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Structure of the Space of Ground States in Systems with Non-Amenable Symmetries

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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.

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Authors and Affiliations

  1. Laboratoire de Mathematiques et Physique Theorique, CNRS/UMR 6083, Université de Tours, Parc de Grandmont, 37200, Tours, France

    M. Niedermaier

  2. Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805, Munich, Germany

    E. Seiler

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  1. M. Niedermaier
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  2. E. Seiler
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Correspondence to M. Niedermaier.

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Communicated by J.Z. Imbrie

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Niedermaier, M., Seiler, E. Structure of the Space of Ground States in Systems with Non-Amenable Symmetries. Commun. Math. Phys. 270, 373–443 (2007). https://doi.org/10.1007/s00220-006-0154-6

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