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.

Phases of matter are usually identified through the lens of spontaneous symmetry breaking, which particularly applies to unconventional superconductivity and the interactions it originates from. In that context, the superconducting state of the quasi-two-dimensional and strongly correlated Sr 2 RuO 4 is uniquely held up as a solid-state analog to superfluid 3 He-A 1, 2 , with an odd-parity vector order parameter that is unidirectional in spin space for all electron momenta and also breaks time-reversal symmetry. This characterization was recently * These authors contributed equally to this work. 1 called into question by a search for, and failure to find, evidence for an expected "split" transition while subjecting a Sr 2 RuO 4 crystal to in-plane uniaxial pressure; instead a dramatic rise and peak in a single transition temperature was observed 3, 4. NMR spectroscopy, which is directly sensitive to the order parameter via the hyperfine coupling to the electronic spin degrees of freedom, is exploited here to probe the nature of superconductivity in Sr 2 RuO 4 and its evolution under strained conditions. A reduction of Knight shifts K is observed for all strain values and temperatures T < T c , consistent with a drop in spin polarization in the superconducting state. In unstrained samples, our results are in contradiction with a body of previous NMR work 5 , and with the most prominent previous proposals for the order parameter. Sr 2 RuO 4 is an extremely clean layered perovskite, and the superconductivity emerges from a strongly correlated Fermi Liquid. The present work imposes tight constraints on the order-parameter symmetry of this archetypal system. The normal state of Sr 2 RuO 4 is based on three bands crossing the Fermi level 6, 7 , with pronounced strong-correlation characteristics linked to Hund's Rule coupling of the partially filled Ru t 2g orbitals dominating the Fermi surface. The transition to a superconducting ground state at T c =1.5 K 8 , with indirect evidence for proximity to ferromagnetism, led to the suggestion that the pair wave functions of the superconducting state likely exhibit a symmetric spin part, i.e., triplet 1. Crucial support for the existence of a triplet order parameter rested on NMR spectroscopy, which showed no change in Knight shift between normal and superconducting states 5. Later, several experiments produced evidence for time-reversal symmetry breaking (TRSB) 9, 10. Together, these reports aligned well to the above-mentioned proposal that Sr 2 RuO 4 is a very clean, quasi two

We study low energy collective modes and transport properties of the "helical metal" on the surface of a topological insulator. At low energies, electrical transport and spin dynamics at the surface are exactly related by an operator identity equating the electric current to the in-plane components of the spin degrees of freedom. From this relation it follows that an undamped spin wave always accompanies the sound mode in the helical metal -thus it is possible to 'hear' the sound of spins. In the presence of long range Coulomb interactions, the surface plasmon mode is also coupled to the spin wave, giving rise to a hybridized "spin-plasmon" mode. We make quantitative predictions for the spin-plasmon in Bi2Se3, and discuss its detection in a spin-grating experiment.PACS numbers: 71.10. Ay, 71.45.Gm, 72.15.Nj, 73.20.Mf, 73.25.+i, 73.43.Lp Introduction -Recently, topological insulators have been theoretically predicted and experimentally observed in both quasi-two dimensional (2D) and three dimensional (3D) systems [1,2,3,4,5,6,7]. The concept of a topological insulator can be defined within the noninteracting topological band theory [8,9] or more generally within the topological field theory[10], which is also valid for interacting systems. The simplest topological insulators such as Bi 2 Se 3 and Bi 2 Te 3 have a full bulk insulating gap and a surface state consisting of a single Dirac cone [5,6,7]. As is the case for the helical edge states of a 2D topological insulator [11,12], the spin and the momentum are intimately locked in the "helical metal" surface state of the 3D topological insulator. This locking effect has been theoretically predicted [6] for Bi 2 Se 3 and Bi 2 Te 3 , and experimentally observed [19] in Bi 2 Se 3 .In this paper, we study the universal surface state properties of the simplest 3D topological insulators, and consider a system governed by a single isotropic Dirac cone at energy scales much lower than the bulk insulating gap. We study the consequences of the helical nature of the metallic states: the coupling between spin and charge excitations, and collective modes of the helical liquid. Our theory here is directly applicable to the case of the Bi 2 Se 3 /Bi 2 Te 3 family, which has a single isotropic Dirac cone that remains isotropic for low dopant concentrations.Spin dynamics and electrical transport -Starting from a low energy effective Hamiltonian for the bulk of a 3D topological insulator in the Bi 2 Se 3 family, the low energy surface Hamiltonian was derived by Zhang et al. [6] by diagonalizing the bulk effective Hamiltonian with open boundary conditions, and by integrating out the high energy bulk degrees of freedom. In Ref.[6], it was shown that for a surface in the xy-plane, the helical states at low energies are governed by

Using an asymptotically exact weak coupling analysis of a multi-orbital Hubbard model of the electronic structure of Sr2RuO4, we show that the interplay between spin and charge fluctuations leads unequivocally to triplet pairing which originates in the quasi-one dimensional bands. The resulting superconducting state spontaneously breaks time-reversal symmetry and is of the form ∆ ∼ (px + ipy)ẑ with sharp gap minima and a d-vector that is only weakly pinned. The superconductor is topologically trivial and hence lacks robust chiral Majorana fermion modes along the boundary. The absence of topologically protected edge modes could explain the surprising absence of experimentally detectable edge currents in this system.