Vacuum orbit and spontaneous symmetry breaking in hyperbolic sigma-models. (English) Zbl 1194.81144
Summary: We present a detailed study of quantized noncompact, nonlinear \(\text{SO}(1,N)\) 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 non-normalizable ground states transforming irreducibly under a nontrivial representation of \(\text{SO}(1,N)\);
(ii) the \(\text{SO}(1,N)\) 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.
The main results in all dimensions \(\geq 2\) are:
(i) on a finite lattice the systems have infinitely many non-normalizable ground states transforming irreducibly under a nontrivial representation of \(\text{SO}(1,N)\);
(ii) the \(\text{SO}(1,N)\) 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.
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