Topological Triply Degenerate Points Induced by Spin-Tensor-Momentum Couplings

Intrinsic Triple Degeneracy Point Bounded by Nodal Surfaces in Chiral Photonic Crystal

In periodic systems, band degeneracies are typically protected and classified by spatial symmetries. However, in photonic systems, the Γ point at zero frequency is an intrinsic degeneracy due to the polarization degree of freedom of electromagnetic waves. For chiral photonic crystals, such an intrinsic degeneracy carries ±2 chiral topological charge while having linear band dispersions, different from the general perception of charge-2 nodes being associated with quadratic dispersions. Here, we show that these topological characters originate from the spin-1 Weyl point at zero frequency node of triple degeneracy, due to the existence of an electrostatic flat band. Such a topological charge at zero frequency is usually buried in bulk band projections and has never been experimentally observed. To address this challenge, we introduce space-group screw symmetries in the design of chiral photonic crystal, which makes the Brillouin zone boundary an oppositely charged nodal surface enclosing the Γ point. As a result, the emergent Fermi arcs are forced to connect the projections of these topological singularities, enabling their experimental observation. The number of Fermi arcs then directly reveals the embedded topological charge at zero frequency.

Topological Surface States in a Gyroid Acoustic Crystal

The acoustic properties of an acoustic crystal consisting of acoustic channels designed according to the gyroid minimal surface embedded in a 3D rigid material are investigated. The resulting gyroid acoustic crystal is characterized by a spin-1 Weyl and a charge-2 Dirac degenerate points that are enforced by its nonsymmorphic symmetry. The gyroid geometry and its symmetries produce multi-fold topological degeneracies that occur naturally without the need for ad hoc geometry designs. The non-trivial topology of the acoustic dispersion produces chiral surface states with open arcs, which manifest themselves as waves whose propagation is highly directional and remains confined to the surfaces of a 3D material. Experiments on an additively manufactured sample validate the predictions of surface arc states and produce negative refraction of waves at the interface between adjoining surfaces. The topological surface states in a gyroid acoustic crystal shed light on nontrivial bulk and edge physics in symmetry-based compact continuum materials, whose capabilities augment those observed in ad hoc designs. The continuous shape design of the considered acoustic channels and the ensuing anomalous acoustic performance suggest this class of phononic materials with semimetal-like topology as effective building blocks for acoustic liners and load-carrying structural components with sound proofing functionality.

Observation of Spin-Tensor Induced Topological Phase Transitions of Triply Degenerate Points with a Trapped Ion

Triply degenerate points (TDPs), which correspond to new types of topological semimetals, can support novel quasiparticles possessing effective integer spins while preserving Fermi statistics. Here by mapping the momentum space to the parameter space of a three-level system in a trapped ion, we experimentally explore the transitions between different types of TDPs driven by spin-tensor-momentum couplings. We observe the phase transitions between TDPs with different topological charges by measuring the Berry flux on a loop surrounding the gap-closing lines, and the jump of the Berry flux gives the jump of the topological charge (up to a 2π factor) across the transitions. For the Berry flux measurement, we employ a new method by examining the geometric rotations of both spin vectors and tensors, which lead to a generalized solid angle equal to the Berry flux. The controllability of a multilevel ion offers a versatile platform to study high-spin physics, and our Letter paves the way to explore novel topological phenomena therein.

Optical conductivities in triple fermions with different monopole charges

We investigate the linear optical conductivities of the newly-discovered triple-component semimetals. Due to the exactly flat band, the optical conductivity relates to the transition between the zero band and the conduction band directly reflecting the band structure of the conduction electrons in contrast to the other materials. For the low-energy models with various monopole charges, the diagonal conductivities show strong anisotropy. Theω-dependence of interband conductivities for a general low-energy model is deduced. The real part of the interbandσ_xxalways linearly depends on the optical frequency, while the one ofσ_zzis proportional toω^2/n-1. This can be a unique fingerprint of the monopole charge. For the lattice models, there also exists the optical anomalous Hall conductivity, where a sign change may appear. The characteristic frequencies of the kink structures are calculated, strictly. Our work will help us to establish the basic picture of linear optical response in topological triple-component semimetals and identify them from other materials.

Topology and symmetry of circular photogalvanic effect in the chiral multifold semimetals: a review

The circular photogalvanic effect (CPGE) is the only possible quantized signal in chiral Weyl and multifold semimetals with inversion and mirror symmetries broken. Here, we review CPGE in the chiral multifold semimetals in terms of classification of CPGE tensor, the quantization of CPGE fromk⋅peffective model and topological semimetal RhSi family. Firstly, we give complete symmetric analysis of CPGE tensors for all nonmagnetic point groups, and get a table classifying matrix of response tensors. Secondly, the CPGE becomes a quantized response in the noncentrosymmetric topological semimetals, and depends on the Chern number of multifold fermions. Based onk⋅peffective model with linear dispersion, detailed derivations about the quantization of CPGE are given. Finally, according toab initioanalysis for the quantized CPGE based on noninteracting electronic structures, we review previous reports and make new calculations for the chiral topological semimetals in RhSi family, which can be separated into two groups. The first group, including RhSi, PtAl and CoSi, can be the promising candidates to exhibit a quantized CPGE trace, while the second group includes PdGa, PtGa and RhSn without a quantization.

