We show how in principle to construct analogs of quantum Hall edge states in "photonic crystals" made with non-reciprocal (Faraday-effect) media. These form "one-way waveguides" that allow electromagnetic energy to flow in one direction only.PACS numbers: 42.70. Qs, 03.65.Vf In this letter, we describe a novel effect involving an interface between two magneto-optic photonic crystals (periodic "metamaterials" that transmit electromagnetic waves) which can theoretically act as a "one-way waveguide", i.e., a channel along which electromagnetic energy can propagate in only a single direction, with no possibility of being back-scattered at bends or imperfections. The unidirectional photonic modes confined to such interfaces are the direct analogs of the "chiral edge-states" of electrons in the quantum Hall effect (QHE) [1,2]. The key enabling ingredient is the presence of "non-reciprocal" (Faraday-effect) media that breaks time-reversal symmetry in the metamaterial.Just as in the electronic case, every two dimensional photonic band is characterized by a topological invariant known as the Chern number [5], an integer that vanishes identically unless time-reversal symmetry is broken. If the material contains a photonic band gap (PBG), the Chern number, summed over all bands below the gap, plays a role similar to that of the same quantity summed over all occupied bands in the electronic case. In particular, if the total Chern number changes across an interface separating two PBG media, there necessarily will occur states localized to the interface having a non-zero net current along the interface [1,2]. In the photonic case, such states would comprise our "one-way waveguide".Such an interface between two PBG media can be realized as a domain wall in a 2D periodic photonic metamaterial, across which the direction of the Faraday axis reverses. Unidirectional edge states are guaranteed in this system provided that the Faraday effect generates photonic bands with non-zero Chern numbers. Here, we construct photonic bands with non-zero Chern invariants in a hexagonal array of dielctric rods with a Faraday effect. We then show that as a consequence of topology of the single-particle photon bands in the Brillouin zone, the edge states of light occur along domain walls (where the Faraday effect vanishes).It may seem surprising that the physics of the QHE can have analogs in photonic systems. The QHE is exhibited by incompressible quantum fluid states of electrons -conserved strongly-interacting charged fermionsin high magnetic fields, while photons are non-conserved neutral bosons which do not interact in linear media; furthermore, photonic bands can be described classically, in terms of Maxwell's equations. However, the integer QHE can in principle occur without any uniform magnetic flux density (just with broken time-reversal symmetry) as has explicitly shown by one of us in a graphene-like model of non-interacting Bloch electrons [6]; thus Landau-level quantization is not an essential requirement for the quantum Hall effec...

"Photonic crystals" built with time-reversal-symmetry-breaking Faraday-effect media can exhibit "chiral" edge modes that propagate unidirectionally along boundaries across which the Faraday axis reverses. These modes are precise analogs of the electronic edge states of quantum Hall effect (QHE) systems, and are also immune to backscattering and localization by disorder. The "Berry curvature" of the photonic bands plays a role analogous to that of the magnetic field in the QHE. Explicit calculations demonstrating the existence of such unidirectionally-propagating photonic edge states are presented.

We consider extended Hubbard models with repulsive interactions on a honeycomb lattice, and the transitions from the semimetal to Mott insulating phases at half-filling. Because of the frustrated nature of the second-neighbor interactions, topological Mott phases displaying the quantum Hall and the quantum spin Hall effects are found for spinless and spin fermion models, respectively. The mean-field phase diagram is presented and the fluctuations are treated within the random phase approximation. Renormalization group analysis shows that these states can be favored over the topologically trivial Mott insulating states.

We construct time-reversal invariant topological superconductors and superfluids in two and three dimensions. These states have a full pairing gap in the bulk, gapless counterpropagating Majorana states at the boundary, and a pair of Majorana zero modes associated with each vortex. The superfluid 3He B phase provides a physical realization of the topological superfluidity, with experimentally measurable surface states protected by the time-reversal symmetry. We show that the time-reversal symmetry naturally emerges as a supersymmetry, which changes the parity of the fermion number associated with each time-reversal invariant vortex and connects each vortex with its superpartner.

Following the discovery of the Fe-pnictide superconductors, LDA band structure calculations showed that the dominant contributions to the spectral weight near the Fermi energy came from the Fe 3d orbitals. The Fermi surface is characterized by two hole surfaces around the Γ point and two electron surfaces around the M point of the 2 Fe/cell Brillouin zone. Here, we describe a 2-band model that reproduces the topology of the LDA Fermi surface and exhibits both ferromagnetic and q = (π, 0) spin density wave (SDW) fluctuations. We argue that this minimal model contains the essential low energy physics of these materials.PACS numbers: 71.10. Fd, 71.18.+y, 74.20.Mn, 74.25.Ha, 75.30.Fv Introduction -The recent discovery of superconductivity in a family of Fe-based oxypnictides with large transition temperatures [1,2,3,4,5,6] has led to tremendous activity aimed at identifying the mechanism for superconductivity in these materials. Preliminary experimental results including specific heat [7], point-contact spectroscopy [8] and high-field resistivity [9, 10] measurements suggest the existence of unconventional superconductivity in these materials. Furthermore, transport [11] and neutron scattering [12] measurements have shown the evidence of magnetic order below T = 150K. An experimental determination of the orbital and spin state of the Cooper pairs, however, has not yet been made.