Energy landscape of the finite-size mean-field 2-spin spherical model and topology trivialization

Motivated by the recently observed phenomenon of topology trivialization of potential energy landscapes (PELs) for several statistical mechanics models, we perform a numerical study of the finite-size 2-spin spherical model using both numerical polynomial homotopy continuation and a reformulation via non-Hermitian matrices. The continuation approach computes all of the complex stationary points of this model while the matrix approach computes the real stationary points. Using these methods, we compute the average number of stationary points while changing the topology of the PEL as well as the variance. Histograms of these stationary points are presented along with an analysis regarding the complex stationary points. This work connects topology trivialization to two different branches of mathematics: algebraic geometry and catastrophe theory, which is fertile ground for further interdisciplinary research.

Figure 1

Mean number of stationary points as a function of $\gamma$.

Figure 2

Mean number of stationary points as a function of $\kappa$.

Figure 3

Variance of the number of stationary points as a function of $\gamma$.

Figure 4

Variance of the number of stationary points as a function of $\kappa$.

Figure 5

Probability densities of the number of stationary points for different values of $\gamma$.

Figure 6

Probability density of $E_{min}$ for $\gamma =2$ and $N=50$.

Figure 7

Probability density of $E_{min}$ for $\sigma =1$ and $N=100$.

Figure 8

Probability density of $E_{min}$ near the critical energy, again with $\sigma =1$ and $N=100$. (a) corresponds to [4] and (b) to [16], with both estimates diverging at $E_{\mathrm{c}}$.

Figure 9

Probability density of $E_{min}$ for $\kappa =1$ and $N=50$.

Figure 10

Plots of the real and imaginary parts of the energy function $E_h\left(\mathbf{x}\right)$ evaluated at the complex stationary points for values of $\gamma$ corresponding to (a) 0.1, (b) 1, (c) 3, (d) 5 and values of $\kappa$ corresponding to (e) 0.1, (f) 1, (g) 2, (h) 4.