The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems

TL;DR

This paper provides a comprehensive overview of tensor network simulation techniques for many-body quantum lattice systems, focusing on algorithms, data structures, and applications in low-dimensional physics.

Contribution

It offers a detailed compilation of tensor network methods, including algorithms and data structures, for simulating low-dimensional quantum lattice systems, serving as both a review and a practical guide.

Findings

Discusses advantages of loop-free network geometries.

Details algorithms for simulating low-energy states.

Provides insights into data structures and performance.

Abstract

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-dimension 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.