Vacuum orbit and spontaneous symmetry breaking in hyperbolic sigma models

TL;DR

This paper investigates noncompact SO(1,N) sigma-models, revealing spontaneous symmetry breaking and vacuum orbit structures, with numerical and analytical results showing symmetry evasion of the Mermin-Wagner theorem in two dimensions.

Contribution

It provides a comprehensive analysis of symmetry breaking, vacuum structure, and mass generation in noncompact sigma-models across multiple dimensions, including numerical and saddle-point methods.

Findings

SO(1,N) symmetry is spontaneously broken in all dimensions ≥ 2.

In 2D, the models evade the Mermin-Wagner theorem.

Numerical results confirm the absence of explicit symmetry breaking.

Abstract

We present a detailed study of quantized noncompact, nonlinear SO(1,N) sigma-models in arbitrary space-time dimensions D \geq 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 \geq 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 \sqrt{\ln V}, while SO(1,N) invariant quantities remain finite.