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

M. Niedermaier·E. Seiler
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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.

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M. Niedermaier, E. Seiler. 2006-10-04. Structure of the space of ground states in systems with non-amenable symmetries. https://doi.org/10.1007/s00220-006-0154-6

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