L lines, C points and Chern numbers: understanding band structure topology using polarization fields

Abstract

Topology has appeared in different physical contexts. The most prominent application is topologically protected edge transport in condensed matter physics. The Chern number, the topological invariant of gapped Bloch Hamiltonians, is an important quantity in this field. Another example of topology, in polarization physics, are polarization singularities, called L lines and C points. By establishing a connection between these two theories, we develop a novel technique to visualize and potentially measure the Chern number: it can be expressed either as the winding of the polarization azimuth along L lines in reciprocal space, or in terms of the handedness and the index of the C points. For mechanical systems, this is directly connected to the visible motion patterns.

Topology of rotating stratified fluids with and without background shear flow

“…The Chern number equals the negative of the total winding within a closed domain so ∆C = ν − − ν + , where ν ± is the winding number of the positive/negative frequency Poincaré mode and a vortex/anti-vortex corresponds to a winding number ±1 [55]. Thus ∆C + = 2 for f > 0 and ∆C − = −2 for f < 0, in agreement with the Chern numbers found for the f-plane [1].…”

“…As an alternative, we instead look for singularities in the phase of the wavefunctions which appear as vortices in wavevector space [54]. In the context of polarization physics, it has been shown that the winding of the polarization azimuth, or the wavefunction phase, equals the enclosed Chern number [55]. We set the Coriolis parameter to its bulk value, f = 1, and examine the phase of the wavefunctions in (k x , k y , f ) to check whether there is a vortex or antivortex in the phase.…”

Direct observation of topological edge states in silicon photonic crystals: Spin, dispersion, and chiral routing

“…We demonstrated that the pseudospin of the states is inherently encoded in the circular polarization of their far fields, and used this strong spin-orbit coupling to selectively excite the states with a focused laser beam. The employed Fourier spectroscopy technique makes it possible to quantify the inherent spinspin scattering of the system, opening doors to further understand and control topological protection in QSHE systems, as well as the connections between chirality and topology [45]. It establishes a straightforward yet versatile path for testing and optimizing novel photonic topological systems for a wide variety of applications including components for integrated photonic chips, quantum optical interfaces, enantiomeric sensing, and lasing at the nanoscale.…”

Collective dynamics of strain-coupled nanomechanical pillar resonators

“…In addition we envisage to implement in-situ thermal tuning of the pillar eigenfrequencies in an array using a spatial light modulator 24 . The resulting nanomechanical network offers inherent, acoustically mediated nearest-neighbor coupling 25 . The large vibration amplitudes of singly-clamped nanopillars in the range of tens or hundreds of nanometers allow for the microscopic imaging of their response and thus for the direct visualization of the many-body dynamics in an all-mechanical array.…”