  The density matrix renormalization group is one of the most powerful
numerical methods for computing ground-state properties of two-dimensional (2D)
quantum lattice systems. Here we show its finite-temperature extensions are
also viable for 2D, using the following strategy: At high temperatures, we
combine density-matrix purification and numerical linked-cluster expansions to
extract static observables directly in the thermodynamic limit. At low
temperatures inaccessible to purification, we use the minimally entangled
typical thermal state (METTS) algorithm on cylinders. We consider the
triangular Heisenberg antiferromagnet as a first application, finding excellent
agreement with other state of the art methods. In addition, we present a
METTS-based approach that successfully extracts critical temperatures, and
apply it to a frustrated lattice model. On a technical level, we compare two
different schemes for performing imaginary-time evolution of 2D clusters,
finding that a Suzuki-Trotter decomposition with swap gates is currently the
most accurate and efficient.
