Vacuum orbit and spontaneous symmetry breaking in hyperbolic sigma-models

Abstract

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 / ( 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.