Structure of the space of ground states in systems with non-amenable symmetries

TL;DR

This paper demonstrates that systems with non-amenable symmetries like ${ m SO}(1,N)$ exhibit spontaneous symmetry breaking even in low dimensions and possess infinitely many non-normalizable ground states, contrasting with systems having compact symmetries.

Contribution

It establishes universal properties of systems with non-amenable symmetries, including symmetry breaking at finite volume and the existence of a rich structure of ground states.

Findings

Spontaneous symmetry breaking occurs in 1D and 2D systems.

Existence of infinitely many non-normalizable ground states.

Ground states form a unitary representation of ${ m SO}(1,N)$.

Abstract

We investigate classical spin systems in $d\geq 1$ dimensions whose transfer operator commutes with the action of a nonamenable unitary representation of a symmetry group, here ${\rm SO}(1,N)$; these systems may alternatively be interpreted as systems of interacting quantum mechanical particles moving on hyperbolic spaces. In sharp contrast to the analogous situation with a compact symmetry group the following results are found and proven: (i) Spontaneous symmetry breaking already takes place for finite spatial volume/finitely many particles and even in dimensions $d=1,2$. The tuning of a coupling/temperature parameter cannot prevent the symmetry breaking. (ii) The systems have infinitely many non-invariant and non-normalizable generalized ground states. (iii) the linear space spanned by these ground states carries a distinguished unitary representation of ${\rm SO}(1,N)$, the limit of the spherical principal series. (iv) The properties (i)--(iii) hold universally, irrespective of the details of the interaction.