Density-matrix renormalization group: a pedagogical introduction

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