena, including the (anomalous) Hall effect, Faraday rotations, non-reciprocal directional dichroism, and off-diagonal linear magnetoelectricity, all within the framework of magnetotoroidicity. Here, we introduce a novel class of transverse electromagnetic responses originating from electrotoroidicity in ferro-rotational (FR) systems with preserved I and T symmetries, distinct from magnetotoroidicity. We discover a high-order off-diagonal magnetic susceptibility of FR domains and a reduced linear diagonal magnetic susceptibility at FR domain walls in doped ilmenite FeTiO3. The non-trivial "Hall-like" effect of the former corresponds to an anomalous transverse susceptibility in the presence of spontaneous electrotoroidal moments in FR materials. Our findings unveil an emergent type of transverse electromagnetic responses even in I and T symmetry-conserved conditions and illustrate new functionalities of abundant FR materials.

Oxides are legendary for their complex energy landscapes, sensitivity to external stimuli, and property control through chemical substitution. Of these, the binary oxide HfO2 is one of the most fascinating because of the extraordinary number competing phases and opportunities to stabilize unique and useful properties. In this work, we combined synchrotron-based infrared absorbance and Raman scattering spectroscopies with diamond anvil cell techniques and first-principles calculations to explore the properties of polar orthorhombic hafnia (chemical formula HfO2:xY, where x = 12%) under pressure. Compression drives this system to the tetragonal form above 22 GPa-quite different from the more conventional phase diagram derived from pressurization of monoclinic HfO2 where the tetragonal phase resides at elevated temperatures. In addition to evidence for a complex energy landscape, we unveil a wide coexistence region, order-of-magnitude differences in phonon lifetimes, and an A 1g symmetry phonon in the tetragonal phase with a negative mode Gruneisen parameter that drives the system toward the cubic phase. Similar pressure pathways may connect other metastable phases in this family of materials.

We present a numerical implementation, based on Wannier interpolation, of a Kubo-Greenwood formalism for computing the spatially dispersive optical conductivity in crystals at first order in the wave vector of light. This approach is more efficient than direct ab initio methods because, with less computational cost, it allows for a much finer sampling of reciprocal space, resulting in better-resolved spectra. Moreover, Wannier interpolation avoids errors arising from truncation of the sums over conduction bands when evaluating the spatially dispersive optical matrix elements. We validate our method by computing the optical activity spectrum of selected crystals, both polar (GaN) and chiral (trigonal Te, trigonal Se, and alpha-quartz), and comparing it with existing literature.

Bands with non-trivial topological indices have a topological obstruction preventing them from being represented by exponentially localized Wannier states. Here, we propose a procedure to construct exponentially localized Wannier functions that span a subspace of topologically obstructed bands through the use of the projection method. These Wannier functions form what we refer to as a "reduced Wannier representation," indicating that the Wannier functions necessarily do not span the full topological manifold. By constructing supercells, we obtain reduced Wannier functions that break the primitive translational symmetry while capturing a substantial portion of the topologically obstructed manifold. This approach effectively decomposes the topological manifold into two subspaces: an itinerant subspace inheriting the topology and a localized trivial subspace represented by reduced Wannier functions. We consider the Haldane and Kane-Mele tight-binding models as the platforms for our investigation.

In loop-current states, interacting electronic degrees of freedom collectively establish interatomic currents, in a rare example of magnetism in which spin degrees of freedom do not play the primary role. The main impact of such states on the electronic spectrum is not via the standard Zeeman term, but via the kinetic energy, in which hopping parameters develop non-trivial phases that break time-reversal symmetry. The recent proposal of loop-current states in kagome superconductors has stimulated renewed interest in this exotic type of magnetism. In this perspective, we use kagome materials as a scaffolding to frame the basic phenomenology of loop-current states. We provide an overview of the group-theoretical properties of loop currents, as well as of relevant microscopic models and ab initio methods. Particular emphasis is given to the comparison with spin-density waves in the presence of spin-orbit coupling, as well as to the anharmonic coupling with charge-density waves, which is present in systems with threefold rotational symmetry. We also provide a brief overview of the current status of loop-current order in kagome metals and discuss open challenges including their experimental detection and interplay with other orders.

