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 nonnormalizable 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/(V ln V) in the volume, the 0-component of the spin field diverges as √ ln V , while SO(1, N) invariant quantities remain finite.