The quantum Hall effects, σ-models at Θ = π and quantum spin chains

Ian K. Affleck

Ian K. Affleck

The critical behaviour of SU(n) quantum ''spin'' chains, Wess-Zumino-Witten sigma-models and grassmanian sigma-models at topological angle theta = ..pi.. (of possible relevance to the quantum Hall effect) is reexamined. It is argued that an additional Z/sub n/ symmetry is generally necessary to stabilize the massless phase. This symmetry is not present for the sigma-models for n>2 and is only present for certain representations of ''spin'' chains.

Ian K. Affleck

Martin R. Zirnbauer

2D nonlinear sigma models with Hermitian symmetric target admit a theta-term, which couples the field theory to the topological charge of its instanton gas. At the special coupling theta = pi, by what is nowadays attributed to a coupling-constant anomaly of Lieb-Schultz-Mattis type, such models have a degenerate ground state. Yet, the details of their non-trivial infrared limit have remained open in general. Here we suggest that non-perturbative renormalization group flow into the strong-coupling regime induces strong fluctuations of the theta-parameter, with the consequence that the instanton density is suppressed, the target-space topology effectively altered, and the target-space metric driven off reality and into geometrostasis. Assuming this heuristic scenario and combining it with a Cauchy process of target-space deformation, we present a detailed argument that the O(3) nonlinear sigma model at theta = pi, known to be the effective field theory for critical antiferromagnetic quantum spin chains with large half-integer spin, renormalizes to the conformal field theory of a U(1) boson compactified at a special radius with SU(2) symmetry. A closely related scenario applies to Pruisken's nonlinear sigma model for the integer quantum Hall transition.

Ian K. Affleck

We study the strong-coupling expansion of a Euclidean lattice formulation of the two-dimensional O(3) \ensuremath{\sigma}-model at nonzero topological angle \ensuremath{\theta}, mapping it onto a solid-on-solid model with well-known critical behavior. The results are consistent with those predicted by the mapping of the Hamiltonian formulation of the \ensuremath{\sigma} model onto quantum spin chains. The model has a gapless phase at \ensuremath{\theta}=\ensuremath{\pi} and weak coupling in which a condensate of Coleman's ``half-asymptotic'' particles exists.