Eu5In2Sb6 is an orthorhombic nonsymmorphic small band gap semiconductor with three distinct Eu2+ sites and two low-temperature magnetic phase transitions. The material displays one of the greatest (negative) magnetoresistances of known stoichiometric antiferromagnets and belongs to a family of Zintl materials that may host an axion insulator. Using single crystal neutron diffraction, we show that the TN1 = 14 K second-order phase transition is associated with long-range antiferromagnetic order within the chemical unit cell (k1 = (000)). Upon cooling below TN1, the relative sublattice magnetizations of this structure vary until a second-order phase transition at TN2 = 7 K that doubles the unit cell along the e axis (k2 = (0021)). We show the anisotropic susceptibility and our magnetic neutron diffraction data are consistent with magnetic structures described by the I'3 irreducible representation with the staggered magnetization of the k1 and k2 components polarized along the b and a axis, respectively. As the k2 component develops, the amplitude of the k1 component is reduced, which indicates a 2k noncollinear magnetic structure. Density functional theory is used to calculate the energies of these magnetic structures and to show the k1 phase is a metal so TN1 is a rare example of a unit-cell-preserving second-order phase transition from a paramagnetic semiconductor to an antiferromagnetic metal. DFT indicates the transition at TN2 to a doubled unit cell reduces the carrier density of the metal.

It was recently argued that Bernal stacked bilayer graphene (BLG) exposed to a 2D superlattice (SL) potential exhibits a variety of intriguing behaviors [Ghorashi et al., Phys. Rev. Lett. 130, 196201 (2023)]. Chief among them is the appearance of flat Chern bands that are favorable to the appearance of fractional Chern insulator states. Here, we explore the application of spatially periodic out-of-plane orbital magnetic fields to the model of Ghorashi et al. to find additional means of inducing flat Chern bands. We focus on fields that vary on length scales much larger than the atomic spacing in BLG, generating what we refer to as magnetic SLs. The magnetic SLs we investigate either introduce no net magnetic flux to the SL unit cell, or a single quantum of flux. We find that magnetic SLs acting on their own can induce topological flat bands, but richer behavior, such as the appearance of flat and generic bands with high Chern numbers, can be observed when the magnetic SLs act in conjunction with commensurate electric SLs. Finally, we propose a method of generating unit-flux-quantum magnetic SLs along with concomitant electric SLs. The magnetic SL is generated by periodic arrays of flux vortices originating from type II superconductors, while the electric SL arises due to a magnetic SL-induced charge density on the surface of a magnetoelectric material. Tuning the vortex lattice and the magnetoelectric coupling permits control of both SLs, and we study their effects on the band structure of BLG.

Electric currents have the intriguing ability to induce magnetization in nonmagnetic crystals with sufficiently low crystallographic symmetry. Some associated phenomena include the non-linear anomalous Hall effect in polar crystals and the nonreciprocal directional dichroism in chiral crystals when magnetic fields are applied. In this work, we demonstrate that the same underlying physics is also manifested in the electronic tunneling process between the surface of a nonmagnetic chiral material and a magnetized scanning probe. In the paramagnetic but chiral metallic compound Co1/3NbS2, the magnetization induced by the tunneling current is shown to become detectable by its coupling to the magnetization of the tip itself. This results in a contrast across different chiral domains, achieving atomic-scale spatial resolution of structural chirality. To support the proposed mechanism, we used first-principles theory to compute the chirality-dependent current-induced magnetization and Berry curvature in the bulk of the material. Our demonstration of this magnetochiral tunneling effect opens up an avenue for investigating atomic-scale variations in the local crystallographic symmetry and electronic structure across the structural domain boundaries of low-symmetry nonmagnetic crystals.

Electric currents have the intriguing ability to induce magnetization in nonmagnetic crystals with sufficiently low crystallographic symmetry. Some associated phenomena include the non-linear anomalous Hall effect in polar crystals and the nonreciprocal directional dichroism in chiral crystals when magnetic fields are applied. In this work, we demonstrate that the same underlying physics is also manifested in the electronic tunneling process between the surface of a nonmagnetic chiral material and a magnetized scanning probe. In the paramagnetic but chiral metallic compound Co 1/3 NbS 2 , the magnetization induced by the tunneling current is shown to become detectable by its coupling to the magnetization of the tip itself. This results in a contrast across different chiral domains, achieving atomic-scale spatial resolution of structural chirality. To support the proposed mechanism, we used first-principles theory to compute the chirality-dependent current-induced magnetization and Berry curvature in the bulk of the material. Our demonstration of this magnetochiral tunneling effect opens up an avenue for investigating atomic-scale variations in the local crystallographic symmetry and electronic structure across the structural domain boundaries of low-symmetry nonmagnetic crystals.

