Transition matrix Monte Carlo and flat-histogram algorithm

In any valid Monte Carlo sampling that realizes microcanonical property we can collect statistics for a transition matrix in energy. This matrix is used to determine the density of states, from which most of the thermodynamical averages can be calculated, including free energy and entropy. We discuss single-spin-flip algorithms, such as flat-histogram and equal-hit algorithms, that can be used for simulations. The flat-histogram algorithm realizes multicanonical ensemble. We demonstrate the use of the method with applications to Ising model and show its efficiency of search for spin-glass ground states.