The energy landscape has been widely applied to many physical and chemical systems. Transition pathways on the energy landscape represent the most probable routes of phase transition between different stable states, and index-1 saddle points on the transition pathways correspond to the transition states. How do we search for a comprehensive tree of possible transition states on a complicated energy landscape, without unwanted random guesses of transition states or prior knowledge of transition pathways? Here I will introduce a general numerical method that constructs the solution landscape, which is a pathway map of stationary points, starting from a saddle point on the energy landscape all the way down to multiple energy minima. The key idea of downward search provides convenience for finding transition states systematically. The solution landscape guides our understanding of how a physical system moves on the energy landscape and identifies multiple transition states between energy minima.

Several examples will be presented to illustrate this method. First, we solve the Landau-de Gennes energy to model nematic liquid crystals confined in a square well. We further compare the results of the Ericksen energy. We also identify non-axisymmetric critical points of the Onsager model with different potential kernels. We also consider phase transition between crystals and quasicrystals with a Lifshitz-Petrich free energy; Two possible transition pathways are found to connect two-dimensional hexagonal crystalline and dodecagonal quasicrystalline phases. As a constrained example, we identify vortex states of two-dimensional rotational Bose-Einstein condensates and reveal four excitation mechanisms. The successive quantum phase transitions with increasing rotational frequencies can be explained in our results.