Rigorous renormalization group at first-order phase transitions

The density matrix renormalization group (DMRG) has been a tremendously powerful method for computing the ground state of one-dimensional or quasi-one-dimensional quantum many-body systems. However, there can be situations that are particularly challenging for DMRG to solve owing to its local optimization procedure. One such example are first-order phase transitions where globally different states lie very close in energy so that DMRG may not converge to the true ground state, but to a local minimum in the energy landscape. Recently, a rigorous renormalization group (RRG) algorithm that employs a more global optimization approach, has been introduced. This algorithm targets a set of low-lying states instead of variationally searching for a single ground state. We compare the performance of both algorithms for typical spin chain models as well as near first-order phase transitions, where we observe improved reliability for RRG in certain cases.