Figure 1

Two generic examples for the evolution of the dispersion relation (4) during a symmetry-breaking phase transition; see also 10. Initially (dotted lines), all $k$ values are stable, ${\omega }^2(k)>0$. At the critical point (dashed lines), the dispersion relation touches the $k$ axis, and after the transition (solid lines), modes in a finite $k$ interval become unstable, ${\omega }^2(k)<0$. The left panel corresponds to a case where ${\omega }^2(k=0)=m^2$ in Eq. (4) remains positive while $c^2$ changes sign (see, e.g., 21); whereas, in the right panel, ${\omega }^2(k=0)=m^2$ becomes negative. In both cases, however, there is a dominant wave vector $k_{*}$ (for large $N$), as indicated by the vertical arrows.Reuse & Permissions