Since the well-known PT symmetry has its fundamental significance and implication in physics, where PT denotes a joint operation of space-inversion P and time-reversal T, it is important and intriguing to explore exotic PT-invariant topological metals and to physically realize them. Here we develop a theory for a new type of topological metals that are described by a two-band model of PTinvariant topological nodal loop states in a three-dimensional Brillouin zone, with the topological stability being revealed through the PT-symmetry-protected nontrivial Z2 topological charge even in the absence of both P and T symmetries. Moreover, the gapless boundary modes are demonstrated to be originated from the nontrivial topological charge of the bulk nodal loop. Based on these exact results, we propose an experimental scheme to realize and to detect tunable PT-invariant topological nodal loop states with ultracold atoms in an optical lattice, in which atoms with two hyperfine spin states are loaded in a spin-dependent three-dimensional optical lattice and two pairs of Raman lasers are used to create out-of-plane spin-flip hopping with site-dependent phase. It is shown that such a realistic cold-atom setup can yield topological nodal loop states, having a tunable band-touching ring with the two-fold degeneracy in the bulk spectrum and non-trivial surface states. The nodal loop states are actually protected by the combined PT symmetry and are characterized by a Z2-type invariant (or topological charge), i.e., a quantized Berry phase. Remarkably, we demonstrate with numerical simulations that (i) the characteristic nodal ring can be detected by measuring the atomic transfer fractions in a Bloch-Zener oscillation; (ii) the topological invariant may be measured based on the time-of-flight imaging; and (iii) the surface states may be probed through Bragg spectroscopy. The present proposal for realizing topological nodal loop states in cold atom systems may provide a unique experimental platform for exploring exotic PT-invariant topological physics.
With formate as ligand, two 1-D 3d-4f compounds (linear and zigzag) based on pyramidal {TbCu(4)} unit were obtained. Chair-like [(H(2)O)(2)(ClO(4))(2)](2-) clusters and μ(5)-η(1):η(4) bridging mode of formate were observed in the linear one which also displays slow relaxation of the magnetization.
Optical chirality enhancement is highly demanded for enantioselective interaction of circularly polarized light with chiral molecules. The chirality enhancement in the coaxial air hole of a hollow silicon disk depends on three aspects, namely, the enhancements of electric and magnetic fields and a factor determined by the phases of their field components. In the spectral regime of dipole resonances, maximum chirality enhancement with sign consistency and uniform spatial distribution in the air hole can be obtained in association with both magnetic dipole resonance and anapole. Due to dipolar interference, the chirality is nulled at their coincidence, around which the sign of chirality is reversed. Maximum chirality with both positive and negative signs can be found between magnetic dipole resonance and anapole in the vicinity of their coincidence. This situation is maintained under size scaling so that the operation wavelength can be broadly tuned. The optical chirality can be further improved by merely adjusting the hole radius, by which the optimal spatially averaged optical chirality enhancement factor can reach 39 and −23. The simple strategy for optimizing Mie resonators presented in this work may benefit the design of Mie resonator‐based achiral metasurfaces for chirality detection application.
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