Structure of the Space of Ground States in Systems with Non-Amenable Symmetries |
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Authors: | M. Niedermaier E. Seiler |
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Affiliation: | 1.Laboratoire de Mathematiques et Physique Theorique,CNRS/UMR 6083, Université de Tours,Tours,France;2.Max-Planck-Institut für Physik (Werner-Heisenberg-Institut),F?hringer Ring 6,Munich,Germany |
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Abstract: | We investigate classical spin systems in d ≥ 1 dimensions whose transfer operator commutes with the action of a nonamenable unitary representation of a symmetry group, here 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 SO(1, N), the limit of the spherical principal series. (iv) The properties (i)–(iii) hold universally, irrespective of the details of the interaction. Membre du CNRS |
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