On-Site Potential Creates Complexity in Systems with Disordered Coupling

We calculate the average number of critical points N¯ of the energy landscape of a many-body system with disordered two-body interactions and a weak on-site potential. We find that introducing a weak nonlinear on-site potential dramatically increases N¯ to exponential in system size and give a complete picture of the organization of critical points. Our results extend solvable spin-glass models to physically more realistic models and are of relevance to glassy systems, nonlinear oscillator networks, and many-body interacting systems.