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It is shown that several different order parameters can be used to characterize a type of P -and T -violating state for spin systems, that we call chiral spin states.There is a closely related, precise notion of chiral spin liquid states. We construct soluble models, based on P and T symmetric local spin Hamiltonians, with chiral spin ground states. Mean field theories leading to chiral spin liquids are proposed. Frustration is essential in stabilizing these states. The quantum numbers of quasiparticles around the chiral spin liquids are analyzed. They generally obey fractional statistics. Based on these ideas, it is speculated that superconducting states with unusual values of the flux quantum may exist.

We discuss some new quantum numbers (spin vector) for the Hall fluid, representing orbital spin degrees of freedom. We show that the spin vectors are quantized. In the absence of impurities, two Hall fluids with different spin vectors cannot change into each other without a phase transition and closing of the energy gap. In principle the spin vector can be measured through its coupling to the curvature of space. Our formalism may be described picturesquely as a unification of electromagnetism and "gravity" in condensed-matter physics.PACS numbers: 73.20.Dx, 02.40.+m It has become clear that the order in the quantum Hall fluid (and also in the closely related chiral spin fluid and anyon fluid) is not associated with broken symmetries, but is topological in character [1]. In a series of papers [2] we have shown that the order may be characterized by a symmetric integer-valued matrix K and an integervalued charge vector /. In this paper we introduce and discuss additional topological quantum numbers, namely, a quantity £ which we will call the shift and a spin vector

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Invoking a duality picture we used earlier to describe semion superconductivity, we suggest an explanation of the universal conductance measured in thin films at the superconductor-insulator phase transition.