General topological features and instanton vacuum in quantum Hall and spin liquids

Abstract

We introduce the concept of super universality in quantum Hall and spin liquids which has emerged from previous studies. It states that all the fundamental features of the quantum Hall effect are generically displayed as general topological features of the $\theta$ parameter in nonlinear sigma models in two dimensions. To establish super universality in spin liquids we revisit the mapping by Haldane who argued that the anti ferromagnetic Heisenberg spin $s$ chain is effectively described by the O(3) nonlinear sigma model with a $\theta$ term. By combining the path integral representation for the dimerized spin $s=1/2$ chain with renormalization group decimation techniques we generalise the Haldane approach to include a more complicated theory, the fermionic rotor chain, involving four different renormalization group parameters. We show how the renormalization group calculation technique can be used to lay the bridge between the fermionic rotor chain and the sigma model. As an integral and fundamental aspect of the mapping we establish the topological significance of the dangling spin at the edge of the chain which is in all respects identical to the massless chiral edge excitations in quantum Hall liquids. We consider various different geometries of the spin chain and show that for each of the different geometries correspond to a topologically equivalent quantum Hall liquid.

Title changed, Section 2 and Appendix expanded, an error in the expression for theta corrected

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• Spin-1 chain with spin-1/2 excitations in the bulk
• Exact Haldane mapping for all $S$ and super universality in spin chains
• Renormalization of the vacuum angle in quantum mechanics, Berry phase and continuous measurements
• Gapless points of dimerized quantum spin chains: analytical and numerical studies
• Renormalization of the vacuum angle for a particle on a ring
• Super universality of dimerised $SU(N+M)$ spin chains