Physics – Mathematical Physics
Scientific paper
2006-01-24
Commun.Math.Phys.270:373-443,2007
Physics
Mathematical Physics
85 pages, to appear in Commun. Math. Phys.. Some typographical and other small errors corrected; remarks about supermanifold v
Scientific paper
10.1007/s00220-006-0154-6
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.
Niedermaier Max
Seiler Erhard
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