Electric currents have the intriguing ability to induce magnetization in nonmagnetic crystals with sufficiently low crystallographic symmetry. Some associated phenomena include the non-linear anomalous Hall effect in polar crystals and the nonreciprocal directional dichroism in chiral crystals when magnetic fields are applied. In this work, we demonstrate that the same underlying physics is also manifested in the electronic tunneling process between the surface of a nonmagnetic chiral material and a magnetized scanning probe. In the paramagnetic but chiral metallic compound Co$_{1/3}$NbS$_2$, the magnetization induced by the tunneling current is shown to become detectable by its coupling to the magnetization of the tip itself. This results in a contrast across different chiral domains, achieving atomic-scale spatial resolution of structural chirality. To support the proposed mechanism, we used first-principles theory to compute the chirality-dependent current-induced magnetization and Berry curvature in the bulk of the material. Our demonstration of this magnetochiral tunneling effect opens up a new avenue for investigating atomic-scale variations in the local crystallographic symmetry and electronic structure across the structural domain boundaries of low-symmetry nonmagnetic crystals.

We combine synchrotron-based infrared absorption and Raman scattering spectroscopies with diamond anvil cell techniques and first-principles calculations to explore the properties of hafnia under compression. We find that pressure drives HfO 2 :7%Y from the mixed monoclinic ( P 2 1 / c ) + antipolar orthorhombic ( Pbca ) phase to pure antipolar orthorhombic ( Pbca ) phase at approximately 6.3 GPa. This transformation is irreversible, meaning that upon release, the material is kinetically trapped in the Pbca metastable state at 300 K. Compression also drives polar orthorhombic ( P c a 2 1 ) hafnia into the tetragonal ( P 4 2 / n m c ) phase, although the latter is not metastable upon release. These results are unified by an analysis of the energy landscape. The fact that pressure allows us to stabilize targeted metastable structures with less Y stabilizer is important to preserving the flat phonon band physics of pure HfO 2 .

In conventional ab initio methodologies, phonons are calculated by solving equations of motion involving static interatomic force constants and atomic masses. The Born-Oppenheimer approximation, where all electronic degrees of freedom are assumed to adiabatically follow the nuclear dynamics, is also adopted. This approach does not fully account for the effects of broken time-reversal symmetry in systems with magnetic order. Recent attempts to rectify this involve the inclusion of the velocity dependence of the interatomic forces in the equations of motion, which accounts for time-reversal symmetry breaking, and can result in chiral phonon modes with nonzero angular momentum even at the zone center. However, since the energy ranges of phonons and magnons typically overlap, the spins cannot be treated as adiabatically following the lattice degrees of freedom. Instead, phonon and spins must be treated on a similar footing. Focusing on zone-center modes, we propose a method involving Hessian matrices and Berry curvature tensors in terms of both phonon and spin degrees of freedom, and describe a first-principles methodology for calculating these. We then solve Lagrange’s equations of motion to determine the energies and characters of the mixed excitations, allowing us to quantify, for example, the energy splittings between chiral pairs of phonons in some cases, and the degree of magnetically induced mixing between infrared and Raman modes in others. The approach is general and can be applied to determine the adiabatic dynamics of any mixed set of slow variables.

In single sheets of graphene, vacancy-induced states have been shown to host an effective spin-1/2 hole that can be Kondo screened at low temperatures. Here, we show how these vacancy-induced impurity states survive in twisted bilayer graphene (TBG), which thus provides a tunable system to probe the critical destruction of the Kondo effect in pseudogap hosts. Ab initio calculations and atomic-scale modeling are used to determine the nature of the vacancy states in the vicinity of the magic angle in TBG, demonstrating that the vacancy can be treated as a quantum impurity. Utilizing this insight, we construct an Anderson impurity model with a TBG host that we solve using the numerical renormalization group combined with the kernel polynomial method. We determine the phase diagram of the model and show how there is a strict dichotomy between vacancies in the AA/BB versus AB/BA tunneling regions. In AB/BA vacancies, the Kondo temperature at the magic angle develops a broad distribution with a tail to vanishing temperatures due to multifractal wave functions at the magic angle. We argue that scanning tunneling microscopy in the vicinity of the vacancy can act as a probe of both the critical single-particle states and the underlying many- body ground state in magic-angle TBG.

