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The surface of a topological insulator is a closed two dimensional manifold. The surface states are described by the Dirac Hamiltonian in curved two dimensional spaces. For a slab-like sample with a magnetic field perpendicular to its top and bottom surfaces, there are chiral states delocalized on the four side faces. These "chiral sheets" carry both charge and spin currents. In strong magnetic fields the quantized charge Hall effect ($\s_{xy}=(2n+1)e^2/h$) will coexist with spin Hall effect.
Comment: Main results are the same, but the presentation is significantly modified. To appear in Physical Review Letters
Comment: Talk given at MOS2002
We obtain the field-theory representations of several network models that are relevant to 2D transport in high magnetic fields. Among them, the simplest one, which is relevant to the plateau transition in the quantum Hall effect, is equivalent to a particular representation of an antiferromagnetic SU(2N) ($N\to 0$) spin chain. Since the later can be mapped onto a $\theta\ne 0$, $U(2N)/U(N)\times U(N)$ sigma model, and since recent numerical analyses of the corresponding network give a delocalization transition with $\nu\approx 2.3$, we conclude that the same exponent is applicable to the sigma model.
Comment: An error in Eq.(11) is corrected
The purpose of this paper is to identify an unsettled issue in the theory of the half-filled Landau level, and state our point of view.
This work is motivated by a specific point of view: at short distances and high energies the undoped and underdoped cuprates resemble the $\pi$-flux phase of the t-J model. The purpose of this paper is to present a mechanism by which pairing grows out of the doped $\pi$-flux phase. According to this mechanism pairing symmetry is determined by a parameter controlling the quantum tunneling of gauge flux quanta. For zero tunneling the symmetry is $d_{x^2-y^2}+id_{xy}$, while for large tunneling it is $d_{x^2-y^2}$. A zero-temperature critical point separates these two limits.
Comment: Two additional references added
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