Non-amenability and spontaneous symmetry breaking - The hyperbolic spin-chain -

Abstract

The hyperbolic spin chain is used to elucidate the notion of spontaneous symmetry breaking for a non-amenable internal symmetry group, here SO(1, 2). The noncompact symmetry is shown to be spontaneously broken - something which would be forbidden for a compact group by the Mermin-Wagner theorem. Expectation functionals are defined through the L → ∞ limit of a chain of length L; the functional measure is found to have its weight mostly on configurations boosted by an amount increasing at least powerlike with L. This entails that despite the non-amenability a certain subclass of noninvariant functions is averaged to an SO(1, 2) invariant result. Outside this class symmetry breaking is generic. Performing an Osterwalder-Schrader reconstruction based on the infinite volume averages one finds that the reconstructed quantum theory is different from the original one. The reconstructed Hilbert space is nonseparable and contains a separable subspace of ground states of the reconstructed transfer operator on which SO(1, 2) acts in a continuous, unitary, and irreducible way.