  The physical properties of a quantum many-body system can, in principle, be
determined by diagonalizing the respective Hamiltonian, but the dimensions of
its matrix representation scale exponentially with the number of degrees of
freedom. Hence, only small systems that are described through simple models can
be tackled via exact diagonalization. To overcome this limitation, numerical
methods based on the renormalization group paradigm that restrict the quantum
many-body problem to a manageable subspace of the exponentially large full
Hilbert space have been put forth. A striking example is the density-matrix
renormalization group (DMRG), which has become the reference numerical method
to obtain the low-energy properties of one-dimensional quantum systems with
short-range interactions. Here, we provide a pedagogical introduction to DMRG,
presenting both its original formulation and its modern tensor-network-based
version. This colloquium sets itself apart from previous contributions in two
ways. First, didactic code implementations are provided to bridge the gap
between conceptual and practical understanding. Second, a concise and
self-contained introduction to the tensor network methods employed in the
modern version of DMRG is given, thus allowing the reader to effortlessly cross
the deep chasm between the two formulations of DMRG without having to explore
the broad literature on tensor networks. We expect this pedagogical review to
find wide readership amongst students and researchers who are taking their
first steps in numerical simulations via DMRG.
