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

Abstract

We introduce the concept of super universality in quantum Hall liquids and spin liquids. This concept has emerged from previous studies of the quantum Hall effect and states that all the fundamental features of the quantum Hall effect are generically displayed as general topological features of the θ parameter in nonlinear σ 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 in 1 + 1 space-time dimensions is effectively described by the O(3) nonlinear σ model with a θ term. By combining the path integral representation for the dimerized spin s = 1/2 chain with renormalization group decimation techniques we generalize 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 O(3) nonlinear σ model with the θ term.As an integral and fundamental aspect of the mapping we establish the topological significance of the dangling spin at the edge of the chain. The edge spin in spin liquids is in all respects identical to the massless chiral edge excitations in quantum Hall liquids. We consider various different geometries of the spin chain such as open and closed chains, chains with an even and odd number of sides. We show that for each of the different geometries the θ term has a distinctly different physical meaning. We compare each case with a topologically equivalent quantum Hall liquid.

Super Universality of the Quantum Hall Effect and the “Large N Picture” of the ϑ Angle

“…non-linear sigma model in two dimensions it was shown that the ϑ vacuum generally displays massless excitations that propagate along the confining "edge" of the system [1][2][3][4][5]. This new aspect of the sigma model has come in many ways as a welcome surprise.…”

“…In applications to quantum A.M.M. Pruisken ( ) Institute for Theoretical Physics, University of Amsterdam, Valckenierstraat 65, 1018 XE Amsterdam, The Netherlands e-mail: pruisken@science.uva.nl spin liquids [6,7], for example, the edge excitations describe the dynamics of the "dangling" quantum spins located at the "edges" of the spin chain [4,5]. In the context of quantum Hall liquids [8][9][10][11][12], on the other hand, these excitations are identically the same as those described by the theory of "chiral edge bosons" [2,3].…”

“…Within the grassmannian sigma model one finds, for example, that the SU(M + N) symmetry is spontaneously broken at the "edge" of the system whereas the critical correlations along the "edge" are identically the same for all values of M and N [2]. These unexpected features have motivated several studies where the idea of super universality has emerged [4,5,[20][21][22][23]. The essence of this idea is that the ϑ vacuum displays all the basic aspects of the quantum Hall effect for all non-negative values of M and N .…”

“…The remarkable and surprising feature of these critical edge correlations is that they are completely independent of M and N . Moreover, if we interpret the edge coordinate x as the imaginary time τ then one recognizes (7) as the bosonic path integral of an SU(M + N) spin with quantum number s = k/2 in a magnetic field B = ρ edge ω [4].…”

Gapless points of dimerized quantum spin chains: Analytical and numerical studies

“…Haldane's analysis used a non-linear σ-model (NLSM) which is a field theoretic description of the long-distance and low-energy modes of the spin system [3,4,5,6,7].…”