A two-dimensional condensed-matter lattice model is presented which exhibits a nonzero quantization of the Hall conductance a" in the absence of an external magnetic field. Massless fermions without spectral doubling occur at critical values of the model parameters, and exhibit the so-called "parity anomaly" of (2+1)-dimensional field theories.

The continuum field theory describing the low-energy dynamics of the large-spin onedimensional Heisenberg Bntiferromagnet is found to be the O(3) nonlinear sigma model. When weak easy-axis anisotropy is present, soliton solutions of the equations of motion are obtained and semiclassically quantized. Integer and half-integer spin systems are distinguished.

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We study the "entanglement spectrum" (a presentation of the Schmidt decomposition analogous to a set of "energy levels") of a many-body state, and compare the Moore-Read model wave function for the nu=5/2 fractional quantum Hall state with a generic 5/2 state obtained by finite-size diagonalization of the second-Landau-level-projected Coulomb interactions. Their spectra share a common "gapless" structure, related to conformal field theory. In the model state, these are the only levels, while in the "generic" case, they are separated from the rest of the spectrum by a clear "entanglement gap", which appears to remain finite in the thermodynamic limit. We propose that the low-lying entanglement spectrum can be used as a "fingerprint" to identify topological order.

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...

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