The van der Waals layered ferroelectric CuInP$_2$S$_6$ has been found to exhibit a variety of intriguing properties arising from the fact that the Cu ions are unusually mobile in this system. While the polarization switching mechanism is usually understood to arise from Cu ion motion within the monolayers, a second switching path involving Cu motion across the van der Waals gaps has been suggested. In this work, we perform zero-temperature first-principles calculations on such switching paths, focusing on two types that preserve the periodicity of the primitive unit cell: ``cooperative" paths preserving the system's glide mirror symmetry, and ``sequential" paths in which the two Cu ions in the unit cell move independently of each other. We find that CuInP$_2$S$_6$ features a rich and varied energy landscape, and that sequential paths are clearly favored energetically both for cross-gap and through-layer paths. Importantly, these segments can be assembled to comprise a globally insulating cycle with the out-of-plane polarization evolving by a quantum as the Cu ions shift to neighboring layers. In this sense, we argue that CuInP$_2$S$_6$ embodies the physics of a quantized adiabatic charge pump.

Conventional approaches for lattice dynamics based on static interatomic forces do not fully account for the effects of time-reversal-symmetry breaking in magnetic systems. Recent approaches to rectify this involve incorporating the first-order change in forces with atomic velocities under the assumption of adiabatic separation of electronic and nuclear degrees of freedom. In this Letter, we develop a first -principles method to calculate this velocity-force coupling in extended solids and show via the example of ferromagnetic CrI3 that, due to the slow dynamics of the spins in the system, the assumption of adiabatic separation can result in large errors for splittings of zone-center chiral modes. We demonstrate that an accurate description of the lattice dynamics requires treating magnons and phonons on the same footing.

Eu_5In_2Sb_6 is an orthorhombic non-symmorphic small band gap semiconductor with three distinct Eu^2+ sites and two low-temperature magnetic phase transitions. The material displays one of the greatest (negative) magnetoresistances of known stoichiometric antiferromagnets and belongs to a family of Zintl materials that may host an axion insulator. Using single crystal neutron diffraction, we show that the T_N1=14 K second-order phase transition is associated with long-range antiferromagnetic order within the chemical unit cell ( k_1 = (000) ). Upon cooling below T_N1, the relative sublattice magnetizations of this structure vary until a second-order phase transition at T_N2=7 K that doubles the unit cell along the ĉ axis ( k_2 = (001/2) ). We show the anisotropic susceptibility and our magnetic neutron diffraction data are consistent with magnetic structures described by the Γ_3 irreducible representation with the staggered magnetization of the k_1 and k_2 components polarized along the b̂ and â axis, respectively. As the k_2 component develops, the amplitude of the k_1 component is reduced, which indicates a 2k non-collinear magnetic structure. Density functional theory is used to calculate the energies of these magnetic structures and to show the k_1 phase is a metal so T_N1 is a rare example of a unit-cell-preserving second-order phase transition from a paramagnetic semiconductor to an antiferromagnetic metal. DFT indicates the transition at T_N2 to a doubled unit cell reduces the carrier density of the metal, which is consistent with resistivity data.

Calcium carbonate is one of the important building components in organisms, especially the two most common polymorphs, calcite and aragonite. Here, to understand the difference in bonding state, the two polymorphs are characterized by valence (low-loss) electron energy loss spectroscopy. It is found that the difference in Ca M23 edge originating from 3p to 3d states is consistent with the change of Ca-O bonds in the two studied polymorphs. Surprisingly, the measured Ca M23 edge is in qualitative agreement with the calculated partial density of states (PDOS) of Ca-d states in contrast to their L edges (from 2p to 3d states) which are strongly influenced by atomic multiplet effect (spin-orbit coupling). This is because the atomic multiplet effect is much reduced for the Ca 3p orbital, which permits the corresponding Ca M23 edge to be compared with the PDOS results. Our findings show insights that PDOS can potentially be used to interpret the M23 edge of lighter 3d transition metals such as scandium, titanium, vanadium and chromium when such interpretation may not be achieved for their L edges.