  Expanding and improving the repertoire of numerical methods for studying
quantum lattice models is an ongoing focus in many-body physics. While the
density matrix renormalization group (DMRG) has been established as a
practically useful algorithm for finding the ground state in 1D systems, a
provably efficient and accurate algorithm remained elusive until the
introduction of the rigorous renormalization group (RRG) by Landau et al.
[Nature Physics 11, 566 (2015)]. In this paper, we study the accuracy and
performance of a numerical implementation of RRG at first order phase
transitions and in symmetry protected topological phases. Our study is motived
by the question of when RRG might provide a useful complement to the more
established DMRG technique. In particular, despite its general utility, DMRG
can give unreliable results near first order phase transitions and in
topological phases, since its local update procedure can fail to adequately
explore (near) degenerate manifolds. The rigorous theoretical underpinnings of
RRG, meanwhile, suggest that it should not suffer from the same difficulties.
We show this optimism is justified, and that RRG indeed determines accurate,
well-ordered energies even when DMRG does not. Moreover, our performance
analysis indicates that in certain circumstances seeding DMRG with states
determined by coarse runs of RRG may provide an advantage over simply
performing DMRG.
