Identity and Democracy: Linking Individual and Social Reasoning

Following Amartya Sen\u27s approach, John Davis and Solange Regina Marin look at individual and social reasoning when examining the complex relationship between identity and democracy. They characterize democracy as a process of social or public reasoning that combines the individual reasoning of all citizens. Identity is explained in terms of personal identity, social identity, and individual identity. They argue that democracy in combining the individual reasoning of all citizens responds to individuals’ different personal identity concerns and needs, reflects their shared social identity interests and goals, and accords them rights and responsibilities associated with their many different individual identities

Weyl points and line nodes in gapless gyroid photonic crystals

Weyl points and line nodes are three-dimensional linear point- and line-degeneracies between two bands. In contrast to Dirac points, which are their two-dimensional analogues, Weyl points are stable in the momentum space and the associated surface states are predicted to be topologically non-trivial. However, Weyl points are yet to be discovered in nature. Here, we report photonic crystals, based on the double-gyroid structures, exhibiting frequency-isolated Weyl points with intricate phase diagrams. The surface states associated with the non-zero Chern numbers are demonstrated. Line nodes are also found in similar geometries; the associated surface states are shown to be flat bands. Our results are readily experimentally realizable at both microwave and optical frequencies.Comment: 6 figures and 8 pages including the supplementary informatio

Non-Abelian Generalizations of the Hofstadter model: Spin-orbit-coupled Butterfly Pairs

The Hofstadter model, well-known for its fractal butterfly spectrum, describes two-dimensional electrons under a perpendicular magnetic field, which gives rise to the integer quantum hall effect. Inspired by the real-space building blocks of non-Abelian gauge fields from a recent experiment [Science, 365, 1021 (2019)], we introduce and theoretically study two non-Abelian generalizations of the Hofstadter model. Each model describes two pairs of Hofstadter butterflies that are spin-orbit coupled. In contrast to the original Hofstadter model that can be equivalently studied in the Landau and symmetric gauges, the corresponding non-Abelian generalizations exhibit distinct spectra due to the non-commutativity of the gauge fields. We derive the genuine (necessary and sufficient) non-Abelian condition for the two models from the commutativity of their arbitrary loop operators. At zero energy, the models are gapless and host Weyl and Dirac points protected by internal and crystalline symmetries. Double (8-fold), triple (12-fold), and quadrupole (16-fold) Dirac points also emerge, especially under equal hopping phases of the non-Abelian potentials. At other fillings, the gapped phases of the models give rise to $\mathbb{Z}_2$ topological insulators. We conclude by discussing possible schemes for the experimental realizations of the models in photonic platforms

Experimental Observation of Large Chern numbers in Photonic Crystals

Despite great interest in the quantum anomalous Hall phase and its analogs, all experimental studies in electronic and bosonic systems have been limited to a Chern number of one. Here, we perform microwave transmission measurements in the bulk and at the edge of ferrimagnetic photonic crystals. Bandgaps with large Chern numbers of 2, 3, and 4 are present in the experimental results which show excellent agreement with theory. We measure the mode profiles and Fourier transform them to produce dispersion relations of the edge modes, whose number and direction match our Chern number calculations.Comment: This experimental work was accepted to PRL on Oct. 13, 2015. Our theoretical work from PRL http://dx.doi.org/10.1103/PhysRevLett.113.11390