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

Abstract

Abstract. In these lecture notes, we present a pedagogical review of a number of related 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.

Quantum information analysis of the phase diagram of the half-filled extended Hubbard model

“…3,4,5 In this work, we determine the ground-state phase boundaries for the half-filled system using methods based on quantum information, 6,7,8,9,10,11,12 calculated using the density-matrix renormalization group (DMRG). 13,14,15 These methods have a number of advantages over other methods involving gaps, correlation functions, or order parameters. 12,16 First, they involve only the properties of the ground state (or, in practice, a numerical approximation to the true ground state), and so can be calculated easily and accurately.…”

Implementation trade-offs of the density matrix renormalization group algorithm on kilo-processor architectures

“…As optimization steps in both cases are similar, for sake of simplicity, we consider only the infinite-lattice algorithm. The detailed description of the algorithm can be found in the original work [1] and various reviews [2], [3], here only the key steps of an iteration of the infinite-lattice algorithm are summarized in Pseudocode 1 providing the basis of our analysis. Construct the density matrix for the given block from the lowest eigenstate.…”

Automated construction of $U(1)$-invariant matrix-product operators from graph representations

“…For this purpose, we exploit the total magnetization of a system with L lattice sites as the conserved U(1) quantity. For the ground state search, we sweeps through the system (using open boundary conditions) and optimize the site tensors via a standard Lanczos algorithm, see [23][24][25]. All MPS contractions are formulated in two-site representation [5].…”