Introduction to Tensor Network Methods: Numerical simulations of low-dimensional many-body quantum systems

Tao Xiang, Hui‐Hai Zhao, Q. N. Chen +3 more

Ling Wang, Frank Verstraete

Irene Papaefstathiou, J. I. Cirac, Technische Universität München

Pietro Silvi, Ferdinand Tschirsich, Matthias Gerster +4 more

We present a compendium of numerical simulation techniques, based on tensor network methods, aiming to address problems of many-body quantum mechanics on a classical computer. The core setting of this anthology are lattice problems in low spatial dimension at finite size, a physical scenario where tensor network methods, both Density Matrix Renormalization Group and beyond, have long proven to be winning strategies. Here we explore in detail the numerical frameworks and methods employed to deal with low-dimensional physical setups, from a computational physics perspective. We focus on symmetries and closed-system simulations in arbitrary boundary conditions, while discussing the numerical data structures and linear algebra manipulation routines involved, which form the core libraries of any tensor network code. At a higher level, we put the spotlight on loop-free network geometries, discussing their advantages, and presenting in detail algorithms to simulate low-energy equilibrium states. Accompanied by discussions of data structures, numerical techniques and performance, this anthology serves as a programmer’s companion, as well as a self-contained introduction and review of the basic and selected advanced concepts in tensor networks, including examples of their applications.

Gonçalo Catarina, Bruno Murta

Abstract 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 among students and researchers who are taking their first steps in numerical simulations via DMRG. Graphic abstract

Shi-Ju Ran, Emanuele Tirrito, Cheng Peng +4 more