Relating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers

Quantum geometry has emerged as a central and ubiquitous concept in quantum sciences, with direct consequences on quantum metrology and many-body quantum physics. In this context, two fundamental geometric quantities are known to play complementary roles:~the Fubini-Study metric, which introduces a notion of distance between quantum states defined over a parameter space, and the Berry curvature associated with Berry-phase effects and topological band structures. In fact, recent studies have revealed direct relations between these two important quantities, suggesting that topological properties can, in special cases, be deduced from the quantum metric. In this work, we establish general and exact relations between the quantum metric and the topological invariants of generic Dirac Hamiltonians. In particular, we demonstrate that topological indices (Chern numbers or winding numbers) are bounded by the quantum volume determined by the quantum metric. Our theoretical framework, which builds on the Clifford algebra of Dirac matrices, is applicable to topological insulators and semimetals of arbitrary spatial dimensions, with or without chiral symmetry. This work clarifies the role of the Fubini-Study metric in topological states of matter, suggesting unexplored topological responses and metrological applications in a broad class of quantum-engineered systems.

The flow of the Berry curvature vector field

The concept of Berry phase and Berry curvature has become ubiquitous in solid state physics as it relates to variety of phenomena, such as topological insulators, polarization, and various Hall effects. It is well known that large Berry curvatures arise from close proximity of hybridizing bands, however, the vectorial nature of the Berry curvature is not utilized in current research. On bulk bcc Fe, we demonstrate the flow of the Berry curvature vector field which features not only monopoles but also higher dimensional structures with its own topological features. They can provide a novel unique view on the electronic structure in all three dimensions. This knowledge is also used to quantify particular contributions to the intrinsic anomalous Hall effect in a simple analytical form.

Nanometric transverse displacement metrology in range of hundreds of micrometers with polarization-encoded metasurface

A long range, high precision and compact transverse displacement metrology method is of crucial importance in many research areas. We propose and experimentally demonstrate the first prototype polarization-encoded metasurface for ultrasensitive transverse displacement metrology. The transverse displacement of the metasurface is encoded into the polarization direction of the outgoing light via the Pancharatnam-Berry phase. By measuring the output light polarization direction, the metasurface’s position can be readout directly according to the Malus law. We experimentally demonstrate nanometer displacement resolution with the uncertainty on the order of 100 pm for a large measurement range of 200 µm with the total area of the metasurface being within 900 µm x 900 µm. The measurement range can be extended further using a larger metasurface. Our work largely broadens the existing application areas of metasurface and opens new avenue of applying metasurface in the field of ultrasensitive optical transverse displacement metrology.

Investigation of the Pancharatnam-Berry phase element for the generation of the top-hat beam

Within optics, the Pancharatnam-Berry phase enables the design and creation of various flat special optical elements such as top-hat converters. We present a study on engineering efficient vectorial top-hat converters inscribed in glass by high-power femtosecond laser pulses. We phase-encode a top-hat converter and demonstrate how its quality is influenced by various parameters. We investigate theoretically the generation of the top-hat beam under imperfect conditions such as the mismatch of the incident beam width or the misalignment of the center of the converter. Experimental verification of the concept is also presented.

Plasmonic metasurfaces manipulating the two spin components from spin–orbit interactions of light with lattice field generations

Artificial nanostructures in metasurfaces induce strong spin–orbit interactions (SOIs), by which incident circularly polarized light can be transformed into two opposite spin components. The component with an opposite helicity to the incident light acquires a geometric phase and is used to realize the versatile functions of the metasurfaces; however, the other component, with an identical helicity, is often neglected as a diffused background. Here, by simultaneously manipulating the two spin components originating from the SOI in plasmonic metasurfaces, independent wavefields in the primary and converted spin channels are achieved; the wavefield in the primary channel is controlled by tailoring the dynamic phase, and that in the converted channel is regulated by designing the Pancharatnam–Berry phase in concurrence with the dynamic phase. The scheme is realized by generating optical lattice fields with different topologies in two spin channels, with the metasurfaces composed of metal nanoslits within six round-apertures mimicking the multi-beam interference. This study demonstrates independent optical fields in a dual-spin channel based on the SOI effect in the metasurface, which provides a higher polarization degree of freedom to modify optical properties at the subwavelength scale.

Higher-order topological superconductivity from repulsive interactions in kagome and honeycomb systems

We discuss a pairing mechanism in interacting two-dimensional multipartite lattices that intrinsically leads to a second order topological superconducting state with a spatially modulated gap. When the chemical potential is close to Dirac points, oppositely moving electrons on the Fermi surface undergo an interference phenomenon in which the Berry phase converts a repulsive electron-electron interaction into an effective attraction. The topology of the superconducting phase manifests as gapped edge modes in the quasiparticle spectrum and Majorana Kramers pairs at the corners. We present symmetry arguments which constrain the possible form of the electron-electron interactions in these systems and classify the possible superconducting phases which result. Exact diagonalization of the Bogoliubov-de Gennes Hamiltonian confirms the existence of gapped edge states and Majorana corner states, which strongly depend on the spatial structure of the gap. Possible applications to vanadium-based superconducting kagome metals AV$_3$Sb$_3$ (A=K,Rb,Cs) are discussed.

Electromagnetic response in spiral magnets and emergent inductance

Emergent electromagnetism in magnets originates from the strong coupling between conduction electron spins and those of noncollinear ordered moments and the consequent Berry phase. This offers possibilities to develop new functions of quantum transport and optical responses. The emergent inductance in spiral magnets is an example recently proposed and experimentally demonstrated, using the emergent electric field induced by alternating currents. However, the microscopic theory of this phenomenon is missing, which should reveal factors to determine the magnitude, sign, frequency dependence, and nonlinearity of the inductance L. Here we theoretically study electromagnetic responses of spiral magnets by taking into account their collective modes. In sharp contrast to collinear spin-density wave, the system remains metallic even in one dimension, and the canonical conjugate relation of uniform magnetization and phason coordinate plays an essential role, determining the properties of L. This result opens a way to design the emergent inductance of desired properties.