Non-Abelian Tensor Berry Connections in Multiband Topological Systems

Here, we introduce and apply non-Abelian tensor Berry connections to topological phases in multiband systems. These gauge connections behave as non-Abelian antisymmetric tensor gauge fields in momentum space and naturally generalize Abelian tensor Berry connections and ordinary non-Abelian (vector) Berry connections. We build these novel gauge fields from momentum-space Higgs fields, which emerge from the degenerate band structure of multiband models. First, we show that the conventional topological invariants of two-dimensional topological insulators and three-dimensional Dirac semimetals can be derived from the winding number associated with the Higgs field. Second, through the non-Abelian tensor Berry connections we construct higher-dimensional Berry-Zak phases and show their role in the topological characterization of several gapped and gapless systems, ranging from two-dimensional Euler insulators to four-dimensional Dirac semimetals. Importantly, through our new theoretical formalism, we identify and characterize a novel class of models that support space-time inversion and chiral symmetries. Our work provides a unifying framework for different multiband topological systems and sheds new light on the emergence of non-Abelian gauge fields in condensed matter physics, with direct implications on the search for novel topological phases in solid-state and synthetic systems.

Optical conductivity of triple point fermions

As a low-energy effective theory on non-symmorphic lattices, we consider a generic triple point fermion Hamiltonian, which is parameterized by an angular parameterλ. We find strongλdependence in both Drude and interband optical absorption of these systems. The deviation of theT2coefficient of the Drude weight from Dirac/Weyl fermions can be used as a quick way to optically distinguish the triple point degeneracies from the Dirac/Weyl degeneracies. At the particularλ=π/6 point, we find that the 'helicity' reversal optical transition matrix element is identically zero. Nevertheless, deviating from this point, the helicity reversal emerges as an absorption channel.

Hidden-symmetry-enforced nexus points of nodal lines in layer-stacked dielectric photonic crystals

It was recently demonstrated that the connectivities of bands emerging from zero frequency in dielectric photonic crystals are distinct from their electronic counterparts with the same space groups. We discover that in an AB-layer-stacked photonic crystal composed of anisotropic dielectrics, the unique photonic band connectivity leads to a new kind of symmetry-enforced triply degenerate points at the nexuses of two nodal rings and a Kramers-like nodal line. The emergence and intersection of the line nodes are guaranteed by a generalized 1/4-period screw rotation symmetry of Maxwell's equations. The bands with a constant k _z and iso-frequency surfaces near a nexus point both disperse as a spin-1 Dirac-like cone, giving rise to exotic transport features of light at the nexus point. We show that spin-1 conical diffraction occurs at the nexus point, which can be used to manipulate the charges of optical vortices. Our work reveals that Maxwell's equations can have hidden symmetries induced by the fractional periodicity of the material tensor components and hence paves the way to finding novel topological nodal structures unique to photonic systems.

Parity anomaly of lattice Maxwell fermions in two spatial dimensions

Unconventional lattice fermions with high degeneracies that are not Weyl or Dirac fermions have attracted increased attention in recent years. In this paper, we consider pseudospin-1 Maxwell fermions and the (2 + 1)-dimensional parity anomaly, which are not constrained by the fermion doubling theorem. We derive the Hall conductivity of a single Maxwell fermion and explain how each Maxwell fermion has a quantized Hall conductance ofe2/h. Parity is spontaneously broken in the effective theory of lattice Maxwell fermions interacting with an (auxiliary) U(1) gauge field, leading to an effective anomaly-induced Chern-Simons theory. An interesting observation about the parity anomaly is that the lattice Maxwell fermions are not constrained by the fermion doubling theorem, so a single Maxwell fermion can exist in a lattice. In addition, our work considers the quantum anomaly in odd-dimensional spinor space.

Topological Bands and Triply Degenerate Points in Non-Hermitian Hyperbolic Metamaterials

Hyperbolic metamaterials (HMMs), an unusual class of electromagnetic metamaterials, have found important applications in various fields due to their distinctive properties. A surprising feature of HMMs is that even continuous HMMs can possess topological edge modes. However, previous studies based on equal-frequency surface (analogy of Fermi surface) may not correctly capture the topology of entire bands. Here we develop a topological band description for continuous HMMs that can be described by a non-Hermitian Hamiltonian formulated from Maxwell's equations. We find two types of three-dimensional non-Hermitian triply degenerate points with complex linear dispersions and topological charges ±2 and 0 induced by chiral and gyromagnetic effects. Because of the photonic nature, the vacuum band plays an important role for topological edge states and bulk-edge correspondence in HMMs. The topological band results are numerically confirmed by direct simulation of Maxwell's equations. Our work presents a general non-Hermitian topological band treatment of continuous HMMs, paving the way for exploring interesting topological phases in photonic continua and device implementations of topological HMMs.