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

TL;DR

This paper investigates spontaneous symmetry breaking in a hyperbolic spin chain with non-amenable SO(1,2) symmetry, revealing unique features of noncompact symmetry breaking and its implications for quantum theory reconstruction.

Contribution

It demonstrates spontaneous symmetry breaking for a non-amenable group in a spin chain and explores the resulting nonseparable quantum theory after reconstruction.

Findings

Non-amenable SO(1,2) symmetry is spontaneously broken.

Expectation functionals are dominated by configurations with large boosts.

Reconstructed quantum theory has a nonseparable Hilbert space with a continuous, unitary SO(1,2) action.

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 \to \infty 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.