  In a previous paper (hep-th/0407256) local scalar QFT (in Weyl algebraic
approach) has been constructed on degenerate semi-Riemannian manifolds
$\bS^1\times \Sigma$ corresponding to the extension of Killing horizons by
adding points at infinity to the null geodesic forming the horizon. It has been
proved that the theory admits a natural representation of $PSL(2,\bR)$ in terms
of $*$-automorphisms and this representation is unitarily implementable if
referring to a certain invariant state $\lambda$. Among other results it has
been proved that the theory admits a class of inequivalent algebraic (coherent)
states $\{\lambda_\zeta\}$, with $\zeta\in L^2(\Sigma)$, which break part of
the symmetry, in the sense that each of them is not invariant under the full
group $PSL(2,\bR)$ and so there is no unitary representation of whole group
$PSL(2,\bR)$ which leaves fixed the cyclic GNS vector. These states, if
restricted to suitable portions of $\bM$ are invariant and extremal KMS states
with respect a surviving one-parameter group symmetry. In this paper we clarify
the nature of symmetry breakdown. We show that, in fact, {\em spontaneous}
symmetry breaking occurs in the natural sense of algebraic quantum field
theory: if $\zeta \neq 0$, there is no unitary representation of whole group
$PSL(2,\bR)$ which implements the $*$-automorphism representation of
$PSL(2,\bR)$ itself in the GNS representation of $\lambda_\zeta$ (leaving fixed
or not the state).
