https://www.selleckchem.com/products/brigatinib-ap26113.html How do we search for the entire family tree of possible intermediate states, without unwanted random guesses, starting from a stationary state on the energy landscape all the way down to energy minima? Here we introduce a general numerical method that constructs the pathway map, which guides our understanding of how a physical system moves on the energy landscape. The method identifies the transition state between energy minima and the energy barrier associated with such a state. As an example, we solve the Landau-de Gennes energy incorporating the Dirichlet boundary conditions to model a liquid crystal confined in a square box; we illustrate the basic concepts by examining the multiple stationary solutions and the connected pathway maps of the model.We predict the existence of a novel interaction-induced spatial localization in a periodic array of qubits coupled to a waveguide. This localization can be described as a quantum analogue of a self-induced optical lattice between two indistinguishable photons, where one photon creates a standing wave that traps the other photon. The localization is caused by the interplay between on-site repulsion due to the photon blockade and the waveguide-mediated long-range coupling between the qubits.Time crystals form when arbitrary physical states of a periodically driven system spontaneously break discrete time-translation symmetry. We introduce one-dimensional time-crystalline topological superconductors, for which time-translation symmetry breaking and topological physics intertwine-yielding anomalous Floquet Majorana modes that are not possible in free-fermion systems. Such a phase exhibits a bulk magnetization that returns to its original form after two drive periods, together with Majorana end modes that recover their initial form only after four drive periods. We propose experimental implementations and detection schemes for this new state.We introduce, for each