Diagonalization- and Numerical Renormalization-Group-Based Methods for Interacting Quantum Systems

TL;DR

This paper reviews numerically exact methods like exact diagonalization, NRG, and DMRG for solving interacting quantum systems, highlighting recent advances and generalizations to broader systems and dynamic properties.

Contribution

It provides a comprehensive pedagogical overview of existing numerical methods for quantum many-body problems, including recent developments in matrix product states and time evolution techniques.

Findings

Survey of numerical methods for quantum systems

Discussion of extensions to dynamical and finite-temperature properties

Overview of recent theoretical advances in DMRG and matrix product states

Abstract

In these lecture notes, we present a pedagogical review of a number of related {\it numerically exact} approaches to quantum many-body problems. In particular, we focus on methods based on the exact diagonalization of the Hamiltonian matrix and on methods extending exact diagonalization using renormalization group ideas, i.e., Wilson's Numerical Renormalization Group (NRG) and White's Density Matrix Renormalization Group (DMRG). These methods are standard tools for the investigation of a variety of interacting quantum systems, especially low-dimensional quantum lattice models. We also survey extensions to the methods to calculate properties such as dynamical quantities and behavior at finite temperature, and discuss generalizations of the DMRG method to a wider variety of systems, such as classical models and quantum chemical problems. Finally, we briefly review some recent developments for obtaining a more general formulation of the DMRG in the context of matrix product states as well as recent progress in calculating the time evolution of quantum systems using the DMRG and the relationship of the foundations of the method with quantum information theory.