Vacuum orbit and spontaneous symmetry breaking in hyperbolic sigma-models

We present a detailed study of quantized noncompact, nonlinear SO(1,N) sigma-models in arbitrary space-time dimensions D⩾2, with the focus on issues of spontaneous symmetry breaking of boost and rotation elements of the symmetry group. The models are defined on a lattice both in terms of a transfer matrix and by an appropriately gauge-fixed Euclidean functional integral. The main results in all dimensions ⩾2 are: (i) on a finite lattice the systems have infinitely many non-normalizable ground states transforming irreducibly under a nontrivial representation of SO(1,N); (ii) the SO(1,N) symmetry is spontaneously broken. For D=2 this shows that the systems evade the Mermin-Wagner theorem. In this case in addition: (iii) Ward identities for the Noether currents are derived to verify numerically the absence of explicit symmetry breaking; (iv) numerical results are presented for the two-point functions of the spin field and the Noether current as well as a new order parameter; (v) in a large N saddle-point analysis the dynamically generated squared mass is found to be negative and of order 1/(VlnV) in the volume, the 0-component of the spin field diverges as lnV, while SO(1,N) invariant quantities remain finite.