Fig. 7.

Spectrum. The real part of the most unstable eigenvalue, ${\lambda }_m^{*\prime }$, of the first five modes f _m calculated from Eq. (15) for J = 0.5. Notice that they are all is positive for all K. Each ${\lambda }_m^{*\prime }$ was calculated by approximating the integral of the R.H.S. of (15) using Gaussian quadrature with N′ = 200 grid points and diagonalizing the resulting matrix. The upper limit of integration R = 1.15 was measured from simulations. The radial density g(r) was determined numerically as discussed in Supplementary Note 1. The kernels H _m in Eq. (15) for m > 1 were calculated numerically. The dashed line marks the approximation to the critical coupling strength (18)